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It makes sense that the statement P→Q is true when P = "x is divisible by 4" and Q = "x is even", because no matter what I plug in for x, either the premise P is false, or P is true and I can prove Q is true using basic number theory.

But the statement P→Q doesn't make sense to me when x is a specific number instead of left as a variable. For example, "If 3 is divisible by 4, then 3 is even" and "If 6 is divisible by 4, then 6 is even" and "If 4 is divisible by 4, then 4 is even" don't feel meaningful in any sense. Yet according to the truth table for implication, these statements all have truth values.

Is the statement P→Q that depends on some x considered true if the implication is true for all values of x?

Mauro ALLEGRANZA
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Isaac Wachsman
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  • If you don't like $p\to q$, try the contraposition $\neg q\to \neg p$ – Quý Nhân May 25 '25 at 05:00
  • The symbol $p\to q$ represents the logical function which returns the value "false" if $p$ is true and $q$ is false, and returns the value "true" in all other cases. People who work with mathematical logic find this function useful, even indispensible. Your problem is not about "deep understanding" of that function. Your problem is that you don't like to call it "implication" because of associations that word has for you. Other people don't like the terms "imaginary number" and "black hole" for similar reasons. Get used to it. "A rose by any other name would smell as sweet." – user14111 May 25 '25 at 05:51
  • @IsaacWachsman I'm struggling to understand your final sentence (it sounds like you are asking, "Is X true if X is true?"). $\quad$ $\quad$ The difference between the first and subsequent examples is that the former is implicitly universally quantified whereas the latter are just clinical exercises in applying the definition of material implication. Related: understanding implication. – ryang May 25 '25 at 07:20
  • Your problem is that you don't like to call it "implication" because of associations that word has for you. @user14111 The OP has the same issue even if the statement is phrased as "If X, then Y", so it isn't about the connotations of the word "implication". – ryang May 25 '25 at 07:21
  • @ryang I think the OP's problem is prior associations with words like "implies" and "if" and "then". It's just a truth function. I suspect OP wouldn't have a problem if it was called something like $f(p,q)$ and we didn't try to put it into English words. I could be wrong; I'm not a psychologist. So you think the OP would object to the function defined by the truth table for $p\to q$ no matter what it was called? – user14111 May 25 '25 at 07:36
  • @user14111 Strawman. – ryang May 25 '25 at 07:56

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