Arnon Avron

The value of the four values

In his well-known paper “How computer should think” Belnap (1977) argues that four-valued semantics is a very suitable setting for computerized reasoning. In this paper we vindicate this thesis by showing that the logical role that the four-valued structure has among Ginsbergs bilattices is similar to the role that the two-valued algebra has among Boolean algebras. Specifically, we provide several theorems that show that the most useful bilattice-valued logics can actually be characterized as four-valued inference relations. In addition, we compare the use of three-valued logics with the use of four-valued logics, and show that at least for the task of handling inconsistent or uncertain information, the comparison is in favor of the latter.

Reasoning with logical bilattices

The notion of bilattice was introduced by Ginsberg, and further examined by Fitting, as a general framework for many applications. In the present paper we develop proof systems, which correspond to bilattices in an essential way. For this goal we introduce the notion of logical bilattices. We also show how they can be used for efficient inferences from possibly inconsistent data. For this we incorporate certain ideas of Kifer and Lozinskii, which happen to suit well the context of our work. The outcome are paraconsistent logics with a lot of desirable properties.A preliminary version of this paper appears in Arieli and Avron (1994).

Natural 3-valued logic—characterization and proof theory

Many-valued logics in general and 3-valued logic in particular is an old subject which had its beginning in the work of Łukasiewicz [Łuk]. Recently there is a revived interest in this topic, both for its own sake (see, for example, [Ho]), and also because of its potential applications in several areas of computer science, such as proving correctness of programs [Jo], knowledge bases [CP] and artificial intelligence [Tu]. There are, however, a huge number of 3-valued systems which logicians have studied throughout the years. The motivation behind them and their properties are not always clear, and their proof theory is frequently not well developed. This state of affairs makes both the use of 3-valued logics and doing fruitful research on them rather difficult. Our first goal in this work is, accordingly, to identify and characterize a class of 3-valued logics which might be called natural . For this we use the general framework for characterizing and investigating logics which we have developed in [Av1]. Not many 3-valued logics appear as natural within this framework, but it turns out that those that do include some of the best known ones. These include the 3-valued logics of Łukasiewicz, Kleene and Sobocinski, the logic LPF used in the VDM project, the logic RM 3 from the relevance family and the paraconsistent 3-valued logic of [dCA]. Our presentation provides justifications for the introduction of certain connectives in these logics which are often regarded as ad hoc. It also shows that they are all closely related to each other. It is shown, for example, that Łukasiewicz 3-valued logic and RM 3 (the strongest logic in the family of relevance logics) are in a strong sense dual to each other, and that both are derivable by the same general construction from, respectively, Kleene 3-valued logic and the 3-valued paraconsistent logic.

Hypersequents, logical consequence and intermediate logics for concurrency

The existence of simple semantics and appropriate cut-free Gentzen-type formulations are fundamental intrinsic criteria for the usefulness of logics. In this paper we show that by using hypersequents (which are multisets of ordinary sequents) we can provide such Gentzen-type systems to many logics. In particular, by using a hypersequential generalization of intuitionistic sequents we can construct cut-free systems for some intermediate logics (including Dummetts LC) which have simple algebraic semantics that suffice, e.g., for decidability. We discuss the possible interpretations of these logics in terms of parallel computation and the role that the usual connectives play in them (which is sometimes different than in the sequential case).

Using typed lambda calculus to implement formal systems on a machine

Much research has been devoted in building computer systems for checking proofs or for developing interactively correct proofs in specific logical systems. However, implementing a proof environment for a specific logical system is both complex and time-consuming, this-together with the proliferation of logics-suggests that a uniform and reliable alternative is desirable. One such alternative is the Edinburgh Logical Framework (LF), developed in the late eighties at the LFCS (Laboratory for Foundations of Computer Science). The LF is a logic-independent tool which, given a specification for a logical system, synthesizes a proof editor and checker for that system. Its specification language is based on a general theory of logics, which enables one to capture uniformities and idiosyncrasies of a large class of logics without sacrificing generality for tractability. Peculiarities (such as side conditions on rule application, variable occurrence or formula formation) are expressed at the level of the specification. In this paper we are going to provide a broad illustration of its applicability and discuss to what extent it is successful. The analysis (of the formal presentation) of a system carried out through encoding often illuminates the system itself. This paper will also deal with this phenomenon.

A constructive analysis of RM

The system RM is the most well-understood (and to our opinion, also the most important) system among the logics developed by the Anderson and Belnap school. In this paper we investigate RM from a constructive point of view. For example, we give a new proof of the completeness of RM relative to the Sugihara matrix (first shown by Meyer), a proof in which a p.r. procedure is presented, applying which to a sentence A in RM language yields either a proof of it in RM or a refuting valuation for it in the Sugihara matrix S Z . Two topics dealt with in this work deserve a special attention. a) The admissibility of γ . This is a famous theorem of Meyer and Dunn. In [1] Anderson and Belnap emphasize that “the Meyer-Dunn argument … guarantees the existence of a proof of B , but there is no guarantee that the proof of B is related in any sort of plausible way to the proofs of A and Ā ∨ B .” In §2 we provide such a guarantee for the RM -case. In fact, we give there a direct method of obtaining a proof of B from given proofs of A and Ā ∨ B . b) The relationships between RM and its full negation-implication fragment . RM is known ([1, pp. 148–149], and [3]) to be a conservative extension of (Sobocinski 3-valued logic; see [4]). Anderson and Belnap admit [1, p. 149] that this fact came to them as a distinct surprise, since RM as a whole is far from being three-valued. In this paper, however, this “surprising” fact appears quite natural (see III.3). In fact, we show that , is the “hard core” of RM , since our proof of the completeness of RM is based in an essential way on the completeness of relative to the Sobocinski matrix, and since the Gentzen-type calculus we develop for RM is a direct extension of a similar (but much simpler) calculus for . Because of the importance has in this work, we devote the first section to a constructive investigation of it. We note, finally, that the Gentzen-type calculus mentioned above admits cut-elimination and normal-form techniques. (Such calculi were found till now only for RM without distribution.)

Canonical Propositional Gentzen-Type Systems

Canonical propositional Gentzen-type systems are systems which in addition to the standard axioms and structural rules have only pure logical rules which have the subformula property, introduce exactly one occurrence of a connective in their conclusion, and no other occurrence of any connective is mentioned anywhere else in their formulation. We provide a constructive coherence criterion for the non-triviality of such systems, and show that a system of this kind admits cut elimination iff it is coherent. We show also that the semantics of such systems is provided by non-deterministic two-valued matrices (2-Nmatrices). 2- Nmatrices form a natural generalization of the classical two-valued matrix, and every coherent canonical system is sound and complete for one of them. Conversely, with any 2-Nmatrix it is possible to associate a coherent canonical Gentzen-type system which has for each connective at most one introduction rule for each side, and is sound and complete for that 2-Nmatrix. We show also that every coherent canonical Gentzen-type system either defines a fragment of the classical two-valued logic, or a logic which has no finite characteristic matrix.

Gentzen-type systems, resolution and tableaux

We show that both the tableaux and the resolution methods can be understood as attempts to exploit the power of cut-elimination theorems in Gentzen-type calculi. Another, related goal is to provide a purely syntactic basis for both methods (in contrast to the semantic proofs concerning resolution that can be found in the textbooks). This allows the use of a fruitful combination of the methods and might be helpful in generalizing them to other logics.

Logical bilattices and inconsistent data

The notion of a bilattice was first proposed by Ginsberg (1988) as a general framework for many applications. This notion was further investigated and applied for various goals by Fitting (1989, 1990, 1991, 1993). In this paper, we develop proof systems which correspond to bilattices in an essential way. We then show how to use those bilattices for efficient inferences from possibly inconsistent data. For this, we incorporate certain ideas of Kifer and Lozinskii (1992) concerning inconsistencies, which happen to well suit the framework of bilattices. The outcome is a paraconsistent logic with many desirable properties.<<ETX>>

The structure of interlaced bilattices

Bilattices were introduced and applied by Ginsberg and Fitting for a diversity of applications, such as truth maintenance systems, default inferences and logic programming. In this paper we investigate the structure and properties of a particularly important class of bilattices called interlaced bilattices, which were introduced by Fitting. The main results are that every interlaced bilattice is isomorphic to the Ginsberg-Fitting product of two bounded lattices and that the variety of interlaced bilattices is equivalent to the variety of bounded lattices with two distinguishable distributive elements, which are complements of each other. This implies that interlaced bilattices can be characterized using a finite set of equations. Our results generalize to interlaced bilattices some results of Ginsberg, Fitting and Jonsson for distributive bilattices.