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Why is it sensical for a proposition with a false antecedent to validate to true? [duplicate]
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This question already has an answer here:
In classical logic, why is $(pRightarrow q)$ True if both $p$ and $q$ are False?
21 answers
In propositional logic, the statement "If pigs can fly, then elephants can lay eggs." validates to true because the antecedent validates to false.
In other words, given $a rightarrow b$, if a is false, the entire statement is true. Why?
Just because the antecendent is false doesn't mean that another fact depends on it, right?
logic
asked Apr 28 '12 at 3:05
David FauxDavid Faux
1,56762342
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marked as duplicate by quid♦, Jyrki Lahtonen Oct 12 '15 at 8:08
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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show 3 more comments
$begingroup$
This question already has an answer here:
In classical logic, why is $(pRightarrow q)$ True if both $p$ and $q$ are False?
21 answers
In propositional logic, the statement "If pigs can fly, then elephants can lay eggs." validates to true because the antecedent validates to false.
In other words, given $a rightarrow b$, if a is false, the entire statement is true. Why?
Just because the antecendent is false doesn't mean that another fact depends on it, right?
logic
asked Apr 28 '12 at 3:05
David FauxDavid Faux
1,56762342
$endgroup$
marked as duplicate by quid♦, Jyrki Lahtonen Oct 12 '15 at 8:08
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
7
$begingroup$
In informal speech, "if $A$ then $B$" and "$A$ implies $B$" are mainly used when it is believed there is a causal connection between $A$ and $B$. The truth-functional connective $rightarrow$ does not capture this feature of "implies."
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– André Nicolas
Apr 28 '12 at 3:24
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I tend to think of it this way: when you draw out a truth table, a statement is considered false statements if and only if it is incompatible with the truth values of $a$ and $b$. For example, $a to b$ is compatible with $neg a, (neg) b$ and $a, b$ but not $a, neg b$.
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– Brett Frankel
Apr 28 '12 at 3:26
1
$begingroup$
The answers to this prior question should prove enlightening.
$endgroup$
– Bill Dubuque
Apr 28 '12 at 3:58
1
$begingroup$
See xkcd
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– Tim S.
Mar 28 '14 at 17:24
1
$begingroup$
Also this earlier question
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– Jyrki Lahtonen
Oct 11 '15 at 15:21
|
show 3 more comments
$begingroup$
This question already has an answer here:
In classical logic, why is $(pRightarrow q)$ True if both $p$ and $q$ are False?
21 answers
In propositional logic, the statement "If pigs can fly, then elephants can lay eggs." validates to true because the antecedent validates to false.
In other words, given $a rightarrow b$, if a is false, the entire statement is true. Why?
Just because the antecendent is false doesn't mean that another fact depends on it, right?
logic
asked Apr 28 '12 at 3:05
David FauxDavid Faux
1,56762342
$endgroup$
This question already has an answer here:
In classical logic, why is $(pRightarrow q)$ True if both $p$ and $q$ are False?
21 answers
In propositional logic, the statement "If pigs can fly, then elephants can lay eggs." validates to true because the antecedent validates to false.
In other words, given $a rightarrow b$, if a is false, the entire statement is true. Why?
Just because the antecendent is false doesn't mean that another fact depends on it, right?
This question already has an answer here:
In classical logic, why is $(pRightarrow q)$ True if both $p$ and $q$ are False?
21 answers
logic
logic
asked Apr 28 '12 at 3:05
David FauxDavid Faux
1,56762342
asked Apr 28 '12 at 3:05
David FauxDavid Faux
1,56762342
asked Apr 28 '12 at 3:05
David FauxDavid Faux
1,56762342
asked Apr 28 '12 at 3:05
David FauxDavid Faux
1,56762342
asked Apr 28 '12 at 3:05
David FauxDavid Faux
1,56762342
1,56762342
marked as duplicate by quid♦, Jyrki Lahtonen Oct 12 '15 at 8:08
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by quid♦, Jyrki Lahtonen Oct 12 '15 at 8:08
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
7
$begingroup$
In informal speech, "if $A$ then $B$" and "$A$ implies $B$" are mainly used when it is believed there is a causal connection between $A$ and $B$. The truth-functional connective $rightarrow$ does not capture this feature of "implies."
$endgroup$
– André Nicolas
Apr 28 '12 at 3:24
$begingroup$
I tend to think of it this way: when you draw out a truth table, a statement is considered false statements if and only if it is incompatible with the truth values of $a$ and $b$. For example, $a to b$ is compatible with $neg a, (neg) b$ and $a, b$ but not $a, neg b$.
$endgroup$
– Brett Frankel
Apr 28 '12 at 3:26
1
$begingroup$
The answers to this prior question should prove enlightening.
$endgroup$
– Bill Dubuque
Apr 28 '12 at 3:58
1
$begingroup$
See xkcd
$endgroup$
– Tim S.
Mar 28 '14 at 17:24
1
$begingroup$
Also this earlier question
$endgroup$
– Jyrki Lahtonen
Oct 11 '15 at 15:21
|
show 3 more comments
7
$begingroup$
In informal speech, "if $A$ then $B$" and "$A$ implies $B$" are mainly used when it is believed there is a causal connection between $A$ and $B$. The truth-functional connective $rightarrow$ does not capture this feature of "implies."
$endgroup$
– André Nicolas
Apr 28 '12 at 3:24
$begingroup$
I tend to think of it this way: when you draw out a truth table, a statement is considered false statements if and only if it is incompatible with the truth values of $a$ and $b$. For example, $a to b$ is compatible with $neg a, (neg) b$ and $a, b$ but not $a, neg b$.
$endgroup$
– Brett Frankel
Apr 28 '12 at 3:26
1
$begingroup$
The answers to this prior question should prove enlightening.
$endgroup$
– Bill Dubuque
Apr 28 '12 at 3:58
1
$begingroup$
See xkcd
$endgroup$
– Tim S.
Mar 28 '14 at 17:24
1
$begingroup$
Also this earlier question
$endgroup$
– Jyrki Lahtonen
Oct 11 '15 at 15:21
7
7
$begingroup$
In informal speech, "if $A$ then $B$" and "$A$ implies $B$" are mainly used when it is believed there is a causal connection between $A$ and $B$. The truth-functional connective $rightarrow$ does not capture this feature of "implies."
$endgroup$
– André Nicolas
Apr 28 '12 at 3:24
$begingroup$
In informal speech, "if $A$ then $B$" and "$A$ implies $B$" are mainly used when it is believed there is a causal connection between $A$ and $B$. The truth-functional connective $rightarrow$ does not capture this feature of "implies."
$endgroup$
– André Nicolas
Apr 28 '12 at 3:24
$begingroup$
I tend to think of it this way: when you draw out a truth table, a statement is considered false statements if and only if it is incompatible with the truth values of $a$ and $b$. For example, $a to b$ is compatible with $neg a, (neg) b$ and $a, b$ but not $a, neg b$.
$endgroup$
– Brett Frankel
Apr 28 '12 at 3:26
$begingroup$
I tend to think of it this way: when you draw out a truth table, a statement is considered false statements if and only if it is incompatible with the truth values of $a$ and $b$. For example, $a to b$ is compatible with $neg a, (neg) b$ and $a, b$ but not $a, neg b$.
$endgroup$
– Brett Frankel
Apr 28 '12 at 3:26
1
1
$begingroup$
The answers to this prior question should prove enlightening.
$endgroup$
– Bill Dubuque
Apr 28 '12 at 3:58
$begingroup$
The answers to this prior question should prove enlightening.
$endgroup$
– Bill Dubuque
Apr 28 '12 at 3:58
1
1
$begingroup$
See xkcd
$endgroup$
– Tim S.
Mar 28 '14 at 17:24
$begingroup$
See xkcd
$endgroup$
– Tim S.
Mar 28 '14 at 17:24
1
1
$begingroup$
Also this earlier question
$endgroup$
– Jyrki Lahtonen
Oct 11 '15 at 15:21
$begingroup$
Also this earlier question
$endgroup$
– Jyrki Lahtonen
Oct 11 '15 at 15:21
|
show 3 more comments
1 Answer
1
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There are some plausible arguments for having "if $a$ then $b$" true when $a$ is false (like suggested ex falso quodlibet). But the fact is $rightarrow$ doesn't even try to capture the if-then relation between propositions. $a rightarrow b$ is defined as $neg a vee b$, and it's obvious why that's true when $a$ is false.
The actual if-then relation can be more appropriately captured by, for example, $a Rightarrow b$. This is not propositional logic statement (rather metalogical), it says "it's impossible for $a$ to be true when $b$ is false".
Or better yet, use modal logics with modalities of necessity (physical, metaphysical, logical etc.): $square (a rightarrow b)$. This is much closer to capturing if-then relation of everyday use. Interpretation is "it's (physically/metaphysically/logically/...) impossible that $a$ is, but $b$ isn't". In fact trying to formalize if-then was perhaps the main reason why alethic modal logic was invented in the first place.
answered Jun 29 '12 at 22:28
Luka MikecLuka Mikec
495317
$endgroup$
1
$begingroup$
You can read more about the modal version of implication Luka mentions here: en.wikipedia.org/wiki/Strict_conditional
$endgroup$
– Qiaochu Yuan
Jun 29 '12 at 22:28
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
There are some plausible arguments for having "if $a$ then $b$" true when $a$ is false (like suggested ex falso quodlibet). But the fact is $rightarrow$ doesn't even try to capture the if-then relation between propositions. $a rightarrow b$ is defined as $neg a vee b$, and it's obvious why that's true when $a$ is false.
The actual if-then relation can be more appropriately captured by, for example, $a Rightarrow b$. This is not propositional logic statement (rather metalogical), it says "it's impossible for $a$ to be true when $b$ is false".
Or better yet, use modal logics with modalities of necessity (physical, metaphysical, logical etc.): $square (a rightarrow b)$. This is much closer to capturing if-then relation of everyday use. Interpretation is "it's (physically/metaphysically/logically/...) impossible that $a$ is, but $b$ isn't". In fact trying to formalize if-then was perhaps the main reason why alethic modal logic was invented in the first place.
answered Jun 29 '12 at 22:28
Luka MikecLuka Mikec
495317
$endgroup$
1
$begingroup$
You can read more about the modal version of implication Luka mentions here: en.wikipedia.org/wiki/Strict_conditional
$endgroup$
– Qiaochu Yuan
Jun 29 '12 at 22:28
add a comment |
$begingroup$
There are some plausible arguments for having "if $a$ then $b$" true when $a$ is false (like suggested ex falso quodlibet). But the fact is $rightarrow$ doesn't even try to capture the if-then relation between propositions. $a rightarrow b$ is defined as $neg a vee b$, and it's obvious why that's true when $a$ is false.
The actual if-then relation can be more appropriately captured by, for example, $a Rightarrow b$. This is not propositional logic statement (rather metalogical), it says "it's impossible for $a$ to be true when $b$ is false".
Or better yet, use modal logics with modalities of necessity (physical, metaphysical, logical etc.): $square (a rightarrow b)$. This is much closer to capturing if-then relation of everyday use. Interpretation is "it's (physically/metaphysically/logically/...) impossible that $a$ is, but $b$ isn't". In fact trying to formalize if-then was perhaps the main reason why alethic modal logic was invented in the first place.
answered Jun 29 '12 at 22:28
Luka MikecLuka Mikec
495317
$endgroup$
1
$begingroup$
You can read more about the modal version of implication Luka mentions here: en.wikipedia.org/wiki/Strict_conditional
$endgroup$
– Qiaochu Yuan
Jun 29 '12 at 22:28
add a comment |
$begingroup$
There are some plausible arguments for having "if $a$ then $b$" true when $a$ is false (like suggested ex falso quodlibet). But the fact is $rightarrow$ doesn't even try to capture the if-then relation between propositions. $a rightarrow b$ is defined as $neg a vee b$, and it's obvious why that's true when $a$ is false.
The actual if-then relation can be more appropriately captured by, for example, $a Rightarrow b$. This is not propositional logic statement (rather metalogical), it says "it's impossible for $a$ to be true when $b$ is false".
Or better yet, use modal logics with modalities of necessity (physical, metaphysical, logical etc.): $square (a rightarrow b)$. This is much closer to capturing if-then relation of everyday use. Interpretation is "it's (physically/metaphysically/logically/...) impossible that $a$ is, but $b$ isn't". In fact trying to formalize if-then was perhaps the main reason why alethic modal logic was invented in the first place.
answered Jun 29 '12 at 22:28
Luka MikecLuka Mikec
495317
$endgroup$
There are some plausible arguments for having "if $a$ then $b$" true when $a$ is false (like suggested ex falso quodlibet). But the fact is $rightarrow$ doesn't even try to capture the if-then relation between propositions. $a rightarrow b$ is defined as $neg a vee b$, and it's obvious why that's true when $a$ is false.
The actual if-then relation can be more appropriately captured by, for example, $a Rightarrow b$. This is not propositional logic statement (rather metalogical), it says "it's impossible for $a$ to be true when $b$ is false".
Or better yet, use modal logics with modalities of necessity (physical, metaphysical, logical etc.): $square (a rightarrow b)$. This is much closer to capturing if-then relation of everyday use. Interpretation is "it's (physically/metaphysically/logically/...) impossible that $a$ is, but $b$ isn't". In fact trying to formalize if-then was perhaps the main reason why alethic modal logic was invented in the first place.
answered Jun 29 '12 at 22:28
Luka MikecLuka Mikec
495317
answered Jun 29 '12 at 22:28
Luka MikecLuka Mikec
495317
answered Jun 29 '12 at 22:28
Luka MikecLuka Mikec
495317
answered Jun 29 '12 at 22:28
Luka MikecLuka Mikec
495317
495317
1
$begingroup$
You can read more about the modal version of implication Luka mentions here: en.wikipedia.org/wiki/Strict_conditional
$endgroup$
– Qiaochu Yuan
Jun 29 '12 at 22:28
add a comment |
1
$begingroup$
You can read more about the modal version of implication Luka mentions here: en.wikipedia.org/wiki/Strict_conditional
$endgroup$
– Qiaochu Yuan
Jun 29 '12 at 22:28
1
1
$begingroup$
You can read more about the modal version of implication Luka mentions here: en.wikipedia.org/wiki/Strict_conditional
$endgroup$
– Qiaochu Yuan
Jun 29 '12 at 22:28
$begingroup$
You can read more about the modal version of implication Luka mentions here: en.wikipedia.org/wiki/Strict_conditional
$endgroup$
– Qiaochu Yuan
Jun 29 '12 at 22:28
add a comment |
7
$begingroup$
In informal speech, "if $A$ then $B$" and "$A$ implies $B$" are mainly used when it is believed there is a causal connection between $A$ and $B$. The truth-functional connective $rightarrow$ does not capture this feature of "implies."
$endgroup$
– André Nicolas
Apr 28 '12 at 3:24
$begingroup$
I tend to think of it this way: when you draw out a truth table, a statement is considered false statements if and only if it is incompatible with the truth values of $a$ and $b$. For example, $a to b$ is compatible with $neg a, (neg) b$ and $a, b$ but not $a, neg b$.
$endgroup$
– Brett Frankel
Apr 28 '12 at 3:26
1
$begingroup$
The answers to this prior question should prove enlightening.
$endgroup$
– Bill Dubuque
Apr 28 '12 at 3:58
1
$begingroup$
See xkcd
$endgroup$
– Tim S.
Mar 28 '14 at 17:24
1
$begingroup$
Also this earlier question
$endgroup$
– Jyrki Lahtonen
Oct 11 '15 at 15:21