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Question:

Do you believe that the material implication correctly models the kind of conditional reasoning necessary in mathematics to prove a theorem?

Example:

If x > y and y > 0, then x > 0;

x > y and y > 0;

So, x > 0.

We can also express this idea like this, where A and B are mathematical expressions which are assumed as either true or false:

"A ⊃ B is a tautology" is equivalent to "If A, then B".

If you believe that, can you justify your belief?

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  • Comments are not for extended discussion; this conversation has been moved to chat.
    – Geoffrey Thomas
    Jan 1 at 11:19
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    Focusing questions on "you" as in "do you believe that?" is unsuitable for this site per philosophy.meta.stackexchange.com/questions/474/…
    – Fizz
    Apr 12 at 22:18
  • @Fizz No. Your reference says: "you should only ask questions that are practical, answerable ones that you run into while studying philosophy. Answers should address the question directly, with rigor and depth. Both should be framed in clear and neutral language." My question does exactly that. Still, I'll edit to suit your taste for being explicit. Apr 13 at 11:11
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First, a pro-material conditional case.

Mathematicians rarely prove theorems which are genuinely of the form "If P then Q." Instead, results which are phrased colloquially as conditionals generally contain a universal quantifier, e.g. "If x is a prime >2 then x is odd" is shorthand for "For all x (if x is a prime >2 then x is odd)."

Now here are some relevant points about how we use, and hence what we mean by, quantifiers and implication in mathematics:

  • A statement of the form "For all x(P(x))" is true iff P(a) is true for each individual a (in the relevant domain of discourse which is presumably implied by the context of the assertion).

  • A statement of the form "For all x(if U(x) then V(x))" is false iff for each a such that U(a), it is also the case that V(a) (ditto).

The second is the more subtle one, but this is plainly shown in how we understand statements like the above-mentioned "For all x(if x is a prime >2 then x is odd)." This statement is true, notwithstanding individuals such as 36 (whose corresponding instance has false antecedent and consequent) or 39 (whose corresponding instance has false antecedent but true consequent).

The point is that by combining these two we see that the only way to have a statement of the form "If U(a) then V(a)" be false is if U(a) is true but V(a) is false: just set P to "If U then V." So when we say "If --- then ---," we are indeed using the material conditional.


The above argument does have a weakness, however. While there are no instances I'm aware of of an accepted theorem whose accepted natural-language expression involves an "if/then" clause but whose accepted proof is invalid when we interpret that expression using the material conditional, there are instances where a reasonable mathematician would feel weird asserting an "if/then" statement despite its being true according to the material conditional.

For example, consider the claim

X: If 2+2=4, then Fermat's Last Theorem is true.

By a short argument of Wiles, together with some careful arithmetic, we can whip up a proof of the formalization of X via the material conditional. However, most mathematicians would feel a bit weird claiming X as true. So in principle there is a "market" for alternative interpretations of the conditional in mathematics.


However, this ultimately doesn't seem to carry much weight with the community. There are plenty of situations where a conditional result "If conjecture A then conjecture B" is proved, and later rendered obsolete by either a disproof of conjecture A (often using that conditional result!) or a proof of conjecture B, but is not "retroactively rejected." Ultimately the response of the mathematical community to weirdness like X above is not that it reveals the inappropriateness of the material conditional for mathematics, but rather that it reveals an inconsistency in our own natural language use which we should take pains to remove from mathematical discourse - with the result that ultimately the vast majority of mathematicians accept X as true, albeit silly to the point of disreputability. So ultimately I claim:

Yes, the material conditional successfully models mathematical reasoning.


Of course it's difficult to prove a claim like this without making a gigantic poll of mathematicians, which I don't have the resources to do. Moreover, even if every mathematician on the planet were in lockstep agreement with me, one could always argue that we're "doing mathematics wrong" in some sense. But at a certain point the onus is on the skeptic to provide something: how do they think natural-language conditional statements, or more relevantly statements incorporating conditionals, about mathematical topics be analyzed in mathematics? Barring a specific claim at some point there's little more to say in my opinion.

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    +1 Your example of "If 2+2=4, then Fermat's Last Theorem is true" falls under the principles of conversational implicature. Grice would say that such a sentence is true, but it is an inappropriate thing to assert because it violates one of the rules of the cooperative principle, namely: be relevant. It would be like saying, "if I'm wearing odd socks tomorrow, there will be a solar eclipse" when I know there will be a solar eclipse whether I wear odd socks or not.
    – Bumble
    Jan 1 at 2:38
  • @NoahSchweber Thank you for addressing the question; The point is whether you believe this is true. I am not asking you to prove it. You could perhaps cut your answer down to bare bones: The point is not to give a lecture on logic. Some elements in your answer are effectively irrelevant. All I hope is to get enough answers to have a rough estimate of how prevailing your view is. A shorter answer would help. Thanks. Jan 1 at 11:13
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    @Speakpigeon I don't think opinion polls are appropriate for this site; I'm happy to restructure my answer and may do so later today, but I'm not interested in shortening it to "yes." Jan 1 at 18:51
  • I would say for most of mathematics it makes little difference which conditional is "actually" used, if this "actual use" is even something cogent. Most proofs simply apply prior results in sequence, so all one needs is transitivity. The use of cases and reductio extends the results to what is materially derivable, but people do not "actually" think in terms of conditionals when using them. So the material conditional is more of an interpolating fiction that reproduces the desired body of theorems than something with a distinct cognitive counterpart.
    – Conifold
    Jan 2 at 10:19
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    @DanChristensen Yes, I am aware of how the material conditional works. My point is that that's an implication which most mathematicians in practice feel a bit odd about construing as a material conditional; regardless of one's feelings about the material conditional, this is a sociological fact. (Hence discussions such as this one.) As I say in my answer, though, I don't ultimately find this compelling - I am after all a proponent of the material conditional. Mar 8 at 3:51
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Do you believe that the material implication correctly models the kind of conditional reasoning necessary in mathematics to prove a theorem?

Yes. Material implication would seem to be consequence of classical logic.

Here I will justify both the standard truth table, as well as the logical equivalence,

A => B = ~[A & ~B]

Both are often used to introduce material implication in introductory textbooks.


1. The Truth Table

enter image description here

We can formally prove each line of the truth table from first principles using a form of natural deduction.

Line 1 tells us that if A is true and B is true, then A => B is true.

Proof:

enter image description here

Line 2 tells us that if A is true and B is false, then A => B is false.

Proof:

enter image description here

Lines 3 and 4 tell us that if A is false, then A => B is true regardless of the truth value of B. Also known as the Principle of Vacuous Truth, it can be used in proofs about empty sets (hence the name?), and in proofs by cases in which one or more cases being considered are known to be false. I would think that it is not much used outside of mathematical proofs.

Proof:

enter image description here

Thus we have justified each line of the standard truth table.


2. The Equivalence: A => B = ~[A & ~B]

We can also prove that if A => B is true, then ~[A & ~B] must be true.

Proof:

enter image description here

Conversely, we can also prove if ~[A & ~B] is true, then A => B must be true.

Proof:

enter image description here

Thus we have established a fundamental theorem of classical logic, the logical equivalence:

A => B = ~[A & ~B]

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