Logical closure properties of propositional proof systems - (Extended abstract)

In this paper we define and investigate basic logical closure properties of propositional proof systems such as closure of arbitrary proof systems under modus ponens or substitutions. As our main result we obtain a purely logical characterization of the degrees of schematic extensions of EF in terms of a simple combination of these properties. This result underlines the empirical evidence that EF and its extensions admit a robust definition which rests on only a few central concepts from propositional logic

Modal logic S4 as a paraconsistent logic with a topological semantics

In this paper the propositional logic LTop is introduced, as an extension of classical propositional logic by adding a paraconsistent negation. This logic has a very natural interpretation in terms of topological models. The logic LTop is nothing more than an alternative presentation of modal logic S4, but in the language of a paraconsistent logic. Moreover, LTop is a logic of formal inconsistency in which the consistency and inconsistency operators have a nice topological interpretation. This constitutes a new proof of S4 as being "the logic of topological spaces", but now under the perspective of paraconsistency

An Abstract Approach to Consequence Relations

We generalise the Blok-J\'onsson account of structural consequence relations, later developed by Galatos, Tsinakis and other authors, in such a way as to naturally accommodate multiset consequence. While Blok and J\'onsson admit, in place of sheer formulas, a wider range of syntactic units to be manipulated in deductions (including sequents or equations), these objects are invariably aggregated via set-theoretical union. Our approach is more general in that non-idempotent forms of premiss and conclusion aggregation, including multiset sum and fuzzy set union, are considered. In their abstract form, thus, deductive relations are defined as additional compatible preorderings over certain partially ordered monoids. We investigate these relations using categorical methods, and provide analogues of the main results obtained in the general theory of consequence relations. Then we focus on the driving example of multiset deductive relations, providing variations of the methods of matrix semantics and Hilbert systems in Abstract Algebraic Logic

The Deduction Theorem for Strong Propositional Proof Systems

This paper focuses on the deduction theorem for propositional logic. We define and investigate different deduction properties and show that the presence of these deduction properties for strong proof systems is powerful enough to characterize the existence of optimal and even polynomially bounded proof systems. We also exhibit a similar, but apparently weaker condition that implies the existence of complete disjoint NPUnknown control sequence '\mathsf' -pairs. In particular, this yields a sufficient condition for the completeness of the canonical pair of Frege systems and provides a general framework for the search for complete NPUnknown control sequence '\mathsf' -pairs

Explicit Evidence Systems with Common Knowledge

Justification logics are epistemic logics that explicitly include justifications for the agents' knowledge. We develop a multi-agent justification logic with evidence terms for individual agents as well as for common knowledge. We define a Kripke-style semantics that is similar to Fitting's semantics for the Logic of Proofs LP. We show the soundness, completeness, and finite model property of our multi-agent justification logic with respect to this Kripke-style semantics. We demonstrate that our logic is a conservative extension of Yavorskaya's minimal bimodal explicit evidence logic, which is a two-agent version of LP. We discuss the relationship of our logic to the multi-agent modal logic S4 with common knowledge. Finally, we give a brief analysis of the coordinated attack problem in the newly developed language of our logic

Propositional semantics for default logic

We present new semantics for propositional default logic based on the notion of meta-interpretations - truth functions that assign truth values to clauses rather than letters. This leads to a propositional characterization of default theories: for each such finite theory, we show a classical propositional theory such that there is a one-to-one correspondence between models for the latter and extensions of the former. This means that computing an extension and answering questions about coherence, set-membership, and set-entailment are reducible to propositional satisfiability. The general transformation is exponential but tractable for a subset which we call 2-DT which is a superset of network default theories and disjunction-free default theories. This leads to the observation that coherence and membership for the class 2-DT is NP-complete and entailment is co-NP-complete.Since propositional satisfiability can be regarded as a constraint satisfaction problem (CSP), this work also paves the way for applying CSP techniques to default reasoning. In particular, we use the taxonomy of tractable CSP to identify new tractable subsets for Reiter's default logic. Our procedures allow also for computing stable models of extended logic programs