Arnon Avron

We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF . It allows the use of set terms, but provides a static check of their validity. Like the inconsistent “ideal calculus” for set theory, it is essentially based on just two set-theoretical principles: extensionality and comprehension (to which we add ∈-induction and optionally the axiom of choice). Comprehension is formulated as: x ∈ {x | ϕ} ↔ ϕ, where {x | ϕ} is a legal set term of the theory. In order for {x | ϕ} to be legal, ϕ should be safe with respect to {x}, where safety is a relation between..

A paraconsistent logic is a logic which allows non-trivial inconsistent theories. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. da Costa’s approach has led to the family of Logics of Formal (In)consistency (LFIs). In this paper we provide non-deterministic semantics for a very large family of first-order LFIs (which includes da Costa’s original system..

We define the notions of a canonical inference rule and a canonical system in the framework of single-conclusion Gentzen-type systems (or, equivalently, natural deduction systems), and prove that such a canonical system is non-trivial iff it is coherent (where coherence is a constructive condition). Next we develop a general non-deterministic Kripke-style semantics for such systems, and show that every constructive canonical system (i.e. coherent canonical single-conclusion system) induces a class of non-deterministic Kripke-style frames for which it is strongly sound and complete. We use this non-deterministic semantics to show that all constructive canonical systems admit a strong form of the cut-elimination theorem. We also use it for providing a decision procedure for every such system. These results identify a large family of basic constructive connectives, each having both a proof-theoretical characterization in terms of a coherent set of canonical rules, as well as a semantic characterization using non-deterministic frames. The family includes the standard intuitionistic connectives, together with many other independent connectives

We introduce a general framework for solving the problem of a computer collecting and combining information from various sources. Unlike previous approaches to this problem, in our framework the sources are allowed to provide information about complex formulae too. This is enabled by the use of a new tool — non-deterministic logical matrices. We also consider several alternative plausible assumptions concerning the framework. These assumptions lead to various logics. We provide strongly sound and complete proof systems for all the basic logics induced in this way

It is well known that every propositional logic which satisfies certain very natural conditions can be characterized semantically using a multi-valued matrix ([Los and Suszko, 1958; W´ ojcicki, 1988; Urquhart, 2001]). However, there are many important decidable logics whose characteristic matrices necessarily consist of an infinite number of truth values. In such a case it might be quite difficult to find any of these matrices, or to use one when it is found. Even in case a logic does have a finite characteristic matrix it might be difficult to discover this fact, or to find such a matrix. The deep reason for these difficulties is that in an ordinary multi-valued semantics the rules and axioms of a system should be considered as a whole, and there is no method for separately determining the semantic effects of each rule alone

We show by way of example how one can provide in a lot of cases simple modular semantics for rules of inference, so that the semantics of a system is obtained by joining the semantics of its rules in the most straightforward way. Our main tool for this task is the use of finite Nmatrices, which are multi-valued structures in which the value assigned by a valuation to a complex formula can be chosen non-deterministically out of a certain nonempty set of options. The method is applied in the area of logics with a formal consistency operator (known as LFIs), allowing us to provide in a modular way effective, finite semantics for thousands of different LFIs

An (n, k)-ary quantifier is a generalized logical connective, binding k variables and connecting n formulas. Canonical systems with (n, k)-ary quantifiers form a natural class of Gentzen-type systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a quantifier is introduced. The semantics for these systems is provided using two-valued non-deterministic matrices, a generalization of the classical matrix. In this paper we use a constructive syntactic criterion of coherence to characterize strong cutelimination in such systems. We show that the following properties of a canonical system G with arbitrary (n, k)-ary quantifiers are equivalent: (i) G is coherent, (ii) G admits strong cut-elimination, and (iii) G has a strongly characteristic two-valued generalized non-deterministic matrix. In addition, we define simple calculi, an important subclass of canonical calculi, for which coherence is equivalent to the weaker, standard cut-elimination property.

We provide a general framework for constructing natural consequence relations for paraconsistent and plausible nonmonotonic reasoning. The framework is based on preferential systems whose preferences are based on the satisfaction of formulas in models. We show that these natural preferential In the research on paraconsistency, preferential systems systems that were originally designed for for paraconsistent reasoning fulfill a key condition (stopperedness or smoothness) from the theoretical research of nonmonotonic reasoning. Consequently, the nonmonotonic consequence relations that they induce fulfill the desired conditions of plausible consequence relations. Hence our framework encompasses different types of preferential systems that were developed from different motivations of paraconsistent reasoning and non-monotonic reasoning, and reveals an important link between them.

An (n, k)-ary quantifier is a generalized logical connective, binding k variables and connecting n formulas. Canonical systems with (n, k)-ary quantifiers form a natural class of Gentzen-type systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a quantifier is introduced. The semantics for these systems is provided using two-valued non-deterministic matrices, a generalization of the classical matrix. In this paper we use a constructive syntactic criterion of coherence to characterize strong cutelimination in such systems. We show that the following properties of a canonical system G with arbitrary (n, k)-ary quantifiers are equivalent: (i) G is coherent, (ii) G admits strong cut-elimination, and (iii) G has a strongly characteristic two-valued generalized non-deterministic matrix.

Propositional canonical Gentzen-type systems, introduced in [2], are systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a connective is introduced and no other connective is mentioned. [2] provides a constructive coherence criterion for the non-triviality of such systems and shows that a system of this kind admits cut-elimination iff it is coherent. The semantics of such systems is provided using two-valued non-deterministic matrices (2Nmatrices). [23] extends these results to systems with unary quantifiers of a very restricted form. In this paper we substantially extend the characterization of canonical systems to (n, k)-ary quantifiers, which bind k distinct variables and connect n formulas, and show that the coherence criterion remains constructive for such systems. Then we focus on the case of k ∈ {0, 1} and show that the following statements concerning a canonical calculus G are equivalent: (i) G is coherent, (ii) G has a strongly characteristic 2Nmatrix, and (iii) G admits strong cut-elimination. We also show that coherence is not a necessary condition for standard cut-elimination, and then characterize a subclass of canonical systems for which this property does hold.