Principia Logico-Mathematica

https://mally.stanford.edu/principia.pdf

Zalta uses second-order logic extended with what might be called 'third-order' predicates which express the encoding relation between $n$ first-order objects and $n$-ary predicates. These he calls the 'encoding' relation (inspired by Meinong and Mally). To us the great interest of the work of Zalta and his collaborators is that it offers a solid example of 'formal philosophy', philosophy carried out in entirely in a formal language standing in for natural language. But the real significance of Zalta's work is not so much a setting up a particular metaphysical-philosophical system or giving plausible charitable (re)interpretations of Plato, Leibniz and Frege but rather furnishing a framework for purely formal dialectics, argumentation and debate. The challenge is to formalize and express in this context the process of debate (we could take for instance the formal game outlined in Aristotle's Topics). This will involve a certain theory $T$ which both players must accept (which will include the core axiomatic-deductive system), a sequence of choices of assumptions which one player or both players must be forced either to accept or reject.  Perhaps the resulting logic is like temporal logic? We have given an example of the formalization of philosophical dialectic (debate) in our preprint "A formalization of a fragment of Plato's Lysis".