Mathematical Logic Seminar Abstracts Autumn 2011

We present proofs of the completeness of several modal predicate logics, both with and without the Barcan formula. The method is by canonical model, the essential features of which will be briefly reviewed. Time permitting, we will discuss the phenomenon of incompleteness in modal predicate logics. This material corresponds roughly to chapters 13-15 of A New Introduction to Modal Logic by Hughes and Cresswell.

Kant's theory of judgements is a subject of extensive and active studies. Kant's formal logic, on the contrary, is studied insufficiently and usually dismissed as 'terrifyingly narrow-minded and mathematically trivial'. Recent work by Theodora Achourioti and Michiel van Lambalgen, A formalization of Kant's transcendental logic, The Review of Symbolic Logic, v.4 no 2, 2011, 254-289 ([AvL] below), seems to refute this verdict. They propose a translation of the philosophical language of Kant's theory of judgements into the language of elementary logic and provide a convincing justification of their view. In formal terms Kant's logic is identified with geometric logic, a subsystem of ordinary first order logic that has been isolated long ago in mainstream mathematics. The model has to elucidate a vast array of statements by Kant like the following:

"Thus, if, e.g., I make the empirical intuition of a house into perception through the apprehension of its manifold, my ground is the necessary unity of space and of outer sensible intuition in general, and I as it were draw its shape in agreement with this synthetic unity of the manifold in space."

[AvL] analyzes Kant's logic in terms of inverse limits of models, a construction widely used in mathematics that reminds one of Kripke models (of ``possible worlds'') or forcing, but inverts the direction in certain sense.

We present basic definitions from [AvL] and translations of Kantian terms (as many as time permits) into logical language. The talk next week by Ulrik Buchholtz contains proofs of technical results.

This talk will summarize much of the material from chapters 16 and 17 of Hughes & Cresswell's A NEW INTRODUCTION TO MODAL LOGIC. Topics covered will include: expanding languages, possibilist quantification, Kripke-style systems of modal predicate logic, identity in LPC, as well as the treatment of definite descriptions, individual constants, and function symbols

Quine's misgivings concerning the modalities started already in his undergraduate thesis in 1928, when he was twenty years old. We will follow the development of his arguments against the modalities, which culminated in Word and Object in 1961. He there argued that quantification into modal contexts leads to a collapse of modal distinctions. We will see how some of his arguments can be overcome, while others give us insights of lasting value.

The notion of an intension (attributable - at least when taken in the precise sense we are interested in - to Rudolf Carnap) and the notion of a counterpart (due to David Lewis) are both tools for rendering precise candidate solutions to philosophical problems concerning the modality of identity claims, and the distinction between de re and de dicto modal claims. Briefly, each notion provides for a logical framework where individual terms can denote different objects in different possible worlds. We consider two versions of first-order modal logic that, respectively, incorporate intensions and counterparts into the semantics. For the former, we follow Hughes and Cresswell in A New Introduction to Modal Logic (1996, Routledge) Chapter 18 and, for the latter, we follow Ghilardi in The Handbook of Modal Logic (2007, Elsevier) Chapter 9, Part II. Our discussion will include a comparison of the two frameworks and a sketch of a soundness and completeness proof for each.

In the late 1930s, Tarski showed that the propositional modal logic S4 can be interpreted in topological spaces. This talk develops a closely-related measure-based semantics for S4, introduced in the last several years by Dana Scott. In a given measure (or probability) model, formulas acquire not just a truth value, but a probability value between 0 and 1. Formally, we do this by evaluating sentences of the language to the Lebesgue measure algebra, or algebra of Borel subsets of the real interval [0, 1], modulo sets of measure zero. This semantics differs from the more standard semantics for modal logics (i.e. topological, and Kripke semantics) in two important ways: (1) the semantics is probabilistic, and (2) there is no notion of truth at a point or world. I develop the formal framework for the probabilistic semantics, and show that it is complete for S4. I then show how this semantics can be extended to dynamic topological logics, in which we have not just the -modality, but also a temporal modality (or next). Finally, I suggest some possible philosophical applications.

The well-known van Benthem characterization theorem shows that propositional modal logic can be viewed as a fragment of classical first-order logic, namely the fragment that is invariant with respect to bisimulations. In our talk, we introduce a new notion of asimulation as an asymmetric version of bisimulation attuned to the specific features of intuitionistic propositional logic.

It is proved then as the main result of the talk that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula iff it is invariant with respect to asimulations and satisfiable.