Melvin Fitting

Proof methods for modal and intuitionistic logics

One / Background.- Two / Analytic Modal Tableaus and Consistency Properties.- Three / Logical Consequence, Compactness, Interpolation, and Other Topics.- Four / Axiom Systems and Natural Deduction.- Five / Non-Analytic Logics.- Six / Non-Normal Logics.- Seven / Quantifiers.- Eight / Prefixed Tableau Systems.- Nine / Intuitionistic Logic.- Special Notation.

Bilattices and the semantics of logic programming

Abstract Bilattices, due to M. Ginsberg, are a family of truth-value spaces that allow elegantly for missing or conflicting information. The simplest example is Belnaps four-valued logic, based on classical two-valued logic. Among other examples are those based on finite many-valued logics and on probabilistic-valued logic. A fixed-point semantics is developed for logic programming, allowing any bilattice as the space of truth values. The mathematics is little more complex than in the classical two-valued setting, but the result provides a natural semantics for distributed logic programs, including those involving confidence factors. The classical two-valued and the Kripke-Kleene three-valued semantics become special cases, since the logics involved are natural sublogics of Belnaps logic, the logic given by the simplest bilattice.

The logic of proofs, semantically

Abstract A new semantics is presented for the logic of proofs (LP), (Technical Report MSI 95-29, Cornell University (1995), Bull. Symbolic Logic 7 (2001) 1) based on the intuition that it is a logic of explicit knowledge. This semantics is used to give new proofs of several basic results concerning LP. In particular, the realization of S4 into LP is established in a way that carefully examines and explicates the role of the + operator. Finally connections are made with the conventional approach, via soundness and completeness results.

Fixpoint semantics for logic programming a survey

The variety of semantical approaches that have been invented for logic programs is quite broad, drawing on classical and many-valued logic, lattice theory, game theory, and topology. One source of this richness is the inherent non-monotonicity of its negation, something that does not have close parallels with the machinery of other programming paradigms. Nonetheless, much of the work on logic programming semantics seems to exist side by side with similar work done for imperative and functional programming, with relatively minimal contact between communities. In this paper we summarize one variety of approaches to the semantics of logic programs: that based on fixpoint theory. We do not attempt to cover much beyond this single area, which is already remarkably fruitful. We hope readers will see parallels with, and the divergences from the better known fixpoint treatments developed for other programming methodologies.

Kleene's three valued logics and their children

Kleene’s strong three-valued logic extends naturally to a four-valued logic proposed by Belnap. We introduce a guard connective into Belnap’s logic and consider a few of its properties. Then we show that by using it four-valued analogs of Kleene’s weak three-valued logic, and the asymmetric logic of Lisp are also available. We propose an extension of these ideas to the family of distributive bilattices. Finally we show that for bilinear bilattices the extensions do not produce any new equivalences.

Kleene's Logic, Generalized

Kleenes well-known strong three-valued logic is shown to be one of a family of logics with similar mathematical properties. These logics are produced by an intuitively natural construction. The resulting logics have direct relationships with bilattices. In addition they possess mathematical features that lend themselves well to semantical constructions based on fixpoint procedures, as in logic programming.

Bilattices in logic programming

Bilattices are a family of multiple-valued logics. Those meeting certain natural conditions have provided the basis for the semantics of a family of logic programming languages. Consideration is given to further restrictions on bilattices in order to narrow things down to logic programming languages that can, at least in principle, be implemented. Appropriate bilattice background information is presented, so the work is relatively self-contained. The backgrounds of logic programming and bilattices are given. Logic programming syntax is discussed, along with fixpoint semantics. Smullyan-style propositional rules are discussed.<<ETX>>

First-order modal tableaux

We describe simple semantic tableau based theorem provers for four standard modal logics, in both propositional and first-order versions. These theorem provers are easy to implement in Prolog, have a behavior that is straightforward to understand, and provide natural places for the incorporation of heuristics.

The family of stable models

Abstract The family of all stable models for a logic program has a surprisingly simple overall structure, once two naturally occuring orderings are made explicit. In a so-called knowledge ordering based on degree of definedless, every logic program P has a smallest stable model sk P —it is the well-founded model. There is also a dual largest stable model Sk P , which has not been considered before. There is another ordering based on degree of truth. Taking the meet and the join, in the truth ordering, of the two extreme stable models sk P and Sk P just mentioned yields the alternating fixed points of Van Gelder, denoted st P and St P here. From st P and St P in turn, sk P and Sk P can be produced again, using the meet and joint of the knowledge ordering. All stable models are bounded by these four valuations. Further, the methods of proof apply not just to logic programs considered classically, but to logic programs over any bilattice meeting certain conditions, and thus apply in a vast range of settings. The methods of proof are largely algebraic.

Tableau methods of proof for modal logics.

1 Introduction: In [1] Fitch proposed a new proof proceedure for several standard modal logics. The chief characteristic of this was the inclusion in the object language, of symbols representing worlds in Kripke models. In this paper we incorporate the device into a tableau proof system and it is seen that the resulting (propositional) proof system is highly analogous to a classical first order tableau system, with the modal operators behaving like quantifiers. Exploiting this similarity, a tableau completeness proof for first order logic directly becomes a Kripke completeness proof for modal logic, and Smullyans fundamental theorem of quantification theory (a Herbrand-like theorem) [7] has its analog. Indeed, more than analogy is at work here; from an appropriate abstract point of view certain modal logics, first order classical and intuitionistic logic, and various infinitary logics may be treated simultaneously, an approach due to R. Smullyan and developed in a forthcoming monograph (see [8] for a preliminary version). We will treat only tableau proof systems and some familiarity with [7] is presumed. In addition to being metatheoretically interesting, specific tableau systems we give for S5, S4, T, B, DS4, DT, and K are quite simple to use. The extension of these systems to first order systems is straightforward , and is discussed briefly in the last section.