Varieties of two-dimensional cylindric algebras.¶Part I: Diagonal-free case

Bezhanishvili, Nick

doi:10.1007/s00012-002-8203-2

Cited by 6 publications

(4 citation statements)

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“…However, we might be able to establish the analog to the central lemma that every quasi-normal extension of 2D has the finite model property via a different route than by local tabularity. In particular, we have reason to hope that all extensions of S5 2 @ do in fact have the finite model property, since all normal extensions of S5 2 have the finite model property; see the reference to Bezhanishvili (2002) in Bezhanishvili & Marx ( 2003, p. 367). §5.…”

Section: S5 2 @ and Informal Completenessmentioning

confidence: 99%

What Is the Correct Logic of Necessity, Actuality and Apriority?

Fritz

2014

The Review of Symbolic Logic

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This paper is concerned with a propositional modal logic with operators for necessity, actuality and apriority. The logic is characterized by a class of relational structures defined according to ideas of epistemic two-dimensional semantics, and can therefore be seen as formalizing the relations between necessity, actuality and apriority according to epistemic two-dimensional semantics. We can ask whether this logic is correct, in the sense that its theorems are all and only the informally valid formulas. This paper gives outlines of two arguments that jointly show that this is the case. The first is intended to show that the logic is informally sound, in the sense that all of its theorems are informally valid. The second is intended to show that it is informally complete, in the sense that all informal validities are among its theorems. In order to give these arguments, a number of independently interesting results concerning the logic are proven. In particular, the soundness and completeness of two proof systems with respect to the semantics is proven (Theorems 2.11 and 2.15), as well as a normal form theorem (Theorem 3.2), an elimination theorem for the actuality operator (Corollary 3.6), and the decidability of the logic (Corollary 3.7). It turns out that the logic invalidates a plausible principle concerning the interaction of apriority and necessity; consequently, a variant semantics is briefly explored on which this principle is valid. The paper concludes by assessing the implications of these results for epistemic two-dimensional semantics. §1. Introduction. Epistemic two-dimensional semantics as proposed by David Chalmers, e.g., in Chalmers (2004), provides an account of meaning that allows a possible world semantics of necessity as well as apriority. The notions of necessity and apriority intended here are those distinguished by Kripke (1972); the first is sometimes called metaphysical necessity. They can roughly be paraphrased by saying that necessary is what could not have failed to be the case, and a priori is what can be known in an a priori way. In Fritz (2013), a propositional modal logic with operators for necessity, actuality and apriority is defined which captures the relevant ideas of epistemic two-dimensional semantics. In particular, a class of relational structures is defined, and it is argued that it represents the evaluation of sentences according to epistemic two-dimensional semantics, and that therefore, the logic characterized by this class captures the relations of the three modalities according to epistemic two-dimensional semantics. Epistemic two-dimensional semantics is a controversial theory, and philosophers who do not want to commit themselves to it can't justify the correctness of this logic using its semantics. This raises the question whether there is a way of arguing for its correctness that is not based on epistemic two-dimensional semantics. The main aim of this paper is to outline such an argument.

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Section: S5 2 @ and Informal Completenessmentioning

confidence: 99%

What Is the Correct Logic of Necessity, Actuality and Apriority?

Fritz

2014

The Review of Symbolic Logic

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“…The poly-size model property of every L ⊃ S5 2 is proven in [3,Corollary 9]. (1) implies that the problem G ∈ F L can be decided in polynomial time in the size of G. The result follows.…”

Section: Proofmentioning

confidence: 76%

“…An S5 2 -frame F is called an Lframe if F validates all formulas in L. Let F L be the set of all L-frames in F S5 2 . Then L is complete with respect to F L [1]. Thus, for our purposes it suffices to consider only finite rooted S5 2 -frames.…”

Section: Preliminariesmentioning

confidence: 99%

All Normal Extensions of S5-squared Are Finitely Axiomatizable

Bezhanishvili

Hodkinson

2004

Stud Logica

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We prove that every normal extension of the bi-modal system S5 2 is finitely axiomatizable and that every proper normal extension has NP-complete satisfiability problem.It is well known that S5 2 has the exponential size model property, and that its satisfiability problem is NEXPTIME-complete [6]. In this paper, by the complexity of a logic we will mean the complexity of its satisfiability problem. It is shown in [3] that in contrast to S5 2 , every proper normal extension L of S5 2 has the poly-size model property. That means that there is a polynomial P (n) such that any L-consistent formula ϕ (that is, ¬ϕ / ∈ L) has a model over a frame validating L and with at most P (|ϕ|) points, where |ϕ| is the length of ϕ.It was conjectured in [3] that every proper normal extension of S5 2 is finitely axiomatizable and NP-complete. In this paper we prove this conjecture. In fact, we show that for every proper normal extension L of S5 2 , there is a finite set M L of finite S5 2 -frames such that an arbitrary finite S5 2frame is a frame for L iff it does not have any frame in M L as a p-morphic image. This condition yields a finite axiomatization of L. We also show that the condition is decidable in deterministic polynomial time. This, together with the poly-size model property, implies NP-completeness of (satisfiability for) L.Finally, we note that general complexity results for (uni)modal logics were investigated before. Bull and Fine proved that every normal extension of S4.3 has the finite model property, is finitely axiomatizable and therefore is decidable (see [4, Theorems 4.96, 4.101]). Hemaspaandra strengthened the second result by showing that every normal extension of S4.3 is NP-complete [4, Theorem 6.41]. The proof of finite axiomatizability uses Kruskal's theorem on well-quasi-orderings [4, Theorem 4.99]. Kracht uses the same technique for showing that every extension of the intermediate logic of leptonic strings is finitely axiomatizable [8, Theorem 14, Proposition 15].This paper takes the same line of research beyond unimodal logics. However, as we will see below, the theory of well-quasi-orderings does not suffice for our purposes; instead, we will use better-quasi-orderings.

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“…In this paper we give answers to cases (1) and (2). In more detail we show that the mentioned logics in (1) and (2) have both the syntactic and semantic Gödel's incompleteness properties.…”

Section: Introductionmentioning

confidence: 86%