15 research outputs found
A Note on Isomorphisms Between Canonical Frames
No full textThis paper looks at the question of what kinds of automorphisms and isomorphisms can exist within and between canonical frames for modal logics. Specifically, this paper is able to partially answer the question posed by the author in [7] of whether F L ! ¸ = F L !1 in the presence of the extra-ZFC axiom "2 ! = 2 !1 ." We answer the question in the negative for all logics below S5 and for all logics satisfying a natural condition. In addition we look at automorphisms and show that while canonical frames for S5 in particular, and logics of bounded alternative in general, have many non-standard automorphisms, the logics satisfying the aforementioned natural condition have none. A Note on Isomorphisms Between Canonical Frames Timothy J. Surendonk January 1, 1997 Abstract This paper looks at the question of what kinds of automorphisms and isomorphisms can exist within and between canonical frames for modal logics. Specifically, this paper is able to partially answer the question ..
Revising Some Basic Proofs in Belief Revision
No full textThis paper reobtains GROVE's results [3] through means which are hopefully clearer and more illustrative of the underlying notions. Also, the proofs are worked through in enough details that possible errors in GROVE's proof are clearly avoided. The paper concludes with a discussion on elementary (or closed) systems of spheres and it notes that each revision function can be given by such a system. Revising Some Basic Proofs in Belief Revision Timothy J. Surendonk May 7, 1997 Abstract This paper reobtains GROVE's results [3] through means which are hopefully clearer and more illustrative of the underlying notions. Also, the proofs are worked through in enough details that possible errors in GROVE's proof are clearly avoided. The paper concludes with a discussion on elementary (or closed) systems of spheres and it notes that each revision function can be given by such a system. 1 Introduction In 1988 ADAM GROVE [3] published his paper which gives the sphere semantics for simple belie..
Expressing Sets With Ultrafilters and the Canonicity of the Sahlqvist Logics
No full textThis paper will look at Sahlqvist's theorem in a new way. We introduce and investigate a "skewed" type of semantics for modal logics over their canonical frames. We will use this semantics, to describe every subset of a canonical frame, we will show that the semantics respects all Sahlqvist formulae, and we then conclude that the Sahlqvist logics are canonical. This will be accomplished without refering to any first order properties of the canonical accessibility relation. Expressing Sets With Ultrafilters and the Canonicity of the Sahlqvist Logics Timothy J. Surendonk September 27, 1996 Abstract This paper will look at Sahlqvist's theorem in a new way. We introduce and investigate a "skewed" type of semantics for modal logics over their canonical frames. We will use this semantics, to describe every subset of a canonical frame, we will show that the semantics respects all Sahlqvist formulae, and we then conclude that the Sahlqvist logics are canonical. This will be accomplished wi..
Semantically Constrained Condensed Detachment is Incomplete
No full textIn reporting on the theorem prover SCOTT (Slaney, SCOTT: A Semantically Guided Theorem Prover, Proc. IJCAI, 1993) we suggested semantic constraint as as an appropriate mechanism for guiding proof searches in propositional systems where the rule of inference is condensed detachment---a generalisation of Modus Ponens. Such constrained condensed detachment is closely analogous to semantic resolution. This paper exhibits an example which shows that semantically constrained condensed detachment is incomplete. That is, there are formulae deducible by means of condensed detachment which are not deducible when the semantic constraint is imposed. This answers an open question from our 1993 paper. Semantically Constrained Condensed Detachment is Incomplete John Slaney and Timothy J. Surendonk August 8, 1995 Abstract In reporting on the theorem prover SCOTT (Slaney, SCOTT: A Semantically Guided Theorem Prover, Proc. IJCAI, 1993) we suggested semantic constraint as as an appropriate mechanism f..
A far too short overview of Set Theory
No full textthis document brief we will gloss over a number of non-central, but still important issues; anyone wishing to think more deeply about sets should refer to any one of the texts noted at the end of this paper. Before we get started, a few remarks about our notation. Essentially we will be using the standard notation inherited from first order logic. One type of notation with which you might not be familiar is the use of an overstrike to represent a finite sequence of variables or objects. In particula
"Does EK4 have the Finite Model Property?" and related open questions
No full textIn this short paper we pose the question of whether EK4 has a canonical neighborhood frame or even has the finite frame property. We also comment on the difficulties in and around this area. 1 Introduction The modal logic K4 is well known to have the Finite Model Property (FMP) and be canonical, the proofs, being rather straightforward, highlight the central role played by relations/normality. Unfortunately classical logics without normality are not so easy to deal with, and the problems of FMP/Canonicity for EK4, the nonnormal counterpart K4 appears very difficult. It is hoped that any solution to this problem will shed light on the field of neighborhood frames, NFs (the structures underlying these systems) which are both straightforward, since satisfaction in canonical models is immediate, and complicated, since each underlying set can have a greater variety of frames defined on it. We will underline this by mentioning further open problems surrounding the notions of canonicity expl..
Canonicity for Intensional Logics without Iterative Axioms
No full textDAVID LEWIS proved in 1974 that all logics without iterative axioms are weakly complete. In this paper we extend LEWIS's ideas and provide a proof that such logics are canonical and so strongly complete. This paper also discusses the differences between relational and neighborhood frame semantics and poses a number of open questions about the latter
