The reason for Aristotle’s treatment of (traditional) fourth figure syllogisms as first figure syllogisms with inverted terms in the conclusion is the following: To disprove the conclusiveness of a premiss pair Aristotle formulates two triplets of true propositions such that two of them correspond to the premiss pair in question and that the third proposition corresponding to a conclusion is an a-proposition in the first case, an e-proposition in the other. Since the truth of an a-proposition grants the falsity of (...) the contrary e- and of the contradictory o-proposition, the first triplet offers two counter-instances for invalid syllogisms with true premisses and false conclusions. Similarly the true e-proposition grants the falsity of an a- and an i-conclusion. Since an a-proposition can be converted to an i-proposition and an e-proposition is equivalent to its converse, these first figure triplets also disprove any first figure syllogism with converted conclusions, with the exception of o-conclusions. The invalidity of the latter ones, however, can be shown by using premiss conversions of (invalid) second and third figure syllogisms. The proposed explanation also makes clear why there are no rejection proofs for invalid syllogisms of (traditional) fourth figure syllogisms in the Analytics. (shrink)

In the present article we attempt to show that Aristotle's syllogistic is an underlying logiC which includes a natural deductive system and that it isn't an axiomatic theory as had previously been thought. We construct a mathematical model which reflects certain structural aspects of Aristotle's logic. We examine the relation of the model to the system of logic envisaged in scattered parts of Prior and Posterior Analytics. Our interpretation restores Aristotle's reputation as a logician of consummate imagination and skill. Several (...) attributions of shortcomings and logical errors to Aristotle are shown to be without merit. Aristotle's logic is found to be self-sufficient in several senses: his theory of deduction is logically sound in every detail. (His indirect deductions have been criticized, but incorrectly on our account.) Aristotle's logic presupposes no other logical concepts, not even those of propositional logic. The Aristotelian system is seen to be complete in the sense that every valid argument expressible in his system admits of a deduction within his deductive system: every semantically valid argument is deducible. (shrink)