Working on a paper on a priority and analyticity which developed out of a commentary on a chapter of a book by Robert Hanna. I hope to expound in greater detail my theory of the formal verification principle and the 'core analytic' logic presupposed by human reason. There are some very interesting connections to the philosophy of Hilbert brought to my attention by an upcoming paper by C. Ortiz Hill.
Going to continue the topos theoretic interpretation of Zalta's Object Logic. Will write a preliminary version of the interpretation of Plato based on a concrete topos of countable sets and computable functions.
Have written a new paper entitled 'Aristotle's Second-Order Logic' which follows up on my previous papers on ancient logic. I show that the metalogic of the Organon is second-order logic with third-order extensions and provide both additional evidence for ancient competency in the logic of multiple generality and a template for a formalization of the Topics.
In the course of a project on Hegel and Category Theory I realized that my approach (which is very heavily focused on topology and infinitesimals) may well have a very deep and natural relationship to the philosophy of Leibniz (to which Hegel is plainly enormously debted). The three levels correspond to the theory of representational continua, the theory of systems relations and the system of axiomatic-deductive systems and their reflections and interpretations.
We have a metaphilosophical perspective which views ancient philosophy as whole, each part having value, and which integrates Hume, Leibniz and Kant in the light of subsequent philosophy.
To habitually think all our thoughts never without the thought that our thoughts are thoughts thought by the thinker that thinks them...
A computational system* consists of a triple $(S,G)$ where $S$ is finite and non-empty and $G$ is a finite set of pairs (rules) $(P,Q)$ with $P,Q \in S^*$. What is the complexity of the problem: given a computational system and a pair $(w,v)$ with $w,v \in S^*$, can $v$ be derived from $w$ ? (use for instance the sequent calculus presentation of predicate logic to show it is undecidable). This was Leibniz's logic (cf. the simple proof of $4 = 2 + 2$ discussed by the Kneales) and perhaps the most elegant embodiment of the pure idea of computation.
*I did not just think up this concept, it already is known as a (semi)-Thue system or term-rewriting system. The word-problem is undecidable. That means, that all axiomatic-deductive systems can be expressed in semi-Thue systems.
To carry out derivations and reason about computational systems we need at the least the following: the ability to seize as a whole a sequence of indefinite length of an indefinite number of characters and a collection of pairs of such sequences. To be able to survey all the elements of a sequence. To compare two sequences and to find the matches of a given sequence with subsequences of another sequence, then seize these matches as a whole and to choose from among them them. To be able to replace a subsequence of a sequence by another sequence according to the data provided by a pair of sequences. To compare a sequence with another one held in memory. To seize derivations.
