# the “vacuously true” matter [duplicate]

I know the mathematical conditional's truth table ($P \Rightarrow Q$) is set this way : "the conditional" truth table

the talk is about the time when P is false. why did we choose the conditional to be true when P is false? I mean we could have said that it is T/F (or we don't know). because, when P is false, it is an argument that the conditional isn't related to. it is related to the argument where P is true. "because it isn't false" isn't a satisfying argument. and I can't see a logical proof of ($P \Rightarrow Q$) being equivalent to ($\lnot P \lor Q$) (that doesn't involve the truth table)(since the truth table is under question)

edit: I have read the" In classical logic, why is $(p\Rightarrow q)$ True if $p$ is False and $q$ is True? " and it didn't have a solution to my question and it doesn't address the possibility of T/F

## marked as duplicate by anomaly, Shailesh, Namaste, Claude Leibovici, Matthew ConroyNov 13 '16 at 19:10

• It makes the extension to predicate logic work right. That's the best motivation I have seen. – Ian Nov 12 '16 at 20:33
• Can we set up a FAQ for questions like this? It's asked so often, and I don't think there's an answer that satisfies the questioners. – anomaly Nov 12 '16 at 20:33
• See also the post implication and ordinary language. – Mauro ALLEGRANZA Nov 12 '16 at 20:44
• @ Mauro ALLEGRANZA , I see your point ,but then why did mathematicians choose the truth functional definition ? it is something that isn't a postulate nor proved. they could have chosen the non-truth functional definition which has the T/F when P is false link .. note: I am no expert. I'm just trying to understand. – someone Nov 13 '16 at 13:37

If we have to treat the conditional as a truth-funtional operator, then the one we picked is, for various reasons, the best one. Here is one quick argument for picking the truth-table definition that we do:

P is true, Q is false: this should of course be false

P is true, Q is true / P is false, Q is false: we want that $P \rightarrow P$ is a tautology, so we have to set both these entries to true to make that work.

P is false, Q is true: we would like to be able to show that $P \rightarrow Q$, therefore $Q \rightarrow P$ is not a valid inference ( that is, we don't want the conditinal to be 'symmetric' or commutative as a mathematician would say). Given the other values, the only way to make the conditinal non-commutativ would be to make the P False Q True entry True.

But you are right, there is no requirement to say that the conditional is truth-functional. That is, it is far from clear that the truth-value of a natural language conditional is a function of the truth-values of the antecedent and consequent, except in the case of the true antecedent plus false consequent of course: that should clearly be false. But take something like 'if snow is white, then bananas are yellow': do we consider that true just because both parts are true? Not clear. Or how about this: if Bob lives in Paris, then Bob lives in Germany. Well, if Bob does not live in Paris, then the material conditional would set this to true, yet any normal person would say this is false.

So yes, there are clear mismatches between the way conditionals are used in natural language, and the mathematically defined truth-function, called the material conditional, that we usually use to analyze such conditionals. It is really not unlike any other case where we try to capture something real using mathematical definitions: the definitions are not always perfect, but are often 'good enough', e.g. If I use euclidian geometry on the surface of the earth, it is usually 'good enough', even though it is only a very close approximation.

Still, the mismatch between the natural language conditional and the material conditional can lead to some really counterintuitive results, so this is called the paradox of te material conditional. Here is an example: according to truth-functional logic, we have the following equivalence $(P \land Q) \rightarrow R \Leftrightarrow (P \rightarrow R) \lor (Q \rightarrow R)$. But if you fill in 'Pat is male' for P, 'Pat is unmarried' for Q, and 'Pat is a bachelor' for R, you will find yourself agreeing with the left side, but not with the right side.

Now, for reasons like these, we could indeed decide to leave the truth-value of the conditional undetermined, and that is what some logicians do. But for many practical purposes, the material conditionalis 'good enough', and it certainly makes our mathematical lives easier: as long as you understand the situations in which the truth-functional analysis is 'good enough', then go ahead and use it. But don't use it when it is not good enough, just like there are times when you should not use euclidian geometry or newtonian physics or statistical analyses, when the assumptions or axioms that go into those methods, definitions, or theorems don't apply to the situation you are analyzing.

So, my best advice is to regard truth-functional logic as a tool. And, just as we learn when it is a good time to use a hammer, and when it is not (and maybe should use a screwdriver instead), an experienced logician will know when it is appropriate to use truth-functional analysis, and when it is not.

• Thank you for this great explanation ,but for what reasons, the ($\lnot P \lor Q$) was picked as the best definition for the conditional ($P \Rightarrow Q$) ? – someone Nov 13 '16 at 18:29
• @someone I added one quick argument. The comments and answers to questions that are seen as duplicate or related to your question will probably have many more arguments. – Bram28 Nov 13 '16 at 19:09