Does (If not P then Q) imply (If P then Q)? My truth table says yes but I want verification
up vote
2
down vote
favorite
As the title says, is this true?
$$(lnot P to lnot Q) to (P to Q)$$
The truth table is
begin{array}{rrrrrr}
P & Q & lnot P & lnot Q & lnot P to lnot Q & P to Q & (lnot P to lnot Q) to (P to Q) \ hline
T & T & F & F & T & T & T \
T & F & F & T & T & F & F \
F & T & T & F & F & T & T \
F & F & T & T & T & T & T \
end{array}
It seems like it's true from the table.
If it is true, is it true because $$(lnot P to lnot Q) to (P to Q)$$ has the same truth table corresponding to the $to$ connective which is false only when the antecedent is T but the consequent is F?
Or is it true because the statement is true when the premises of $lnot P to lnot Q$ and $P to Q$ are true?
If it's not true, why not?
logic
asked Nov 20 at 22:05
000
154
add a comment |
up vote
2
down vote
favorite
As the title says, is this true?
$$(lnot P to lnot Q) to (P to Q)$$
The truth table is
begin{array}{rrrrrr}
P & Q & lnot P & lnot Q & lnot P to lnot Q & P to Q & (lnot P to lnot Q) to (P to Q) \ hline
T & T & F & F & T & T & T \
T & F & F & T & T & F & F \
F & T & T & F & F & T & T \
F & F & T & T & T & T & T \
end{array}
It seems like it's true from the table.
If it is true, is it true because $$(lnot P to lnot Q) to (P to Q)$$ has the same truth table corresponding to the $to$ connective which is false only when the antecedent is T but the consequent is F?
Or is it true because the statement is true when the premises of $lnot P to lnot Q$ and $P to Q$ are true?
If it's not true, why not?
logic
asked Nov 20 at 22:05
000
154
2
Why would it seem to you like it is true from the truth table? You have a F in the last column. That indicates that it is not a tautology.
– Graham Kemp
Nov 20 at 22:11
The title suggests the formula you want is $(neg Pto Q)to(Pto Q)$ not $(neg Ptoneg Q)to(Pto Q)$.
– Derek Elkins
Nov 21 at 4:38
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
As the title says, is this true?
$$(lnot P to lnot Q) to (P to Q)$$
The truth table is
begin{array}{rrrrrr}
P & Q & lnot P & lnot Q & lnot P to lnot Q & P to Q & (lnot P to lnot Q) to (P to Q) \ hline
T & T & F & F & T & T & T \
T & F & F & T & T & F & F \
F & T & T & F & F & T & T \
F & F & T & T & T & T & T \
end{array}
It seems like it's true from the table.
If it is true, is it true because $$(lnot P to lnot Q) to (P to Q)$$ has the same truth table corresponding to the $to$ connective which is false only when the antecedent is T but the consequent is F?
Or is it true because the statement is true when the premises of $lnot P to lnot Q$ and $P to Q$ are true?
If it's not true, why not?
logic
asked Nov 20 at 22:05
000
154
As the title says, is this true?
$$(lnot P to lnot Q) to (P to Q)$$
The truth table is
begin{array}{rrrrrr}
P & Q & lnot P & lnot Q & lnot P to lnot Q & P to Q & (lnot P to lnot Q) to (P to Q) \ hline
T & T & F & F & T & T & T \
T & F & F & T & T & F & F \
F & T & T & F & F & T & T \
F & F & T & T & T & T & T \
end{array}
It seems like it's true from the table.
If it is true, is it true because $$(lnot P to lnot Q) to (P to Q)$$ has the same truth table corresponding to the $to$ connective which is false only when the antecedent is T but the consequent is F?
Or is it true because the statement is true when the premises of $lnot P to lnot Q$ and $P to Q$ are true?
If it's not true, why not?
logic
logic
asked Nov 20 at 22:05
000
154
asked Nov 20 at 22:05
000
154
asked Nov 20 at 22:05
000
154
asked Nov 20 at 22:05
000
154
asked Nov 20 at 22:05
000
154
154
2
Why would it seem to you like it is true from the truth table? You have a F in the last column. That indicates that it is not a tautology.
– Graham Kemp
Nov 20 at 22:11
The title suggests the formula you want is $(neg Pto Q)to(Pto Q)$ not $(neg Ptoneg Q)to(Pto Q)$.
– Derek Elkins
Nov 21 at 4:38
add a comment |
2
Why would it seem to you like it is true from the truth table? You have a F in the last column. That indicates that it is not a tautology.
– Graham Kemp
Nov 20 at 22:11
The title suggests the formula you want is $(neg Pto Q)to(Pto Q)$ not $(neg Ptoneg Q)to(Pto Q)$.
– Derek Elkins
Nov 21 at 4:38
2
2
Why would it seem to you like it is true from the truth table? You have a F in the last column. That indicates that it is not a tautology.
– Graham Kemp
Nov 20 at 22:11
Why would it seem to you like it is true from the truth table? You have a F in the last column. That indicates that it is not a tautology.
– Graham Kemp
Nov 20 at 22:11
The title suggests the formula you want is $(neg Pto Q)to(Pto Q)$ not $(neg Ptoneg Q)to(Pto Q)$.
– Derek Elkins
Nov 21 at 4:38
The title suggests the formula you want is $(neg Pto Q)to(Pto Q)$ not $(neg Ptoneg Q)to(Pto Q)$.
– Derek Elkins
Nov 21 at 4:38
add a comment |
2 Answers
2
active
oldest
votes
up vote
3
down vote
accepted
$(lnot Ptolnot Q)to(Pto Q)$ is not a tautology because it is not true when $P$ is true but $Q$ is false. That is shown in the second row of your truth table.
Likewise, it is not a contradiction. The statement is conditionally true.
The statement is logically equivalent to $lnot(Plandlnot Q)$, also to $(lnot Plor Q)$.
Now $(lnot Ptolnot Q)to(Qto P)$ is a tautology in classical logic. Notice the order of the terms.
Indeed $lnot Pto lnot Q$ is the contrapositive of $Qto P$, and the two are logically equivalent.
answered Nov 20 at 22:14
Graham Kemp
84.6k43378
Is it conditionally true because (¬P→¬Q) is a premise and (P→Q) is a conclusion, and the conclusion is false when the premise is true when P is true and Q is false? In other words whether this statement is true or not doesn't have anything to do with the evaluation of (¬P→¬Q)→(P→Q) itself i.e. I can remove the last column and be able to evaluate the truth table
– 000
Nov 20 at 22:27
"In other words whether this statement is true or not doesn't have anything to do with the evaluation of $(neg Pto neg Q)to (Pto Q)$ itself", except that truth is defined exactly by the result of valuations.
– Git Gud
Nov 20 at 22:29
@GitGud Ok so the last column is necessary because it gives us the final statement to evaluate it's truthfulness correct?
– 000
Nov 20 at 22:32
I have a question. Is (¬P→¬Q) the premise and (P→Q) the conclusion? Or are they both premises with (¬P→¬Q)→(P→Q) as the conclusion?
– 000
Nov 20 at 22:40
@000 We say $lnot Ptolnot Q$ is the antecedant, and $Pto Q$ is the consequent, of the statement $(lnot Ptolnot Q)to(Pto Q)$. Premise and conclusion reference parts of a logical argument.
– Graham Kemp
Nov 20 at 22:42
|
show 2 more comments
up vote
1
down vote
No, if we have a statement "$P$ then $Q$", then "$neg P$ then $neg Q$" is the inverse of the statement. The inverse being true does not imply the statement is true.
For instance consider a class where the cutoff for an $A$ is $90%$. Consider the statement $$
text{"If you have above an }80%text{, then you will receive an }Atext{."}
$$
This statement is not true. However its inverse is true.
$$
text{"If you do not have above an }80%text{, then you will not receive an }Atext{."}
$$
answered Nov 20 at 22:14
Joey Kilpatrick
1,183422
Thank you. Is (¬P→¬Q) the premise and (P→Q) the conclusion? Or are they both premises with (¬P→¬Q)→(P→Q) as the conclusion?
– 000
Nov 20 at 22:40
($neg Prightarrow neg Q$) is the premise or hypothesis and ($Prightarrow Q$) is the conclusion.
– Joey Kilpatrick
Nov 20 at 22:42
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
$(lnot Ptolnot Q)to(Pto Q)$ is not a tautology because it is not true when $P$ is true but $Q$ is false. That is shown in the second row of your truth table.
Likewise, it is not a contradiction. The statement is conditionally true.
The statement is logically equivalent to $lnot(Plandlnot Q)$, also to $(lnot Plor Q)$.
Now $(lnot Ptolnot Q)to(Qto P)$ is a tautology in classical logic. Notice the order of the terms.
Indeed $lnot Pto lnot Q$ is the contrapositive of $Qto P$, and the two are logically equivalent.
answered Nov 20 at 22:14
Graham Kemp
84.6k43378
Is it conditionally true because (¬P→¬Q) is a premise and (P→Q) is a conclusion, and the conclusion is false when the premise is true when P is true and Q is false? In other words whether this statement is true or not doesn't have anything to do with the evaluation of (¬P→¬Q)→(P→Q) itself i.e. I can remove the last column and be able to evaluate the truth table
– 000
Nov 20 at 22:27
"In other words whether this statement is true or not doesn't have anything to do with the evaluation of $(neg Pto neg Q)to (Pto Q)$ itself", except that truth is defined exactly by the result of valuations.
– Git Gud
Nov 20 at 22:29
@GitGud Ok so the last column is necessary because it gives us the final statement to evaluate it's truthfulness correct?
– 000
Nov 20 at 22:32
I have a question. Is (¬P→¬Q) the premise and (P→Q) the conclusion? Or are they both premises with (¬P→¬Q)→(P→Q) as the conclusion?
– 000
Nov 20 at 22:40
@000 We say $lnot Ptolnot Q$ is the antecedant, and $Pto Q$ is the consequent, of the statement $(lnot Ptolnot Q)to(Pto Q)$. Premise and conclusion reference parts of a logical argument.
– Graham Kemp
Nov 20 at 22:42
|
show 2 more comments
up vote
3
down vote
accepted
$(lnot Ptolnot Q)to(Pto Q)$ is not a tautology because it is not true when $P$ is true but $Q$ is false. That is shown in the second row of your truth table.
Likewise, it is not a contradiction. The statement is conditionally true.
The statement is logically equivalent to $lnot(Plandlnot Q)$, also to $(lnot Plor Q)$.
Now $(lnot Ptolnot Q)to(Qto P)$ is a tautology in classical logic. Notice the order of the terms.
Indeed $lnot Pto lnot Q$ is the contrapositive of $Qto P$, and the two are logically equivalent.
answered Nov 20 at 22:14
Graham Kemp
84.6k43378
Is it conditionally true because (¬P→¬Q) is a premise and (P→Q) is a conclusion, and the conclusion is false when the premise is true when P is true and Q is false? In other words whether this statement is true or not doesn't have anything to do with the evaluation of (¬P→¬Q)→(P→Q) itself i.e. I can remove the last column and be able to evaluate the truth table
– 000
Nov 20 at 22:27
"In other words whether this statement is true or not doesn't have anything to do with the evaluation of $(neg Pto neg Q)to (Pto Q)$ itself", except that truth is defined exactly by the result of valuations.
– Git Gud
Nov 20 at 22:29
@GitGud Ok so the last column is necessary because it gives us the final statement to evaluate it's truthfulness correct?
– 000
Nov 20 at 22:32
I have a question. Is (¬P→¬Q) the premise and (P→Q) the conclusion? Or are they both premises with (¬P→¬Q)→(P→Q) as the conclusion?
– 000
Nov 20 at 22:40
@000 We say $lnot Ptolnot Q$ is the antecedant, and $Pto Q$ is the consequent, of the statement $(lnot Ptolnot Q)to(Pto Q)$. Premise and conclusion reference parts of a logical argument.
– Graham Kemp
Nov 20 at 22:42
|
show 2 more comments
up vote
3
down vote
accepted
up vote
3
down vote
accepted
$(lnot Ptolnot Q)to(Pto Q)$ is not a tautology because it is not true when $P$ is true but $Q$ is false. That is shown in the second row of your truth table.
Likewise, it is not a contradiction. The statement is conditionally true.
The statement is logically equivalent to $lnot(Plandlnot Q)$, also to $(lnot Plor Q)$.
Now $(lnot Ptolnot Q)to(Qto P)$ is a tautology in classical logic. Notice the order of the terms.
Indeed $lnot Pto lnot Q$ is the contrapositive of $Qto P$, and the two are logically equivalent.
answered Nov 20 at 22:14
Graham Kemp
84.6k43378
$(lnot Ptolnot Q)to(Pto Q)$ is not a tautology because it is not true when $P$ is true but $Q$ is false. That is shown in the second row of your truth table.
Likewise, it is not a contradiction. The statement is conditionally true.
The statement is logically equivalent to $lnot(Plandlnot Q)$, also to $(lnot Plor Q)$.
Now $(lnot Ptolnot Q)to(Qto P)$ is a tautology in classical logic. Notice the order of the terms.
Indeed $lnot Pto lnot Q$ is the contrapositive of $Qto P$, and the two are logically equivalent.
answered Nov 20 at 22:14
Graham Kemp
84.6k43378
edited Nov 20 at 22:20
answered Nov 20 at 22:14
Graham Kemp
84.6k43378
answered Nov 20 at 22:14
Graham Kemp
84.6k43378
answered Nov 20 at 22:14
Graham Kemp
84.6k43378
84.6k43378
Is it conditionally true because (¬P→¬Q) is a premise and (P→Q) is a conclusion, and the conclusion is false when the premise is true when P is true and Q is false? In other words whether this statement is true or not doesn't have anything to do with the evaluation of (¬P→¬Q)→(P→Q) itself i.e. I can remove the last column and be able to evaluate the truth table
– 000
Nov 20 at 22:27
"In other words whether this statement is true or not doesn't have anything to do with the evaluation of $(neg Pto neg Q)to (Pto Q)$ itself", except that truth is defined exactly by the result of valuations.
– Git Gud
Nov 20 at 22:29
@GitGud Ok so the last column is necessary because it gives us the final statement to evaluate it's truthfulness correct?
– 000
Nov 20 at 22:32
I have a question. Is (¬P→¬Q) the premise and (P→Q) the conclusion? Or are they both premises with (¬P→¬Q)→(P→Q) as the conclusion?
– 000
Nov 20 at 22:40
@000 We say $lnot Ptolnot Q$ is the antecedant, and $Pto Q$ is the consequent, of the statement $(lnot Ptolnot Q)to(Pto Q)$. Premise and conclusion reference parts of a logical argument.
– Graham Kemp
Nov 20 at 22:42
|
show 2 more comments
Is it conditionally true because (¬P→¬Q) is a premise and (P→Q) is a conclusion, and the conclusion is false when the premise is true when P is true and Q is false? In other words whether this statement is true or not doesn't have anything to do with the evaluation of (¬P→¬Q)→(P→Q) itself i.e. I can remove the last column and be able to evaluate the truth table
– 000
Nov 20 at 22:27
"In other words whether this statement is true or not doesn't have anything to do with the evaluation of $(neg Pto neg Q)to (Pto Q)$ itself", except that truth is defined exactly by the result of valuations.
– Git Gud
Nov 20 at 22:29
@GitGud Ok so the last column is necessary because it gives us the final statement to evaluate it's truthfulness correct?
– 000
Nov 20 at 22:32
I have a question. Is (¬P→¬Q) the premise and (P→Q) the conclusion? Or are they both premises with (¬P→¬Q)→(P→Q) as the conclusion?
– 000
Nov 20 at 22:40
@000 We say $lnot Ptolnot Q$ is the antecedant, and $Pto Q$ is the consequent, of the statement $(lnot Ptolnot Q)to(Pto Q)$. Premise and conclusion reference parts of a logical argument.
– Graham Kemp
Nov 20 at 22:42
Is it conditionally true because (¬P→¬Q) is a premise and (P→Q) is a conclusion, and the conclusion is false when the premise is true when P is true and Q is false? In other words whether this statement is true or not doesn't have anything to do with the evaluation of (¬P→¬Q)→(P→Q) itself i.e. I can remove the last column and be able to evaluate the truth table
– 000
Nov 20 at 22:27
Is it conditionally true because (¬P→¬Q) is a premise and (P→Q) is a conclusion, and the conclusion is false when the premise is true when P is true and Q is false? In other words whether this statement is true or not doesn't have anything to do with the evaluation of (¬P→¬Q)→(P→Q) itself i.e. I can remove the last column and be able to evaluate the truth table
– 000
Nov 20 at 22:27
"In other words whether this statement is true or not doesn't have anything to do with the evaluation of $(neg Pto neg Q)to (Pto Q)$ itself", except that truth is defined exactly by the result of valuations.
– Git Gud
Nov 20 at 22:29
"In other words whether this statement is true or not doesn't have anything to do with the evaluation of $(neg Pto neg Q)to (Pto Q)$ itself", except that truth is defined exactly by the result of valuations.
– Git Gud
Nov 20 at 22:29
@GitGud Ok so the last column is necessary because it gives us the final statement to evaluate it's truthfulness correct?
– 000
Nov 20 at 22:32
@GitGud Ok so the last column is necessary because it gives us the final statement to evaluate it's truthfulness correct?
– 000
Nov 20 at 22:32
I have a question. Is (¬P→¬Q) the premise and (P→Q) the conclusion? Or are they both premises with (¬P→¬Q)→(P→Q) as the conclusion?
– 000
Nov 20 at 22:40
I have a question. Is (¬P→¬Q) the premise and (P→Q) the conclusion? Or are they both premises with (¬P→¬Q)→(P→Q) as the conclusion?
– 000
Nov 20 at 22:40
@000 We say $lnot Ptolnot Q$ is the antecedant, and $Pto Q$ is the consequent, of the statement $(lnot Ptolnot Q)to(Pto Q)$. Premise and conclusion reference parts of a logical argument.
– Graham Kemp
Nov 20 at 22:42
@000 We say $lnot Ptolnot Q$ is the antecedant, and $Pto Q$ is the consequent, of the statement $(lnot Ptolnot Q)to(Pto Q)$. Premise and conclusion reference parts of a logical argument.
– Graham Kemp
Nov 20 at 22:42
|
show 2 more comments
up vote
1
down vote
No, if we have a statement "$P$ then $Q$", then "$neg P$ then $neg Q$" is the inverse of the statement. The inverse being true does not imply the statement is true.
For instance consider a class where the cutoff for an $A$ is $90%$. Consider the statement $$
text{"If you have above an }80%text{, then you will receive an }Atext{."}
$$
This statement is not true. However its inverse is true.
$$
text{"If you do not have above an }80%text{, then you will not receive an }Atext{."}
$$
answered Nov 20 at 22:14
Joey Kilpatrick
1,183422
Thank you. Is (¬P→¬Q) the premise and (P→Q) the conclusion? Or are they both premises with (¬P→¬Q)→(P→Q) as the conclusion?
– 000
Nov 20 at 22:40
($neg Prightarrow neg Q$) is the premise or hypothesis and ($Prightarrow Q$) is the conclusion.
– Joey Kilpatrick
Nov 20 at 22:42
add a comment |
up vote
1
down vote
No, if we have a statement "$P$ then $Q$", then "$neg P$ then $neg Q$" is the inverse of the statement. The inverse being true does not imply the statement is true.
For instance consider a class where the cutoff for an $A$ is $90%$. Consider the statement $$
text{"If you have above an }80%text{, then you will receive an }Atext{."}
$$
This statement is not true. However its inverse is true.
$$
text{"If you do not have above an }80%text{, then you will not receive an }Atext{."}
$$
answered Nov 20 at 22:14
Joey Kilpatrick
1,183422
Thank you. Is (¬P→¬Q) the premise and (P→Q) the conclusion? Or are they both premises with (¬P→¬Q)→(P→Q) as the conclusion?
– 000
Nov 20 at 22:40
($neg Prightarrow neg Q$) is the premise or hypothesis and ($Prightarrow Q$) is the conclusion.
– Joey Kilpatrick
Nov 20 at 22:42
add a comment |
up vote
1
down vote
up vote
1
down vote
No, if we have a statement "$P$ then $Q$", then "$neg P$ then $neg Q$" is the inverse of the statement. The inverse being true does not imply the statement is true.
For instance consider a class where the cutoff for an $A$ is $90%$. Consider the statement $$
text{"If you have above an }80%text{, then you will receive an }Atext{."}
$$
This statement is not true. However its inverse is true.
$$
text{"If you do not have above an }80%text{, then you will not receive an }Atext{."}
$$
answered Nov 20 at 22:14
Joey Kilpatrick
1,183422
No, if we have a statement "$P$ then $Q$", then "$neg P$ then $neg Q$" is the inverse of the statement. The inverse being true does not imply the statement is true.
For instance consider a class where the cutoff for an $A$ is $90%$. Consider the statement $$
text{"If you have above an }80%text{, then you will receive an }Atext{."}
$$
This statement is not true. However its inverse is true.
$$
text{"If you do not have above an }80%text{, then you will not receive an }Atext{."}
$$
answered Nov 20 at 22:14
Joey Kilpatrick
1,183422
answered Nov 20 at 22:14
Joey Kilpatrick
1,183422
answered Nov 20 at 22:14
Joey Kilpatrick
1,183422
answered Nov 20 at 22:14
Joey Kilpatrick
1,183422
1,183422
Thank you. Is (¬P→¬Q) the premise and (P→Q) the conclusion? Or are they both premises with (¬P→¬Q)→(P→Q) as the conclusion?
– 000
Nov 20 at 22:40
($neg Prightarrow neg Q$) is the premise or hypothesis and ($Prightarrow Q$) is the conclusion.
– Joey Kilpatrick
Nov 20 at 22:42
add a comment |
Thank you. Is (¬P→¬Q) the premise and (P→Q) the conclusion? Or are they both premises with (¬P→¬Q)→(P→Q) as the conclusion?
– 000
Nov 20 at 22:40
($neg Prightarrow neg Q$) is the premise or hypothesis and ($Prightarrow Q$) is the conclusion.
– Joey Kilpatrick
Nov 20 at 22:42
Thank you. Is (¬P→¬Q) the premise and (P→Q) the conclusion? Or are they both premises with (¬P→¬Q)→(P→Q) as the conclusion?
– 000
Nov 20 at 22:40
Thank you. Is (¬P→¬Q) the premise and (P→Q) the conclusion? Or are they both premises with (¬P→¬Q)→(P→Q) as the conclusion?
– 000
Nov 20 at 22:40
($neg Prightarrow neg Q$) is the premise or hypothesis and ($Prightarrow Q$) is the conclusion.
– Joey Kilpatrick
Nov 20 at 22:42
($neg Prightarrow neg Q$) is the premise or hypothesis and ($Prightarrow Q$) is the conclusion.
– Joey Kilpatrick
Nov 20 at 22:42
add a comment |
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2
Why would it seem to you like it is true from the truth table? You have a F in the last column. That indicates that it is not a tautology.
– Graham Kemp
Nov 20 at 22:11
The title suggests the formula you want is $(neg Pto Q)to(Pto Q)$ not $(neg Ptoneg Q)to(Pto Q)$.
– Derek Elkins
Nov 21 at 4:38