Do we use material conditional in regular mathematics?

Question

I have been confused by the definition and truth table of material conditional (MC) for quite a while and the only responses I have received are variations of 'it is correct' and 'I should just get used to it'.

Today I discovered an interesting MC example, which regular mathematicians would never agree to as true, yet according to MC rules it is for most of natural numbers:

$x$ is prime $\implies$ $x + 2$ is prime

According to the definition of MC, this statement is true for all even numbers (except 2), all non-prime odd numbers, as well as for a small fraction of prime numbers (every first prime in a twin-prime pair). Only for the prime numbers that are not the first prime in a twin prime pair, does it fail to hold. This statement is true for about 99% of natural numbers.

Yet it is clear and understood according to regular mathematics that this is statement is simply wrong. By regular mathematics I mean 99% of mathematics, when we're not studying foundational issues, logic, and so on.

My question is, is it correct to state that MC has practically no relevance to regular mathematics? If yes, is it a good idea to insist on 'getting used to the MC definition', 'just treat MC as a truth table', or 'try not to worry too much about MC' when studying logic?

Edit 1: I would like to expand on what I said, in response to the comments mentioning quantified (or first-order) implication. Let's denote it by $_{\implies}^{\forall}$. The following is false in logic as well as regular mathematics:

$x$ is prime $_{\implies}^{\forall}$ $x+2$ is prime

However, there's agreement here only because we have counterexamples (e.g., $x=7$ and many more). We could reach the same conclusion even if we discarded the true parts of the MC truth table, i.e., the three clauses F$\implies$F, F$\implies$T, and T$\implies$T. We could replace their true values with something like, "N/A", "Unknown", or "Inconclusive", and our first-order logic, and the rest of regular mathematics would still work just fine. (Of course in that case, the MC table would no longer be a complete truth-table).

Quantified-implication does not need the MC truth-table, or even for MC to exist. All it requires is the result of a counterexample search (i.e., a case of $A \land \lnot B$, where, e.g., $A$ is '$x$ is prime' and $B$ is '$x+2$ is prime'). If the counterexample search is complete, and a counterexample is found, $_{\implies}^{\forall}$ is false. If the search is complete and no counterexample found, $_{\implies}^{\forall}$ is true. (If a counterexample is not found, and the search is not complete, or we're not sure if the search is complete, the true values of MC only mislead, never help).

In other words, we introduce true clauses in MC, because (maybe) we like a completed truth-table, at the expense of confusing students, when those true values are useless, and infact harmful, once we start using quantified implication with counterexample search.

Edit 2: I should clarify (and maybe correct) myself. Quantified-implication does need the MC truth table. So I stand corrected. However, MC truth table does not make much sense without the use of a quantifier. So what I'm trying to say is the following:

• Take the MC formula $\lnot [A \land \lnot B]$ and ignore the MC symbol $\implies$.
• The formula $\lnot [A \land \lnot B]$ is combined with $\forall$ to form quantified-implication $_{\implies}^{\forall}$ as follows:

$A(x) {}_{\implies}^{\forall} B(x)$ defined as $\forall x (\lnot [A(x) \land \lnot B(x)])$

• What that means is that the MC formula finds its use through first-order quantifiers in mathematics. However, in my opinion, if we try to ascribe 'implication' or 'if ... then' or any other meaning to the MC formula itself, instead of treating it as a mere formula, that causes needless confusion.

Edit 3: After the answer by Alex Kruckman, I should add that that's another helpful test-case, i.e., prove implication mathematically before invoking MC. However, it means the MC truth-table was only consulted after-the-fact. The implication was established mathematically, not through propositional logic. So the two use-cases so far: (1) as part of a quantifier, (2) as a pre-proved implication. My prime example doesn't fall in either, and hence I'm still of the opinion that there are problematic consequences of MC (and I'm not even talking about the paradoxes).

Answer 1

I have been confused by the definition and truth table of material conditional (MC) for quite a while and the only responses I have received are variations of 'it is correct' and 'I should just get used to it'.

The definition of the material conditional usually given as $A\to B \equiv \neg [A \land \neg B]$ as well as each line of its truth table can be formally derived from first principles using a simple form of natural deduction.

• $A\to B \equiv \neg [A \land \neg B]~~~$ Formal Proof (19 lines)
• $A \land B \to [A\to B]~~~~~~~$ Line 1 of the truth table Formal Proof (6 lines)
• $A \land \neg B \to \neg [A \to B]~$ Line 2 of the truth table Formal Proof (8 lines)
• $\neg B \to [A\to B]~~~~~~~~~~~$ Lines 3 and 4 of the truth table Formal Proof (8 lines)

My question is, is it correct to state that MC has practically no relevance to regular mathematics?

No. Quite the contrary. MC is unavoidable in "regular" mathematics. You seem to be quite misinformed on this matter. If you want to make a name for yourself, find the error in any of the above proofs. The first is probably the most important. (Only 19 lines!)

Answer 2

In the comments to Dan Christensen's answer, you ask for examples in "regular mathematics" where the material conditional finds use without a quantifier in a meaningful way.

Of course this happens - mathematicians prove conditional theorems all the time. Many important theorems in number theory are conditional on difficult open problems. For example, you might prove a theorem like "If the Riemann Hypothesis holds, then we get some bound on the error term in the prime number theorem." See here. Let's abbreviate this example theorem by RH$\rightarrow$EB.

You can see the truth table for the material conditional at work here:

• If someone comes along and proves RH, then because the theorem RH$\rightarrow$EB is true, the error bound EB must be true.
• On the other hand, if RH turns out to be false, then the error bound EB might still be true or it might not. In this situation (when RH is false, and regardless of whether EB is true or not), we do not have to turn around and start looking for a mistake in the proof of our theorem RH$\rightarrow$EB. The theorem RH$\rightarrow$EB is suddenly much less interesting, but it is still true!