Page 294 - Encyclopedia of Philosophy of Language

Language and Logic
chs. 5-8, 10, 11, 14 of De Interpretatione and Prior Analytics, i, 1-7.
In De Interpretatione, after an initial discussion of certain issues in the philosophy of logic, Aristotle investigates the question of which statements can be said to be some sort of denial of some given statement. Almost exclusively, these statements are all general subject-predicate claims. Aristotle's own examples are as follows (1):
thinks of all sorts of deductive inference as demon- strative, but a little imagination should show the way. If one thinks about what goes on in school geometry, then one can get the appropriate flavor. A great deal of what the geometer does is to assert categorically of various classes of things (circles, triangles, lines joining midpoints, parallels, etc.) that they are or are not members of some further class. If one took the reason- ing of the geometer as the central case of reasoning in general, then it would seem reasonably natural to regard the categorical sentences of the square of oppo- sition as central and typical of a type of reasoning (demonstrative) that it was worth capturing.
The doctrine of the syllogism is described in the Prior Analytics. In barest outline it consists in the study of those forms of inference which contain two categorical premises and which validly imply some further categorical conclusion. A typical example of such an inference is (2):
Every dog is a mammal. (2) Every mammal is warm-blooded.
Therefore, every dog is warm-blooded.
Every man is white, No man is white,
Some man is white, Some man is not white.
(1)
The sentences in this example do not have the precise syntactical form of any example of Aristotle's, and this is itself important. For it is no easy matter to translate directly from Greek into fluent English whilst preserving the precise syntactical structure of the Greek. In the Greek of Aristotle and Plato there is often a complicated and clumsy use of pronouns in connection with general statements. Perhaps because of the awkwardness of categorical statements in Greek (especially when studying inferences between them) Aristotle introduced (but without explanation) the use of letters as variables standing for the terms in cat- egorical sentences. Thus, the above syllogism is said to have the following structure of terms (3):
A—B (3) B—C
A—C
This use of symbols is the first of its kind in logic, and it represents the beginnings of the formalization of inference. As used of categorical sentences and syllo- gisms, this formalization may not seem revolutionary, but this is perhaps because we have come to take such a use of symbols for granted. What Aristotle achieved here was truly remarkable: he presented a pattern of inference in a precise enough way for it to become an object of purely formal or syntactical study. Thus, by thinking of the A's, B's, and C's as forming patterns, it is possible to study the validity of syllogistic infer- ences by reference to these patterns and, of course, by reference to the quantifier words (every, some, none) which modify each line of the pattern.
The logicians of the Middle Ages carried this study much further—some might even say to extremes. And
Additionally, Aristotle introduces and discusses vari- ous further relations between statements of the above forms, relations we would now consider to be infer- ential. Thus, he discusses the issue of whether 'Every man is white' implies 'Some man is white' (a relation- ship which has been called 'subalternation') and whether 'No man is white' is equivalent to 'Every man is nonwhite.' This latter relation is called 'obversion' and Aristotle decides that it is valid in one direction, but is not an equivalence relation.
When the four forms of general statement are arranged spatially as in Fig. 1, the result is the famous square of opposition. This was the starting point for all teaching of Aristotelian logic in the middle ages and beyond, though in this precise form it is not in Aristotle's text. The details of the square of opposition are not important here, but what is crucial to recognize is that Aristotle's whole further development of logic is based upon sentences of the above forms—sentences which are known as categorical. The syllogistic, which will be surveyed shortly, is a restricted system of infer- ences all of whose premises and conclusions are drawn from the set of four types of statement that figure in the square of opposition.
Why did Aristotle find these sorts of categorical general statement so central to the study of inference? There is no fixed view about this, but it does seem likely that his interest in what was described above as 'demonstrative' inference was crucial here. From a modern day perspective this can seem odd, since one
272