Showing papers in "Studia Logica in 1987"
TL;DR: This paper is able to prove a strengthen form of the firstorder interpolation theorem as well as provide a correct description of Skolem functions and the Herbrand Universe.
Abstract: A structure which generalizes formulas by including substitution terms is used to represent proofs in classical logic. These structures, called expansion trees, can be most easily understood as describing a tautologous substitution instance of a theorem. They also provide a computationally useful representation of classical proofs as first-class values. As values they are compact and can easily be manipulated and transformed. For example, we present an explicit transformations between expansion tree proofs and cut-free sequential proofs. A theorem prover which represents proofs using expansion trees can use this transformation to present its proofs in more human-readable form. Also a very simple computation on expansion trees can transform them into Craig-style linear reasoning and into interpolants when they exist. We have chosen a sublogic of the Simple Theory of Types for our classical logic because it elegantly represents substitutions at all finite types through the use of the typed λ-calculus. Since all the proof-theoretic results we shall study depend heavily on properties of substitutions, using this logic has allowed us to strengthen and extend prior results: we are able to prove a strengthen form of the firstorder interpolation theorem as well as provide a correct description of Skolem functions and the Herbrand Universe. The latter are not straightforward generalization of their first-order definitions.
132 citations
TL;DR: The extensions of Gödel's completeness theorems are proved which confirm that the first-order fuzzy logic is also semantically complete.
Abstract: This paper is an attempt to develop the many-valued first-order fuzzy logic. The set of its truth, values is supposed to be either a finite chain or the interval 〈0, 1〉 of reals. These are special cases of a residuated lattice 〈L, ∨, ∧, ⊗, →, 1, 0〉. It has been previously proved that the fuzzy propositional logic based on the same sets of truth values is semantically complete. In this paper the syntax and semantics of the first-order fuzzy logic is developed. Except for the basic connectives and quantifiers, its language may contain also additional n-ary connectives and quantifiers. Many propositions analogous to those in the classical logic are proved. The notion of the fuzzy theory in the first-order fuzzy logic is introduced and its canonical model is constructed. Finally, the extensions of Godel's completeness theorems are proved which confirm that the first-order fuzzy logic is also semantically complete.
86 citations
TL;DR: A class of normal modal calculi PFD, whose syntax is endowed with operators Mr (and their dual ones, Lr), one for each r ε [0,1]: if a is α sentence, Mrα is to he evaluated as true or false in every world of a F-restricted probabilistic kripkean model.
Abstract: We present a class of normal modal calculi PFD, whose syntax is endowed with operators Mr (and their dual ones, Lr), one for each r e [0,1]: if a is α sentence, Mrα is to he read “the probability that a is true is strictly greater than r” and to he evaluated as true or false in every world of a F-restricted probabilistic kripkean model. Every such a model is a kripkean model, enriched by a family of regular (see below) probability evaluations with range in a fixed finite subset F of [0,1]: there is one such a function for every world w, PF(w,-), and this allows to evaluate Mra as true in the world w iff pF(w, α) 〉 r.
40 citations
TL;DR: The purpose of this note is to formulate some weaker versions of the so called Ramsey test that do not entail the following unacceptable consequence: if A and C are already accepted in K, then “if A, then C” is also accepted inK.
Abstract: The purpose of this note is to formulate some weaker versions of the so called Ramsey test that do not entail the following unacceptable consequence If A and C are already accepted in K, then “if A, then C” is also accepted in K and to show that these versions still lead to the same triviality result when combined with a preservation criterion
30 citations
TL;DR: The main purpose of this paper is to show that both OQL and PCL cannot satisfy the Lindenbaum property.
Abstract: This paper will take into account the Lindenbaum property in Orthomodular Quantum Logic (OQL) and Partial Classical Logic (PCL) The Lindenbaum property has an interest both from a logical and a physical point of view since it has to do with the problem of the completeness of quantum theory and with the possibility of extending any semantically non-contradictory set of formulas to a semantically non-contradictory complete set of formulas The main purpose of this paper is to show that both OQL and PCL cannot satisfy the Lindenbaum property
29 citations
TL;DR: It follows that all countably-categorical elementary theories of Boolean algebras with distinguished ideals are finite-axiomatizable, decidable and, consequently, their countable models are strongly constructivizable.
Abstract: In the paper all countable Boolean algebras with m distinguished. ideals having countably-categorical elementary theory are described and constructed. From the obtained characterization it follows that all countably-categorical elementary theories of Boolean algebras with distinguished ideals are finite-axiomatizable, decidable and, consequently, their countable models are strongly constructivizable.
22 citations
TL;DR: The Joint Non-Trivialization Theorem, two Definability Theorems and the generalized Quantifier Elimination Theorem are proved for J3-theories, which are three-valued with more than one distinguished truth-value, reflect certain aspects of model type logics and can be paraconsistent.
Abstract: The Joint Non-Trivialization Theorem, two Definability Theorems and the generalized Quantifier Elimination Theorem are proved for J3-theories. These theories are three-valued with more than one distinguished truth-value, reflect certain aspects of model type logics and can. be paraconsistent. J3-theories were introduced in the author's doctoral dissertation.
21 citations
TL;DR: The modal completeness proofs of Guaspari and Solovay (1979) for their systems R and R− are improved and the relationship between R andR− is clarified.
Abstract: The modal completeness proofs of Guaspari and Solovay (1979) for their systems R and R− are improved and the relationship between R and R− is clarified.
19 citations
TL;DR: In this article, it was shown that the class of all isomorphic images of Boolean Products of members of SR is the same as that of all archimedean W-algebras.
Abstract: We show that the class of all isomorphic images of Boolean Products of members of SR [1] is the class of all archimedean W-algebras. We obtain this result from the characterization of W-algebras which are isomorphic images of Boolean Products of CW-algebras.
19 citations
TL;DR: Those formulas which are valid in every Kripke model having constant domain whose base is the ordered set R of real numbers (or, theordered set Q of rational numbers) are characterized syntactically.
Abstract: Those formulas which are valid in every Kripke model having constant domain whose base is the ordered set R of real numbers (or, the ordered set Q of rational numbers) are characterized syntactically.
15 citations
TL;DR: It is proved that for a syntactical function F there is a semantical function Г correlated with F iff F preserves propositional connectives up to logical equivalence.
Abstract: We say that a semantical function Г is correlated with a syntactical function F iff for any structure A and any sentence ϕ we have A ⊧ Fϕ ↔ ΓA ⊧ ϕ.
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TL;DR: It is shown that the class of all semi-Post algebras is closed under these products and that every semi- Post algebra is a semi- post product of some generalized Post algeBRas.
Abstract: In this paper, semi-Post algebras are introduced and investigated. The generalized Post algebras are subcases of semi-Post algebras. The so called primitive Post constants constitute an arbitrary partially ordered set, not necessarily connected as in the case of the generalized Post algebras examined in [3]. By this generalization, semi-Post products can be defined. It is also shown that the class of all semi-Post algebras is closed under these products and that every semi-Post algebra is a semi-Post product of some generalized Post algebras.
TL;DR: A Hilbert's style axiomatization is proved complete for this logic, as well as for countable sublogics and subtheories, and it is shown that the logic has the interpolation property.
Abstract: A logic with normal modal operators and countable infinite conjunctions and disjunctions is introduced. A Hilbert's style axiomatization is proved complete for this logic, as well as for countable sublogics and subtheories. It is also shown that the logic has the interpolation property.
TL;DR: A system LR of modal propositional logic is axiomatized, a modal completeness theorem is established for LR and, after that, a uniform arithmetical (Solovay-type) completeness criterion with respect to PA is obtained for LR.
Abstract: To the standard propositional modal system of provability logic constants are added to account for the arithmetical fixed points introduced by Bernardi-Montagna in [5]. With that interpretation in mind, a system LR of modal propositional logic is axiomatized, a modal completeness theorem is established for LR and, after that, a uniform arithmetical (Solovay-type) completeness theorem with respect to PA is obtained for LR.
TL;DR: Characterization of semi-Post subalgebras and homomorphisms, relationships between subalgee and homology, and of generalized Post algebra relationships are examined.
Abstract: Semi-Post algebras have been introduced and investigated in [6]. This paper is devoted to semi-Post subalgebras and homomorphisms. Characterization of semi-Post subalgebras and homomorphisms, relationships between subalgebras and homomorphisms of semi-Post algebras and of generalized Post algebras are examined.
TL;DR: It is proved that there exist infinitely indeterminate L-subsets with no “more precise” decidable versions and classical subsets whose unique shaded decidable version are the L- Subsets almost-everywhere indeterminates.
Abstract: If X is set and L a lattice, then an L-subset or fuzzy subset of X is any map from X to L, [11] In this paper we extend some notions of recursivity theory to fuzzy set theory, in particular we define and examine the concept of almost decidability for L-subsets Moreover, we examine the relationship between imprecision and decidability Namely, we prove that there exist infinitely indeterminate L-subsets with no “more precise” decidable versions and classical subsets whose unique shaded decidable versions are the L-subsets almost-everywhere indeterminate
TL;DR: Here it is proved that Cut Theorems for these systems are proved, and then it is shown that modus ponens is admissible — which is not so trivial as one normally expects.
Abstract: This paper is a study of four subscripted Gentzen systems GuR+, GuT+, GuRW+ and GuTW+. [16] shows that the first three are equivalent to the semilattice relevant logics uR+, uT+ and uRW+ and conjectures that GuTW+ is, equivalent to uTW+. Here we prove Cut Theorems for these systems, and then show that modus ponens is admissible — which is not so trivial as one normally expects. Finally, we give decision procedures for the contractionless systems, GuTW+ and GuRW+.
TL;DR: The elementary ontology with the axiom ∃ S (S ɛ S) is strictly embeddable into monadic second-order calculus of predicates which provides a formalization of the classes of all formulas valid in all non-empty domains.
Abstract: There is given the proof of strict embedding of Leśniewski's elementary ontology into monadic second-order calculus of predicates providing a formalization of the class of all formulas valid in all domains (including the empty one). The elementary ontology with the axiom ∃ S (S ɛ S) is strictly embeddable into monadic second-order calculus of predicates which provides a formalization of the classes of all formulas valid in all non-empty domains.
TL;DR: In this article the first-order language of [2] is extended by the addition of the operator ‘the story ... says that ...’, as in ‘The story Flashman among the Redskins says that Flashman met Sitting Bull’.
Abstract: In [2] a semantics for implication is offered that makes use of ‘stories’ — sets of sentences assembled under various constraints. Sentences are evaluated at an ‘actual’ world and in each member of a set of stories. A sentence B is true in a story s just when B e s. A implies B iff for all stories and the actual world, whenever A is true, B is true. In this article the first-order language of [2] is extended by the addition of the operator ‘the story ... says that ...’, as in ‘The story Flashman among the Redskins says that Flashman met Sitting Bull’. The resulting language is shown to be sound and complete.
TL;DR: It is shown that the mapping ϕ which was introduced by V. A. Smirnov is an embedding of EOA into LS and an embeddings of LS into EOA are given.
Abstract: Let EOA be the elementary ontology augmented by an additional axiom ∃S (S ɛ S), and let LS be the monadic second-order predicate logic. We show that the mapping ϕ which was introduced by V. A. Smirnov is an embedding of EOA into LS. We also give an embedding of LS into EOA.
TL;DR: The intuitionistic implication with full strength is definable in the second order versions of these systems WLJ and SI of the intuitionistic propositional logic LJ because they are justifiable by purely constructive semantics.
Abstract: We introduce subsystems WLJ and SI of the intuitionistic propositional logic LJ, by weakening the intuitionistic implication. These systems are justifiable by purely constructive semantics. Then the intuitionistic implication with full strength is definable in the second order versions of these systems. We give a relationship between SI and a weak modal system WM. In Appendix the Kripke-type model theory for WM is given.
TL;DR: The aim of the paper is to formalize I. Bellert's (McGill) proposal to characterize distinct nonequivalent readings of quantificationally ambiguous sentences with help of two features: absoluteness and distributiveness.
Abstract: The aim of the paper is to formalize I. Bellert's (McGill) proposal to characterize distinct nonequivalent readings of quantificationally ambiguous sentences with help of two features: absoluteness and distributiveness. The formalisation makes use of set theoretical and model theoretical standard notions. Foundamental rules, proposed by Bellert, govering the interpretation of cooccurring quantifiers are quoted and outlines of proofs of important derived rules are given.
TL;DR: It is proved, among other things, that there are eight interpretations from the variety of Monadic algebras into itself.
Abstract: In [3], O. C. Garcia and W. Taylor make an in depth study of the lattice of interpretability types of varieties first introduced by W. Neumann [5]. In this lattice several varieties are identified so in order to distinguish them and understand the fine structure of the lattice, we propose the study of the interpretations between them, in particular, how many there are and what these are. We prove, among other things, that there are eight interpretations from the variety of Monadic algebras into itself.
TL;DR: It is shown that the implicational fragment of Anderson and Belnap's R, i.e. Church's weakimplicational calculus, is not uniquely characterized by MP (modus ponens), US (uniform substitution), and WDT (Church's weak deduction theorem), but that no unique logic is characterized by these.
Abstract: It is shown that the implicational fragment of Anderson and Belnap's R, i.e. Church's weak implicational calculus, is not uniquely characterized by MP (modus ponens), US (uniform substitution), and WDT (Church's weak deduction theorem). It is also shown that no unique logic is characterized by these, but that the addition of further rules results in the implicational fragment of R. A similar result for E is mentioned.
TL;DR: The aim of this paper is to give a detailed description of the logics C1 and C2 in a lattice-theoretical language and to show that the logic CJ is 2-uniform.
Abstract: The present paper is to be considered as a sequel to [1], [2]. It is known that Johansson's minimal logic is not uniform, i.e. there is no single matrix which determines this logic. Moreover, the logic CJ is 2-uniform. It means that there are two uniform logics C1, C2 (each of them is determined by a single matrix) such that the infimum of C1 and C2 is CJ. The aim of this paper is to give a detailed description of the logics C1 and C2. It is performed in a lattice-theoretical language.
TL;DR: In the paper A. I. Malcev's problem on the characterization of axioms for classes with strong homomorphisms is being solved, it is shown that the axiomatic consistency of these classes is improved.
Abstract: In the paper A. I. Malcev's problem on the characterization of axioms for classes with strong homomorphisms is being solved.
TL;DR: Using the semantic embedding technique the theorem announced by the title is proved: the theorem announcing the theorem about semantic embeddings is proved.
Abstract: Using the semantic embedding technique the theorem announced by the title is proved.
TL;DR: It is proved that Gergonne's syllogistic is isomorphic to closed elements algebra of a proper approximation relation algebra and this isomorphism permits to evaluate Aristotle’s syllogisms and also the laws of conversion and relations in the “square of oppositions” by means of regular computations with Boolean matrices.
Abstract: A connection between Aristotle's syllogistic and the calculus of relations is investigated. Aristotle's and Gergonne's syllogistics are considered as some algebraic structures. It is proved that Gergonne's syllogistic is isomorphic to closed elements algebra of a proper approximation relation algebra. This isomorphism permits to evaluate Gergonne's syllogisms and also Aristotle's syllogisms, laws of conversion and relations in the “square of oppositions” by means of regular computations with Boolean matrices.
TL;DR: This paper defines the relation ≺t of elementary extension of topological models in the language Ltand shows a Back and Forth criterion for ≹t and introduces some new operations on partial homeomorphisms preserving back and Forth properties.
Abstract: In this paper we define the relation ≺t of elementary extension of topological models in the language Ltand show a Back and Forth criterion for ≺t. We introduce some new operations on partial homeomorphisms preserving Back and Forth properties. Some properties of ≺t are proved by the Back and Forth technique.