The Structure of Aristotle’s logic

The easiest way to unravel the complex structure of Aristotle’s logic is to start with the definition of possibility that he gives in Chapter 13 of Book I of the First Analytics. The possibility that Aristotle uses as the basis for his logic is two-sided. It is a possibility and only a possibility: it is opposed not only to impossibility but also to necessity. Aristotle considers this understanding of possibility to be “ordinary and inherent in the nature of things.” An alternative interpretation of possibility, when it is recognized as one-sided, opposed only to impossibility but subordinate to necessity, Aristotle rejects it, considering it inconsistent with the nature of things, purely homonymous and irrelevant to his logic. In the same chapter 13, Aristotle writes that all premises about the possible are mutually reversible in the sense that if the presence of something is possible, then its absence is always possible. In other words, the Aristotelian possibility is such that the premises “it is possible that p” and “it is possible that not-p” are equivalent. Thus, it turns out that the basis of the structure of Aristotelian logic is the “triangle of opposites”, in the corners of which are the modalities “it is necessary that p”, “it is necessary that not-p” and “it is possible that p” (“it is possible that not-p”). All three modalities are incompatible, but their incompatibility is “one-sided”. If any one of them is true, then the other two are false, but if any one is false, then uncertainty arises: it is not known which of the remaining modalities is true and which is false. This is because the law of the excluded middle does not apply here: after all, the “third” is not excluded here. Aristotle, however, does not limit himself to considering only the two above-mentioned modalities. In Chapter 2 of Book 1 of the First Analytics, he writes that for him every premise is a premise about what is inherent, or about what is necessarily inherent, or about what is possibly inherent. The premisses about what is inherent are p and not-p. They are incompatible with each other, but how do they relate to the premisses about what is possibly inherent and what is necessarily inherent? It is clear that the premisses about what is inherent cannot be incompatible with the premise about what is possible. After all, if this were to happen, then a repetition of the triangle considered above would arise: p would coincide with “it is necessary that p”, and not-p would coincide with “it is necessary that not-p”. A careful examination of the proofs of syllogisms with mixed premises shows that Aristotle postulates an implicative relation between the inherent and the possibly inherent. He accepts the following propositions: (1) if p, then it is possible that p, and it is possible that not-p; (2) if not-p, then it is possible that p, and it is possible that not-p. As a result, it turns out that the modality “it is possible that p” and “it is possible that not-p” turns out to be true both when p is true and when not-p is true. In other words, it becomes always true and ceases to influence what happens in the area of p and not-p. It becomes the “third” that is excluded, and in the area of p and not-p the law of the excluded middle begins to operate. In Chapter 9 of the treatise “On Interpretation,” Aristotle speaks of “conditional necessity.” It arises when one of the possibilities p or not-p is realized, and the other is not realized. But this is not the unconditional necessity that opposes possibility in the triangle of opposites. Conditional necessity remains a possibility, albeit a realized possibility. It is the necessity of an accomplished fact. What is unconditionally necessary always exists and cannot not exist, and what is conditionally necessary arises only when some possibility becomes a fact. According to Aristotle, everything that simply exists is one of the realized possibilities and could not exist, in contrast to what is unconditionally necessary and is not capable of not existing.