Arnon Avron

Hypersequents are finite sets of ordinary sequents. We show that multiple-conclusion sequents and single-conclusion hypersequents represent two different natural methods of switching from a single-conclusion calculus to a multiple-conclusion one. The use of multiple-conclusion sequents corresponds to using a multiplicative disjunction, while the use of single-conclusion hypersequents corresponds to using an additive one. Moreover: each of the two methods is usually based on a different natural semantic idea and accordingly leads to a different class of algebraic structures. In the cases we consider here the use of multiple-conclusion sequents corresponds to focusing the attention on structures in which there is a full symmetry between the sets of designated and antidesignated elements. The use of single-conclusion hypersequents, on the other hand, corresponds to the use of structures in which all elements except one are designated. Not surprisingly, the use of multiple-conclusion hypersequents corresponds to the use of structures which are both symmetrical and with a single nondesignated element

There is a long tradition (See e.g. [9, 10]) starting from [12], according to which the meaning of a connective is determined by the introduction and elimination rules which are associated with it. The supporters of this thesis usually have in mind natural deduction systems of a certain ideal type (explained in Section 3 below). Unfortunately, already the handling of classical negation requires rules which are not of that type. This problem can be solved in the framework of multiple-conclusion Gentzen-type systems (also first introduced in [12]), where instead of introduction and elimination rules there are left introduction rules and right introduction rules.

Non-deterministic matrices are multiple-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. We consider two different types of semantics which are based on Nmatrices: the dynamic one and the static one . We use the Rasiowa-Sikorski decomposition methodology to get sound and complete proof systems employing finite sets of mv-signed formulas for all propositional logics based on such structures with either of the above types of semantics. Later we demonstrate how these systems can be converted into cut-free ordinary Gentzen calculi which are also sound and complete for the corresponding non-deterministic semantics. As a by-product, we get new semantic characterizations for some well-known logics

We suggest a new framework for the Weyl-Feferman predicativist program by constructing a formal predicative set theory P ZF which resembles ZF , and is suitable for mechanization. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an absolute way, independent of the extension of the “surrounding universe”. The language of P ZF is type-free, and it reflects real mathematical practice in making an extensive use of statically defined abstract set terms. Another important feature of P ZF is that its underlying logic is ancestral logic (i.e. the extension of first-order logic with a transitive closure operation)

We show that the rule that allows the inference of A from A ⊗ B is admissible in many of the basic multiplicative systems. By adding this rule to these systems we get, therefore, conservative extensions in which the tensor behaves as classical conjunction. Among the systems obtained in this way the one derived from RMIm has a particular interest. We show that this system has a simple infinite-valued semantics, relative to which it is strongly complete, and a nice cut-free Gentzen-type formulation which employs hypersequents . Moreover: classical logic has a simple, strong translation into this logic. This translation uses definable connectives and preserves the consequence relation of classical logic . Similar results, but with a 3-valued semantics, obtain if instead of RMIm we use RMm

We provide a general investigation of Logic in which the notion of a simple consequence relation is taken to be fundamental. Our notion is more general than the usual one since we give up monotonicity and use multisets rather than sets. We use our notion for characterizing several known logics (including Linear Logic and non-monotonic logics) and for a general, semantics-independent classi cation of standard connectives via equations on consequence relations (these include Girard's \multiplicatives" and \additives"). We next investigate the standard methods for uniformly representing consequence relations: Hilbert type, Natural Deduction and Gentzen type. The advantages and disadvantages of using each system and what should be taken as..

In advanced books and courses on logic (e.g. Sm], BM]) Gentzen-type systems or their dual, tableaux, are described as techniques for showing validity of formulae which are more practical than the usual Hilbert-type formalisms. People who have learnt these methods often wonder why the Automated Reasoning community seems to ignore them and prefers instead the resolution method. Some of the classical books on AD (such as CL], Lo]) do not mention these methods at all. Others (such as Ro]) do, but the connections and reasons for preference remain unclear after reading them (at least to the present author, and obviously also the authors of OS], in which a theorem-prover, based exclusively on tableaux, is described). The confusion becomes greater when the reader is introduced to Kowalski's form of a clause ( Ko], Bu]), which is nothing but a Gentzen's sequent of atomic formulae, and when he realizes that resolution is just a form of a Cut, and so that while the elimination of cuts is the principal tool in proof-theory, its use is the main technique in AD!

We show that the elimination rule for the multiplicative conjunction $\wedge$ is admissible in many important multiplicative substructural logics. These include LL$_m$ and RMI$_m$ An exception is R$_m$. Let SLL$_m$ and SR$_m$ be, respectively, the systems which are obtained from LL$_m$ and R$_m$ by adding this rule as a new rule of inference. The set of theorems of SR$_m$ is a proper extension of that of R$_m$, but a proper subset of the set of theorems of RMI$_m$. Hence it still has the variable-sharing property. SR$_m$ has also the interesting property that classical logic has a strong translation into it. We next introduce general algebraic structures, called strong multiplicative structures, and prove strong soundness and completeness of SLL$_m$ relative to them. We show that in the framework of these structures, the addition of the weakening axiom to SLL$_m$ corresponds to the condition that there will be exactly one designated element, while the addition of the contraction axiom corresponds to the condition that there will be exactly one nondesignated element. Various other systems in which multiplicative conjunction functions as a true conjunction are studied, together with their algebraic counterparts.

Until not too many years ago, all logics except classical logic (and, perhaps, intuitionistic logic too) were considered to be things esoteric. Today this state of a airs seems to have completely been changed. There is a growing interest in many types of nonclassical logics: modal and temporal logics, substructural logics, paraconsistent logics, non-monotonic logics { the list is long. The diversity of systems that have been proposed and studied is so great that a need is felt by many researchers to try to put some order in the present logical jungle. Thus Cl91], Ep90] and Wo88] are three recent books in which an attempt is made to develop a general theoretical framework for the study of logics. On the more pragmatic side, several systems have been developed with the goal of providing a computerized logical framework in which many di erent logical systems can be implemented in a uniform way. An example is the Edinburgh LF( HHP91])

One of the most signi cant drawbacks of classical logic is its being useless in the presence of an inconsistency. Nevertheless, the classical calculus is a very convenient framework to work with. In this work we propose means for drawing conclusions from systems that are based on classical logic, although the informationmightbe inconsistent. The idea is to detect those parts of the knowledge-base that \cause" the inconsistency, and isolate the parts that are \recoverable". We do this by temporarily switching into Ginsberg/Fitting multi-valued framework of bilattices (which is a common framework for logic programmingand nonmonotonic reasoning). Our method is conservative in the sense that it considers the contradictory data as useless and regards all the remaining information una ected. The resulting logic is nonmonotonic, paraconsistent, and a plausibility logic in the sense of Lehmann

We describe a method for obtaining classical logic from intuitionistic logic which does not depend on any proof system, and show that by applying it to the most important implicational relevance logics we get relevance logics with nice semantical and proof-theoretical properties. Semantically all these logics are sound and strongly complete relative to classes of structures in which all elements except one are designated. Proof-theoretically they correspond to cut-free hypersequential Gentzen-type calculi. Another major property of all these logics is that the classical implication can faithfully be translated into them.

In order to handle inconsistent knowledge bases in a reasonable way, one needs a logic which allows nontrivial inconsistent theories. Logics of this sort are called paraconsistent. 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 in a modular way simple non-deterministic semantics for 64 of the most important logics from this family. Our semantics is 3-valued for some of the systems, and infinite-valued for the others. We prove that these results cannot be improved: neither of the systems with a three-valued non-deterministic semantics has either a finite characteristic ordinary matrix or a two-valued characteristic non-deterministic matrix, and neither of the other systems we investigate has a finite characteristic non-deterministic matrix. Still, our semantics provides decision procedures for all the systems investigated, as well as easy proofs of important proof-theoretical properties of them

The notion of a bilattice was rst introduced by Ginsburg (see Gin]) as a general framework for a diversity of applications (such as truth maintenance systems, default inferences and others). The notion was further investigated and applied for various purposes by Fitting (see Fi1]- Fi6]). The main idea behind bilattices is to use structures in which there are two (partial) order relations, having di erent interpretations. The two relations should, of course, be connected somehow in order for the mathematical structure to be useful. It is not clear, however, what this connection should be. Ginsberg, for example, has made the connection through an extra operation of negation. Fitting, on the other hand, has investigated connections in the form of conditions on the structure (such as being interlaced { see below). These conditions are independent of the existence, or even the possibility to de ne, operations like Ginsberg's negation.* Fitting de nes, accordingly, notions like \an interlaced bilattice", \a distributive bilattice", \a bilattice with negation" and others. He does not provide, however, any de nition of the notion of bilattice itself (without an extra modi er). I was unable to nd anywhere, in fact, a de nition which will cover all the structures which were called \bilattice" in the literature

Around 1950, B.A. Trakhtenbrot proved an important undecidability result (known, by a pure accident, as \Trakhtenbrot's theorem"): there is no algorithm to decide, given a rst-order sentence, whether the sentence is satis able in some nite model. The result is in fact true even if we restrict ourselves to languages that has only one binary relation Tra63]. It is hardly conceivable that at that time Prof. Trakhtenbrot expected his result to in uence the development of the theory of relational databases query languages, but it did. This perhaps is not surprising in view of the following two facts: 1) The theory of relational databases is strongly rooted in logic and can easily be abstracted to make it a branch of mathematical logic. 2) The main interest in the theory of relational databases is in nite relations. In the rst section we explain those two points in more detail. Then, in section 2 of the present paper, we discuss the question: \What constitutes a reasonable query to a database?" The discussion naturally leads to a certain class of queries, known as the \domain independent" queries, as an ideal query language. At this point Trakhtenbrot's theorem appears as a major obstacle, since it easily implies that this ideal class is undecidable. Real query languages cannot be allowed to have this property. Section 3 outlines then several possible ways to circumvent this di culty. The following section explains the..