sunnuntai 11. syyskuuta 2011

Christian Wolff: Reasonable thoughts on the capacities of the human understanding and their correct use in knowing truth - Syllogistic 102: go figure

”That which belongs to all things of a kind must also belong to this that is of the same kind.”
”What is denied of a whole kind must also be denied of anyone of the same kind.”

These somewhat complex sentences Wolff calls the principles of syllogisms. They are supposedly not the final axioms of syllogistic, because Wolff thinks they are themselves based on the so-called principle of contradiction: a thing cannot both have and not have a characteristic.

The two principles could also be stated through three statements:

”A property C belongs or does not belong to all things of a kind B.”
”A is a thing of kind B.”
”Thus, the property C belongs or does not belong to A.”

or in a symbolic form:

B – C
A – B
Thus, A – C

Such a combination of three sentences is what has been traditionally called a syllogism. Actually all the sentences in a syllogism have their own traditional names. The first statement is known as a major proposition, or as Wolff calls it, an upper proposition (Ober-Satz), while the second statement is known as a minor proposition (in Wolff, Unter-Satz or lower proposition). A major proposition here characterises a certain kind or species and it often does describe a general law connecting two concepts. The minor proposition here states that a certain thing belongs to a certain kind and it often presents an example of a general species. The major and minor proposition together are called premises (in Wolff, Förder-Sätze or front propositions), while the third proposition is then the conclusion of the syllogism (in Wolff, Hinter-Satz or back proposition).

In the previous text I noted that syllogistic logic required only two divisions of judgements: to universal and particular and to affirmative and negative. Clearly then there are four different judgement types to consider: universal-affirmative (all As are Bs), particular-affirmative (some As are Bs), universal-negative (no As are Bs) and particular-negative (some As are not Bs). Now, Aristotle had painstaikingly investigated all the different possible combinations of two premisses and noted which combinations could be used as premisses of syllogisms. We need not bother with the details, but we may note that at least one premiss must be universal and affirmative.

Besides the judgements, the words or concepts in the syllogism have also traditional names. The subject of the conclusion, which in the example is also the subject of the minor proposition, is called the minor term (in Wolff, Förder-Glied or front term), and similarly predicate of the conclusion, which in the example is the predicate of the major proposition, is called the major term (in Wolff, Hinter-Glied or back term). The third concept, which, as it were, connects the minor and the major term, but vanishes when we come to the conclusion of the syllogism, is then called the middle term (Mittel-Glied).

In the example above, the middle term is in the middle of the syllogism in a very concrete sense, as it is the predicate of one and the subject of the other premiss. But we could also change the places of the three terms. For instance, we could place the middle term as the subject of both premisses:

B – C
B – A
Thus, A – C

Now, in this case the premisses tell that a certain species of objects is a common subspecies for two other species – and nothing else. Hence, the conclusion can at most be a particular judgement, some As (those that are Bs) are Cs. For instance, because bats are both mammals and flying animals, some mammals can fly.

We could also place the middle term as the predicate of both premisses:

C – B
A – B
Thus, A – C

In this case two affirmative premisses would tell that A and C share some predicate or are subspecies of the same genus. This does not by itself tell us anything new: two species of the same genus might have no common elements (like tigers and lions), but they also can have common elements (like teachers and writers, because a person can be both). More results are gained when one of the premisses is negative – one things has a predicate, the other does not, therefore, we cannot identify these things or even connect them in a judgement. Thus, because apples are plants, but bats are not, apples cannot be bats.

Aristotle classified the different syllogisms into three figures according to the three different positions the middle term could take. After Aristotle, people noted that there is actually a fourth possible figure. That is, we could reverse the positions of the minor and major terms in the first figure like this:

C – B
B – A
Thus, A – C.

Logicians quickly noted that the syllogisms of the fourth figure were not very helpful. It is then no wonder that some philosophers, like Wolff and even Aristotle himself, simply ignored it, and that Hegel mentioned it merely to make ridicule of the unnecessary complexity of syllogistic. Another piece of complexity one might also want to make fun of is the medieval invention of giving all the individual syllogisms a name of their own. Each of the four possible types of judgements was assigned its own vowel, and as every syllogism comes with three judgements, a name with just these three vowels was given to each syllogism. A famous example is Baroco, that is, a syllogism of the sort:

All gold is malleable,
But some people are not malleable,
So some people are not golden.

It seems unbelievable that one would try to argue for such an insignifanct conclusion with such complexities. And indeed, the name Baroco, or its modification, baroque, acquired later a meaning of unnecessary extravagancies. Indeed, as even Aristotle noted, all the other syllogisms could actually be based on the syllogisms of the first figure – and with his pragmatic nature Wolff instructs his students to ignore the other figures.

Despite the extravagant and unnecessary intricacy of syllogistic, we should not disvalue syllogistic completely. Syllogisms were the one form of argumentation by which from premisses known to be true one could infallibly deduce further truths. Of course, this infallibility is also based on knowing some truths beforehand: from false premisses syllogisms can produce both true and false conclusions. In the traditional terms, the truth of the premisses is what makes syllogisms into demonstrations. There have been various suggestions as to how one can find true premisses – we shall see how Wolff answers the question in the next text.

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