Completeness of Certain Bimodal Logics for Subset Spaces

Subset Spaces were introduced by L. Moss and R. Parikh in [8]. These spaces model the reasoning about knowledge of changing states.In [2] a kind of subset space called intersection space was considered and the question about the existence of a set of axioms that is complete for the logic of intersection spaces was addressed. In [9] the first author introduced the class of directed spaces and proved that any set of axioms for directed frames also characterizes intersection spaces.We give here a complete axiomatization for directed spaces. We also show that it is not possible to reduce this set of axioms to a finite set.     

A Loop-Free Decision Procedure for Modal Propositional Logics K4, S4 and S5

The aim of this paper is to present a loop-free decision procedure for modal propositional logics K4, S4 and S5. We prove that the procedure terminates and that it is sound and complete. The procedure is based on the method of Socratic proofs for modal logics, which is grounded in the logic of questions IEL.     

Taking Degrees of Truth Seriously

This is a contribution to the discussion on the role of truth degrees in manyvalued logics from the perspective of abstract algebraic logic. It starts with some thoughts on the so-called Suszko’s Thesis (that every logic is two-valued) and on the conception of semantics that underlies it, which includes the truth-preserving notion of consequence. The alternative usage of truth values in order to define logics that preserve degrees of truth is presented and discussed. Some recent works studying these in the particular cases of Łukasiewicz’s many-valued logics and of logics associated with varieties of residuated lattices are also presented. Finally the extension of this paradigm to other, more general situations is discussed, highlighting the need for philosophical or applied motivations in the selection of the truth degrees, due both to the interpretation of the idea of truth degree and to some mathematical difficulties.     

How True It Is = Who Says It’s True

This is a largely expository paper in which the following simple idea is pursued. Take the truth value of a formula to be the set of agents that accept the formula as true. This means we work with an arbitrary (finite) Boolean algebra as the truth value space. When this is properly formalized, complete modal tableau systems exist, and there are natural versions of bisimulations that behave well from an algebraic point of view. There remain significant problems concerning the proper formalization, in this context, of natural language statements, particularly those involving negative knowledge and common knowledge. A case study is presented which brings these problems to the fore. None of the basic material presented here is new to this paper—all has appeared in several papers over many years, by the present author and by others. Much of the development in the literature is more general than here—we have confined things to the Boolean case for simplicity and clarity. Most proofs are omitted, but several of the examples are new. The main virtue of the present paper is its coherent presentation of a systematic point of view—identify the truth value of a formula with the set of those who say the formula is true.     

Proof Systems for Reasoning about Computation Errors

In the paper we examine the use of non-classical truth values for dealing with computation errors in program specification and validation. In that context, 3-valued McCarthy logic is suitable for handling lazy sequential computation, while 3-valued Kleene logic can be used for reasoning about parallel computation. If we want to be able to deal with both strategies without distinguishing between them, we combine Kleene and McCarthy logics into a logic based on a non-deterministic, 3-valued matrix, incorporating both options as a non-deterministic choice. If the two strategies are to be distinguished, Kleene and McCarthy logics are combined into a logic based on a 4-valued deterministic matrix featuring two kinds of computation errors which correspond to the two computation strategies described above. For the resulting logics, we provide sound and complete calculi of ordinary, two-valued sequents. Presented by Yaroslav Shramko and Heinrich Wansing     

Knowledge on Treelike Spaces

This paper presents a bimodal logic for reasoning about knowledge during knowledge acquisitions. One of the modalities represents (effort during) non-deterministic time and the other represents knowledge. The semantics of this logic are tree-like spaces which are a generalization of semantics used for modeling branching time and historical necessity. A finite system of axiom schemes is shown to be canonically complete for the formentioned spaces. A characterization of the satisfaction relation implies the small model property and decidability for this system.     

Characterizing Equivalential and Algebraizable Logics by the Leibniz Operator

In [14] we used the term finitely algebraizable for algebraizable logics in the sense of Blok and Pigozzi [2] and we introduced possibly infinitely algebraizable, for short, p.i.-algebraizable logics. In the present paper, we characterize the hierarchy of protoalgebraic, equivalential, finitely equivalential, p.i.-algebraizable, and finitely algebraizable logics by properties of the Leibniz operator. A Beth-style definability result yields that finitely equivalential and finitely algebraizable as well as equivalential and p.i.-algebraizable logics can be distinguished by injectivity of the Leibniz operator. Thus, from a characterization of equivalential logics we obtain a new short proof of the main result of [2] that a finitary logic is finitely algebraizable iff the Leibniz operator is injective and preserves unions of directed systems. It is generalized to nonfinitary logics. We characterize equivalential and, by adding injectivity, p.i.-algebraizable logics.     

The elimination of de re formulas

It is shown that de re formulas are eliminable in the modal logic S5 extended with the axiom scheme x x.     

康德哲学与广松涉"交互主体的共同主观性"理论

广松涉认为,对近代认识论基础的问题式及其构成机制和局限进行反思,首先需要将着眼点放在康德哲学的问题式及其构成机制上。康德哲学的总的问题式就是在以先验逻辑学为基础的逻辑学、认识论和本体论“三位一体”的哲学构架中,探讨人的认识以及纯粹理性的先天综合判断及其能力何以可能。这一问题式潜含着一个意义非常重大的认识论问题——即“主体际共同主观性”问题．需要我们认真地加以研究。