Semantics For First-order Logic Research Articles

Game semantics for first-order logic

We refine HO/N game semantics with an additional notion of pointer (mu-pointers) and extend it to first-order classical logic with completeness results. We use a Church style extension of Parigot's lambda-mu-calculus to represent proofs of first-order classical logic. We present some relations with Krivine's classical realizability and applications to type isomorphisms.

Presenting functors on many-sorted varieties and applications

This paper studies several applications of the notion of a presentation of a functor by operations and equations. We show that the technically straightforward generalisation of this notion from the one-sorted to the many-sorted case has several interesting consequences. First, it can be applied to give equational logic for the binding algebras modelling abstract syntax. Second, it provides a categorical approach to algebraic semantics of first-order logic. Third, this notion links the uniform treatment of logics for coalgebras of an arbitrary type T with concrete syntax and proof systems. Analysing the many-sorted case is essential for modular completeness proofs of coalgebraic logics.

Harmonious logic: Craig’s interpolation theorem and its descendants

Though deceptively simple and plausible on the face of it, Craig’s interpolation theorem (published 50 years ago) has proved to be a central logical property that has been used to reveal a deep harmony between the syntax and semantics of first order logic. Craig’s theorem was generalized soon after by Lyndon, with application to the characterization of first order properties preserved under homomorphism. After retracing the early history, this article is mainly devoted to a survey of subsequent generalizations and applications, especially of many-sorted interpolation theorems. Attention is also paid tomethodological considerations, since the Craig theorem and its generalizations were initially obtained by proof-theoretic arguments while most of the applications are model-theoretic in nature. The article concludes with the role of the interpolation property in the quest for “reasonable” logics extending first-order logic within the framework of abstract model theory.

Understanding SPKI/SDSI using first-order logic

SPKI/SDSI is a language for expressing distributed access control policy, derived from SPKI and SDSI. We provide a first-order logic (FOL) semantics for SDSI, and show that it has several advantages over previous semantics. For example, the FOL semantics is easily extended to additional policy concepts and gives meaning to a larger class of access control and other policy analysis queries. We prove that the FOL semantics is equivalent to the string rewriting semantics used by SDSI designers, for all queries associated with the rewriting semantics. We also provide a FOL semantics for SPKI/SDSI and use it to analyze the design of SPKI/SDSI. This reveals some problems. For example, the standard proof procedure in RFC 2693 is semantically incomplete. In addition, as noted before by other authors, authorization tags in SPKI/SDSI are algorithmically problematic, making a complete proof procedure unlikely. We compare SPKI/SDSI with RT 1 C, which is a language in the RTRole-based Trust-management framework that can be viewed as an extension of SDSI. The constraint feature of RT 1 C, based on Constraint Datalog, provides an alternative mechanism that is expressively similar to SPKI/SDSI tags, semantically natural, and algorithmically tractable.

Axiomatization of a Denotational Semantics for First-order Logic

Axiomatization of a Denotational Semantics for First-order Logic

Provability with Finitely Many Variables

AbstractFor every finite n ≥ 4 there is a logically valid sentence φn with the following properties: φn contains only 3 variables (each of which occurs many times); φn contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol); φn has a proof in first-order logic with equality that contains exactly n variables, but no proof containing only n − 1 variables. This result was first proved using the machinery of algebraic logic developed in several research monographs and papers. Here we replicate the result and its proof entirely within the realm of (elementary) first-order binary predicate logic with equality. We need the usual syntax, axioms, and rules of inference to show that φn has a proof with only n variables. To show that φn has no proof with only n − 1 variables we use alternative semantics in place of the usual, standard, set-theoretical semantics of first-order logic.

Provability with Finitely Many Variables

For every finite n ≥ 4 there is a logically valid sentence φ n with the following properties: φ n contains only 3 variables (each of which occurs many times); φ n contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol); φ n has a proof in first-order logic with equality that contains exactly n variables, but no proof containing only n − 1 variables. This result was first proved using the machinery of algebraic logic developed in several research monographs and papers. Here we replicate the result and its proof entirely within the realm of (elementary) first-order binary predicate logic with equality. We need the usual syntax, axioms, and rules of inference to show that φ n has a proof with only n variables. To show that φ n has no proof with only n − 1 variables we use alternative semantics in place of the usual, standard, set-theoretical semantics of first-order logic.

Xlinkit

xlinkit is a lightweight application service that provides rule-based link generation and checks the consistency of distributed Web content. It leverages standard Internet technologies, notably XML, XPath, and XLink. xlinkit can be used as part of a consistency management scheme or in applications that require smart link generation, including portal construction and management of large document repositories. In this article we show how consistency constraints can be expressed and checked. We describe a novel semantics for first-order logic that produces links instead of truth values and give an account of our content management strategy. We present the architecture of our service and the results of two substantial case studies that use xlinkit for checking course syllabus information and for validating UML models supplied by industrial partners.

Model-Checking View-Based Partial Specifications

We develop foundations for the view-based specification of software artifacts in first-order logic. Standard notions of models and semantics of first-order logic are generalized to partial models that do not have access to the entire global signature. At the model-theoretic level, this is achieved via the Smyth powerdomain over the semantic universe of elements. At the logical level, we accomplish this via the Smyth powerdomain over the standard two-valued booleans. A refinement notion is developed and its soundness is proved for a fixed set of semantic elements. Standard models and semantics of first-order logic are subsumed by our framework as models in which all signature information is present (“complete models”). We decompose our semantics into a consensus (the union of the Smyth powerdomain) of an optimistic and a pessimistic semantics. That way, we may compute this semantics as a standard model check in first-order logic over a model lifted by the Smyth powerdomain.

Safety, domain independence and generalized quantification

We define QLGQ, a query language with generalized quantifiers, which uses only conjuction to combine formulas, the rest of its expressive power coming from generalized quantification. However, the language is not safe. A safe (and domain independent) sublanguage ( SQLGQ) is defined; the definition uses properties of generalized quantifiers as well as (minimal) syntax limitations on formulas, and is motivated on semantical grounds, unlike most past approaches which are purely syntactic. We prove that SQLGQ subsumes RC/ S *, one of the largest classes of safe languages proposed in the literature. Given these results, we propose that the most useful type of quantification for query languages is relativized quantification, in which the scope of the quantifier is made explicit as part of the query, and give this idea a technical form using generalized quantifiers. Finally, we extend work by Hull and Su on alternative semantics for first-order logic (FOL) to this new framework and propose some open questions for further research.

Compositional semantics for a language of imperfect information

We describe a logic which is the same as first-order logic except that it allows control over the information that passes down from formulas to subformulas. For example the logic is adequate to express branching quantifiers. We describe a compositional semantics for this logic; in particular this gives a compositional meaning to formulas of the 'information-friendly' language of Hintikka and Sandu. For first-order formulas the semantics reduces to Tarski's semantics for first-order logic. We prove that two formulas have the same interpretation in all structures if and only if replacing an occurrence of one by an occurrence of the other in a sentence never alters the truth-value of the sentence in any structure. Keywords: game semantics, compositional, imperfect information, independence-friendly, branching quantifier, generalised quantifier, trump

A logic for category theory

MacLane and Feferman have argued that the traditional set theories of Zermelo—Fraenkel and Gödel—Bernays are not suitable foundations for category theory because of the requirement for self-referencing abstractions. The necessity for distinguishing between small and large categories reflects this unsuitability. The purpose of this paper is to demonstrate that a formalization within a natural-deduction-based logic and set theory called NaDSet avoids the difficulties that arise with the use of the traditional set theories. Definitions are provided within NaDSet for most of the fundamental concepts and constructs in category theory. The main result of the paper is a proof that the set of all functors forms a category under appropriate definitions of composition and equivalence. Additional definitions and discussions on products, comma categories, universals, limits and adjoints are presented. They provide evidence to support the claim that any construct, not only in categories, but also in toposes, sheaves, triples and similar theories can be formalized within NaDSet. NaDSet succeeds as a logic and set theory for category theory because the resolution of the paradoxes provided for it is based on an extension of Tarski's reductionist semantics for first-order logic. Self-membership and self-reference is not explicitly excluded. The reductionist semantics is most simply presented as a natural-deduction logic. A sketch of the elementary and logical syntax or proof theory of the logic is provided. A consistency proof for NaDSet is provided elsewhere.

Robert Bull and Krister Segerberg. Basic modal logic. Handbook of philosophical logic, Volume II, Extensions of classical logic, edited by D. Gabbay and F. Guenthner, Synthese library, vol. 165, D. Reidel Publishing Company, Dordrecht, Boston, and Lancaster, 1984, pp. 1–88. - John P. Burgess. Basic tense logic. Handbook of philosophical logic, Volume II, Extensions of classical logic, edited by D.

This handbook provides a systematic survey of the central areas of philosophical logic. It is divided into four volumes, each devoted to a major sub-field within the disciplines. The articles should be of value to general philosophers, linguists, logicians, mathematicians and computer scientists. Volume 1, Elements of Classical Logic, deals with the background to what has come to be considered the standard formulation of predicate logic - both as far as its semantics and proof theory are concerned. The central chapter on predicate logic is followed by chapters outlining various alternative, but essentially equivalent ways of constructing the semantics for first-order logic as well as its proof theory. In addition, this volume contains a discussion of higher-order extensions of first-order logic and a compendium of the algorithmic and decision-theoretic prerequisites in the study of logical systems. Volume 2, Extensions of Classical Logic, surveys the most significant extensions of predicate logic and their applications to various philosophical fields of inquiry. The 12 chapters in this volume together provide a succinct introduction to a variety of intensional frameworks, a discussion of the most well-known logical systems, as well as an overview of major applications and of the open problems in the respective fields. Volume 3, Alternatives to Classical Logic, consists of a series of surveys of some of the alternatives to the basic assumptions of classical logic. These include many-valued logic, partial logic, free logic, relevance and entailment logics, dialogue logic, quantum logic and intuitionism. Volume 4, Topics in the Philosophy of Language, presents a panorama of the applications of logical tools and methods in the formal analysis of natural language. Since a number of developments in philosophical logic were originally stimulated by concern arising in the semantic analysis of natural language discourse, the chapters in this volume provide some criteria of evaluation of the applications of work in philosophical logic. In revealing both the adequacies and inadequacies of logical investigations in the semantic structures of natural discourse, these chapters also point the way to future developments in philosophical logic in general and thus close again the circle of inquiry relating logic and language.

Nonfinite axiomatizability results for cylindric and relation algebras

AbstractThe set of equations which use only one variable and hold in all representable relation algebras cannot be derived from any finite set of equations true in all representable relation algebras. Similar results hold for cylindric algebras and for logic with finitely many variables. The main tools are a construction of nonrepresentable one-generated relation algebras, a method for obtaining cylindric algebras from relation algebras, and the use of relation algebras in defining algebraic semantics for first-order logic.

Stone duality for first order logic

Publisher Summary This chapter presents a general theory on the relationship of syntax and semantics of first-order logic. The theory has a close-formal relationship to the Stone-duality theory for Boolean algebras. It subsumes the Godel completeness theorem that occupies a place in it and is analogous to that of the Stone representation inside Stone duality. The theory makes an essential use of ultraproducts. The theory is formulated in the language of category theory. Categories appear in three ways: (1) (first order) theories themselves are made into categories (pretoposes), (2) the collection of models of a fixed theory is made into a category and endowed with an additional structure derived from ultraproducts, resulting in the “ultracategory” of models, and (3) pretoposes on the one hand, and ultracategories on the other, are organized into categories (actually: two categories), and the main result in its final form is stated, in terms of a comparison between these two categories.

A new semantics for first-order logic, multivalent and mostly intensional

Cited by many as distinctive of first-order logic are the bivalence of its statements and the extensionality of its operators, among them the three operators '~', '&', and 'V'. It is to that bivalence and that extensionality, we are told, that the logical entailments and, hence, logical truths peculiar to the logic are due. But first-order logic is a far more diverse thing than its exponents usually allow. As proof I submit here a new semantics for it, one in which statements are susceptible of up to 2 t~~ values, the operators '&' and 'V' are intensional, and yet the logical entailments and logical truths that first-order logic acknowledges are all preserved. Credit should go to Karl Popper, from whom I largely borrowed Constraints D1-D6 on p. 57; to Kent Bendall, from whom I borrowed Constraint D7 on that page; and to Nicholas Rescher, who pointed out in 1963 that '&' - as interpreted here - is intensional in the presence of '~'.

A ?definitive? probabilistic semantics for first-order logic

A ?definitive? probabilistic semantics for first-order logic

Urn models: A classical exposition

Urn models were developed by Veikko Rantala to provide a non-standard semantics for first-order logic in which the domains, over which the quantifiers range, are allowed to vary. Rantala uses game-theoretical semantics in his presentation, and the present paper is a study of urn models from a more classical, truth-conditional point of view. An axiomatic system for urn logic is set out and completeness is proved by the method of maximal consistent sets.

Belief-theoretic formal semantics for first-order logic and probability

Belief-theoretic formal semantics for first-order logic and probability