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In classical logic, why is$ (p\Rightarrow q)$ True if both p and q are False?
How to interpret material conditional and explain it to freshmen?

We're told that Mathematicians regard the propositions “P implies Q” and “if P then Q” as synonymous, so both have the same truth table. One example is:

“If Goldbach’s Conjecture is true, then $x^2$ ≥ 0 for every real number x.”

Since we have no idea whether Goldbach's conjecture is true, we are either looking at

TT or FT as far as PQ in the truth table. Either way, since Q is definitely true, the proposition as a whole is true.

But I think I'm missing something basic here. In what sense is this "if this then that"? In ordinary english, at least, "if this then that" means that the truth of "this" is what the truth of "that" depends on. But that "$x^2$ ≥ 0 for every real number x." is true regardless of the truth of Goldbach's Conjecture.

Like I said, I'm clearly missing something here. I just don't understand how either "P implies Q" or "if P then Q" corresponds in any way to what I see in the truth table. Could some explain this?


2 Answers 2


One way to think of conditional implication is as a promise.

"If you get an A in my course, then I promise to pay for your tuition."

When is the promise broken? It is broken only if you get an A in my course, but I do not pay for your tuition (i.e. $T \rightarrow F$).

The case that's giving you some pause is the case where you do not get an A in my course, yet I still pay for your tuition ($F \rightarrow T$). Can it be said that I've broken my promise in this instance? No, I've just been exceptionally generous for some reason.

Since the promise is broken only in the case $T \rightarrow F$, the statement as a whole is only false in that particular case.

Your example could be read as:

"If Goldbach's Conjecture is true, then Mathematics promises $x^2 \geq 0$ for every real number $x$."

This statement is true as a whole because, regardless of whether Goldbach's Conjecture is true, Mathematics is keeping its end of the bargain regarding the statement about real numbers.

As an aside, you are right that trivially true statements of this kind are a little strange and uninteresting. The kind of conditional statements people actually care about in practice are the ones where it actually matters whether the hypothesis is true. In such a case, you would be justified in saying the that conclusion depends on the hypothesis.

  • $\begingroup$ But if Mathematics is keeping its end of the bargain regarding the statement about real numbers regardless of whether Goldbach's Conjecture is true, then it what sense is it "THEN mathematics promises"? How is this conditional? $\endgroup$
    – LuxuryMode
    Nov 25, 2012 at 0:18
  • 1
    $\begingroup$ @LuxuryMode You are overloading the word "conditional". In this setting, it is just a particular type of mathematical statement. It does not mean "the truth of the conclusion depends on the truth of hypothesis"; these are just the only interesting kinds of conditionals. $\endgroup$ Nov 25, 2012 at 0:23
  • $\begingroup$ Gotcha. Thanks Austin. Thinking about it like a promise is certainly helpful. $\endgroup$
    – LuxuryMode
    Nov 25, 2012 at 0:26
  • $\begingroup$ @LuxuryMode: Related. $\endgroup$ Nov 25, 2012 at 14:16
  • $\begingroup$ @CameronBuie very helpful, thanks! $\endgroup$
    – LuxuryMode
    Nov 26, 2012 at 1:04

First, re the claim that

'In ordinary english, at least, "if this then that" means that the truth of "this" is what the truth of "that" depends on.'

Not so.

Suppose you recall going to the party at Jack and Jill's back in February, and recall the party was a birthday party for one of them, but not which one.

You are then asked when Jill's birthday is. Then you are of course entitled to reply, 'Well, if Jack doesn't have a February birthday, then Jill does'. You have excellent evidence the conditional is true. And that isn't defeated by the fact that Jill's being born in February, if she was, doesn't depend on which month Jack was born in unless they are twins or such like: there is of course (almost certainly) no connection at all between the facts of their birth dates.

In sum: any ordinary language indicative conditional commit us to no more than a truth-functional association between the antecedent and consequent.

Second, re the claim

'Mathematicians regard the propositions “P implies Q” and “if P then Q” as synonymous'.

No they don't. Or at least not in practice, whatever they sometimes carelessly say, especially when "if" is read -- as it often is in mathematical contexts -- as officially truth-functional.

Who would say that, e.g. Fermat's Last Theorem implies the Hawking singularity theorem, even though the corresponding material conditional is true? Or choose some other example of two true propositions $P$ and $Q$ chosen from unrelated bits of maths. The (material) conditional $P \to Q$ may be true, but even though neither proposition implies the other in any normal sense.

  • $\begingroup$ Hi, Peter, I'm glad you answered. I just didn't have the "wind" to address "implies" and/or "if...then" again, save for pointing to some earlier posts. I would like to add some "frequently asked questions" (adding "logic" as a topic/category) on the meta site, and in particular, to address the material conditional, as it seems to be a frequently encountered question. See this post and my "answer": Feel free to "chime in" (edit my post, add links to questions with good answers), comment, etc. $\endgroup$
    – amWhy
    Nov 25, 2012 at 0:50
  • $\begingroup$ Great points Peter. Thanks! Could you elaborate on what you mean by "truth-functional"? $\endgroup$
    – LuxuryMode
    Nov 25, 2012 at 4:30

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