Page 273 - Encyclopedia of Philosophy of Language

tinct from other meaningful composite expressions like prayers, commands, etc. Aristotle divides cat- egorical sentences into affirmative and negative. Each affirmative sentence has its own opposite negative, and each negative has its affirmative opposite. Every such pair of sentences is called contradictory— assuming that the terms involved are the same and used in the same sense. In De Interpretatione the div- ision between primary and secondary substances is reflected in the division between universal and indi- vidual terms. Universal terms correspond to sec- ondary substances, individual terms correspond to primary substances. A sentence is called universal whenithastheformEveryAisBorNoAisB(or, equivalently, Every A is not B). A sentence is called particular when it has the form Some A is B, Not every A is B (or, equivalently,Some A is not B).
The combination of universal and affirmative sen- tences yields the well-knownopposition schema:
(a) A pair of affirmative and negative universal sentences, involving the same terms used in the same way, is called contrary. Contraries cannot both be true at one time, but they can both be false.
(b) Pairs consisting of a universal affirmative sen- tence and a particular negative sentence or of a universal negative sentence and a particular affirmative sentence, involving the same terms used in the same way, are called contra- dictories. Contradictory sentences cannot both be true at one time, nor can they both be false at the same time.
3. The Prior Analytics and the Theory of Syllogisms Initially, Aristotle did not restrict the term 'syllogism' to arguments having only two premises and three terms. He defines a syllogism as a discourse in which certain things being posited, something other than what is posited follows of necessity from their being so. However, the core of the Prior Analytics consists of an analysis of arguments with two premises relating three terms. Aristotle argues that every argument can be expressed as a series of syllogistic inferences. This is the so-called 'syllogistic thesis': every argument is a syllogism. The point is that Aristotle assumes that the conclusion of every nonformal deductive argument is a categorical sentence. He then argues that the only way such a conclusion can be derived is through prem- ises which link the terms of the conclusion through a middle term relating them.
Aristotle's strategy in treating syllogistic conse- quence is the following. He considers 48 possible pairs of premises. Besides the so-called perfect syllogisms, Aristotle is able to eliminate by counterexample all but 10 other pairs of premises as having no syllogistic consequences. The remaining syllogisms are perfected (reduced) to first-figure syllogisms by using deduct- ions. His method is this. He selects a few valid infer-
ence patterns and justifies the remaining valid patterns by showing that they are a conservative extension of the original core: one can move from the premises to the conclusion without them. The invalid inference patterns are rejected by using counterexamples. The traditional practices of using a formal deduction to show that an inference pattern is valid and using a counterexample to show that it is not are both intro- duced by Aristotle. The valid inference patterns which Aristotle employs are:
(a) The rules of conversion.
(b) Thereductioadabsurdum.
(c) The perfect syllogisms.
The conversion rules are inference patterns based
on special properties of the expressions Some and No: Some A is B entails Some B is A and No A is B entails No B is A.
The reductio ad absurdum is, in Aristotle's system, a kind of indirect proof: if from a set of assumptions S and a sentence P a contradiction follows, then one can derive the contradictory of P.
Aristotle divides syllogisms into perfect and imper- fect ones. A perfect syllogism is a trustworthy prin- ciple of inference in that it needs nothing to prove that the conclusion follows from the premises: it can be seen directly that this is the case. These syllogisms have the following form (their traditional names are used here):
Barbara: Every A B, Every C A =*• Every C B. Darii: Every A B, Some C A => SomeCB. Celarent: NoAB, EveryCA => NoCB. Ferio: N o A B , Some C A => Not all C B.
A syllogism is imperfect if it needs additional dis- course in order to prove that the conclusion follows from the premises. Central to Aristotle's logical pro- gram is the proof that all the valid imperfect syllogisms are perfectible. The perfection of an imperfect syl- logism consists in showing how one can move from the premises to the conclusion by using the rules of conversion, the reductio rule, and the perfect syllo- gisms. This means that Aristotle has to prove in the evident part of his system that each argument that uses nonevident syllogistic principles to prove a certain conclusion C can be replaced by a proof of C that uses only the evident principles. This goal is achieved by Aristotle in the Prior Analytics, which made him the founder of logic and metalogic at the same time.
4. Conclusion
The modern interpretation and assessment of Ari- stotle's logical theory started with Lukasiewicz (1957) in which Aristotle's method of perfecting syllogisms is cast in axiomatic form. According to Lukasiewicz, Aristotle used but did not develop a logic of propositions.
A more accurate interpretation of Aristotle's strat- egy was offered by Corcoran (1974). In this work
Aristotle and Logic
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