PhilLogic@CMU: Philosophical Logic at Carnegie Mellon

Workshop on Modal Logic Homepage

Friday 28 April 2006, 11:00 AM - 6:00 PM
Wean Hall 5328, CMU Campus
Speakers: Horacio Arlo-Costa, Steve Awodey, Nuel Belnap, Kevin Kelly
Guests: Rohit Parikh, Johan van Benthem
Attendance is free and open to the public

Schedule

Abstracts

"How simplicity helps you find the truth without pointing at it"
Kevin T. Kelly
Department of Philosophy
Carnegie Mellon University

It seems that a fixed bias toward simplicity should help one find the truth, since scientific theorizing and our most recent data-mining procedures are guided by such a bias. But it also seems that a fixed bias toward simplicity cannot indicate or point at the truth, since an indicator has to be sensitive to what it indicates. I argue that both views are correct. It is demonstrated, for a broad range of cases, that the Ockham strategy of favoring the simplest hypothesis, together with the strategy of never dropping the simplest hypothesis until it is no longer simplest, uniquely minimizes reversals of opinion and the times at which the reversals occur prior to convergence to the truth. Thus, simplicity guides one down the straightest path to the truth, even though that path may involve twists and turns along the way. The optimality proof does not appeal to prior probabilities biased toward simplicity. Instead, it is based upon minimization of worst-case reversals of prior opinions over complexity classes of possibilities modeled as branching futures. Empirical complexity is analyzed in terms of forcing properties in infinite games against nature.

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"Topological semantics for first-order modal logic"
Steve Awodey
Department of Philosophy
Carnegie Mellon University

Tarski showed in the 1940s how to give semantics for the propositional modal logic S4 using the operation of interior in a topological space as the interpretation of the necessity operator. Here it is shown how to extend that interpretation to all of first-order S4 using sheaves on the space to interpret arbitrary predicates. Completeness is established by sheaf-theoretic methods generalizing the Stone representation theorem. Joint work with Kohei Kishida.

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"Quantified classical modal logic: The family of free classical systems"
Horacio Arlo-Costa
Department of Philosophy
Carnegie Mellon University

Recent research in the area of quantified modal logic has focused on presenting a unified semantic treatment of the family of quantified modal logics. Much of this work is nevertheless limited to the extent that it applies only to quantified normal systems. Some recent proposals, like (Corsi, JSL, 67(4), 2002) or (Garson, JPL, forthcoming) are also affected by incompleteness results. In addition, the completeness property for various salient systems remains unknown in various cases (like Q^0.K + BF in Corsi's notation).

A different proposal (Arlo-Costa and Pacuit, Studia Logica, forthcoming; Arlo-Costa, Studia Logica, 2002; and Arlo-Costa, JPL, 2005) offers a general completeness result for the entire family of classical first order modal systems encompassing both normal and non-normal systems (via the use of general neighborhood frames with constant domains). This works covers various important systems that only admit relational semantics via frames with varying domains, but it does not consider quantified extensions of classical modal systems with free logic (which are also typically modeled via relational semantics with varying domains). This talk focuses on these systems and therefore it presents preliminary work towards building a unified semantic theory for an even more comprehensive family of quantified classical systems. We propose a generalization of both neighborhood frames with constant and varying domains and use them to offer solutions for open problems presented in the literature (like the one mentioned parenthetically at the end of the first paragraph). Time permitting some applications as well as some foundational problems presented in the seminar will be discussed.

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"Bressan’s type-theoretical combination of quantification and modality" [PDF]
Nuel Belnap
Department of Philosophy
University of Pittsburgh

The Lindstrom and Segerberg 2005 essay, “Modal logic and philosophy”, provides a comprehensive discussion of the most substantial researchers in this field. There is, however, at least one figure that could be added to this discussion, namely Aldo Bressan. While Bressen lacks the same degree of historical influence, his ideas ought to be a major force for good in modal logic. Bressan is a physicist working and publishing on standard physical topics, including continuous mechanics and general relativity, and his work in modal logic aims to be a framework for axiomatic calculus of physics. An important vehicle for Bressan’s logical ideas is the type-theoretical modal language ML^v described in his 1972 work, A General Interpreted Modal Calculus, which Bressan developed partly in order to be able to use the idea of an “absolute concept” in formulating physical laws and statements. This language is significantly like Montague’s later IL (Montague 1974), and while the two are mathematically equivalent, they also exhibit significant conceptual differences. My exposition will concentrate on Bressan’s ML^v, with remarks on Montague’s IL. First I describe (1) the type hierarchy itself, and then give an account of how it guides us (2) grammatically, (3) ontologically, and (4) semantically.

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