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I was watching a video lecture on logical implication. A part of Prof. Keith Devlin's Introduction to Mathematical Thinking course. As the lecture was going, and after the professor gave the rationale for the truth value of implication in the first two cases where the antecedent (hypothesis) is true, he started discussing the truth value of implication in the other two cases by the following paraphrased sentence: "Since we have no intuition about the truth value of implication whenever the antecedent is false, we discuss its truth value using the negation form of it. Namely, when the antecedent does not imply the consequence." After that, however; he derived the case where the the negation of implication is true, by the following sentence: "Even though the antecedent is T, the consequence is nonetheless F." And proceeded: "In all other circumstances, the negation of the implication is false."

What I'm scratching my head over is the following:

What the professor discussed when dealing with the negation form was the case (T implies F), which we already had a good intuition for using the original form. And proceeded by just saying that in all other circumstances the negation form is false (which I do not understand how he assumed as well). How did the negation form help us at all build any kind of intuition for the false antecedent cases when all it discussed was a true antecedent case?

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closed as unclear what you're asking by Mauro ALLEGRANZA, 5xum, YCor, egreg, Mark Bennet Nov 14 '17 at 19:11

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Maybe we can try this way...

For a possible justification of the truth table for the conditional, we can use the classical equivalence of $¬(A → B)$ and $(A ∧ ¬B)$.

This is based on the reading of the conditional as: "$B$ is a necessary condition for $A$".

We have that $(A ∧ ¬B)$ is FALSE when either $A$ is FALSE or $¬B$ is FALSE.

Thus, the conditional $A → B$ has the value TRUE either when $A$ has the value FALSE or $B$ has the value TRUE, otherwise $A → B$ has the value FALSE.

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