In Defense of Material Implication

[revised April 2009]

Jimmy wrote:

> And of course, it's even worse than that. Under material implication,
> every true statement is implied by anything whatsoever. There
> need be no causal or reality-oriented connection at all.
>
> If Castro is a dictator, then my pants are brown!
> Castro is a dictator,
> therefore my pants are brown!
>
> If Bill Clinton is a liar, then my computer runs Linux!
> Bill Clinton is a liar,
> therefore my computer runs Linux!
>
> These are both "valid" arguments in terms of strict logical form.
> Furthermore, it turns out that the premises are all true, and
> the conclusion is true!
>
> But something is deeply wrong with this picture.
>
> Rob, do you grant that something is deeply wrong with this picture?

Indeed. Something is deeply wrong here, but it has mostly to do with failures to understand material implication and very little to do with anything wrong with material implication itself.

There are three points to keep in mind. The first is that material implication -- as you and many others have noted -- issn't the same as what people are talking about in ordinary speech when they say that one thing is implied by another. It should be completely unsurprising that one of two statements is materially implied by the other when we wouldn't ordinarily say it is implied by the other. All that's required is that we keep clearly in mind which we're talking about.

If it's not the same notion, though, it might be wondered why it should be called "implication." The short answer is that it captures features that are present in any inference that deserves to be called implication. Specifically,

a) it requires that Q be true whenever P and P → Q are. That's necessary to support Modus Ponens inferences.
b) it requires that P → Q can be true while both P and Q are false. That's necessary to support Modus Tollens inferences.
c) it requires that P → Q is false when P is true but Q is false.

No sensible account of implication denies any of those. Material implication doesn't give us the whole story about what we normally call implication, but it gives the common core that has to be present in any more developed account.

Consider a tabular arrangement of the requirements (a) through (c).

P	Q	P → Q	Justification
True	True	True	(a)
False	True	?	?
True	False	False	(c)
False	False	True	(b)

They amount to most of the truth-table for material implication. There's only one line that is not settled by the need to accommodate well-known and uncontroversial forms of inference. Specifically, shall we say that "P → Q” is true or false, when P is false and Q is true? If we say that it's true, then we have the standard truth-table for material implication, so let's look around a bit.

Suppose we say it's false. The problem is that this does not escape admitting material implication. Let's use "*" for this supposed connective:

P	Q	P * Q
True	True	True
False	True	False
True	False	False
False	False	True

That is just the truth-table for the biconditional, "P if and only if Q," normally symbolized with either ≡ or ↔. But the important point is that the biconditional, P ≡ Q, is equivalent to the conjunction of two material conditionals, P → Q and Q → P. Thus,

P	Q	P → Q	Q → P	(P → Q) & (Q → P)
True	True	True	True	True
False	True	True	False	False
True	False	False	True	False
False	False	True	True	True

Now, there’s nothing wrong with the biconditional—but it’s not a way of avoiding material implication!

Can we instead just leave the matter undecided? That is, can we say that the truth or falsity of P → Q is not settled when P is false and Q is true? Surely, we can, but there are two further problems, depending on which way we want to go. The first is that if we do that and stick to it, we will only have a partial function for our implication operator, and with only a partial function, we will not be able to construct or avail ourselves of the standard proofs of consistency and completeness for first-order logic. The second is that it is really very hard to stick to a resolve not to use material implication. That's because it is equivalent to other formulations:

(P → Q) ≡ (~P v Q) ≡ ~(P & ~Q)

All of those have identical truth-tables, so it's easier to get rid of the words, "material implication" than to get rid of the thing itself. Once that is recognized, it seems, on the one hand, churlish to insist that logicians not introduce an explicitly defined connective for the job (and why not "→"?), and, on the other hand, it would involve truly heroic self-denial, not to mention a crippling of one's capacities for logical inference to avoid the thing itself rather than just the name.

There are three further points worth brief exploration. The first is that it is true that one can place additional restrictions upon the notion of implication. This will typically involve defining some other implication operator. "A implies B” may be understood to assert (say) a logical connection between A and B, such that it is not logically possible for A to be true and B to be false. E.g., A may be "this ball is round” and B may be "this ball has a shape." Unless there is that kind of logical connection between the two, you will not (on this proposal) admit that "A implies B" is true. But wait! Within the definition of "implication” appeared the phrase, "not logically possible.” What is that supposed to mean? It means that, using the ordinary logical tools, one can derive a contradiction from the assertions "A implies B," "A," and "~B.” Deriving the contradiction may require further definitions of A and B (as of "round” and "has a shape"), but once you have those in hand, you don't need anything but ordinary truth-functional logic to do it. In other words, the "alternatives” to material implication are usually not alternatives at all. They presuppose it.

The second point is that no one is going to be misled by material implications such as the one from Castro's being a dictator to the color of Jimmy's pants. It's true that one materially implies the other -- in fact, if the pants are brown, anything, whether true or false, materially implies that they are -- but this doesn't get us anywhere, unless we already know that the pants are brown. No exponent of standard formal logic is going to be led to make crazy inferences, because the truth of the conditional premises needed can only be known if the truth of the conclusions to be derived is already known.

The third is the most general. Do we need a truth-functional logic? That is, do we need some kind of logic that can identify tautologies (statements that are true, no matter what else is true), contradictions (statements that are false, no matter what else is true), and contingent statements (which are neither tautologies nor contradictions) by looking only to the form of the statements and the truth-values of their component parts?

I think the answer to that is -- or should be -- obvious. That's just what logic, at the most basic level, is -- the truths that hold for everything and don't depend on any particular content. If there is no respectable logic in that sense, then we wouldn't even be able to tell that "P & ~P” is always false, without knowing what "P" stands for.

Rob
_____
Rob Bass
rhbass@gmail.com
http://oocities.com/amosapient