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Logicians,

Can you explain a seemingly curious facet of Modus Ponens?

Given Modus Ponens is expressed as:

$$p \to q $$

Supported by this Truth Table:

Line $p$ $q$ $p \to q$
1 T T T
2 T F F
3 F T T
4 F F T

QUESTION: Why does Line 3 resolve to $T$? It seems that $F \to T$ should resolve to $F$. No?

Your assistance in understanding this apparent contradiction is requested with gratitude.

Plane Wryter
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  • If you assume something false, anything can be proven true. So, the implication that a false thing implies anything is always true. That's why the last two rows are true. For example, if $0 = 1$, then I can add 1 to both sides to get $1 = 2$. – J126 Nov 02 '21 at 22:43
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    You're right: to some degree, this is a matter of convention, not some physical principal. It does allow a simpler set-up for propositional logic, and is not completely crazy... so there we are. – paul garrett Nov 02 '21 at 22:50
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    Note the distinction between the phrases "$q$ is a necessary condition for $p$ to occur" and "$q$ is both a necessary and sufficient condition for $p$ to occur"... the distinction between the expression $p\implies q$ and the expression $p\iff q$ – JMoravitz Nov 02 '21 at 22:53
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    @JMoravitz, good point. That does connect the issue with colloquial usage, too. :) – paul garrett Nov 02 '21 at 22:55
  • @paulgarrett It's why we argue the empty set is a subset of every set, so it's importance in mathematical reasoning is makes it more like an axiom/definition. But regarding your comment, you might enjoy "The Meanings of Implication" by John Corcoran. One realizes quickly all the subtleties in the natural language uses of the term, "implication". – amWhy Nov 02 '21 at 22:59
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    @amWhy, thanks for the recommendation! :) – paul garrett Nov 02 '21 at 23:06
  • $p \to q$ is not Modus Ponens. – Mauro ALLEGRANZA Nov 03 '21 at 07:09

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An implication is always true if the antecedent (the $p$ in the $p\to q$) is false. This is just true by definition. To offer an intuitive explanation, let's say a statement is true if I don't lie, and false if I lie.

Now consider me telling you this:

"If I race you on Tuesday, then I will win."

Now, if I didn't race you on Tuesday (say we raced on Wednesday and I lost)--did I lie? I say I didn't tell, since we didn't race on Thursday. I would've only been lying if I raced you on Tuesday and lost.

So, a person only lies (an implication is only false) if they make a statement of the form "if $p$, then $q$" and then they do $p$ but not $q$. They didn't lie if they didn't do $p$, even if they went on to do $q$ or not.