I have trouble understanding this fallacy: “If A, then B. Therefore if not-B, then not-A.”












14















About "If A, then B. Therefore, if not-B, then not-A":



From what I understand the conclusion is wrong, because it is not said that A is a sufficient condition for B, (and there may be other conditions required for B, so if they are not present B won't be the case, even if A is the case.).



But I have trouble finding a real life example to this and I'm not sure if it is the consept that I don't understand or it is just the way we express ourselves in natural language that causes the confusion.



Here is an example I could think of, in natural language: "In order to have internet (B), the router must be on (A). So, if there is no internet (not-B), that means the router is not on (not-A)" (Which is not true because there may be a problem with the providers, for example.)



So turning the router on is a necessary condition for having internet but can you then say: "If you turn the router on (A), there will be internet (B)." Saying that is just is not true. So is it just that we don't have a good way to express sufficient conditions (in comparison to necessary conditions) or is it that I just don't understanding some fundamental concept (or both)?










share|improve this question























  • Comments are not for extended discussion; this conversation has been moved to chat.

    – Geoffrey Thomas
    Feb 26 at 11:13











  • You know "A->B". Then you assume "Not-B" is true. If A were true, you would have both "B" and "Not-B", so necessarily, A is false.

    – coredump
    Feb 27 at 9:38
















14















About "If A, then B. Therefore, if not-B, then not-A":



From what I understand the conclusion is wrong, because it is not said that A is a sufficient condition for B, (and there may be other conditions required for B, so if they are not present B won't be the case, even if A is the case.).



But I have trouble finding a real life example to this and I'm not sure if it is the consept that I don't understand or it is just the way we express ourselves in natural language that causes the confusion.



Here is an example I could think of, in natural language: "In order to have internet (B), the router must be on (A). So, if there is no internet (not-B), that means the router is not on (not-A)" (Which is not true because there may be a problem with the providers, for example.)



So turning the router on is a necessary condition for having internet but can you then say: "If you turn the router on (A), there will be internet (B)." Saying that is just is not true. So is it just that we don't have a good way to express sufficient conditions (in comparison to necessary conditions) or is it that I just don't understanding some fundamental concept (or both)?










share|improve this question























  • Comments are not for extended discussion; this conversation has been moved to chat.

    – Geoffrey Thomas
    Feb 26 at 11:13











  • You know "A->B". Then you assume "Not-B" is true. If A were true, you would have both "B" and "Not-B", so necessarily, A is false.

    – coredump
    Feb 27 at 9:38














14












14








14


5






About "If A, then B. Therefore, if not-B, then not-A":



From what I understand the conclusion is wrong, because it is not said that A is a sufficient condition for B, (and there may be other conditions required for B, so if they are not present B won't be the case, even if A is the case.).



But I have trouble finding a real life example to this and I'm not sure if it is the consept that I don't understand or it is just the way we express ourselves in natural language that causes the confusion.



Here is an example I could think of, in natural language: "In order to have internet (B), the router must be on (A). So, if there is no internet (not-B), that means the router is not on (not-A)" (Which is not true because there may be a problem with the providers, for example.)



So turning the router on is a necessary condition for having internet but can you then say: "If you turn the router on (A), there will be internet (B)." Saying that is just is not true. So is it just that we don't have a good way to express sufficient conditions (in comparison to necessary conditions) or is it that I just don't understanding some fundamental concept (or both)?










share|improve this question














About "If A, then B. Therefore, if not-B, then not-A":



From what I understand the conclusion is wrong, because it is not said that A is a sufficient condition for B, (and there may be other conditions required for B, so if they are not present B won't be the case, even if A is the case.).



But I have trouble finding a real life example to this and I'm not sure if it is the consept that I don't understand or it is just the way we express ourselves in natural language that causes the confusion.



Here is an example I could think of, in natural language: "In order to have internet (B), the router must be on (A). So, if there is no internet (not-B), that means the router is not on (not-A)" (Which is not true because there may be a problem with the providers, for example.)



So turning the router on is a necessary condition for having internet but can you then say: "If you turn the router on (A), there will be internet (B)." Saying that is just is not true. So is it just that we don't have a good way to express sufficient conditions (in comparison to necessary conditions) or is it that I just don't understanding some fundamental concept (or both)?







logic fallacies






share|improve this question













share|improve this question











share|improve this question




share|improve this question










asked Feb 24 at 10:25









user18894user18894

77113




77113













  • Comments are not for extended discussion; this conversation has been moved to chat.

    – Geoffrey Thomas
    Feb 26 at 11:13











  • You know "A->B". Then you assume "Not-B" is true. If A were true, you would have both "B" and "Not-B", so necessarily, A is false.

    – coredump
    Feb 27 at 9:38



















  • Comments are not for extended discussion; this conversation has been moved to chat.

    – Geoffrey Thomas
    Feb 26 at 11:13











  • You know "A->B". Then you assume "Not-B" is true. If A were true, you would have both "B" and "Not-B", so necessarily, A is false.

    – coredump
    Feb 27 at 9:38

















Comments are not for extended discussion; this conversation has been moved to chat.

– Geoffrey Thomas
Feb 26 at 11:13





Comments are not for extended discussion; this conversation has been moved to chat.

– Geoffrey Thomas
Feb 26 at 11:13













You know "A->B". Then you assume "Not-B" is true. If A were true, you would have both "B" and "Not-B", so necessarily, A is false.

– coredump
Feb 27 at 9:38





You know "A->B". Then you assume "Not-B" is true. If A were true, you would have both "B" and "Not-B", so necessarily, A is false.

– coredump
Feb 27 at 9:38










16 Answers
16






active

oldest

votes


















44














There are multiple good answers here, including references to modus tollens and the contrapositive (both of which are correct).



What helps me understand this concept is a more intuitive/layman's perspective.



Say that if it's raining, you ALWAYS bring an umbrella outside. (Assume that you have a perfect memory). Therefore we have:




Raining -> Umbrella




However, if you have an umbrella, does it mean that it's raining? Well, no. You might be bringing an umbrella for a trip, using it as a parasol in the scorching summer sun, or even simply returning it to a friend.



But if you don't have an umbrella, is it raining? No, and this is supported by modus tollens/contrapositive. That is to say:




Not(Umbrella) -> Not(Raining)




And that makes sense to us. If it were raining, you would've brought your umbrella. We said that you ALWAYS bring an umbrella if it's raining. So if you don't have it, it can't possibly be raining.



Let's apply it to your example:




"In order to have internet (B), the router must be on (A). So, if
there is no internet (not-B), that means the router is not on (not-A)"




This is actually not formulated properly; A and B have been reversed. This is probably why it's confusing. (Intuitively, it's in line with what you've been saying: if we don't have Internet, it's not necessarily because of the router.)



So let's word it differently:




If we have an Internet connection (A), the router must be on (B).



InternetConnected -> RouterOn



A -> B




But if the router isn't on, then we can't possibly connect to the Internet. (How would we even do that?) We know this is the case, because we said that if we're connected to the Internet, the router MUST be on.




Not(RouterOn) -> Not(InternetConnected)



Not(B) -> Not(A)




Hope this helps! :)



Edit: My "rain" example was criticized for clarity, since the real-life relationship isn't as straightforward. (Fair enough.) We can apply the same thing to dogs, though, and it works the same way but exploits the IS-A/inheritance relationship you're probably familiar with if you have a CS background.




Corgi => Dog




All corgis are dogs. But are all dogs corgis? No! However, we DO know that if it's not a dog, it's clearly not a Corgi (since a Corgi is a type of dog).




Not(Dog) => Not(Corgi)




Also, to echo the other users: this is not a fallacy. You can find more about logical fallacies here.






share|improve this answer

































    14














    ‘If not B then not A’ is the contrapositive of ‘If A then B’ and is logically valid.



    By saying ‘if A then B’, the author is saying that whenever A happens, B will definitely happen. Hence, if B does not happen, it is clear that A did not happen (if not B then not A).






    share|improve this answer































      13














      You are correct: The statement does not show a sufficient condition!



      Consider this statement: "If (A) it rains, then (B) the street will be wet."



      Assuming the street is always wet after it rained (because it is not covered by a roof or something), the contraposition would hold true!



      Contraposition: "If the street is not wet, it did not rain"



      I think what is giving you troubles is that the street in this example can be wet without rain (someone sprayed water onto it), therefore the street is not always wet just because of rain.
      That's called an inverse and differentiation can sometimes be hard.






      share|improve this answer



















      • 1





        That's a strange way of contradicting yourself. A ==> B strictly forces !B ==> !A . That's what the OP was asking... unless you're just pointing out that he contradicted his own question.

        – Carl Witthoft
        Feb 26 at 15:33











      • @CarlWitthoft I was saying that the statement is not complete. The street ALWAYS has to be wet when it rained. The internet (in OPs example) works ONLY because of that router. In real life, conversations are not really always fully logical statements. The example he used, shows that well, I think.

        – Moritz K
        Feb 27 at 17:56





















      8














      Your confusion might stem from the fact that implications in propositional logic don't always correspond intuitively to cause and effect in the real world. I'll get back to your example later; for now, let's focus on a more intuitive implication:




      If [a: there is fire], then [b: there is smoke].




      This is a natural process in which the cause (fire) always produces the effect (smoke), and that's a situation that can be described with an implication. (In the real world, there are substances that burn without smoke, so let's constrain ourselves to wood fires.) In this situation, you probably agree that the contrapositive is true as well:




      If [not(b): there is no smoke], then [not(a): there is no fire].




      This makes intuitive sense - we're not observing the effect that is always produced by the cause, so the cause can't be occurring.



      We can formally prove why the contrapositive is equivalent to the original statement. Here is the truth table that defines implications (T for true and F for false):



      | a | b | a -> b |
      +---+---+--------+
      | F | F | T |
      | F | T | T |
      | T | F | F |
      | T | T | T |


      So an implication is only false if the left hand side is true and the right hand side is false. Let's interpret this via the example. Our implication is intended to be a description of the real world, so all the true rows should correspond to situations that can happen in the real world, and all the false rows should correspond to situations that can not:




      1. It can happen that there is no fire and no smoke.

      2. It can happen that there is no fire, but there is smoke (which might be produced by something else than fire).

      3. It can not happen that there is fire but no smoke.

      4. It can happen that there is both fire and smoke.


      Now, let's add the contrapositive to the truth table:



      | a | b | a -> b | not(b) | not(a) | not(b) -> not(a) |
      +---+---+--------+--------+--------+------------------+
      | F | F | T | T | T | T |
      | F | T | T | F | T | T |
      | T | F | F | T | F | F |
      | T | T | T | F | F | T |


      The not(b) -> not(a) is identical to the a -> b column, so the two statements are logically equivalent.



      In your example, you seem to think that the router is the cause that produces the effect of internet. However, unlike in the fire and smoke example, internet isn't a guaranteed effect of the router alone, and therefore there is not an implication from router to internet. Perhaps surprisingly, there is an implication, but it goes the other way. The router is just one of several necessary ingredients for having internet, and in an implication, the necessary part goes on the right hand side - so even though the router is one of the technical "causes" of the internet, the implication looks like this:




      If [a: you have internet], then [b: the router is on].




      This unintuitive order is easier to understand if you stop thinking about an implication as the left hand side being a physical cause of the right hand side, and just think of it as the statement "it is not the case that the left hand side is true while the right hand side is false". Let's confirm that this is a correct description of the real world (the only thing that shouldn't happen is that the left hand side is true and the right hand side is false):




      1. It can happen that you don't have internet and the router is off.

      2. It can happen that you don't have internet, but the router is on (so you've got a tech problem somewhere).

      3. It can not happen that you've got internet but the router is off.

      4. It can happen that you've got internet and the router is on.


      In this example, the contrapositive is actually the more intuitive one:




      If [not(b): the router is off], then [not(a): you don't have internet].






      (My source for everything in this answer is the first chapter of "Discrete Mathematics and Its Applications" by Kenneth H. Rosen, but I've added a few Wikipedia links.)






      share|improve this answer

































        7














        No fallacy




        Router necessary for internet




        Can be restated




        Internet sufficient for router




        Let's restate the second more elaborately




        Internet (found to be) working is sufficient (evidence)
        that the router (has to be) working






        In addition to contrapositive suggested by @virmalor you may like to see also modus tollens






        share|improve this answer

































          5














          As others have said, there is no fallacy in a contrapositive argument, i.e. "if A then B. Not B, therefore not A". But your example is faulty:




          Here is an example I could think of, in natural language: "In order to have internet (B), the router must be on (A). So, if there is no internet (not-B), that means the router is not on (not-A)" (Which is not true because there may be a problem with the providers, for example.)




          What you have written is "if B then A. Not B, therefore not A"--which is of course invalid reasoning. A better way of putting an example of the contrapositive into natural language would be "If we have an internet [connection], then the router is on. The router is off [not on], therefore we do not have an internet [connection]." Which is a valid argument of the contrapositive form.






          share|improve this answer

































            3














            It's not a fallacy, but it's only guaranteed to hold in Aristotelian logic, where we have the law of the excluded middle, which states that every proposition is either true or false.



            Example: Anyone who is a Christian believes that Jesus is their savior.



            A is the proposition that my dog is a Christian.



            B is the proposition that my dog has a belief in Jesus as her savior.



            Now B is false, because my dog has no belief in Jesus as her savior. If you're working in Aristotelian logic, then it's a perfectly valid logical conclusion to say that my dog is not a Christian: if B is false, then A must be false.



            But it's at least as reasonable to apply non-Aristotelian logic here. We can say that Christian and non-Christian are silly categories to apply to a dog. Then we say that neither A nor not-A is true. The truth-value of A is undefined. The fact that B is false tells us that A is not true, but it doesn't tell us that not-A is true.



            Most mathematicians spend most or all of their time working within Aristotelian logic, but that doesn't mean that non-Aristotelian logic is invalid or not accepted by the mathematical community. Similarly, some people play basketball, some play tennis, and some play both. There is no controversy over which set of rules is the right set of rules, and everyone understands implicitly which set of rules is agreed upon when they're playing.



            In most cases in everyday life, Aristotelian logic just isn't a good model of how we reason about the world. If a friend asserts that jazz is more fun than pop music, I would probably answer, "Well, yes and no..."






            share|improve this answer

































              3














              "In order to have internet, the router must be on" corresponds to the implication "if there is internet, the router is on." The contrapositive is "if the router is not on, there is no internet."



              The router being on is a necessary condition for internet, while having internet is a sufficient condition for the router being on. (It can be tricky translating statements to implications, especially when there are modals like "must"! The translation reduces the statement to its barest essence with a material implication, and the translation is usually lossy.)






              share|improve this answer































                3














                Your translation from natural English to formal logic is where you go wrong.



                "If-A then B" precisely means that A is a sufficient condition for B. - Ie if we have case A we can know that we also have case B.



                "In order to have internet (B), the router must be on (A)." describes a necessary condition. This can be written as "Only if A then B" or "If B then A". The contrapositive would be "If Not-A then Not-B"; Ie. "If the router is not on then we do not have internet."






                share|improve this answer



















                • 1





                  This is the crux of the matter. "if A then B" does not "require other conditions for B". It simply states that if A holds, then B must also hold.

                  – HolKann
                  Feb 26 at 19:09



















                1














                Yes, you are confused due to the use of some natural language which are by nature all ambiguous/not precise (but that makes them very powerful!). Ancient philosophers were strongly interested on this problem and that leads to formal logic.



                For contrapositive the real life example could be:




                if it's raining then the ground is wet.




                And its contrapositive formulæ :




                if the ground is not wet then it is not raining.




                Sorry for @eurieka's answer but his example is not so good due to the fact that the real life relation in between rain and umbrella use is not so direct (I never use umbrella raining or not). While it is more easy to agree on rain and wet ground.



                In formal logic you can't repudiate a contrapositive, if you think the contrapositive is not correct then the initial proposition is also wrong.



                In formal logic, you can't state something like : router on implies internet accessible, and then after add another condition about provider state. If it is the case, then you need to reformulate your problem taking into account everything.






                share|improve this answer































                  1














                  "If A, then B. Therefore, if not-B, then not-A":



                  A == Duck



                  B == Swim.



                  If A, then B == If it's a Duck then it can swim.



                  if not-B, then not-A == If this little guy can't swim, then it cannot be a duck.



                  Why? Because a duck can swim! how can it be a duck but cannot swim?



                  Isn't this logic easy to see?






                  share|improve this answer
























                  • It might be a sick duck? But I think your example works. If you have references for the reader for more information that would strengthen your answer and direct the readers who want more information to places you have selected for them. Welcome!

                    – Frank Hubeny
                    Feb 25 at 18:08





















                  1














                  As others have said, this is not a fallacy.



                  Visuals can make this easier without the need to follow potentially confusing logical progressions, or the need to know any logical terms. Here's a Venn diagram:



                  Venn FTW




                  • "If A then B" means that A is a subset of B.

                  • Anything outside of B ("not-B") is clearly "not-A"

                  • For good measure, if you also want to consider "not-A", this means "everything outside of A", which could be B, but could also mean "not-B".




                  Extending this to the router & internet example in the question:



                  Router and internet Venn




                  • The internet is only available to you if your router is on

                  • It is very possible to have no internet availability, even when your router is on

                  • The internet is certainly not available to you if your router/modem is not on






                  share|improve this answer

































                    1














                    "If A, then B. Therefore, if not-B, then not-A"



                    This is not a fallacy because what it is stating is equivalent to this:



                    "If there is A, then there must be a B--every time. Therefore, if there is no B, there could not have been an A, because had there been an A the rule is that there must be a B. So in any case where there is no B, there must have been no A."






                    share|improve this answer































                      1














                      You probably have mistranslated the statement :




                      "In order to have internet (B), the router must be on (A). So, if
                      there is no internet (not-B), that means the router is not on (not-A)"




                      It is not a fallacy, but rather a mistranslation.



                      If you have something that comes before therefore, it is already a premise.



                      Your premise:




                      If A then B




                      Is already supposed to be the case. It does not say that there are other premises that contribute to the conditional.



                      It is analogous to: If it is raining then there are clouds.



                      And we can infer from it that if there are no clouds then there is not raining.



                      So, there is no fallacy here.



                      But remember this : Deductive reasoning is about the logical form of your argument.



                      If we suppose (just suppose) that the conditional if A then B is true, then it follows that the conditional If not-B then not-A is true.



                      Suppose that the conditional if A then B lacks C, which is also needed for B to take place.



                      Well, that's not deductive problem (not a non-sequitur), but it is a problem related either to the truth or reliability of the premise, or simply a mistranslation as I suggested.



                      If I say that If you worked then you did make money , then it is the same as if I say if you did not make money, then you did not work.



                      But whether work is sufficient to make money or not, that's another subject. Deductive arguments are only about the truth of the conclusion, given the premises are true.



                      Now back to your example :




                      "In order to have internet (B), the router must be on (A). So, if
                      there is no internet (not-B), that means the router is not on (not-A)"




                      In this case in order to have internet (B) the router must be on (A) should not be translated to if A then B , but instead if B then A.



                      If there is internet -B- then the router is ON -A-.



                      That is the correct conditional here :




                      • If there is internet then the router is On

                      • therefore, if the router is not-On then there is no-Internet.


                      So, your premise was mistranslated .



                      Consider the clouds example, to see what I mean : Even though clouds can cause rain (with other factors), it does not mean that clouds should be the antecedent in a conditional, so if there are clouds then it is raining is a false premise.



                      But it means that if it rains then there are clouds, so the cause should be a consequent, not an antecedent. Think about it.






                      share|improve this answer





















                      • 1





                        Are you sure?

                        – JBentley
                        Feb 26 at 13:09











                      • Clouds do not have to be necessarily right about one's head, it is possible for the clouds to be in a distant area, and wind does the rest. So, it rains therefore there are clouds (either here or somewhere else)

                        – SmootQ
                        Feb 26 at 13:32











                      • True, but then that complicates the logic. E.g. no clouds -> no rain, do we mean clouds that you can see? Because the clouds can be elsewhere and causing rain. But if you mean clouds anywhere (whether visible or not), there is always a cloud somewhere (and indeed, probably always rain somewhere). Also, I'm not a meteorologist, but it seems to me we can have a situation where the cloud has just finished evaporating, and the last drops of rain are still falling.

                        – JBentley
                        Feb 26 at 13:42













                      • No, the conditional means "there exists a cloud" whether we see it or not, even if not right above us. There is no rain without clouds. This premise is inductive in nature: so there is an implied "probably".

                        – SmootQ
                        Feb 26 at 15:05



















                      0














                      Another way of looking at this is with a diagram.



                      In your example, “If A, then B” is consistent with A and B both being true, or A being false and B being anything — but not with A being true and B being false.  You could draw that as:



                          A:║ true │ false
                      B ║ │
                      ══════╬══════╪══════
                      true ║ ✔️ │ ✔️
                      ──────╫──────┼──────
                      false ║ ✘✘✘ │ ✔️


                      But that's exactly the same diagram you'd get for “If not-B then not-A.”  So the two are equivalent.



                      (As per other answers, this assumes classical two-valued logic, of course.  This is basically a Karnaugh map, though that's mainly used for simplifying boolean expressions.  You could alternatively use a truth table, as per other answers, but I find this diagram easier to follow.)






                      share|improve this answer

































                        0














                        I think that you need to express the problem as so: If A->B implies that B will happen ALWAYS when A has occurred. Then, if B has not occurred, you are sure that A has not occurred.






                        share|improve this answer
























                        • but will B only happen if A?

                          – Mr. Kennedy
                          Mar 1 at 19:37










                        protected by Eliran Feb 25 at 18:35



                        Thank you for your interest in this question.
                        Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).



                        Would you like to answer one of these unanswered questions instead?














                        16 Answers
                        16






                        active

                        oldest

                        votes








                        16 Answers
                        16






                        active

                        oldest

                        votes









                        active

                        oldest

                        votes






                        active

                        oldest

                        votes









                        44














                        There are multiple good answers here, including references to modus tollens and the contrapositive (both of which are correct).



                        What helps me understand this concept is a more intuitive/layman's perspective.



                        Say that if it's raining, you ALWAYS bring an umbrella outside. (Assume that you have a perfect memory). Therefore we have:




                        Raining -> Umbrella




                        However, if you have an umbrella, does it mean that it's raining? Well, no. You might be bringing an umbrella for a trip, using it as a parasol in the scorching summer sun, or even simply returning it to a friend.



                        But if you don't have an umbrella, is it raining? No, and this is supported by modus tollens/contrapositive. That is to say:




                        Not(Umbrella) -> Not(Raining)




                        And that makes sense to us. If it were raining, you would've brought your umbrella. We said that you ALWAYS bring an umbrella if it's raining. So if you don't have it, it can't possibly be raining.



                        Let's apply it to your example:




                        "In order to have internet (B), the router must be on (A). So, if
                        there is no internet (not-B), that means the router is not on (not-A)"




                        This is actually not formulated properly; A and B have been reversed. This is probably why it's confusing. (Intuitively, it's in line with what you've been saying: if we don't have Internet, it's not necessarily because of the router.)



                        So let's word it differently:




                        If we have an Internet connection (A), the router must be on (B).



                        InternetConnected -> RouterOn



                        A -> B




                        But if the router isn't on, then we can't possibly connect to the Internet. (How would we even do that?) We know this is the case, because we said that if we're connected to the Internet, the router MUST be on.




                        Not(RouterOn) -> Not(InternetConnected)



                        Not(B) -> Not(A)




                        Hope this helps! :)



                        Edit: My "rain" example was criticized for clarity, since the real-life relationship isn't as straightforward. (Fair enough.) We can apply the same thing to dogs, though, and it works the same way but exploits the IS-A/inheritance relationship you're probably familiar with if you have a CS background.




                        Corgi => Dog




                        All corgis are dogs. But are all dogs corgis? No! However, we DO know that if it's not a dog, it's clearly not a Corgi (since a Corgi is a type of dog).




                        Not(Dog) => Not(Corgi)




                        Also, to echo the other users: this is not a fallacy. You can find more about logical fallacies here.






                        share|improve this answer






























                          44














                          There are multiple good answers here, including references to modus tollens and the contrapositive (both of which are correct).



                          What helps me understand this concept is a more intuitive/layman's perspective.



                          Say that if it's raining, you ALWAYS bring an umbrella outside. (Assume that you have a perfect memory). Therefore we have:




                          Raining -> Umbrella




                          However, if you have an umbrella, does it mean that it's raining? Well, no. You might be bringing an umbrella for a trip, using it as a parasol in the scorching summer sun, or even simply returning it to a friend.



                          But if you don't have an umbrella, is it raining? No, and this is supported by modus tollens/contrapositive. That is to say:




                          Not(Umbrella) -> Not(Raining)




                          And that makes sense to us. If it were raining, you would've brought your umbrella. We said that you ALWAYS bring an umbrella if it's raining. So if you don't have it, it can't possibly be raining.



                          Let's apply it to your example:




                          "In order to have internet (B), the router must be on (A). So, if
                          there is no internet (not-B), that means the router is not on (not-A)"




                          This is actually not formulated properly; A and B have been reversed. This is probably why it's confusing. (Intuitively, it's in line with what you've been saying: if we don't have Internet, it's not necessarily because of the router.)



                          So let's word it differently:




                          If we have an Internet connection (A), the router must be on (B).



                          InternetConnected -> RouterOn



                          A -> B




                          But if the router isn't on, then we can't possibly connect to the Internet. (How would we even do that?) We know this is the case, because we said that if we're connected to the Internet, the router MUST be on.




                          Not(RouterOn) -> Not(InternetConnected)



                          Not(B) -> Not(A)




                          Hope this helps! :)



                          Edit: My "rain" example was criticized for clarity, since the real-life relationship isn't as straightforward. (Fair enough.) We can apply the same thing to dogs, though, and it works the same way but exploits the IS-A/inheritance relationship you're probably familiar with if you have a CS background.




                          Corgi => Dog




                          All corgis are dogs. But are all dogs corgis? No! However, we DO know that if it's not a dog, it's clearly not a Corgi (since a Corgi is a type of dog).




                          Not(Dog) => Not(Corgi)




                          Also, to echo the other users: this is not a fallacy. You can find more about logical fallacies here.






                          share|improve this answer




























                            44












                            44








                            44







                            There are multiple good answers here, including references to modus tollens and the contrapositive (both of which are correct).



                            What helps me understand this concept is a more intuitive/layman's perspective.



                            Say that if it's raining, you ALWAYS bring an umbrella outside. (Assume that you have a perfect memory). Therefore we have:




                            Raining -> Umbrella




                            However, if you have an umbrella, does it mean that it's raining? Well, no. You might be bringing an umbrella for a trip, using it as a parasol in the scorching summer sun, or even simply returning it to a friend.



                            But if you don't have an umbrella, is it raining? No, and this is supported by modus tollens/contrapositive. That is to say:




                            Not(Umbrella) -> Not(Raining)




                            And that makes sense to us. If it were raining, you would've brought your umbrella. We said that you ALWAYS bring an umbrella if it's raining. So if you don't have it, it can't possibly be raining.



                            Let's apply it to your example:




                            "In order to have internet (B), the router must be on (A). So, if
                            there is no internet (not-B), that means the router is not on (not-A)"




                            This is actually not formulated properly; A and B have been reversed. This is probably why it's confusing. (Intuitively, it's in line with what you've been saying: if we don't have Internet, it's not necessarily because of the router.)



                            So let's word it differently:




                            If we have an Internet connection (A), the router must be on (B).



                            InternetConnected -> RouterOn



                            A -> B




                            But if the router isn't on, then we can't possibly connect to the Internet. (How would we even do that?) We know this is the case, because we said that if we're connected to the Internet, the router MUST be on.




                            Not(RouterOn) -> Not(InternetConnected)



                            Not(B) -> Not(A)




                            Hope this helps! :)



                            Edit: My "rain" example was criticized for clarity, since the real-life relationship isn't as straightforward. (Fair enough.) We can apply the same thing to dogs, though, and it works the same way but exploits the IS-A/inheritance relationship you're probably familiar with if you have a CS background.




                            Corgi => Dog




                            All corgis are dogs. But are all dogs corgis? No! However, we DO know that if it's not a dog, it's clearly not a Corgi (since a Corgi is a type of dog).




                            Not(Dog) => Not(Corgi)




                            Also, to echo the other users: this is not a fallacy. You can find more about logical fallacies here.






                            share|improve this answer















                            There are multiple good answers here, including references to modus tollens and the contrapositive (both of which are correct).



                            What helps me understand this concept is a more intuitive/layman's perspective.



                            Say that if it's raining, you ALWAYS bring an umbrella outside. (Assume that you have a perfect memory). Therefore we have:




                            Raining -> Umbrella




                            However, if you have an umbrella, does it mean that it's raining? Well, no. You might be bringing an umbrella for a trip, using it as a parasol in the scorching summer sun, or even simply returning it to a friend.



                            But if you don't have an umbrella, is it raining? No, and this is supported by modus tollens/contrapositive. That is to say:




                            Not(Umbrella) -> Not(Raining)




                            And that makes sense to us. If it were raining, you would've brought your umbrella. We said that you ALWAYS bring an umbrella if it's raining. So if you don't have it, it can't possibly be raining.



                            Let's apply it to your example:




                            "In order to have internet (B), the router must be on (A). So, if
                            there is no internet (not-B), that means the router is not on (not-A)"




                            This is actually not formulated properly; A and B have been reversed. This is probably why it's confusing. (Intuitively, it's in line with what you've been saying: if we don't have Internet, it's not necessarily because of the router.)



                            So let's word it differently:




                            If we have an Internet connection (A), the router must be on (B).



                            InternetConnected -> RouterOn



                            A -> B




                            But if the router isn't on, then we can't possibly connect to the Internet. (How would we even do that?) We know this is the case, because we said that if we're connected to the Internet, the router MUST be on.




                            Not(RouterOn) -> Not(InternetConnected)



                            Not(B) -> Not(A)




                            Hope this helps! :)



                            Edit: My "rain" example was criticized for clarity, since the real-life relationship isn't as straightforward. (Fair enough.) We can apply the same thing to dogs, though, and it works the same way but exploits the IS-A/inheritance relationship you're probably familiar with if you have a CS background.




                            Corgi => Dog




                            All corgis are dogs. But are all dogs corgis? No! However, we DO know that if it's not a dog, it's clearly not a Corgi (since a Corgi is a type of dog).




                            Not(Dog) => Not(Corgi)




                            Also, to echo the other users: this is not a fallacy. You can find more about logical fallacies here.







                            share|improve this answer














                            share|improve this answer



                            share|improve this answer








                            edited Feb 26 at 10:37

























                            answered Feb 24 at 14:55









                            euriekaeurieka

                            54114




                            54114























                                14














                                ‘If not B then not A’ is the contrapositive of ‘If A then B’ and is logically valid.



                                By saying ‘if A then B’, the author is saying that whenever A happens, B will definitely happen. Hence, if B does not happen, it is clear that A did not happen (if not B then not A).






                                share|improve this answer




























                                  14














                                  ‘If not B then not A’ is the contrapositive of ‘If A then B’ and is logically valid.



                                  By saying ‘if A then B’, the author is saying that whenever A happens, B will definitely happen. Hence, if B does not happen, it is clear that A did not happen (if not B then not A).






                                  share|improve this answer


























                                    14












                                    14








                                    14







                                    ‘If not B then not A’ is the contrapositive of ‘If A then B’ and is logically valid.



                                    By saying ‘if A then B’, the author is saying that whenever A happens, B will definitely happen. Hence, if B does not happen, it is clear that A did not happen (if not B then not A).






                                    share|improve this answer













                                    ‘If not B then not A’ is the contrapositive of ‘If A then B’ and is logically valid.



                                    By saying ‘if A then B’, the author is saying that whenever A happens, B will definitely happen. Hence, if B does not happen, it is clear that A did not happen (if not B then not A).







                                    share|improve this answer












                                    share|improve this answer



                                    share|improve this answer










                                    answered Feb 24 at 12:01









                                    danielloiddanielloid

                                    2412




                                    2412























                                        13














                                        You are correct: The statement does not show a sufficient condition!



                                        Consider this statement: "If (A) it rains, then (B) the street will be wet."



                                        Assuming the street is always wet after it rained (because it is not covered by a roof or something), the contraposition would hold true!



                                        Contraposition: "If the street is not wet, it did not rain"



                                        I think what is giving you troubles is that the street in this example can be wet without rain (someone sprayed water onto it), therefore the street is not always wet just because of rain.
                                        That's called an inverse and differentiation can sometimes be hard.






                                        share|improve this answer



















                                        • 1





                                          That's a strange way of contradicting yourself. A ==> B strictly forces !B ==> !A . That's what the OP was asking... unless you're just pointing out that he contradicted his own question.

                                          – Carl Witthoft
                                          Feb 26 at 15:33











                                        • @CarlWitthoft I was saying that the statement is not complete. The street ALWAYS has to be wet when it rained. The internet (in OPs example) works ONLY because of that router. In real life, conversations are not really always fully logical statements. The example he used, shows that well, I think.

                                          – Moritz K
                                          Feb 27 at 17:56


















                                        13














                                        You are correct: The statement does not show a sufficient condition!



                                        Consider this statement: "If (A) it rains, then (B) the street will be wet."



                                        Assuming the street is always wet after it rained (because it is not covered by a roof or something), the contraposition would hold true!



                                        Contraposition: "If the street is not wet, it did not rain"



                                        I think what is giving you troubles is that the street in this example can be wet without rain (someone sprayed water onto it), therefore the street is not always wet just because of rain.
                                        That's called an inverse and differentiation can sometimes be hard.






                                        share|improve this answer



















                                        • 1





                                          That's a strange way of contradicting yourself. A ==> B strictly forces !B ==> !A . That's what the OP was asking... unless you're just pointing out that he contradicted his own question.

                                          – Carl Witthoft
                                          Feb 26 at 15:33











                                        • @CarlWitthoft I was saying that the statement is not complete. The street ALWAYS has to be wet when it rained. The internet (in OPs example) works ONLY because of that router. In real life, conversations are not really always fully logical statements. The example he used, shows that well, I think.

                                          – Moritz K
                                          Feb 27 at 17:56
















                                        13












                                        13








                                        13







                                        You are correct: The statement does not show a sufficient condition!



                                        Consider this statement: "If (A) it rains, then (B) the street will be wet."



                                        Assuming the street is always wet after it rained (because it is not covered by a roof or something), the contraposition would hold true!



                                        Contraposition: "If the street is not wet, it did not rain"



                                        I think what is giving you troubles is that the street in this example can be wet without rain (someone sprayed water onto it), therefore the street is not always wet just because of rain.
                                        That's called an inverse and differentiation can sometimes be hard.






                                        share|improve this answer













                                        You are correct: The statement does not show a sufficient condition!



                                        Consider this statement: "If (A) it rains, then (B) the street will be wet."



                                        Assuming the street is always wet after it rained (because it is not covered by a roof or something), the contraposition would hold true!



                                        Contraposition: "If the street is not wet, it did not rain"



                                        I think what is giving you troubles is that the street in this example can be wet without rain (someone sprayed water onto it), therefore the street is not always wet just because of rain.
                                        That's called an inverse and differentiation can sometimes be hard.







                                        share|improve this answer












                                        share|improve this answer



                                        share|improve this answer










                                        answered Feb 24 at 13:45









                                        Moritz KMoritz K

                                        1392




                                        1392








                                        • 1





                                          That's a strange way of contradicting yourself. A ==> B strictly forces !B ==> !A . That's what the OP was asking... unless you're just pointing out that he contradicted his own question.

                                          – Carl Witthoft
                                          Feb 26 at 15:33











                                        • @CarlWitthoft I was saying that the statement is not complete. The street ALWAYS has to be wet when it rained. The internet (in OPs example) works ONLY because of that router. In real life, conversations are not really always fully logical statements. The example he used, shows that well, I think.

                                          – Moritz K
                                          Feb 27 at 17:56
















                                        • 1





                                          That's a strange way of contradicting yourself. A ==> B strictly forces !B ==> !A . That's what the OP was asking... unless you're just pointing out that he contradicted his own question.

                                          – Carl Witthoft
                                          Feb 26 at 15:33











                                        • @CarlWitthoft I was saying that the statement is not complete. The street ALWAYS has to be wet when it rained. The internet (in OPs example) works ONLY because of that router. In real life, conversations are not really always fully logical statements. The example he used, shows that well, I think.

                                          – Moritz K
                                          Feb 27 at 17:56










                                        1




                                        1





                                        That's a strange way of contradicting yourself. A ==> B strictly forces !B ==> !A . That's what the OP was asking... unless you're just pointing out that he contradicted his own question.

                                        – Carl Witthoft
                                        Feb 26 at 15:33





                                        That's a strange way of contradicting yourself. A ==> B strictly forces !B ==> !A . That's what the OP was asking... unless you're just pointing out that he contradicted his own question.

                                        – Carl Witthoft
                                        Feb 26 at 15:33













                                        @CarlWitthoft I was saying that the statement is not complete. The street ALWAYS has to be wet when it rained. The internet (in OPs example) works ONLY because of that router. In real life, conversations are not really always fully logical statements. The example he used, shows that well, I think.

                                        – Moritz K
                                        Feb 27 at 17:56







                                        @CarlWitthoft I was saying that the statement is not complete. The street ALWAYS has to be wet when it rained. The internet (in OPs example) works ONLY because of that router. In real life, conversations are not really always fully logical statements. The example he used, shows that well, I think.

                                        – Moritz K
                                        Feb 27 at 17:56













                                        8














                                        Your confusion might stem from the fact that implications in propositional logic don't always correspond intuitively to cause and effect in the real world. I'll get back to your example later; for now, let's focus on a more intuitive implication:




                                        If [a: there is fire], then [b: there is smoke].




                                        This is a natural process in which the cause (fire) always produces the effect (smoke), and that's a situation that can be described with an implication. (In the real world, there are substances that burn without smoke, so let's constrain ourselves to wood fires.) In this situation, you probably agree that the contrapositive is true as well:




                                        If [not(b): there is no smoke], then [not(a): there is no fire].




                                        This makes intuitive sense - we're not observing the effect that is always produced by the cause, so the cause can't be occurring.



                                        We can formally prove why the contrapositive is equivalent to the original statement. Here is the truth table that defines implications (T for true and F for false):



                                        | a | b | a -> b |
                                        +---+---+--------+
                                        | F | F | T |
                                        | F | T | T |
                                        | T | F | F |
                                        | T | T | T |


                                        So an implication is only false if the left hand side is true and the right hand side is false. Let's interpret this via the example. Our implication is intended to be a description of the real world, so all the true rows should correspond to situations that can happen in the real world, and all the false rows should correspond to situations that can not:




                                        1. It can happen that there is no fire and no smoke.

                                        2. It can happen that there is no fire, but there is smoke (which might be produced by something else than fire).

                                        3. It can not happen that there is fire but no smoke.

                                        4. It can happen that there is both fire and smoke.


                                        Now, let's add the contrapositive to the truth table:



                                        | a | b | a -> b | not(b) | not(a) | not(b) -> not(a) |
                                        +---+---+--------+--------+--------+------------------+
                                        | F | F | T | T | T | T |
                                        | F | T | T | F | T | T |
                                        | T | F | F | T | F | F |
                                        | T | T | T | F | F | T |


                                        The not(b) -> not(a) is identical to the a -> b column, so the two statements are logically equivalent.



                                        In your example, you seem to think that the router is the cause that produces the effect of internet. However, unlike in the fire and smoke example, internet isn't a guaranteed effect of the router alone, and therefore there is not an implication from router to internet. Perhaps surprisingly, there is an implication, but it goes the other way. The router is just one of several necessary ingredients for having internet, and in an implication, the necessary part goes on the right hand side - so even though the router is one of the technical "causes" of the internet, the implication looks like this:




                                        If [a: you have internet], then [b: the router is on].




                                        This unintuitive order is easier to understand if you stop thinking about an implication as the left hand side being a physical cause of the right hand side, and just think of it as the statement "it is not the case that the left hand side is true while the right hand side is false". Let's confirm that this is a correct description of the real world (the only thing that shouldn't happen is that the left hand side is true and the right hand side is false):




                                        1. It can happen that you don't have internet and the router is off.

                                        2. It can happen that you don't have internet, but the router is on (so you've got a tech problem somewhere).

                                        3. It can not happen that you've got internet but the router is off.

                                        4. It can happen that you've got internet and the router is on.


                                        In this example, the contrapositive is actually the more intuitive one:




                                        If [not(b): the router is off], then [not(a): you don't have internet].






                                        (My source for everything in this answer is the first chapter of "Discrete Mathematics and Its Applications" by Kenneth H. Rosen, but I've added a few Wikipedia links.)






                                        share|improve this answer






























                                          8














                                          Your confusion might stem from the fact that implications in propositional logic don't always correspond intuitively to cause and effect in the real world. I'll get back to your example later; for now, let's focus on a more intuitive implication:




                                          If [a: there is fire], then [b: there is smoke].




                                          This is a natural process in which the cause (fire) always produces the effect (smoke), and that's a situation that can be described with an implication. (In the real world, there are substances that burn without smoke, so let's constrain ourselves to wood fires.) In this situation, you probably agree that the contrapositive is true as well:




                                          If [not(b): there is no smoke], then [not(a): there is no fire].




                                          This makes intuitive sense - we're not observing the effect that is always produced by the cause, so the cause can't be occurring.



                                          We can formally prove why the contrapositive is equivalent to the original statement. Here is the truth table that defines implications (T for true and F for false):



                                          | a | b | a -> b |
                                          +---+---+--------+
                                          | F | F | T |
                                          | F | T | T |
                                          | T | F | F |
                                          | T | T | T |


                                          So an implication is only false if the left hand side is true and the right hand side is false. Let's interpret this via the example. Our implication is intended to be a description of the real world, so all the true rows should correspond to situations that can happen in the real world, and all the false rows should correspond to situations that can not:




                                          1. It can happen that there is no fire and no smoke.

                                          2. It can happen that there is no fire, but there is smoke (which might be produced by something else than fire).

                                          3. It can not happen that there is fire but no smoke.

                                          4. It can happen that there is both fire and smoke.


                                          Now, let's add the contrapositive to the truth table:



                                          | a | b | a -> b | not(b) | not(a) | not(b) -> not(a) |
                                          +---+---+--------+--------+--------+------------------+
                                          | F | F | T | T | T | T |
                                          | F | T | T | F | T | T |
                                          | T | F | F | T | F | F |
                                          | T | T | T | F | F | T |


                                          The not(b) -> not(a) is identical to the a -> b column, so the two statements are logically equivalent.



                                          In your example, you seem to think that the router is the cause that produces the effect of internet. However, unlike in the fire and smoke example, internet isn't a guaranteed effect of the router alone, and therefore there is not an implication from router to internet. Perhaps surprisingly, there is an implication, but it goes the other way. The router is just one of several necessary ingredients for having internet, and in an implication, the necessary part goes on the right hand side - so even though the router is one of the technical "causes" of the internet, the implication looks like this:




                                          If [a: you have internet], then [b: the router is on].




                                          This unintuitive order is easier to understand if you stop thinking about an implication as the left hand side being a physical cause of the right hand side, and just think of it as the statement "it is not the case that the left hand side is true while the right hand side is false". Let's confirm that this is a correct description of the real world (the only thing that shouldn't happen is that the left hand side is true and the right hand side is false):




                                          1. It can happen that you don't have internet and the router is off.

                                          2. It can happen that you don't have internet, but the router is on (so you've got a tech problem somewhere).

                                          3. It can not happen that you've got internet but the router is off.

                                          4. It can happen that you've got internet and the router is on.


                                          In this example, the contrapositive is actually the more intuitive one:




                                          If [not(b): the router is off], then [not(a): you don't have internet].






                                          (My source for everything in this answer is the first chapter of "Discrete Mathematics and Its Applications" by Kenneth H. Rosen, but I've added a few Wikipedia links.)






                                          share|improve this answer




























                                            8












                                            8








                                            8







                                            Your confusion might stem from the fact that implications in propositional logic don't always correspond intuitively to cause and effect in the real world. I'll get back to your example later; for now, let's focus on a more intuitive implication:




                                            If [a: there is fire], then [b: there is smoke].




                                            This is a natural process in which the cause (fire) always produces the effect (smoke), and that's a situation that can be described with an implication. (In the real world, there are substances that burn without smoke, so let's constrain ourselves to wood fires.) In this situation, you probably agree that the contrapositive is true as well:




                                            If [not(b): there is no smoke], then [not(a): there is no fire].




                                            This makes intuitive sense - we're not observing the effect that is always produced by the cause, so the cause can't be occurring.



                                            We can formally prove why the contrapositive is equivalent to the original statement. Here is the truth table that defines implications (T for true and F for false):



                                            | a | b | a -> b |
                                            +---+---+--------+
                                            | F | F | T |
                                            | F | T | T |
                                            | T | F | F |
                                            | T | T | T |


                                            So an implication is only false if the left hand side is true and the right hand side is false. Let's interpret this via the example. Our implication is intended to be a description of the real world, so all the true rows should correspond to situations that can happen in the real world, and all the false rows should correspond to situations that can not:




                                            1. It can happen that there is no fire and no smoke.

                                            2. It can happen that there is no fire, but there is smoke (which might be produced by something else than fire).

                                            3. It can not happen that there is fire but no smoke.

                                            4. It can happen that there is both fire and smoke.


                                            Now, let's add the contrapositive to the truth table:



                                            | a | b | a -> b | not(b) | not(a) | not(b) -> not(a) |
                                            +---+---+--------+--------+--------+------------------+
                                            | F | F | T | T | T | T |
                                            | F | T | T | F | T | T |
                                            | T | F | F | T | F | F |
                                            | T | T | T | F | F | T |


                                            The not(b) -> not(a) is identical to the a -> b column, so the two statements are logically equivalent.



                                            In your example, you seem to think that the router is the cause that produces the effect of internet. However, unlike in the fire and smoke example, internet isn't a guaranteed effect of the router alone, and therefore there is not an implication from router to internet. Perhaps surprisingly, there is an implication, but it goes the other way. The router is just one of several necessary ingredients for having internet, and in an implication, the necessary part goes on the right hand side - so even though the router is one of the technical "causes" of the internet, the implication looks like this:




                                            If [a: you have internet], then [b: the router is on].




                                            This unintuitive order is easier to understand if you stop thinking about an implication as the left hand side being a physical cause of the right hand side, and just think of it as the statement "it is not the case that the left hand side is true while the right hand side is false". Let's confirm that this is a correct description of the real world (the only thing that shouldn't happen is that the left hand side is true and the right hand side is false):




                                            1. It can happen that you don't have internet and the router is off.

                                            2. It can happen that you don't have internet, but the router is on (so you've got a tech problem somewhere).

                                            3. It can not happen that you've got internet but the router is off.

                                            4. It can happen that you've got internet and the router is on.


                                            In this example, the contrapositive is actually the more intuitive one:




                                            If [not(b): the router is off], then [not(a): you don't have internet].






                                            (My source for everything in this answer is the first chapter of "Discrete Mathematics and Its Applications" by Kenneth H. Rosen, but I've added a few Wikipedia links.)






                                            share|improve this answer















                                            Your confusion might stem from the fact that implications in propositional logic don't always correspond intuitively to cause and effect in the real world. I'll get back to your example later; for now, let's focus on a more intuitive implication:




                                            If [a: there is fire], then [b: there is smoke].




                                            This is a natural process in which the cause (fire) always produces the effect (smoke), and that's a situation that can be described with an implication. (In the real world, there are substances that burn without smoke, so let's constrain ourselves to wood fires.) In this situation, you probably agree that the contrapositive is true as well:




                                            If [not(b): there is no smoke], then [not(a): there is no fire].




                                            This makes intuitive sense - we're not observing the effect that is always produced by the cause, so the cause can't be occurring.



                                            We can formally prove why the contrapositive is equivalent to the original statement. Here is the truth table that defines implications (T for true and F for false):



                                            | a | b | a -> b |
                                            +---+---+--------+
                                            | F | F | T |
                                            | F | T | T |
                                            | T | F | F |
                                            | T | T | T |


                                            So an implication is only false if the left hand side is true and the right hand side is false. Let's interpret this via the example. Our implication is intended to be a description of the real world, so all the true rows should correspond to situations that can happen in the real world, and all the false rows should correspond to situations that can not:




                                            1. It can happen that there is no fire and no smoke.

                                            2. It can happen that there is no fire, but there is smoke (which might be produced by something else than fire).

                                            3. It can not happen that there is fire but no smoke.

                                            4. It can happen that there is both fire and smoke.


                                            Now, let's add the contrapositive to the truth table:



                                            | a | b | a -> b | not(b) | not(a) | not(b) -> not(a) |
                                            +---+---+--------+--------+--------+------------------+
                                            | F | F | T | T | T | T |
                                            | F | T | T | F | T | T |
                                            | T | F | F | T | F | F |
                                            | T | T | T | F | F | T |


                                            The not(b) -> not(a) is identical to the a -> b column, so the two statements are logically equivalent.



                                            In your example, you seem to think that the router is the cause that produces the effect of internet. However, unlike in the fire and smoke example, internet isn't a guaranteed effect of the router alone, and therefore there is not an implication from router to internet. Perhaps surprisingly, there is an implication, but it goes the other way. The router is just one of several necessary ingredients for having internet, and in an implication, the necessary part goes on the right hand side - so even though the router is one of the technical "causes" of the internet, the implication looks like this:




                                            If [a: you have internet], then [b: the router is on].




                                            This unintuitive order is easier to understand if you stop thinking about an implication as the left hand side being a physical cause of the right hand side, and just think of it as the statement "it is not the case that the left hand side is true while the right hand side is false". Let's confirm that this is a correct description of the real world (the only thing that shouldn't happen is that the left hand side is true and the right hand side is false):




                                            1. It can happen that you don't have internet and the router is off.

                                            2. It can happen that you don't have internet, but the router is on (so you've got a tech problem somewhere).

                                            3. It can not happen that you've got internet but the router is off.

                                            4. It can happen that you've got internet and the router is on.


                                            In this example, the contrapositive is actually the more intuitive one:




                                            If [not(b): the router is off], then [not(a): you don't have internet].






                                            (My source for everything in this answer is the first chapter of "Discrete Mathematics and Its Applications" by Kenneth H. Rosen, but I've added a few Wikipedia links.)







                                            share|improve this answer














                                            share|improve this answer



                                            share|improve this answer








                                            edited Feb 25 at 17:50

























                                            answered Feb 25 at 10:31









                                            Aasmund EldhusetAasmund Eldhuset

                                            1813




                                            1813























                                                7














                                                No fallacy




                                                Router necessary for internet




                                                Can be restated




                                                Internet sufficient for router




                                                Let's restate the second more elaborately




                                                Internet (found to be) working is sufficient (evidence)
                                                that the router (has to be) working






                                                In addition to contrapositive suggested by @virmalor you may like to see also modus tollens






                                                share|improve this answer






























                                                  7














                                                  No fallacy




                                                  Router necessary for internet




                                                  Can be restated




                                                  Internet sufficient for router




                                                  Let's restate the second more elaborately




                                                  Internet (found to be) working is sufficient (evidence)
                                                  that the router (has to be) working






                                                  In addition to contrapositive suggested by @virmalor you may like to see also modus tollens






                                                  share|improve this answer




























                                                    7












                                                    7








                                                    7







                                                    No fallacy




                                                    Router necessary for internet




                                                    Can be restated




                                                    Internet sufficient for router




                                                    Let's restate the second more elaborately




                                                    Internet (found to be) working is sufficient (evidence)
                                                    that the router (has to be) working






                                                    In addition to contrapositive suggested by @virmalor you may like to see also modus tollens






                                                    share|improve this answer















                                                    No fallacy




                                                    Router necessary for internet




                                                    Can be restated




                                                    Internet sufficient for router




                                                    Let's restate the second more elaborately




                                                    Internet (found to be) working is sufficient (evidence)
                                                    that the router (has to be) working






                                                    In addition to contrapositive suggested by @virmalor you may like to see also modus tollens







                                                    share|improve this answer














                                                    share|improve this answer



                                                    share|improve this answer








                                                    edited Mar 1 at 17:21









                                                    V2Blast

                                                    1094




                                                    1094










                                                    answered Feb 24 at 10:59









                                                    RusiRusi

                                                    25316




                                                    25316























                                                        5














                                                        As others have said, there is no fallacy in a contrapositive argument, i.e. "if A then B. Not B, therefore not A". But your example is faulty:




                                                        Here is an example I could think of, in natural language: "In order to have internet (B), the router must be on (A). So, if there is no internet (not-B), that means the router is not on (not-A)" (Which is not true because there may be a problem with the providers, for example.)




                                                        What you have written is "if B then A. Not B, therefore not A"--which is of course invalid reasoning. A better way of putting an example of the contrapositive into natural language would be "If we have an internet [connection], then the router is on. The router is off [not on], therefore we do not have an internet [connection]." Which is a valid argument of the contrapositive form.






                                                        share|improve this answer






























                                                          5














                                                          As others have said, there is no fallacy in a contrapositive argument, i.e. "if A then B. Not B, therefore not A". But your example is faulty:




                                                          Here is an example I could think of, in natural language: "In order to have internet (B), the router must be on (A). So, if there is no internet (not-B), that means the router is not on (not-A)" (Which is not true because there may be a problem with the providers, for example.)




                                                          What you have written is "if B then A. Not B, therefore not A"--which is of course invalid reasoning. A better way of putting an example of the contrapositive into natural language would be "If we have an internet [connection], then the router is on. The router is off [not on], therefore we do not have an internet [connection]." Which is a valid argument of the contrapositive form.






                                                          share|improve this answer




























                                                            5












                                                            5








                                                            5







                                                            As others have said, there is no fallacy in a contrapositive argument, i.e. "if A then B. Not B, therefore not A". But your example is faulty:




                                                            Here is an example I could think of, in natural language: "In order to have internet (B), the router must be on (A). So, if there is no internet (not-B), that means the router is not on (not-A)" (Which is not true because there may be a problem with the providers, for example.)




                                                            What you have written is "if B then A. Not B, therefore not A"--which is of course invalid reasoning. A better way of putting an example of the contrapositive into natural language would be "If we have an internet [connection], then the router is on. The router is off [not on], therefore we do not have an internet [connection]." Which is a valid argument of the contrapositive form.






                                                            share|improve this answer















                                                            As others have said, there is no fallacy in a contrapositive argument, i.e. "if A then B. Not B, therefore not A". But your example is faulty:




                                                            Here is an example I could think of, in natural language: "In order to have internet (B), the router must be on (A). So, if there is no internet (not-B), that means the router is not on (not-A)" (Which is not true because there may be a problem with the providers, for example.)




                                                            What you have written is "if B then A. Not B, therefore not A"--which is of course invalid reasoning. A better way of putting an example of the contrapositive into natural language would be "If we have an internet [connection], then the router is on. The router is off [not on], therefore we do not have an internet [connection]." Which is a valid argument of the contrapositive form.







                                                            share|improve this answer














                                                            share|improve this answer



                                                            share|improve this answer








                                                            edited Feb 25 at 14:37

























                                                            answered Feb 24 at 23:04









                                                            Mike MaxwellMike Maxwell

                                                            512




                                                            512























                                                                3














                                                                It's not a fallacy, but it's only guaranteed to hold in Aristotelian logic, where we have the law of the excluded middle, which states that every proposition is either true or false.



                                                                Example: Anyone who is a Christian believes that Jesus is their savior.



                                                                A is the proposition that my dog is a Christian.



                                                                B is the proposition that my dog has a belief in Jesus as her savior.



                                                                Now B is false, because my dog has no belief in Jesus as her savior. If you're working in Aristotelian logic, then it's a perfectly valid logical conclusion to say that my dog is not a Christian: if B is false, then A must be false.



                                                                But it's at least as reasonable to apply non-Aristotelian logic here. We can say that Christian and non-Christian are silly categories to apply to a dog. Then we say that neither A nor not-A is true. The truth-value of A is undefined. The fact that B is false tells us that A is not true, but it doesn't tell us that not-A is true.



                                                                Most mathematicians spend most or all of their time working within Aristotelian logic, but that doesn't mean that non-Aristotelian logic is invalid or not accepted by the mathematical community. Similarly, some people play basketball, some play tennis, and some play both. There is no controversy over which set of rules is the right set of rules, and everyone understands implicitly which set of rules is agreed upon when they're playing.



                                                                In most cases in everyday life, Aristotelian logic just isn't a good model of how we reason about the world. If a friend asserts that jazz is more fun than pop music, I would probably answer, "Well, yes and no..."






                                                                share|improve this answer






























                                                                  3














                                                                  It's not a fallacy, but it's only guaranteed to hold in Aristotelian logic, where we have the law of the excluded middle, which states that every proposition is either true or false.



                                                                  Example: Anyone who is a Christian believes that Jesus is their savior.



                                                                  A is the proposition that my dog is a Christian.



                                                                  B is the proposition that my dog has a belief in Jesus as her savior.



                                                                  Now B is false, because my dog has no belief in Jesus as her savior. If you're working in Aristotelian logic, then it's a perfectly valid logical conclusion to say that my dog is not a Christian: if B is false, then A must be false.



                                                                  But it's at least as reasonable to apply non-Aristotelian logic here. We can say that Christian and non-Christian are silly categories to apply to a dog. Then we say that neither A nor not-A is true. The truth-value of A is undefined. The fact that B is false tells us that A is not true, but it doesn't tell us that not-A is true.



                                                                  Most mathematicians spend most or all of their time working within Aristotelian logic, but that doesn't mean that non-Aristotelian logic is invalid or not accepted by the mathematical community. Similarly, some people play basketball, some play tennis, and some play both. There is no controversy over which set of rules is the right set of rules, and everyone understands implicitly which set of rules is agreed upon when they're playing.



                                                                  In most cases in everyday life, Aristotelian logic just isn't a good model of how we reason about the world. If a friend asserts that jazz is more fun than pop music, I would probably answer, "Well, yes and no..."






                                                                  share|improve this answer




























                                                                    3












                                                                    3








                                                                    3







                                                                    It's not a fallacy, but it's only guaranteed to hold in Aristotelian logic, where we have the law of the excluded middle, which states that every proposition is either true or false.



                                                                    Example: Anyone who is a Christian believes that Jesus is their savior.



                                                                    A is the proposition that my dog is a Christian.



                                                                    B is the proposition that my dog has a belief in Jesus as her savior.



                                                                    Now B is false, because my dog has no belief in Jesus as her savior. If you're working in Aristotelian logic, then it's a perfectly valid logical conclusion to say that my dog is not a Christian: if B is false, then A must be false.



                                                                    But it's at least as reasonable to apply non-Aristotelian logic here. We can say that Christian and non-Christian are silly categories to apply to a dog. Then we say that neither A nor not-A is true. The truth-value of A is undefined. The fact that B is false tells us that A is not true, but it doesn't tell us that not-A is true.



                                                                    Most mathematicians spend most or all of their time working within Aristotelian logic, but that doesn't mean that non-Aristotelian logic is invalid or not accepted by the mathematical community. Similarly, some people play basketball, some play tennis, and some play both. There is no controversy over which set of rules is the right set of rules, and everyone understands implicitly which set of rules is agreed upon when they're playing.



                                                                    In most cases in everyday life, Aristotelian logic just isn't a good model of how we reason about the world. If a friend asserts that jazz is more fun than pop music, I would probably answer, "Well, yes and no..."






                                                                    share|improve this answer















                                                                    It's not a fallacy, but it's only guaranteed to hold in Aristotelian logic, where we have the law of the excluded middle, which states that every proposition is either true or false.



                                                                    Example: Anyone who is a Christian believes that Jesus is their savior.



                                                                    A is the proposition that my dog is a Christian.



                                                                    B is the proposition that my dog has a belief in Jesus as her savior.



                                                                    Now B is false, because my dog has no belief in Jesus as her savior. If you're working in Aristotelian logic, then it's a perfectly valid logical conclusion to say that my dog is not a Christian: if B is false, then A must be false.



                                                                    But it's at least as reasonable to apply non-Aristotelian logic here. We can say that Christian and non-Christian are silly categories to apply to a dog. Then we say that neither A nor not-A is true. The truth-value of A is undefined. The fact that B is false tells us that A is not true, but it doesn't tell us that not-A is true.



                                                                    Most mathematicians spend most or all of their time working within Aristotelian logic, but that doesn't mean that non-Aristotelian logic is invalid or not accepted by the mathematical community. Similarly, some people play basketball, some play tennis, and some play both. There is no controversy over which set of rules is the right set of rules, and everyone understands implicitly which set of rules is agreed upon when they're playing.



                                                                    In most cases in everyday life, Aristotelian logic just isn't a good model of how we reason about the world. If a friend asserts that jazz is more fun than pop music, I would probably answer, "Well, yes and no..."







                                                                    share|improve this answer














                                                                    share|improve this answer



                                                                    share|improve this answer








                                                                    edited Feb 24 at 16:54

























                                                                    answered Feb 24 at 16:49









                                                                    Ben CrowellBen Crowell

                                                                    22126




                                                                    22126























                                                                        3














                                                                        "In order to have internet, the router must be on" corresponds to the implication "if there is internet, the router is on." The contrapositive is "if the router is not on, there is no internet."



                                                                        The router being on is a necessary condition for internet, while having internet is a sufficient condition for the router being on. (It can be tricky translating statements to implications, especially when there are modals like "must"! The translation reduces the statement to its barest essence with a material implication, and the translation is usually lossy.)






                                                                        share|improve this answer




























                                                                          3














                                                                          "In order to have internet, the router must be on" corresponds to the implication "if there is internet, the router is on." The contrapositive is "if the router is not on, there is no internet."



                                                                          The router being on is a necessary condition for internet, while having internet is a sufficient condition for the router being on. (It can be tricky translating statements to implications, especially when there are modals like "must"! The translation reduces the statement to its barest essence with a material implication, and the translation is usually lossy.)






                                                                          share|improve this answer


























                                                                            3












                                                                            3








                                                                            3







                                                                            "In order to have internet, the router must be on" corresponds to the implication "if there is internet, the router is on." The contrapositive is "if the router is not on, there is no internet."



                                                                            The router being on is a necessary condition for internet, while having internet is a sufficient condition for the router being on. (It can be tricky translating statements to implications, especially when there are modals like "must"! The translation reduces the statement to its barest essence with a material implication, and the translation is usually lossy.)






                                                                            share|improve this answer













                                                                            "In order to have internet, the router must be on" corresponds to the implication "if there is internet, the router is on." The contrapositive is "if the router is not on, there is no internet."



                                                                            The router being on is a necessary condition for internet, while having internet is a sufficient condition for the router being on. (It can be tricky translating statements to implications, especially when there are modals like "must"! The translation reduces the statement to its barest essence with a material implication, and the translation is usually lossy.)







                                                                            share|improve this answer












                                                                            share|improve this answer



                                                                            share|improve this answer










                                                                            answered Feb 24 at 20:03









                                                                            Kyle MillerKyle Miller

                                                                            1312




                                                                            1312























                                                                                3














                                                                                Your translation from natural English to formal logic is where you go wrong.



                                                                                "If-A then B" precisely means that A is a sufficient condition for B. - Ie if we have case A we can know that we also have case B.



                                                                                "In order to have internet (B), the router must be on (A)." describes a necessary condition. This can be written as "Only if A then B" or "If B then A". The contrapositive would be "If Not-A then Not-B"; Ie. "If the router is not on then we do not have internet."






                                                                                share|improve this answer



















                                                                                • 1





                                                                                  This is the crux of the matter. "if A then B" does not "require other conditions for B". It simply states that if A holds, then B must also hold.

                                                                                  – HolKann
                                                                                  Feb 26 at 19:09
















                                                                                3














                                                                                Your translation from natural English to formal logic is where you go wrong.



                                                                                "If-A then B" precisely means that A is a sufficient condition for B. - Ie if we have case A we can know that we also have case B.



                                                                                "In order to have internet (B), the router must be on (A)." describes a necessary condition. This can be written as "Only if A then B" or "If B then A". The contrapositive would be "If Not-A then Not-B"; Ie. "If the router is not on then we do not have internet."






                                                                                share|improve this answer



















                                                                                • 1





                                                                                  This is the crux of the matter. "if A then B" does not "require other conditions for B". It simply states that if A holds, then B must also hold.

                                                                                  – HolKann
                                                                                  Feb 26 at 19:09














                                                                                3












                                                                                3








                                                                                3







                                                                                Your translation from natural English to formal logic is where you go wrong.



                                                                                "If-A then B" precisely means that A is a sufficient condition for B. - Ie if we have case A we can know that we also have case B.



                                                                                "In order to have internet (B), the router must be on (A)." describes a necessary condition. This can be written as "Only if A then B" or "If B then A". The contrapositive would be "If Not-A then Not-B"; Ie. "If the router is not on then we do not have internet."






                                                                                share|improve this answer













                                                                                Your translation from natural English to formal logic is where you go wrong.



                                                                                "If-A then B" precisely means that A is a sufficient condition for B. - Ie if we have case A we can know that we also have case B.



                                                                                "In order to have internet (B), the router must be on (A)." describes a necessary condition. This can be written as "Only if A then B" or "If B then A". The contrapositive would be "If Not-A then Not-B"; Ie. "If the router is not on then we do not have internet."







                                                                                share|improve this answer












                                                                                share|improve this answer



                                                                                share|improve this answer










                                                                                answered Feb 25 at 9:04









                                                                                TaemyrTaemyr

                                                                                1814




                                                                                1814








                                                                                • 1





                                                                                  This is the crux of the matter. "if A then B" does not "require other conditions for B". It simply states that if A holds, then B must also hold.

                                                                                  – HolKann
                                                                                  Feb 26 at 19:09














                                                                                • 1





                                                                                  This is the crux of the matter. "if A then B" does not "require other conditions for B". It simply states that if A holds, then B must also hold.

                                                                                  – HolKann
                                                                                  Feb 26 at 19:09








                                                                                1




                                                                                1





                                                                                This is the crux of the matter. "if A then B" does not "require other conditions for B". It simply states that if A holds, then B must also hold.

                                                                                – HolKann
                                                                                Feb 26 at 19:09





                                                                                This is the crux of the matter. "if A then B" does not "require other conditions for B". It simply states that if A holds, then B must also hold.

                                                                                – HolKann
                                                                                Feb 26 at 19:09











                                                                                1














                                                                                Yes, you are confused due to the use of some natural language which are by nature all ambiguous/not precise (but that makes them very powerful!). Ancient philosophers were strongly interested on this problem and that leads to formal logic.



                                                                                For contrapositive the real life example could be:




                                                                                if it's raining then the ground is wet.




                                                                                And its contrapositive formulæ :




                                                                                if the ground is not wet then it is not raining.




                                                                                Sorry for @eurieka's answer but his example is not so good due to the fact that the real life relation in between rain and umbrella use is not so direct (I never use umbrella raining or not). While it is more easy to agree on rain and wet ground.



                                                                                In formal logic you can't repudiate a contrapositive, if you think the contrapositive is not correct then the initial proposition is also wrong.



                                                                                In formal logic, you can't state something like : router on implies internet accessible, and then after add another condition about provider state. If it is the case, then you need to reformulate your problem taking into account everything.






                                                                                share|improve this answer




























                                                                                  1














                                                                                  Yes, you are confused due to the use of some natural language which are by nature all ambiguous/not precise (but that makes them very powerful!). Ancient philosophers were strongly interested on this problem and that leads to formal logic.



                                                                                  For contrapositive the real life example could be:




                                                                                  if it's raining then the ground is wet.




                                                                                  And its contrapositive formulæ :




                                                                                  if the ground is not wet then it is not raining.




                                                                                  Sorry for @eurieka's answer but his example is not so good due to the fact that the real life relation in between rain and umbrella use is not so direct (I never use umbrella raining or not). While it is more easy to agree on rain and wet ground.



                                                                                  In formal logic you can't repudiate a contrapositive, if you think the contrapositive is not correct then the initial proposition is also wrong.



                                                                                  In formal logic, you can't state something like : router on implies internet accessible, and then after add another condition about provider state. If it is the case, then you need to reformulate your problem taking into account everything.






                                                                                  share|improve this answer


























                                                                                    1












                                                                                    1








                                                                                    1







                                                                                    Yes, you are confused due to the use of some natural language which are by nature all ambiguous/not precise (but that makes them very powerful!). Ancient philosophers were strongly interested on this problem and that leads to formal logic.



                                                                                    For contrapositive the real life example could be:




                                                                                    if it's raining then the ground is wet.




                                                                                    And its contrapositive formulæ :




                                                                                    if the ground is not wet then it is not raining.




                                                                                    Sorry for @eurieka's answer but his example is not so good due to the fact that the real life relation in between rain and umbrella use is not so direct (I never use umbrella raining or not). While it is more easy to agree on rain and wet ground.



                                                                                    In formal logic you can't repudiate a contrapositive, if you think the contrapositive is not correct then the initial proposition is also wrong.



                                                                                    In formal logic, you can't state something like : router on implies internet accessible, and then after add another condition about provider state. If it is the case, then you need to reformulate your problem taking into account everything.






                                                                                    share|improve this answer













                                                                                    Yes, you are confused due to the use of some natural language which are by nature all ambiguous/not precise (but that makes them very powerful!). Ancient philosophers were strongly interested on this problem and that leads to formal logic.



                                                                                    For contrapositive the real life example could be:




                                                                                    if it's raining then the ground is wet.




                                                                                    And its contrapositive formulæ :




                                                                                    if the ground is not wet then it is not raining.




                                                                                    Sorry for @eurieka's answer but his example is not so good due to the fact that the real life relation in between rain and umbrella use is not so direct (I never use umbrella raining or not). While it is more easy to agree on rain and wet ground.



                                                                                    In formal logic you can't repudiate a contrapositive, if you think the contrapositive is not correct then the initial proposition is also wrong.



                                                                                    In formal logic, you can't state something like : router on implies internet accessible, and then after add another condition about provider state. If it is the case, then you need to reformulate your problem taking into account everything.







                                                                                    share|improve this answer












                                                                                    share|improve this answer



                                                                                    share|improve this answer










                                                                                    answered Feb 25 at 13:24









                                                                                    Jean-Baptiste YunèsJean-Baptiste Yunès

                                                                                    1112




                                                                                    1112























                                                                                        1














                                                                                        "If A, then B. Therefore, if not-B, then not-A":



                                                                                        A == Duck



                                                                                        B == Swim.



                                                                                        If A, then B == If it's a Duck then it can swim.



                                                                                        if not-B, then not-A == If this little guy can't swim, then it cannot be a duck.



                                                                                        Why? Because a duck can swim! how can it be a duck but cannot swim?



                                                                                        Isn't this logic easy to see?






                                                                                        share|improve this answer
























                                                                                        • It might be a sick duck? But I think your example works. If you have references for the reader for more information that would strengthen your answer and direct the readers who want more information to places you have selected for them. Welcome!

                                                                                          – Frank Hubeny
                                                                                          Feb 25 at 18:08


















                                                                                        1














                                                                                        "If A, then B. Therefore, if not-B, then not-A":



                                                                                        A == Duck



                                                                                        B == Swim.



                                                                                        If A, then B == If it's a Duck then it can swim.



                                                                                        if not-B, then not-A == If this little guy can't swim, then it cannot be a duck.



                                                                                        Why? Because a duck can swim! how can it be a duck but cannot swim?



                                                                                        Isn't this logic easy to see?






                                                                                        share|improve this answer
























                                                                                        • It might be a sick duck? But I think your example works. If you have references for the reader for more information that would strengthen your answer and direct the readers who want more information to places you have selected for them. Welcome!

                                                                                          – Frank Hubeny
                                                                                          Feb 25 at 18:08
















                                                                                        1












                                                                                        1








                                                                                        1







                                                                                        "If A, then B. Therefore, if not-B, then not-A":



                                                                                        A == Duck



                                                                                        B == Swim.



                                                                                        If A, then B == If it's a Duck then it can swim.



                                                                                        if not-B, then not-A == If this little guy can't swim, then it cannot be a duck.



                                                                                        Why? Because a duck can swim! how can it be a duck but cannot swim?



                                                                                        Isn't this logic easy to see?






                                                                                        share|improve this answer













                                                                                        "If A, then B. Therefore, if not-B, then not-A":



                                                                                        A == Duck



                                                                                        B == Swim.



                                                                                        If A, then B == If it's a Duck then it can swim.



                                                                                        if not-B, then not-A == If this little guy can't swim, then it cannot be a duck.



                                                                                        Why? Because a duck can swim! how can it be a duck but cannot swim?



                                                                                        Isn't this logic easy to see?







                                                                                        share|improve this answer












                                                                                        share|improve this answer



                                                                                        share|improve this answer










                                                                                        answered Feb 25 at 17:55









                                                                                        George HanGeorge Han

                                                                                        111




                                                                                        111













                                                                                        • It might be a sick duck? But I think your example works. If you have references for the reader for more information that would strengthen your answer and direct the readers who want more information to places you have selected for them. Welcome!

                                                                                          – Frank Hubeny
                                                                                          Feb 25 at 18:08





















                                                                                        • It might be a sick duck? But I think your example works. If you have references for the reader for more information that would strengthen your answer and direct the readers who want more information to places you have selected for them. Welcome!

                                                                                          – Frank Hubeny
                                                                                          Feb 25 at 18:08



















                                                                                        It might be a sick duck? But I think your example works. If you have references for the reader for more information that would strengthen your answer and direct the readers who want more information to places you have selected for them. Welcome!

                                                                                        – Frank Hubeny
                                                                                        Feb 25 at 18:08







                                                                                        It might be a sick duck? But I think your example works. If you have references for the reader for more information that would strengthen your answer and direct the readers who want more information to places you have selected for them. Welcome!

                                                                                        – Frank Hubeny
                                                                                        Feb 25 at 18:08













                                                                                        1














                                                                                        As others have said, this is not a fallacy.



                                                                                        Visuals can make this easier without the need to follow potentially confusing logical progressions, or the need to know any logical terms. Here's a Venn diagram:



                                                                                        Venn FTW




                                                                                        • "If A then B" means that A is a subset of B.

                                                                                        • Anything outside of B ("not-B") is clearly "not-A"

                                                                                        • For good measure, if you also want to consider "not-A", this means "everything outside of A", which could be B, but could also mean "not-B".




                                                                                        Extending this to the router & internet example in the question:



                                                                                        Router and internet Venn




                                                                                        • The internet is only available to you if your router is on

                                                                                        • It is very possible to have no internet availability, even when your router is on

                                                                                        • The internet is certainly not available to you if your router/modem is not on






                                                                                        share|improve this answer






























                                                                                          1














                                                                                          As others have said, this is not a fallacy.



                                                                                          Visuals can make this easier without the need to follow potentially confusing logical progressions, or the need to know any logical terms. Here's a Venn diagram:



                                                                                          Venn FTW




                                                                                          • "If A then B" means that A is a subset of B.

                                                                                          • Anything outside of B ("not-B") is clearly "not-A"

                                                                                          • For good measure, if you also want to consider "not-A", this means "everything outside of A", which could be B, but could also mean "not-B".




                                                                                          Extending this to the router & internet example in the question:



                                                                                          Router and internet Venn




                                                                                          • The internet is only available to you if your router is on

                                                                                          • It is very possible to have no internet availability, even when your router is on

                                                                                          • The internet is certainly not available to you if your router/modem is not on






                                                                                          share|improve this answer




























                                                                                            1












                                                                                            1








                                                                                            1







                                                                                            As others have said, this is not a fallacy.



                                                                                            Visuals can make this easier without the need to follow potentially confusing logical progressions, or the need to know any logical terms. Here's a Venn diagram:



                                                                                            Venn FTW




                                                                                            • "If A then B" means that A is a subset of B.

                                                                                            • Anything outside of B ("not-B") is clearly "not-A"

                                                                                            • For good measure, if you also want to consider "not-A", this means "everything outside of A", which could be B, but could also mean "not-B".




                                                                                            Extending this to the router & internet example in the question:



                                                                                            Router and internet Venn




                                                                                            • The internet is only available to you if your router is on

                                                                                            • It is very possible to have no internet availability, even when your router is on

                                                                                            • The internet is certainly not available to you if your router/modem is not on






                                                                                            share|improve this answer















                                                                                            As others have said, this is not a fallacy.



                                                                                            Visuals can make this easier without the need to follow potentially confusing logical progressions, or the need to know any logical terms. Here's a Venn diagram:



                                                                                            Venn FTW




                                                                                            • "If A then B" means that A is a subset of B.

                                                                                            • Anything outside of B ("not-B") is clearly "not-A"

                                                                                            • For good measure, if you also want to consider "not-A", this means "everything outside of A", which could be B, but could also mean "not-B".




                                                                                            Extending this to the router & internet example in the question:



                                                                                            Router and internet Venn




                                                                                            • The internet is only available to you if your router is on

                                                                                            • It is very possible to have no internet availability, even when your router is on

                                                                                            • The internet is certainly not available to you if your router/modem is not on







                                                                                            share|improve this answer














                                                                                            share|improve this answer



                                                                                            share|improve this answer








                                                                                            edited Feb 25 at 18:18

























                                                                                            answered Feb 25 at 17:56









                                                                                            RussRuss

                                                                                            1112




                                                                                            1112























                                                                                                1














                                                                                                "If A, then B. Therefore, if not-B, then not-A"



                                                                                                This is not a fallacy because what it is stating is equivalent to this:



                                                                                                "If there is A, then there must be a B--every time. Therefore, if there is no B, there could not have been an A, because had there been an A the rule is that there must be a B. So in any case where there is no B, there must have been no A."






                                                                                                share|improve this answer




























                                                                                                  1














                                                                                                  "If A, then B. Therefore, if not-B, then not-A"



                                                                                                  This is not a fallacy because what it is stating is equivalent to this:



                                                                                                  "If there is A, then there must be a B--every time. Therefore, if there is no B, there could not have been an A, because had there been an A the rule is that there must be a B. So in any case where there is no B, there must have been no A."






                                                                                                  share|improve this answer


























                                                                                                    1












                                                                                                    1








                                                                                                    1







                                                                                                    "If A, then B. Therefore, if not-B, then not-A"



                                                                                                    This is not a fallacy because what it is stating is equivalent to this:



                                                                                                    "If there is A, then there must be a B--every time. Therefore, if there is no B, there could not have been an A, because had there been an A the rule is that there must be a B. So in any case where there is no B, there must have been no A."






                                                                                                    share|improve this answer













                                                                                                    "If A, then B. Therefore, if not-B, then not-A"



                                                                                                    This is not a fallacy because what it is stating is equivalent to this:



                                                                                                    "If there is A, then there must be a B--every time. Therefore, if there is no B, there could not have been an A, because had there been an A the rule is that there must be a B. So in any case where there is no B, there must have been no A."







                                                                                                    share|improve this answer












                                                                                                    share|improve this answer



                                                                                                    share|improve this answer










                                                                                                    answered Feb 25 at 21:58









                                                                                                    ChelonianChelonian

                                                                                                    1,16939




                                                                                                    1,16939























                                                                                                        1














                                                                                                        You probably have mistranslated the statement :




                                                                                                        "In order to have internet (B), the router must be on (A). So, if
                                                                                                        there is no internet (not-B), that means the router is not on (not-A)"




                                                                                                        It is not a fallacy, but rather a mistranslation.



                                                                                                        If you have something that comes before therefore, it is already a premise.



                                                                                                        Your premise:




                                                                                                        If A then B




                                                                                                        Is already supposed to be the case. It does not say that there are other premises that contribute to the conditional.



                                                                                                        It is analogous to: If it is raining then there are clouds.



                                                                                                        And we can infer from it that if there are no clouds then there is not raining.



                                                                                                        So, there is no fallacy here.



                                                                                                        But remember this : Deductive reasoning is about the logical form of your argument.



                                                                                                        If we suppose (just suppose) that the conditional if A then B is true, then it follows that the conditional If not-B then not-A is true.



                                                                                                        Suppose that the conditional if A then B lacks C, which is also needed for B to take place.



                                                                                                        Well, that's not deductive problem (not a non-sequitur), but it is a problem related either to the truth or reliability of the premise, or simply a mistranslation as I suggested.



                                                                                                        If I say that If you worked then you did make money , then it is the same as if I say if you did not make money, then you did not work.



                                                                                                        But whether work is sufficient to make money or not, that's another subject. Deductive arguments are only about the truth of the conclusion, given the premises are true.



                                                                                                        Now back to your example :




                                                                                                        "In order to have internet (B), the router must be on (A). So, if
                                                                                                        there is no internet (not-B), that means the router is not on (not-A)"




                                                                                                        In this case in order to have internet (B) the router must be on (A) should not be translated to if A then B , but instead if B then A.



                                                                                                        If there is internet -B- then the router is ON -A-.



                                                                                                        That is the correct conditional here :




                                                                                                        • If there is internet then the router is On

                                                                                                        • therefore, if the router is not-On then there is no-Internet.


                                                                                                        So, your premise was mistranslated .



                                                                                                        Consider the clouds example, to see what I mean : Even though clouds can cause rain (with other factors), it does not mean that clouds should be the antecedent in a conditional, so if there are clouds then it is raining is a false premise.



                                                                                                        But it means that if it rains then there are clouds, so the cause should be a consequent, not an antecedent. Think about it.






                                                                                                        share|improve this answer





















                                                                                                        • 1





                                                                                                          Are you sure?

                                                                                                          – JBentley
                                                                                                          Feb 26 at 13:09











                                                                                                        • Clouds do not have to be necessarily right about one's head, it is possible for the clouds to be in a distant area, and wind does the rest. So, it rains therefore there are clouds (either here or somewhere else)

                                                                                                          – SmootQ
                                                                                                          Feb 26 at 13:32











                                                                                                        • True, but then that complicates the logic. E.g. no clouds -> no rain, do we mean clouds that you can see? Because the clouds can be elsewhere and causing rain. But if you mean clouds anywhere (whether visible or not), there is always a cloud somewhere (and indeed, probably always rain somewhere). Also, I'm not a meteorologist, but it seems to me we can have a situation where the cloud has just finished evaporating, and the last drops of rain are still falling.

                                                                                                          – JBentley
                                                                                                          Feb 26 at 13:42













                                                                                                        • No, the conditional means "there exists a cloud" whether we see it or not, even if not right above us. There is no rain without clouds. This premise is inductive in nature: so there is an implied "probably".

                                                                                                          – SmootQ
                                                                                                          Feb 26 at 15:05
















                                                                                                        1














                                                                                                        You probably have mistranslated the statement :




                                                                                                        "In order to have internet (B), the router must be on (A). So, if
                                                                                                        there is no internet (not-B), that means the router is not on (not-A)"




                                                                                                        It is not a fallacy, but rather a mistranslation.



                                                                                                        If you have something that comes before therefore, it is already a premise.



                                                                                                        Your premise:




                                                                                                        If A then B




                                                                                                        Is already supposed to be the case. It does not say that there are other premises that contribute to the conditional.



                                                                                                        It is analogous to: If it is raining then there are clouds.



                                                                                                        And we can infer from it that if there are no clouds then there is not raining.



                                                                                                        So, there is no fallacy here.



                                                                                                        But remember this : Deductive reasoning is about the logical form of your argument.



                                                                                                        If we suppose (just suppose) that the conditional if A then B is true, then it follows that the conditional If not-B then not-A is true.



                                                                                                        Suppose that the conditional if A then B lacks C, which is also needed for B to take place.



                                                                                                        Well, that's not deductive problem (not a non-sequitur), but it is a problem related either to the truth or reliability of the premise, or simply a mistranslation as I suggested.



                                                                                                        If I say that If you worked then you did make money , then it is the same as if I say if you did not make money, then you did not work.



                                                                                                        But whether work is sufficient to make money or not, that's another subject. Deductive arguments are only about the truth of the conclusion, given the premises are true.



                                                                                                        Now back to your example :




                                                                                                        "In order to have internet (B), the router must be on (A). So, if
                                                                                                        there is no internet (not-B), that means the router is not on (not-A)"




                                                                                                        In this case in order to have internet (B) the router must be on (A) should not be translated to if A then B , but instead if B then A.



                                                                                                        If there is internet -B- then the router is ON -A-.



                                                                                                        That is the correct conditional here :




                                                                                                        • If there is internet then the router is On

                                                                                                        • therefore, if the router is not-On then there is no-Internet.


                                                                                                        So, your premise was mistranslated .



                                                                                                        Consider the clouds example, to see what I mean : Even though clouds can cause rain (with other factors), it does not mean that clouds should be the antecedent in a conditional, so if there are clouds then it is raining is a false premise.



                                                                                                        But it means that if it rains then there are clouds, so the cause should be a consequent, not an antecedent. Think about it.






                                                                                                        share|improve this answer





















                                                                                                        • 1





                                                                                                          Are you sure?

                                                                                                          – JBentley
                                                                                                          Feb 26 at 13:09











                                                                                                        • Clouds do not have to be necessarily right about one's head, it is possible for the clouds to be in a distant area, and wind does the rest. So, it rains therefore there are clouds (either here or somewhere else)

                                                                                                          – SmootQ
                                                                                                          Feb 26 at 13:32











                                                                                                        • True, but then that complicates the logic. E.g. no clouds -> no rain, do we mean clouds that you can see? Because the clouds can be elsewhere and causing rain. But if you mean clouds anywhere (whether visible or not), there is always a cloud somewhere (and indeed, probably always rain somewhere). Also, I'm not a meteorologist, but it seems to me we can have a situation where the cloud has just finished evaporating, and the last drops of rain are still falling.

                                                                                                          – JBentley
                                                                                                          Feb 26 at 13:42













                                                                                                        • No, the conditional means "there exists a cloud" whether we see it or not, even if not right above us. There is no rain without clouds. This premise is inductive in nature: so there is an implied "probably".

                                                                                                          – SmootQ
                                                                                                          Feb 26 at 15:05














                                                                                                        1












                                                                                                        1








                                                                                                        1







                                                                                                        You probably have mistranslated the statement :




                                                                                                        "In order to have internet (B), the router must be on (A). So, if
                                                                                                        there is no internet (not-B), that means the router is not on (not-A)"




                                                                                                        It is not a fallacy, but rather a mistranslation.



                                                                                                        If you have something that comes before therefore, it is already a premise.



                                                                                                        Your premise:




                                                                                                        If A then B




                                                                                                        Is already supposed to be the case. It does not say that there are other premises that contribute to the conditional.



                                                                                                        It is analogous to: If it is raining then there are clouds.



                                                                                                        And we can infer from it that if there are no clouds then there is not raining.



                                                                                                        So, there is no fallacy here.



                                                                                                        But remember this : Deductive reasoning is about the logical form of your argument.



                                                                                                        If we suppose (just suppose) that the conditional if A then B is true, then it follows that the conditional If not-B then not-A is true.



                                                                                                        Suppose that the conditional if A then B lacks C, which is also needed for B to take place.



                                                                                                        Well, that's not deductive problem (not a non-sequitur), but it is a problem related either to the truth or reliability of the premise, or simply a mistranslation as I suggested.



                                                                                                        If I say that If you worked then you did make money , then it is the same as if I say if you did not make money, then you did not work.



                                                                                                        But whether work is sufficient to make money or not, that's another subject. Deductive arguments are only about the truth of the conclusion, given the premises are true.



                                                                                                        Now back to your example :




                                                                                                        "In order to have internet (B), the router must be on (A). So, if
                                                                                                        there is no internet (not-B), that means the router is not on (not-A)"




                                                                                                        In this case in order to have internet (B) the router must be on (A) should not be translated to if A then B , but instead if B then A.



                                                                                                        If there is internet -B- then the router is ON -A-.



                                                                                                        That is the correct conditional here :




                                                                                                        • If there is internet then the router is On

                                                                                                        • therefore, if the router is not-On then there is no-Internet.


                                                                                                        So, your premise was mistranslated .



                                                                                                        Consider the clouds example, to see what I mean : Even though clouds can cause rain (with other factors), it does not mean that clouds should be the antecedent in a conditional, so if there are clouds then it is raining is a false premise.



                                                                                                        But it means that if it rains then there are clouds, so the cause should be a consequent, not an antecedent. Think about it.






                                                                                                        share|improve this answer















                                                                                                        You probably have mistranslated the statement :




                                                                                                        "In order to have internet (B), the router must be on (A). So, if
                                                                                                        there is no internet (not-B), that means the router is not on (not-A)"




                                                                                                        It is not a fallacy, but rather a mistranslation.



                                                                                                        If you have something that comes before therefore, it is already a premise.



                                                                                                        Your premise:




                                                                                                        If A then B




                                                                                                        Is already supposed to be the case. It does not say that there are other premises that contribute to the conditional.



                                                                                                        It is analogous to: If it is raining then there are clouds.



                                                                                                        And we can infer from it that if there are no clouds then there is not raining.



                                                                                                        So, there is no fallacy here.



                                                                                                        But remember this : Deductive reasoning is about the logical form of your argument.



                                                                                                        If we suppose (just suppose) that the conditional if A then B is true, then it follows that the conditional If not-B then not-A is true.



                                                                                                        Suppose that the conditional if A then B lacks C, which is also needed for B to take place.



                                                                                                        Well, that's not deductive problem (not a non-sequitur), but it is a problem related either to the truth or reliability of the premise, or simply a mistranslation as I suggested.



                                                                                                        If I say that If you worked then you did make money , then it is the same as if I say if you did not make money, then you did not work.



                                                                                                        But whether work is sufficient to make money or not, that's another subject. Deductive arguments are only about the truth of the conclusion, given the premises are true.



                                                                                                        Now back to your example :




                                                                                                        "In order to have internet (B), the router must be on (A). So, if
                                                                                                        there is no internet (not-B), that means the router is not on (not-A)"




                                                                                                        In this case in order to have internet (B) the router must be on (A) should not be translated to if A then B , but instead if B then A.



                                                                                                        If there is internet -B- then the router is ON -A-.



                                                                                                        That is the correct conditional here :




                                                                                                        • If there is internet then the router is On

                                                                                                        • therefore, if the router is not-On then there is no-Internet.


                                                                                                        So, your premise was mistranslated .



                                                                                                        Consider the clouds example, to see what I mean : Even though clouds can cause rain (with other factors), it does not mean that clouds should be the antecedent in a conditional, so if there are clouds then it is raining is a false premise.



                                                                                                        But it means that if it rains then there are clouds, so the cause should be a consequent, not an antecedent. Think about it.







                                                                                                        share|improve this answer














                                                                                                        share|improve this answer



                                                                                                        share|improve this answer








                                                                                                        edited Feb 26 at 9:58

























                                                                                                        answered Feb 25 at 13:43









                                                                                                        SmootQSmootQ

                                                                                                        1,651214




                                                                                                        1,651214








                                                                                                        • 1





                                                                                                          Are you sure?

                                                                                                          – JBentley
                                                                                                          Feb 26 at 13:09











                                                                                                        • Clouds do not have to be necessarily right about one's head, it is possible for the clouds to be in a distant area, and wind does the rest. So, it rains therefore there are clouds (either here or somewhere else)

                                                                                                          – SmootQ
                                                                                                          Feb 26 at 13:32











                                                                                                        • True, but then that complicates the logic. E.g. no clouds -> no rain, do we mean clouds that you can see? Because the clouds can be elsewhere and causing rain. But if you mean clouds anywhere (whether visible or not), there is always a cloud somewhere (and indeed, probably always rain somewhere). Also, I'm not a meteorologist, but it seems to me we can have a situation where the cloud has just finished evaporating, and the last drops of rain are still falling.

                                                                                                          – JBentley
                                                                                                          Feb 26 at 13:42













                                                                                                        • No, the conditional means "there exists a cloud" whether we see it or not, even if not right above us. There is no rain without clouds. This premise is inductive in nature: so there is an implied "probably".

                                                                                                          – SmootQ
                                                                                                          Feb 26 at 15:05














                                                                                                        • 1





                                                                                                          Are you sure?

                                                                                                          – JBentley
                                                                                                          Feb 26 at 13:09











                                                                                                        • Clouds do not have to be necessarily right about one's head, it is possible for the clouds to be in a distant area, and wind does the rest. So, it rains therefore there are clouds (either here or somewhere else)

                                                                                                          – SmootQ
                                                                                                          Feb 26 at 13:32











                                                                                                        • True, but then that complicates the logic. E.g. no clouds -> no rain, do we mean clouds that you can see? Because the clouds can be elsewhere and causing rain. But if you mean clouds anywhere (whether visible or not), there is always a cloud somewhere (and indeed, probably always rain somewhere). Also, I'm not a meteorologist, but it seems to me we can have a situation where the cloud has just finished evaporating, and the last drops of rain are still falling.

                                                                                                          – JBentley
                                                                                                          Feb 26 at 13:42













                                                                                                        • No, the conditional means "there exists a cloud" whether we see it or not, even if not right above us. There is no rain without clouds. This premise is inductive in nature: so there is an implied "probably".

                                                                                                          – SmootQ
                                                                                                          Feb 26 at 15:05








                                                                                                        1




                                                                                                        1





                                                                                                        Are you sure?

                                                                                                        – JBentley
                                                                                                        Feb 26 at 13:09





                                                                                                        Are you sure?

                                                                                                        – JBentley
                                                                                                        Feb 26 at 13:09













                                                                                                        Clouds do not have to be necessarily right about one's head, it is possible for the clouds to be in a distant area, and wind does the rest. So, it rains therefore there are clouds (either here or somewhere else)

                                                                                                        – SmootQ
                                                                                                        Feb 26 at 13:32





                                                                                                        Clouds do not have to be necessarily right about one's head, it is possible for the clouds to be in a distant area, and wind does the rest. So, it rains therefore there are clouds (either here or somewhere else)

                                                                                                        – SmootQ
                                                                                                        Feb 26 at 13:32













                                                                                                        True, but then that complicates the logic. E.g. no clouds -> no rain, do we mean clouds that you can see? Because the clouds can be elsewhere and causing rain. But if you mean clouds anywhere (whether visible or not), there is always a cloud somewhere (and indeed, probably always rain somewhere). Also, I'm not a meteorologist, but it seems to me we can have a situation where the cloud has just finished evaporating, and the last drops of rain are still falling.

                                                                                                        – JBentley
                                                                                                        Feb 26 at 13:42







                                                                                                        True, but then that complicates the logic. E.g. no clouds -> no rain, do we mean clouds that you can see? Because the clouds can be elsewhere and causing rain. But if you mean clouds anywhere (whether visible or not), there is always a cloud somewhere (and indeed, probably always rain somewhere). Also, I'm not a meteorologist, but it seems to me we can have a situation where the cloud has just finished evaporating, and the last drops of rain are still falling.

                                                                                                        – JBentley
                                                                                                        Feb 26 at 13:42















                                                                                                        No, the conditional means "there exists a cloud" whether we see it or not, even if not right above us. There is no rain without clouds. This premise is inductive in nature: so there is an implied "probably".

                                                                                                        – SmootQ
                                                                                                        Feb 26 at 15:05





                                                                                                        No, the conditional means "there exists a cloud" whether we see it or not, even if not right above us. There is no rain without clouds. This premise is inductive in nature: so there is an implied "probably".

                                                                                                        – SmootQ
                                                                                                        Feb 26 at 15:05











                                                                                                        0














                                                                                                        Another way of looking at this is with a diagram.



                                                                                                        In your example, “If A, then B” is consistent with A and B both being true, or A being false and B being anything — but not with A being true and B being false.  You could draw that as:



                                                                                                            A:║ true │ false
                                                                                                        B ║ │
                                                                                                        ══════╬══════╪══════
                                                                                                        true ║ ✔️ │ ✔️
                                                                                                        ──────╫──────┼──────
                                                                                                        false ║ ✘✘✘ │ ✔️


                                                                                                        But that's exactly the same diagram you'd get for “If not-B then not-A.”  So the two are equivalent.



                                                                                                        (As per other answers, this assumes classical two-valued logic, of course.  This is basically a Karnaugh map, though that's mainly used for simplifying boolean expressions.  You could alternatively use a truth table, as per other answers, but I find this diagram easier to follow.)






                                                                                                        share|improve this answer






























                                                                                                          0














                                                                                                          Another way of looking at this is with a diagram.



                                                                                                          In your example, “If A, then B” is consistent with A and B both being true, or A being false and B being anything — but not with A being true and B being false.  You could draw that as:



                                                                                                              A:║ true │ false
                                                                                                          B ║ │
                                                                                                          ══════╬══════╪══════
                                                                                                          true ║ ✔️ │ ✔️
                                                                                                          ──────╫──────┼──────
                                                                                                          false ║ ✘✘✘ │ ✔️


                                                                                                          But that's exactly the same diagram you'd get for “If not-B then not-A.”  So the two are equivalent.



                                                                                                          (As per other answers, this assumes classical two-valued logic, of course.  This is basically a Karnaugh map, though that's mainly used for simplifying boolean expressions.  You could alternatively use a truth table, as per other answers, but I find this diagram easier to follow.)






                                                                                                          share|improve this answer




























                                                                                                            0












                                                                                                            0








                                                                                                            0







                                                                                                            Another way of looking at this is with a diagram.



                                                                                                            In your example, “If A, then B” is consistent with A and B both being true, or A being false and B being anything — but not with A being true and B being false.  You could draw that as:



                                                                                                                A:║ true │ false
                                                                                                            B ║ │
                                                                                                            ══════╬══════╪══════
                                                                                                            true ║ ✔️ │ ✔️
                                                                                                            ──────╫──────┼──────
                                                                                                            false ║ ✘✘✘ │ ✔️


                                                                                                            But that's exactly the same diagram you'd get for “If not-B then not-A.”  So the two are equivalent.



                                                                                                            (As per other answers, this assumes classical two-valued logic, of course.  This is basically a Karnaugh map, though that's mainly used for simplifying boolean expressions.  You could alternatively use a truth table, as per other answers, but I find this diagram easier to follow.)






                                                                                                            share|improve this answer















                                                                                                            Another way of looking at this is with a diagram.



                                                                                                            In your example, “If A, then B” is consistent with A and B both being true, or A being false and B being anything — but not with A being true and B being false.  You could draw that as:



                                                                                                                A:║ true │ false
                                                                                                            B ║ │
                                                                                                            ══════╬══════╪══════
                                                                                                            true ║ ✔️ │ ✔️
                                                                                                            ──────╫──────┼──────
                                                                                                            false ║ ✘✘✘ │ ✔️


                                                                                                            But that's exactly the same diagram you'd get for “If not-B then not-A.”  So the two are equivalent.



                                                                                                            (As per other answers, this assumes classical two-valued logic, of course.  This is basically a Karnaugh map, though that's mainly used for simplifying boolean expressions.  You could alternatively use a truth table, as per other answers, but I find this diagram easier to follow.)







                                                                                                            share|improve this answer














                                                                                                            share|improve this answer



                                                                                                            share|improve this answer








                                                                                                            edited Feb 25 at 14:44

























                                                                                                            answered Feb 25 at 14:22









                                                                                                            giddsgidds

                                                                                                            1011




                                                                                                            1011























                                                                                                                0














                                                                                                                I think that you need to express the problem as so: If A->B implies that B will happen ALWAYS when A has occurred. Then, if B has not occurred, you are sure that A has not occurred.






                                                                                                                share|improve this answer
























                                                                                                                • but will B only happen if A?

                                                                                                                  – Mr. Kennedy
                                                                                                                  Mar 1 at 19:37
















                                                                                                                0














                                                                                                                I think that you need to express the problem as so: If A->B implies that B will happen ALWAYS when A has occurred. Then, if B has not occurred, you are sure that A has not occurred.






                                                                                                                share|improve this answer
























                                                                                                                • but will B only happen if A?

                                                                                                                  – Mr. Kennedy
                                                                                                                  Mar 1 at 19:37














                                                                                                                0












                                                                                                                0








                                                                                                                0







                                                                                                                I think that you need to express the problem as so: If A->B implies that B will happen ALWAYS when A has occurred. Then, if B has not occurred, you are sure that A has not occurred.






                                                                                                                share|improve this answer













                                                                                                                I think that you need to express the problem as so: If A->B implies that B will happen ALWAYS when A has occurred. Then, if B has not occurred, you are sure that A has not occurred.







                                                                                                                share|improve this answer












                                                                                                                share|improve this answer



                                                                                                                share|improve this answer










                                                                                                                answered Feb 26 at 11:39









                                                                                                                RodolfoAPRodolfoAP

                                                                                                                1,025412




                                                                                                                1,025412













                                                                                                                • but will B only happen if A?

                                                                                                                  – Mr. Kennedy
                                                                                                                  Mar 1 at 19:37



















                                                                                                                • but will B only happen if A?

                                                                                                                  – Mr. Kennedy
                                                                                                                  Mar 1 at 19:37

















                                                                                                                but will B only happen if A?

                                                                                                                – Mr. Kennedy
                                                                                                                Mar 1 at 19:37





                                                                                                                but will B only happen if A?

                                                                                                                – Mr. Kennedy
                                                                                                                Mar 1 at 19:37





                                                                                                                protected by Eliran Feb 25 at 18:35



                                                                                                                Thank you for your interest in this question.
                                                                                                                Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).



                                                                                                                Would you like to answer one of these unanswered questions instead?



                                                                                                                Popular posts from this blog

                                                                                                                Probability when a professor distributes a quiz and homework assignment to a class of n students.

                                                                                                                Aardman Animations

                                                                                                                Are they similar matrix