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The implication "$A\to B$" is false if $A$ is true and $B$ is false. Otherwise, the implication is true. So, this gives the somewhat surprising implication "False -> True/False" to be true.

I was able to wrap my head around this and prove some examples, but I'm not able to do that with the following implication "$A\to \lnot A$". For example, how can someone make sense of "if $7<4$, then $7\ge 4$"?

amWhy
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billy 9
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  • We have the tautology $\neg A \implies (A\implies B)$ regardless of the truth value of $B$. While $A\implies B$ may be true, we can infer nothing about the truth value of $B$ since the antecedent $A$ is false. More on this topic here: https://dcproof.wordpress.com/2017/12/28/if-pigs-could-fly/ – Dan Christensen Jan 03 '23 at 01:05

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"If you move, I shoot" = "Don't move or I shoot".

Hence, it's pretty natural to say that $A \to B$ is an abbreviation for $\text{not}(A) \,\, \text{or} \,\, B.$ But then if you think of the implication as $\text{not}(A) \,\, \text{or} \,\, B$ you immediately understand why it is true if $A$ is false. Indeed, if $A$ is false, $\text{not}(A)$ is true and therefore $\text{not}(A) \,\, \text{or} \,\, B$ is also true.

user700974
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