Is there always true predicate? [closed]
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Is there such predicate $P(x)$ where the statement is always true, for whatever the $x$ is?
predicate-logic boolean-algebra
edited Dec 27 '18 at 19:02
dantopa
6,64942245
asked Dec 27 '18 at 17:23
Edvards ZakovskisEdvards Zakovskis
354
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closed as off-topic by Namaste, Lord_Farin, Lord Shark the Unknown, Eevee Trainer, Leucippus Dec 28 '18 at 3:48
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Namaste, Lord_Farin, Eevee Trainer, Leucippus
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Is there such predicate $P(x)$ where the statement is always true, for whatever the $x$ is?
predicate-logic boolean-algebra
edited Dec 27 '18 at 19:02
dantopa
6,64942245
asked Dec 27 '18 at 17:23
Edvards ZakovskisEdvards Zakovskis
354
$endgroup$
closed as off-topic by Namaste, Lord_Farin, Lord Shark the Unknown, Eevee Trainer, Leucippus Dec 28 '18 at 3:48
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Namaste, Lord_Farin, Eevee Trainer, Leucippus
If this question can be reworded to fit the rules in the help center, please edit the question.
5
$begingroup$
$(x=x)$........
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– Mauro ALLEGRANZA
Dec 27 '18 at 17:25
2
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Who says that $x$ even needs to appear in a predicate? Just like how $f(x)=5$ is a perfectly valid function (it represents a horizontal line at height $5$), so too is $P(x) equiv text{True}$ a perfectly valid predicate. The predicate "$text{True}$" is always true for every value of $x$.
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– JMoravitz
Dec 27 '18 at 17:30
$begingroup$
See also: Tautology (logic) on wikipedia.
$endgroup$
– JMoravitz
Dec 27 '18 at 17:32
add a comment |
$begingroup$
Is there such predicate $P(x)$ where the statement is always true, for whatever the $x$ is?
predicate-logic boolean-algebra
edited Dec 27 '18 at 19:02
dantopa
6,64942245
asked Dec 27 '18 at 17:23
Edvards ZakovskisEdvards Zakovskis
354
$endgroup$
Is there such predicate $P(x)$ where the statement is always true, for whatever the $x$ is?
predicate-logic boolean-algebra
predicate-logic boolean-algebra
edited Dec 27 '18 at 19:02
dantopa
6,64942245
asked Dec 27 '18 at 17:23
Edvards ZakovskisEdvards Zakovskis
354
edited Dec 27 '18 at 19:02
dantopa
6,64942245
asked Dec 27 '18 at 17:23
Edvards ZakovskisEdvards Zakovskis
354
edited Dec 27 '18 at 19:02
dantopa
6,64942245
edited Dec 27 '18 at 19:02
dantopa
6,64942245
edited Dec 27 '18 at 19:02
dantopa
6,64942245
6,64942245
asked Dec 27 '18 at 17:23
Edvards ZakovskisEdvards Zakovskis
354
asked Dec 27 '18 at 17:23
Edvards ZakovskisEdvards Zakovskis
354
asked Dec 27 '18 at 17:23
Edvards ZakovskisEdvards Zakovskis
354
354
closed as off-topic by Namaste, Lord_Farin, Lord Shark the Unknown, Eevee Trainer, Leucippus Dec 28 '18 at 3:48
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Namaste, Lord_Farin, Eevee Trainer, Leucippus
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Namaste, Lord_Farin, Lord Shark the Unknown, Eevee Trainer, Leucippus Dec 28 '18 at 3:48
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Namaste, Lord_Farin, Eevee Trainer, Leucippus
If this question can be reworded to fit the rules in the help center, please edit the question.
5
$begingroup$
$(x=x)$........
$endgroup$
– Mauro ALLEGRANZA
Dec 27 '18 at 17:25
2
$begingroup$
Who says that $x$ even needs to appear in a predicate? Just like how $f(x)=5$ is a perfectly valid function (it represents a horizontal line at height $5$), so too is $P(x) equiv text{True}$ a perfectly valid predicate. The predicate "$text{True}$" is always true for every value of $x$.
$endgroup$
– JMoravitz
Dec 27 '18 at 17:30
$begingroup$
See also: Tautology (logic) on wikipedia.
$endgroup$
– JMoravitz
Dec 27 '18 at 17:32
add a comment |
5
$begingroup$
$(x=x)$........
$endgroup$
– Mauro ALLEGRANZA
Dec 27 '18 at 17:25
2
$begingroup$
Who says that $x$ even needs to appear in a predicate? Just like how $f(x)=5$ is a perfectly valid function (it represents a horizontal line at height $5$), so too is $P(x) equiv text{True}$ a perfectly valid predicate. The predicate "$text{True}$" is always true for every value of $x$.
$endgroup$
– JMoravitz
Dec 27 '18 at 17:30
$begingroup$
See also: Tautology (logic) on wikipedia.
$endgroup$
– JMoravitz
Dec 27 '18 at 17:32
5
5
$begingroup$
$(x=x)$........
$endgroup$
– Mauro ALLEGRANZA
Dec 27 '18 at 17:25
$begingroup$
$(x=x)$........
$endgroup$
– Mauro ALLEGRANZA
Dec 27 '18 at 17:25
2
2
$begingroup$
Who says that $x$ even needs to appear in a predicate? Just like how $f(x)=5$ is a perfectly valid function (it represents a horizontal line at height $5$), so too is $P(x) equiv text{True}$ a perfectly valid predicate. The predicate "$text{True}$" is always true for every value of $x$.
$endgroup$
– JMoravitz
Dec 27 '18 at 17:30
$begingroup$
Who says that $x$ even needs to appear in a predicate? Just like how $f(x)=5$ is a perfectly valid function (it represents a horizontal line at height $5$), so too is $P(x) equiv text{True}$ a perfectly valid predicate. The predicate "$text{True}$" is always true for every value of $x$.
$endgroup$
– JMoravitz
Dec 27 '18 at 17:30
$begingroup$
See also: Tautology (logic) on wikipedia.
$endgroup$
– JMoravitz
Dec 27 '18 at 17:32
$begingroup$
See also: Tautology (logic) on wikipedia.
$endgroup$
– JMoravitz
Dec 27 '18 at 17:32
add a comment |
1 Answer
1
active
oldest
votes
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Some (not all) presentations of first-order logic include as primitive symbols "$top$" and "$perp$," for "truth" and "falsity" respectively as primitive logical symbols - on the same level as $=$, $forall,exists$, and the Boolean connectives and parentheses. (They can be thought of as nullary relation symbols, similarly to how constant symbols can be thought of as nullary function symbols, but since they're included as part of the logical language this isn't necessarily a helpful interpretation.)
The formula "$x=x$" also does the job, although the situation there may be more complicated in certain nonclassical logics. Similarly, formulas like "$P(x)veeneg P(x)$" will work in classical logic, and formulas like "$neg P(x)vee negneg P(x)$" will work even in intuitionistic logic. Because things do get more complicated when we leave classical logic, I tend to prefer including $top$ and $perp$ as logical primitives; but as long as we stay in the context of classical logic, it's unimportant.
answered Dec 27 '18 at 19:16
Noah SchweberNoah Schweber
127k10151290
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Some (not all) presentations of first-order logic include as primitive symbols "$top$" and "$perp$," for "truth" and "falsity" respectively as primitive logical symbols - on the same level as $=$, $forall,exists$, and the Boolean connectives and parentheses. (They can be thought of as nullary relation symbols, similarly to how constant symbols can be thought of as nullary function symbols, but since they're included as part of the logical language this isn't necessarily a helpful interpretation.)
The formula "$x=x$" also does the job, although the situation there may be more complicated in certain nonclassical logics. Similarly, formulas like "$P(x)veeneg P(x)$" will work in classical logic, and formulas like "$neg P(x)vee negneg P(x)$" will work even in intuitionistic logic. Because things do get more complicated when we leave classical logic, I tend to prefer including $top$ and $perp$ as logical primitives; but as long as we stay in the context of classical logic, it's unimportant.
answered Dec 27 '18 at 19:16
Noah SchweberNoah Schweber
127k10151290
$endgroup$
add a comment |
$begingroup$
Some (not all) presentations of first-order logic include as primitive symbols "$top$" and "$perp$," for "truth" and "falsity" respectively as primitive logical symbols - on the same level as $=$, $forall,exists$, and the Boolean connectives and parentheses. (They can be thought of as nullary relation symbols, similarly to how constant symbols can be thought of as nullary function symbols, but since they're included as part of the logical language this isn't necessarily a helpful interpretation.)
The formula "$x=x$" also does the job, although the situation there may be more complicated in certain nonclassical logics. Similarly, formulas like "$P(x)veeneg P(x)$" will work in classical logic, and formulas like "$neg P(x)vee negneg P(x)$" will work even in intuitionistic logic. Because things do get more complicated when we leave classical logic, I tend to prefer including $top$ and $perp$ as logical primitives; but as long as we stay in the context of classical logic, it's unimportant.
answered Dec 27 '18 at 19:16
Noah SchweberNoah Schweber
127k10151290
$endgroup$
add a comment |
$begingroup$
Some (not all) presentations of first-order logic include as primitive symbols "$top$" and "$perp$," for "truth" and "falsity" respectively as primitive logical symbols - on the same level as $=$, $forall,exists$, and the Boolean connectives and parentheses. (They can be thought of as nullary relation symbols, similarly to how constant symbols can be thought of as nullary function symbols, but since they're included as part of the logical language this isn't necessarily a helpful interpretation.)
The formula "$x=x$" also does the job, although the situation there may be more complicated in certain nonclassical logics. Similarly, formulas like "$P(x)veeneg P(x)$" will work in classical logic, and formulas like "$neg P(x)vee negneg P(x)$" will work even in intuitionistic logic. Because things do get more complicated when we leave classical logic, I tend to prefer including $top$ and $perp$ as logical primitives; but as long as we stay in the context of classical logic, it's unimportant.
answered Dec 27 '18 at 19:16
Noah SchweberNoah Schweber
127k10151290
$endgroup$
Some (not all) presentations of first-order logic include as primitive symbols "$top$" and "$perp$," for "truth" and "falsity" respectively as primitive logical symbols - on the same level as $=$, $forall,exists$, and the Boolean connectives and parentheses. (They can be thought of as nullary relation symbols, similarly to how constant symbols can be thought of as nullary function symbols, but since they're included as part of the logical language this isn't necessarily a helpful interpretation.)
The formula "$x=x$" also does the job, although the situation there may be more complicated in certain nonclassical logics. Similarly, formulas like "$P(x)veeneg P(x)$" will work in classical logic, and formulas like "$neg P(x)vee negneg P(x)$" will work even in intuitionistic logic. Because things do get more complicated when we leave classical logic, I tend to prefer including $top$ and $perp$ as logical primitives; but as long as we stay in the context of classical logic, it's unimportant.
answered Dec 27 '18 at 19:16
Noah SchweberNoah Schweber
127k10151290
answered Dec 27 '18 at 19:16
Noah SchweberNoah Schweber
127k10151290
answered Dec 27 '18 at 19:16
Noah SchweberNoah Schweber
127k10151290
answered Dec 27 '18 at 19:16
Noah SchweberNoah Schweber
127k10151290
127k10151290
add a comment |
add a comment |
5
$begingroup$
$(x=x)$........
$endgroup$
– Mauro ALLEGRANZA
Dec 27 '18 at 17:25
2
$begingroup$
Who says that $x$ even needs to appear in a predicate? Just like how $f(x)=5$ is a perfectly valid function (it represents a horizontal line at height $5$), so too is $P(x) equiv text{True}$ a perfectly valid predicate. The predicate "$text{True}$" is always true for every value of $x$.
$endgroup$
– JMoravitz
Dec 27 '18 at 17:30
$begingroup$
See also: Tautology (logic) on wikipedia.
$endgroup$
– JMoravitz
Dec 27 '18 at 17:32