-4

I am a linguist and not a mathematician. If I was an expert mathematician, I wouldn't need your help.

In response to a query in a comment about when if P, Q is true, I recently was told by @amWhy:

  • "If P then Q" (P→Q) is true whenever P is false, or Q is true (or both).

This was in response to a comment on this question here:

(You will only be able to see this post if you have a very large reputation score)

Now this interpretation is one of many mathematical (not all very simple) interpretations of when If P, Q is true. What I would like to know is whether the following sentence is true, if it is said about a particular number which as been written down on a specific piece of paper:

  • If the number written on the paper is divisible by 9, it is divisible by 3.

In particular, suppose the number which is written on the paper is 4, is the sentence true?

Araucaria - him
  • 307
  • 2
    See https://en.wikipedia.org/wiki/Vacuous_truth -- in particular the example "if $2\gt5$ then $2\gt3$." – Barry Cipra Dec 19 '17 at 00:50
  • 1
    This seems to be duplicate of the question you linked to, which itself was marked a duplicate of an earlier question of yours. – Matthew Leingang Dec 19 '17 at 00:51
  • @MatthewLeingang I don't understand Matthew. How is this question a duplicate of whether a number divisible by 7 and also a square number is divisible by 49? – Araucaria - him Dec 19 '17 at 00:53
  • 1
    yes, if the number that is written, say $x$, is divisible by $9$ then it is also divisible by $3$, if $x$ is not divisible then $x$ may or may not be divisible by $3$. if $x$ is a constant that is not divisible by $9$ then this is vacuous truth, the statement is about an empty set(written on the paper$\land$divisible by 9) so it has to be true – ℋolo Dec 19 '17 at 00:56
  • See: Assumed True until proven False. The Curious Case of the Vacuous Truth – amWhy Dec 19 '17 at 01:39
  • If the number written on the paper is divisible by 9, it is divisible by 3. In your last supposition, "suppose the number which is written on the paper is 4, is the sentence true?" Yes, the sentence is true. The only case in which an implication is false is when the antecedent is true, but the conclusion is false. All the other three truth value assignments make the conditional true. – amWhy Dec 19 '17 at 01:44
  • This is the the definition of the conditional $P\to Q$. Yes, I know we want relevance, cause and effect, etc (as often implied in natural language). But mathematically/logically $P\to Q$ is defined to be true unless P is true, and Q is false. Not necessarily intuitive, but that is the definition. – amWhy Dec 19 '17 at 01:50
  • 1
    You write: "Now this interpretation is one of many mathematical (not all very simple) interpretations of when If P, Q is true." This is incorrect. The statement "If P, Q" has exactly one interpretation in mathematics. Unsurprisingly, it is sometimes interpreted differently in natural language, but that's not at issue here. – Alex Kruckman Dec 19 '17 at 01:52
  • @AlexKruckman Unsurprisingly, it is sometimes interpreted differently in natural language, but that's not at issue here. Except the OP is a linguist and not a mathematician, as they wrote, and there appears to be some crosstalk between the two here. – dxiv Dec 19 '17 at 01:58
  • 3
    @dxiv I understand that, and I think the crosstalk between the two is exactly the problem - in a sense, that was the point of my comment. If the OP is going to ask mathematicians what "if then" statements mean in mathematics, the OP should be prepared to discard their prior notions of what "if then" statements mean in natural language. – Alex Kruckman Dec 19 '17 at 02:07
  • 2
    @Araucaria: I think you are confusing a lot of people because you quote a definition of "if P then Q" which immediately answers your question, so it is unclear what exactly you feel you are missing. I think (but am not sure) that your misconception is that you believe "if P then Q" can have multiple different meanings in mathematics and so are asking about those other meanings, but this is just totally false as Alex Kruckman said. – Eric Wofsey Dec 19 '17 at 03:04
  • You may be interested in this blog post by Tim Gowers. @Araucaria –  Dec 19 '17 at 03:53
  • Hint: We have: $\neg 3\mid 4$ and $\neg 9 \mid 4$ – Dan Christensen Dec 19 '17 at 04:31
  • Hint: By the contrapositive rule, we also have: $9\mid 4 \implies 3\mid 4 \space \equiv \space \neg 3 \mid 4 \implies \neg 9 \mid 4$. Both are true. – Dan Christensen Dec 19 '17 at 04:34

2 Answers2

1

If the number that is written, say $x$, is divisible by $9$ then it is also divisible by $3$. If $x$ is not divisible by $9$, then $x$ may or may not be divisible by $3$, so in this case this is true statement.

If $x$ is a constant that is not divisible by $9$ then this is vacuous truth. The statement is about an empty set (written on the paper $\land$ divisible by 9), so it has to be true.

amWhy
  • 209,954
ℋolo
  • 10,006
1

Suppose you saw a sign one evening:

If you wear jeans then you will not be allowed inside this concert hall.

Would you question the truth of the statement just because no one around you is wearing jeans?

dxiv
  • 76,497
  • Doesn't answer the question. -1 – Araucaria - him Dec 19 '17 at 01:21
  • @Araucaria Guess we'll have to agree to disagree on that. IMHO this directly answers the question as literally written. If you had a different question in mind then by all means spell that out better in the main post. – dxiv Dec 19 '17 at 01:24
  • 4
    @Araucaria The truth value of the statement does not depend on the choice of the free variables, and I believe that's where your confusion lies. In your case, the statement is true regardless of what number is written on the paper. In my answer, the statement is true regardless of who is wearing what, or even whether there are any spectators lined up at all. – dxiv Dec 19 '17 at 01:33
  • 1
    anything follows from something that isn't true Correct, that's for example why inconsistent theories are complete. but you didn't say any of that Right, the point was to put forward an example along the same line as yours, only pushed to the extreme where (I thought) it became self-evident what the answer was and why. – dxiv Dec 19 '17 at 01:54