In this note we assume the reader is familiar with the system of logic expounded in our paper 'Aristotle's Second-Order Logic'. We refer to this system now as second-and-a-half order logic as it sits between second and third-order logic. In the above paper we argue that this is the natural logic to formalize Aristotle's philosophy.
Chapter 1. Aristotle's definition of 'syllogism' here is quite general (and should not be confused with the 'syllogisms' of the Analytics) and a good translation would be 'inference', the kind of inference represented by a sequent in the sequent calculus (with the cut-rule) with one formula on the right $A_1,A_1,....,A_n \vdash B$. Aristotle's 'true' or 'primary' things are sequence of the form $\vdash A$.
Chapter 2. But what is dialectic in the Topics ? Does it investigate the axioms of the particular sciences themselves (which cannot be investigated in those sciences) ? The passage 101b1-3 is mysterious. Dialectic is a critical path having the 'beginning of all methods'.
Chapter 4. The protasis. Each protasis indicates (is made up from) either property, genus or accident. Difference is strangely classed as 'pertaining to genus (generic)'. And 'problems' can be constructed from every protasis by changing the 'mode'. This is very subtle and interesting point regarding intensional logic and, so it seems, a term-formation corresponding to interrogation.
Chapter 5. This is a very important section. We have a 'logos' which 'semainein' (signifies). It is not clear if the logos is meant as a mere signifier or as the sign (signifier + signified). Here logos is contrasted with onoma. Apparently this corresponds to the difference between simple and complex terms. We see that there are definitions of complex terms. An important question is: can we accept a definition consisting of a simple term ? Here Aristotle hesitates but admits that such protasis are at least useful for definition. A fundamental concept is that of 'antikategorein' (to be convertible with). In 'Aristotle's Second-Order Logic' we argue that the Fregean distinctions between Sinn and Bedeutung as well as between concept, object (and extension) are all present in the Topics. A is convertible with B for Aristotle if A and B are predicates with the same extension (but not necessarily with the same meaning). Aristotle's formula in 102b20-23 is : if (it) is A then (it) is B and if (it) is B then (it) is A. Clearly definition and property have the same extension but different meaning (they both do not signify essence). The rest of the discussion is valuable for elucidation of 'accident' and how it overlaps with relative (and temporary) property. The example of the 'only man sitting' (in a group) suggests a connection to definite descriptions of individuals.
In second-and-a-half order logic, all quantifiers should be bound - and how are we to interpret quantification, for instance in chapter 1 of the second book. How did Aristotle distinguish between: 'all men are animals', 'man is an animal', 'animal is the genus of man', between extensional and intensional predication. In the above chapter Aristotle seems to give the rule: from $\Gamma \vdash \forall_{x:Ax} B$ we can derive $\Gamma \vdash \exists_{x:Ax} B$. In the beginning of chapter 6 of the second book we find the Stoic exclusive disjunction. It is interesting that 'connectives' for Aristotle are just as much term operators as operators on propositions.
Chapter 7. Peri Tautou (sameness, identity). There is a kind of homotopy or qualitative sameness considered here and the example of the water drawn from a given spring is noteworthy. The question of identity is examined in our paper and is crucial with regards to extensionalism. What is Aristotle's 'arithmetical identity': the name being many the thing being one (103a9-10). Here Aristotle may be interpreted as postulating that identity is not a primitive notion but a polysemic and it to be defined in terms of either homonymity, definition or property or even accident ! This rules out any extensionality (Frege's Law V).
Chapter 8. Definition consists of genus and differences and these are said to be 'in' the definition. There is still the question, however, of the precise relationship between difference and accident.
Chapter 9. For Aristotle 'kategoria' means 'predicate'. What is the relationship between onoma, pros, protasis, logos and kategoria? Category in the ordinary sense is actually 'genus of predicate'. This chapter is very important and very subtle. The ten classes seem to be genera both of predicates and of things in general - an ontology. In our paper (and in Modern Definition and Ancient Definition) we raise the question of the definition of objects that do not belong to the category of substance. Elsewhere we have inquired about Aristotle's view on statements of the form 'A is A'. Aristotle is stating here that if the thing and predicate belong to the same class then we have an essential predication, otherwise we do not. But how can we accept Aristotle's example of predicating 'man' of a given man being a predication according to essence ? How can 'white is white' signify essence?
Chapter 10. This chapter offers us the rudiments of a new kind of intensional logic: a doxastic logic, and is of considerable interest. We can think of a modal operator Dox(P) satisfying certain logical rules.
Chapter 13. Differences of meaning of a term and a term qua term can be objects themselves of propositions. To formalize the Topics we thus may need a third-order semantic identity relation.
Chapter 15. It would be interesting to investigate formal systems in which each term is assumed to be interpreted as having possibly a set of references (and meanings) rather than one (or none). This is the kind of polysemic logic that looms large in the Topics. A kind of semantic set theory, perhaps. The task is to construct expressions which are singletons and to detect them within the formal logical and grammatical rules of the system. Aristotle must accept that there is a notion of semantic identity (which is not the same as that of 'antikategorein' or extensional equivalence). We tried to formalize this notion in our second-and-a-half order logic. See the previous remark.
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