Mathematics of term (Aristotle) logic I am reading book "Probabilistic Logic Networks" by Ben Goertzel et. al. and large part of this book is based in term logic - logic by Aristotle that was used before discovery of predicate logic. This logic is still in use in Catholic theological high schools.
My question is - are there mathematical works on term logic and is there any perspective, elaboration and development possible in term logics? Is it possible to formalize term logic, what are connections between term logic and predicate logic?
The proponents of the term logic says that the term logic is closer to the natural language and hence it both facilitates the development of formal semantics for natural language and also it opens wider possibilities as natural language is more expressive than predicate calculus.
I am fascinated by the work of Ben Goertzel (opencog.org) in the field of Artificial (General) Intelligence but I would like to stay in mainstream science.
 A: There is certainly a lot of research done on variants/fragments/extensions of first-order logic in the context of natural reasoning, and I presume that there's a lot on term logic in particular. This isn't exactly my field of expertise, unfortunately, but I can give a couple hopefully useful suggestions:
I believe Larry Moss has worked on term logic, and other fragments/variations of first-order logic in the context of natural reasoning. The following stuff of his may be a good starting place for you:

*

*https://iulg.sitehost.iu.edu/moss/overview.pdf


*https://iulg.sitehost.iu.edu/moss/unit1.pdf (this one's about syllogistic logic - I think that's the same as term logic, but I'm not sure)
I also believe Barwise did a lot of work on natural language and natural reasoning, but I don't have a citation at the moment.
A: What you will want to look at first is the traditional square of opposition at the Stanford Encyclopedia of Philosophy.  It will show you the formalized statements corresponding to the quantifiers of term logic.  You should compare the forms with those of restricted quantifiers.
Term logic semantics is based upon part and whole.  There is no sense of "individual" as a first principle (think constants in a language signature), although one certainly has statements involving proper names.  It had actually been Bolzano's attempt to define individuals using term logic that led to the idea that one must have a foundation based upon "undefined language primitives".  What one does have is a genus coordinatized into species.  Aristotle uses the example of "animal" coordinatized into "aquatic", "winged", and "footed".  Species are said to be simultaneous.  A genus is said to be prior to its coordinate species.
Because coordinates are based upon (general) names, the logic is not binary in the usual sense.  Aristotle would call "not-winged" a derivative or secondary name.  It does not describe a logical species.  For this reason, Kant defined a concept as one of two contradictory predicates.  By contrast, modern logic tends to treat predicates and concepts similarly (at least this seems to hold with respect to comprehension).  The import of this comes from how one should try to understand coordinatizations.  The model will be more like partition lattices than chains in Boolean lattices.
Nevertheless, with respect to individuation, one ought to make comparison with  specialization preorders in topology.  This is mentioned in the Wikipedia link,
https://en.wikipedia.org/wiki/Specialization_(pre)order#Definition_and_motivation
But, what is not mentioned is the relationship with Leibniz' ideas concerning logic.  Leibniz describes his own logic as inverting the sense of the part and whole relationships of Aristotelian semantics.  This is how the identity of indiscernibles arises.  If you look at the Wikipedia entry on metric spaces, you will find one of the axioms described as "identity of indiscernibles".  And, with respect to the link above, you might consider Cantor's nested set theorem for closed sets of vanishing diameter.
According to Leibniz, the identity of indiscernbiles originates with Aquinas' opinion that God knows each soul as an individual.  Hence, what is involved is the non-Aristotelian claim that an individual is the lowest species.  But, it originates with the idea of descending chains of coordinatized genera.
Although Aristotle claims that genera are prior to species, he grounds truth in the idea that individuals are primary substance while species and genera are secondary substance.  This is essentially what Russell did with his theory of types, although secondary substance is now a hierarchy of orders brought into the transfinite by Goedel.  And, of course, the many desirable qualities of first-order logic arise from the fact that the terms of first-order logic are intended to denote only individuals.
You might consider actually reading Aristotle up to his book "Posterior Analytics".  The next book is "Topics".  Some of it is worth reading.  But where "Posterior Analytics" is about proof in scientific enterprises, "Topics" deals more with proof used for rhetoric.  Aristotle's ideas permeate mathematics because that is what learned people studied and because the ideas of logical people ought to converge on similar forms if, in fact, logic has any relation at all to "laws of thought".
