Monday, 10 October 2016

Material Logic vs. Formal Logic?

Edwin D. Mares displays the problem (if it is a problem) with a purely formal logic by offering us the following example of a valid argument:

The sky is blue.
there is no integer n greater than or equal to 3 such that for any non-zero                                                    integers x, y, z, xn = yn + zn.

Mares says that the above “is valid, in fact sound, on the classical logician's definition” (609). It's the argument that is valid; whereas the premise and conclusion are sound (i.e., true). In more detail, the “premise cannot be true in any possible circumstance in which the conclusion is false”.

Clearly the content of the premise isn't connected, semantically, to the semantic content of the conclusion. However, the argument is valid and sound.

So what's the point of the above?

Perhaps no logician would state it for real. He would only do so, as Mares himself does, to prove a point about logical validity. But can't we now ask why it's 'valid' even when the premise and conclusion are true?

Perhaps showing the bare bones of this argument will help. Thus:

P
Q

Do that look any better? I suppose so. Even though we aren't given the semantic content, both P and Q must be seen to have a truth-value. (In this case, both P and Q are true.) It is saying: P is true. Therefore Q is true. It isn't saying, Q is a consequence of P; or that P entails Q. Basically, we're being told, by the logic, that two true statements can exist together if they don't contradict each another.

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There can be cases in which the premises of an argument are all true, and the conclusion is also true, yet, as Stephen Read puts it, “there is an obvious sense in which the truth of the premises does not guarantee that of the conclusion” (237). Ordinarily, of course, the truth of the premises is meant to guarantee the truth of the conclusion. So let's look at Read's example, thus:

i) All cats are animals
ii) Some animals have tails
iii) So some cats have tails.

Clearly, premises i) and ii) are true. Indeed iii) is also true. (Not all cats have tails. And, indeed, according to some logicians, saying 'some' also implies saying 'all'.)

So why is the argument invalid?

It's invalid not for the truth-values of the premises and conclusion; but for another reason.

The reason is that the sets in the argument are, as it were, mixed up. Thus we have [animals], [cats] and, indeed, [animals which have tails]. In other words, it doesn't logically follow from “some animals have tails” that “some cats have tails”. If some animals have tails it might have been the case that cats were animals which didn't have tails. Thus iii) doesn't necessarily follow from ii). (And iii) doesn't follow from i) either.) ii) can be taken as an existential quantification over [animals]. iii), on the other hand, is an existential quantification over [cats]. Thus:

ii) ((Ǝx) (Ax) iii) (Ǝx) (Cx))

Clearly Ax and Cx are quantifications over different classes. It doesn't follow, then, that what's true about [animals] generally is true also of [cats]; even though cats are members of the set [animals]. Thus iii) doesn't follow from ii).

To repeat: even though both premises and the conclusion are all true, the above still isn't a valid argument.

Read himself helps to show this by displaying an argument-form with mutually-exclusive sets. Namely, [cats] and [dogs]. Thus:

i) All cats are animals
ii) Some animals are dogs
iii) So some cats are dogs.

This time, however, the conclusion is false; whereas i) and ii) are true. It's the case that the subset [dogs] belongs to the set [animals]. Some animals are indeed dogs. However, because some animals are dogs, it doesn't follow that “some cats are dogs”. In other words, because dogs are members of the set [animals], that doesn't mean that they're also members of the subclass [cats] simply because cats themselves are also members of the set [animals]. Cats and dogs share animalhood; though they are different subsets of the set [animal]. In other words, what's true of dogs isn't automatically true of cats. (Wouldn't iii) above work better if it were 'some dogs are cats', not 'some cats are dogs'?)

The importance of sets, and their relation to subsets, may be expressed in terms of bracketed predicates. Thus:

[animals [cats [cats with tails]]]
not-[animals [cats [dogs]]]

Material and Formal Validity

Stephen Read makes a distinction between formal validity and material validity. He does so using this example:

i) Iain is a bachelor
ii) So Iain in unmarried.

(One doesn't ordinarily find an argument with only a single premise.)

The above is materially valid because there's enough semantic material, as it were, in i) to make the conclusion acceptable. After all, if x is a bachelor, he must also be unmarried. Despite that, it's still formally invalid because there isn't enough content in the premise to bring about the conclusion. That is, one can only move from i) to ii) if one already knows that all bachelors are unmarried. We either recognise the shared semantic content or we know that the word 'unmarried man' is a synonym of 'bachelor'. Thus we have to add semantic content to i) in order to get ii). And it's because of this that the overall argument is said to be formally invalid. Nonetheless, because of what I've already said, it is indeed materially valid.

The material validity of the above can also be shown by its inversion, thus:

i) Iain is unmarried
ii) So Iain is a bachelor.

Read makes a distinction by saying that its “validity depends not on any form it exhibits, but on the content of certain expressions in it” (239) Thus, in terms of logical form, it is invalid. In terms of content (or the expressions used), it is valid. This means, obviously, that the following wouldn't work as either a materially or a formally valid argument. Thus:

i) Iain is a bachelor.
ii) So Iain is a footballer.

There's no semantic content in the word 'bachelor' that can be directly tied to the content of the word 'footballer'. Iain may well be a footballer; though the necessary consequence of him be a footballer doesn't follow from his being a bachelor. As it is, the conclusion is false even though the premise is true.
Another way of explaining the material, not formal, validity of the argument above is in terms of what logicians call a “suppressed premise”. This is more explicit than talk of synonyms or shared contents. What the suppressed premise does, in this case, is show the semantic connections between i) and ii). The actual suppressed premise would be the following:

All bachelors are unmarried.

Thus we would actually have the following argument:

i) Iain is a bachelor.
ii) All bachelors are unmarried.
iii) Therefore Iain is unmarried.

It may now be seen more clearly that

i) Iain is unmarried.
ii) So Iain is a bachelor.

doesn't work formally; though it does work materially.

i) All bachelors are unmarried.
ii) So Iain is unmarried.

To state the obvious, this is clearly a bad argument. (It's called an enthymeme.) Indeed it can't really be said to be an argument at all. Nonetheless, this too can be seen to have a suppressed premise. Thus:

i) All bachelors are unmarried.
[Iain is a bachelor.]
ii) So Iain is unmarried.

Now let's take the classic case of modus ponens:

A, if A then B / so B

That means:

A, if A is the case (or true), then B is the case (or true). A is the case, so B must also be the case.

The obvious question here is: What connects A to B (or B to A)? In terms of this debate, is the connection material or formal? Clearly if the content of both A and B isn't given, then it's impossible to answer this question.

We can treat the above as having the aforesaid suppressed premise. Thus:

[Britain's leading politician is the Prime Minister.]
i) Theresa May is Britain's leading politician.
ii) So she is Prime Minister.

In this instance, the premise and conclusion are both true. Yet they're only contingently, not necessarily, connected.

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*) Stephen Read makes the formalist position on logic very clear when he states the following:

Logic is now seen – now redefined – as the study of formal consequence, those validities resulting not from the matter and content of the constituent expressions, but from the formal structure.” (240)

We can now ask: What is the point of a logic without material or semantic content? Would all premise, predicate, etc. symbols - not the purely logical symbols - simply be autonyms or self-referential in nature? (Thus all the p's, q's, x's, F's, G's etc. would be self-referential/autonyms.) And what would be left of logic if this were the case? Clearly we could no longer really say that it's about argumentation – or could we? That is, we can still learn about argumentation from schemas/argument-forms which are purely formal in nature. The dots don't always - or necessarily - need to be filled in.

References

Mares, Edwin D. (2002) 'Relevance Logic'.
Read, Stephen. (1994) 'Formal and Material Consequence'.