Can a statement be true ( materially) while contradicting a true general rule of mathematics? ( An apparent problem with a conditional statement).

Question

Suppose a student says : "if 17 is even, then 2 is not a divisor of 17".

Surely his teacher would tell him he is wrong, saying that when a number is even, this number has 2 as divisor. The teacher would correct with " if 17 were even, then 2 would be a divisor of 17". In other words, the student's claim contradicts the general rule : " For all number x, if x is even , then x has 2 as divisor." So, if 17 is even...

But, since the sentence uttered by the student is a conditional with a false antecedent, this sentence ( the whole conditional) is true ; in virtue of " ex falso sequitur quodlibet" ( from a false proposition, anything follows).

My question is : what is wrong in the student's claim?

Can this hypothetical case be clarified by saying that

(1) the student's sentence is materially true

(2) the teacher is right in saying that the sentence is false in case it is understood as asserting a consequence relation ( logical consequence) between the antecedent and the consequent?

Or , am I wrong in saying that " if 17 is even , then 2 is not a divisor of 17 " contradicts ( or is incompatible with) " For all number x, if x is even, then x is divisible by 2" ?

Answer 1

First of all, the fact that a material implication is considered true as soon as its antecedent is false, is not the same as ex falso sequitur quodlibet, which says that any statement follows from a contradiction.

But yes, if you interpret the student's claim as a material conditional, then technically the student's claim is considered true. It would be just as true as "If I live in London, then I live in Germany" ... interpreted as a material conditional this is considered true because I don't live in London.

Still, the teacher would say the student is wrong, and offer the correction exactly as you indicated. This is because in practice, the use of mathematics is such that under normal circumstances, when the student makes a statement like this, the student is expected to have used the definition of what it means for a number to be even, rather than that the student is making some kind of smart-aleck claim trying to exploit the paradox of the material implication.

Indeed, note that the teacher does not say that the student's claim is false, but rather that the student did something wrong: the student wrongly applied the definition of even-ness. Thus, the teacher says: "No no, you did that wrong: if 17 is even, then 2 is a divisor of 17".

Answer 2

The student would be right in claiming "if $17$ is even, then $2$ does not divide $17$", as well as in claiming "if $17$ is even, then $2$ divides $17$".

The fact of the matter is that for any proposition $P$, "if $17$ is even then $P$" is correct, no matter how contradictory $P$ is : for instance, one rightfully conclude from these two statements that "if $17$ is even, then $2$ divides and does not divide $17$". One must not forget that there is a "if $17$ is even"-assumption, which of course makes all the rest irrelevant, because $17$ isn't even.

The teacher would be wrong on two accounts: saying that the student is wrong; but also transforming the mathematically relevant "if $17$ is even" into the irrelevant "if $17$ were even", which is not a mathematical statement.

By the way, if you believe that intuitionism has managed to go beyond material implication, then that isn't a property of material implication, just of implication.

Answer 3

You have stated a general rule, which is mathematically true:

For every number $x,$ if $x$ is even, then $x$ has $2$ as divisor.

Since the part following "for every number $x$" is true for every number $x,$ it is true in the particular case where $x=17,$ that is,

If $17$ is even, then $17$ has $2$ as divisor.

Interpreted as mathematical statements, both the statement about the number $17$ and the "if-then" clause of the general rule are material conditionals, which are true whenever the antecedent is false.

The following is also a material conditional, true for the same reason that the previous conditional was (namely, because the antecedent is false):

If $17$ is even, then $2$ is not a divisor of $17$.

Note that this statement does not contradict either the previous conditional or the general rule in any way. In order to contradict a statement, you must assert its negation. You could write the negation of "If $17$ is even, then $17$ has $2$ as divisor" as follows:

$17$ is even and $2$ is not a divisor of $17.$

You could write the negation of the general rule this way:

There exists a number $x$ such that $x$ is even and $2$ is not a divisor of $x$.

The statement "If $17$ is even, then $2$ is not a divisor of $17$" is not much like either of those statements; "if ... then" is nothing like "and".

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The material conditional is not how I would express a logical consequence. If I wanted to make a claim that "$2$ does not divide $17$" arose as a logical consequence of the premise that "$17$ is even", I might write

Assuming that $17$ is even, it follows that $2$ does not divide $17.$

This is a very different sentence from any of the examples I gave above. It includes no material conditional. Instead it asserts the existence of a logical inference which (I believe) is invalid. And indeed if a student were to make such a statement, a teacher should not accept it as true, but might instead ask the student to provide a proof of this statement.

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By the way, the principle "ex falso sequitur quodlibet" is applied to deductions. An implication can be the subject of a deduction, but it is not itself a deduction. So I would not say that a material conditional with false antecedent is true because of "ex falso sequitur quodlibet." The falseness of the antecedent in a material conditional does not cause the consequent to follow (that is, it does not cause the consequent to become true); rather, it allows the consequent to be false without falsifying the entire conditional statement.