Logic Theorem Research Articles

The formal sciences discover the philosophers' stone

Aristotelians deplore the narrow range of examples chosen for discussion in traditional philosophy of mathematics. The traditional diet – numbers, sets, infinite cardinals, axioms, theorems of formal logic – is far from typical of what mathematicians do. It has led to intellectual anorexia, by depriving the philosophy of mathematics of the nourishment it could and should receive from the expansive world of mathematics of the last hundred years. Philosophers have almost completely ignored not only the broad range of pure and applied mathematics and statistics, but a whole suite of ‘formal’ or ‘mathematical’ sciences that have appeared only in the last eighty years. I give here a few brief examples to indicate why these developments are of philosophical interest to those pursuing realist views of mathematics. Of special significance is that they contain many examples of necessities about the real world.

Partial and unsharp quantum logics

The total and the sharp character of orthodox quantum logic has been put in question in different contexts. This paper presents the basic ideas for a unified approach to partial and unsharp forms of quantum logic. We prove a completeness theorem for some partial logics based on orthoalgebras and orthomodular posets. We introduce the notion of unsharp orthoalgebra and of generalized MV algebra. The class of all effects of any Hilbert space gives rise to particular examples of these structures. Finally, we investigate the relationship between unsharp orthoalgebras, generalized MV algebras, and orthomodular lattices.

A Major Failure within Modern Analytic Philosophy

The paper claims that analytic philosophy has failed within the philosophy of science due to the way the dynamic aspect of scientific theories is traditionally treated. On the formal side this failure manifests itself in the first-order logical and the model-theoretic analyses of scientific theories. An amendment of the treatment is sketched. It is based on using model generation, of the kind used in proving the Completeness Theorem for first-order logic, in such a way that some dynamic quantities in the dynamic theory are formally represented as functions relating closed terms and sentences to their interpretations in a generated model.

Finite-model theory - a personal perspective

Finite-model theory is a study of the logical properties of finite mathematical structures. This paper is a very personalized view of finite-model theory, where the author focuses on his own personal history, and results and problems of interest to him, especially those springing from work in his Ph.D. thesis. Among the topics discussed are: 1. Differences between the model theory of finite structures and infinite structures. Most of the classical theorems of logic fail for finite structures, which gives us a challenge to develop new concepts and tools, appropriate for finite structures. 2. The relationship between finite-model theory and complexity theory. Surprisingly enough, it turns out that, in some cases, we can characterize complexity classes (such as NP) in terms of logic, where there is no notion of machine, computation, or time. 3. 0–1 laws. There is a remarkable phenomenon which says that certain properties (such as those expressible in first-order logic) are either almost surely true or almost surely false. 4. Descriptive complexity theory. Here we consider how complex a formula must be to express a given property. In recent years, there has been a re-awakening of interest in finite-model theory. One goal of this paper is to help “fan the flames” of interest, by introducing more researchers to this fascinating area.

Lisa Reidhaar-Olson. A new proof of the fixed-point theorem of provability logic. Notre Dame journal of formal logic, vol. 31 (1990), pp. 37–43.

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Sistemas de cálculo como formas de Logicismo

The logicism may be regarded like a fossil stone that has not utility nowadays. In this sense, logicism took care of the research about the foundations of mathematics but apparently its task arrived at its end many years ago because of sorne results that were eetablished during the century. However it is not wholly right. Understanding logicism as an attempt to reduce classical mathematics to logic means we can distinguish: 1) the idea according to which mathematic is logic in sorne way, and 2) a metaphysical program of research to: a) define mathematical notions as logical notions, and b) show that the mathematical theorems are logical theorems. The failure (if so) concerned to 2), since 1) was assumed by many logicians. Recovering logicism is not easy and there may be several ways. One of them is the one followed by N.B. Cocchiarella whose systems (there are more than one) represent a form of logicism (Frege's or Russell's form). From those systems -though a bit changed from my own point of view- we can define a modal calcule that may have application in computer science, what would not be a stale work. From a common language we take in account two systems in order to show that Cocchiarella's modified system is as powerful deductively as that of Church modified functional second order calcule. We can obtain new systems that represent form of logicism and are more powerful than that of Church enlarging Cochiarella's modified system. These new systems, that becomes modal systems provided that one adds appropiate modal tools (then they may be used in computer science), may be useful to study logicism itself (as historical philosophy of logic and mathematics).

Identity in modal logic theorem proving

THINKER is an automated natural deduction first-order theorem proving program. This paper reports on how it was adapted so as to prove theorems in modal logic. The method employed is an “indirect semantic method”, obtained by considering the semantic conditions involved in being a valid argument in these modal logics. The method is extended from propositional modal logic to predicate modal logic, and issues concerning the domain of quantification and “existence in a world's domain” are discussed. Finally, we look at the very interesting issues involved with adding identity to the theorem prover in the realm of modal predicate logic. Various alternatives are discussed.

Verified functions for generating signed-binary arithmetic hardware

Formally verified metafunctions which synthesize array multipliers and inner product hardware of arbitrary size and structure are presented. The metafunctions operate on signed-binary inputs in general and two's-complement in particular, and are higher order. They are shown to be equivalence-preserving transformations and correctly produce multipliers and inner product hardware of arbitrary size and structure. All the metafunctions, their associated correctness theorems, and their correctness proofs are machine executable within the higher order logic (HOL) theorem prover. The function expressions produced by the metafunctions can be used as hardware synthesis descriptions or as comparison functions for Boolean-comparison-based systems. In addition to the definitions written in higher-order logic, the major definitions are written in a more informal functional programming language-like notation which should facilitate translation of the synthesis functions to other hardware description languages.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A simple proof of a completeness result for leads-to in the UNITY logic

Jutla, Knapp and Rao (in: Proc. 8th Ann. Symp. on Principles of Distributed Computing) proved a completeness result for leads-to in UNITY, using their definition of leads-to by means of a predicate-transformer; the proof relies on theorems of temporal logic and μ-calculus. Sanders ( Formal Aspects of Computing 3 (1991) 189–205) defined leads-to in a version of the UNITY logic that incorporates initial conditions, and derived the completeness of temporal logic. The present paper contains a simple direct proof of the same result.

A new algebraic semantic approach and some adequate connectives for computation with temporal logic over discrete time

ABSTRACT In this paper we present a new semantic approach for propositional linear temporal logic with discrete time, strongly based in the well-order of IN (the set of natural numbers). We consider temporal connectives which express precedence, posteriority and simultaneity, and they provide a family of expressively complete temporal logics. The selection of the new semantics and connectives used in this work was principally to obtain a suitable executable temporal logic, which can be used for the specification and control of process behaviour in discrete time in a similar way to thai presented by D. Gabbay in [GAB 89], Our new approach has two advantages: firstly, the connectives are defined intuitively so that they have interpretations which relate to properties of interest in real systems; secondly, it provides a new semantics that facilitates simpler proofs of many valid formulas and metatheorems. To confirm this second advantage, we use our semantics to give a formal proof of the Separation Theorem for one of these logics (specifically LN3). The great interest of this Separation Theorem for the executable temporal logic is emphasised by D. Gabbay in [GAB 89], [GAB 88].

Programming in temporal-nonmonotonic reasoning

Nonmonotonic reasoning has been proposed as an extension to classical first-order logic. Now people are interested in temporal reasoning with nonmonotonic logic [6]. We combine the monotonic logic [7] with a temporal logic to get a more general reasoning language. We discuss a monotonic logic TML which has predicate formulas, temporal formulas and a special modal formula, and give a completeness theorem of it. We use TH(Δ) to designate the set of theorems of a temporal-nonmonotonic theory Δ which has the same language with TML. The completeness theorem of the temporal-nonmonotonic logic naturally arises. Like the relationship between predicate logic with a practical logic programming language PROLOG, we propose a useful temporal-nonmonotonic reasoning language TN for the temporal-nonmonotonic logic. As an appendix we supply an algorithm for the programming language TN.

An almost general splitting theorem for modal logic

Given a normal (multi-)modal logic Θ a characterization is given of the finitely presentable algebras A whose logics L A split the lattice of normal extensions of Θ. This is a substantial generalization of Rautenberg [10] and [11] in which Θ is assumed to be weakly transitive and A to be finite. We also obtain as a direct consequence a result by Blok [2] that for all cycle-free and finite A L A splits the lattice of normal extensions of K. Although we firmly believe it to be true, we have not been able to prove that if a logic Λ splits the lattice of extensions of Θ then Λ is the logic of an algebra finitely presentable over Θ; in this respect our result remains partial.

The OTTER environment for resolution-based proof of hardware correctness

Efficient proof techniques are vital for the success of formal verification of hardware. First-Order Logic theorem provers satisfy this requirement. This paper deals with experiences in applying such a tool, OTTER, to boolean expression and combinational logic. Preliminary results are reported and future developments are described.

The deduction theorem for quantum logic—some negative results

AbstractWe prove that no logic (i.e. consequence operation) determined by any class of orthomodular lattices admits the deduction theorem (Theorem 2.7). We extend those results to some broader class of logics determined by ortholattices (Corollary 2.6).

A new proof of the fixed-point theorem of provability logic.

We give a simple semantic proof of the fixed-point theorem of the modal system G (also known as GL, PRL, L, and K4W). It is modeled after a syntactic proof of Sambin, yet is simpler than his, due to our taking advantage of the Kripke semantics for G. The advantages of this particular version are that it is les complicated and the fixed-point so obtained has the same general «appearance» as the original formula

A NORMAL FORM THEOREM FOR SECOND-ORDER CLASSICAL LOGIC WITH AN AXIOM OF CHOICE

A cut-elimination theorem for the second-order logic with an axiom of choice of type 0,1 or 1,1 is proved. In the first case the Pappinghaus scheme is applied; in the second the calculus with an epsilon-symbol for predicates is used. Bibliography: 5 titles.

Completeness for systems including real numbers

The usual completeness theorem for first-order logic is extended in order to allow for a natural incorporation of real analysis. Essentially, this is achieved by building in the set of real numbers into the structures for the language, and by adjusting other semantical notions accordingly. We use many-sorted languages so that the resulting formal systems are general enough for axiomatic treatments of empirical theories without recourse to elements of set theory which are difficult to interprete empirically. Thus we provide a way of applying model theory to empirical theories without “tricky” detours. Our frame is applied to axiomatizations of three empirical theories: classical mechanics, phenomenological thermodynamics, and exchange economics.

On the validity of hilbert's nullstellensatz, artin's theorem, and related results in grothendieck toposes

Using formally intuitionistic logic coupled with infinitary logic and the completeness theorem for coherent logic, we establish the validity, in Grothendieck toposes, of a number of well-known, classically valid theorems about fields and ordered fields. Classically, these theorems have proofs by contradiction and most involve higher order notions. Here, the theorems are each given a first-order formulation, and this form of the theorem is then deduced using coherent or formally intuitionistic logic. This immediately implies their validity in arbitrary Grothendieck toposes. The main idea throughout is to use coherent theories and, whenever possible, find coherent formulations of formulas which then allow us to call upon the completeness theorem of coherent logic. In one place, the positive model-completeness of the relevant theory is used to find the necessary coherent formulas.The theorems here deal with polynomials or rational functions (in s indeterminates) over fields. A polynomial over a field can, of course, be represented by a finite string of field elements, and a rational function can be represented by a pair of strings of field elements. We chose the approach whereby results on polynomial rings are reduced to results about the base field, because the theory of polynomial rings in s indeterminates over fields, although coherent, is less desirable from a model-theoretic point of view. Ultimately we are interested in the models.This research was originally motivated by the works of Saracino and Weispfenning [SW], van den Dries [Dr], and Bunge [Bu], each of whom generalized some theorems from algebraic geometry or ordered fields to (commutative, von Neumann) regular rings (with unity).

On the validity of hilbert's nullstellensatz, artin's theorem, and related results in grothendieck toposes

Using formally intuitionistic logic coupled with infinitary logic and the completeness theorem for coherent logic, we establish the validity, in Grothendieck toposes, of a number of well-known, classically valid theorems about fields and ordered fields. Classically, these theorems have proofs by contradiction and most involve higher order notions. Here, the theorems are each given a first-order formulation, and this form of the theorem is then deduced using coherent or formally intuitionistic logic. This immediately implies their validity in arbitrary Grothendieck toposes. The main idea throughout is to use coherent theories and, whenever possible, find coherent formulations of formulas which then allow us to call upon the completeness theorem of coherent logic. In one place, the positive model-completeness of the relevant theory is used to find the necessary coherent formulas.The theorems here deal with polynomials or rational functions (in s indeterminates) over fields. A polynomial over a field can, of course, be represented by a finite string of field elements, and a rational function can be represented by a pair of strings of field elements. We chose the approach whereby results on polynomial rings are reduced to results about the base field, because the theory of polynomial rings in s indeterminates over fields, although coherent, is less desirable from a model-theoretic point of view. Ultimately we are interested in the models.This research was originally motivated by the works of Saracino and Weispfenning [SW], van den Dries [Dr], and Bunge [Bu], each of whom generalized some theorems from algebraic geometry or ordered fields to (commutative, von Neumann) regular rings (with unity).

Completeness Theorems for Universal and Implicational Logics of Algebras Via Congruences

In this paper, simple algebraic proofs are given for the completeness theorems for the implicational and universal logics of algebras. The proofs are obtained by examining congruences, $\theta$, on the algebra of terms, $F(\omega )$, such that $F(\omega )/\theta$ belongs to the given class of algebras. Thus, they are direct analogs of G. Birkhoff’s proof of the completeness theorem for equational logic.