SyllogisticLogic

Long before quantifiers and connectives, arguments were settled by their shape alone: all Greeks are human, all humans are mortal, so all Greeks are mortal. The statements involved say only that one class falls within another, or that two classes overlap, and nothing else about them matters.

This chapter takes that fragment seriously as a logic in its own right. Its statements are built from terms and nothing else; its rules are the traditional figures; and it is complete, in the sense that whatever holds under every reading of the terms can be derived. The proof systems of the previous chapter supply the derivations, so a syllogism here is a tree like any other.

The point of starting so small is that everything after it is an extension. Negation, the connectives and quantifiers are added to this, rather than syllogistic reasoning being recovered as a fragment of them.

Sources: Larry Moss, Three Flavors of Natural Logic: extended syllogisms, logics with variables, and reasoning with polarities (https://web.stanford.edu/~icard/logic&language/Moss.pdf); Robert van Rooij, The propositional and relational syllogistic (https://staff.fnwi.uva.nl/r.a.m.vanrooij/LogAnal.pdf); Jörg Endrullis and Lawrence S. Moss, Syllogistic logic with “Most”, Mathematical Structures in Computer Science 29 (2019) (https://research.vu.nl/ws/portalfiles/portal/235241103/Syllogistic_logic_with_Most_.pdf).