Classical Predicate Logic Research Articles

Craig's theorem in superintuitionistic logics and amalgamable varieties of pseudo-boolean algebras

In recent years there have appeared many studies of an interesting and important property of logical theories, the so-called Craig interpolation theorem. Craig proved the interpolation theorem for classical predicate logic in 1957 [9]. SchHtte [23] proved the interpolation theorem for intuitionistic predicate logic, and Gabbay [12] for certain extensions of this logic. In [I0, 13] the interpolation theorem was proved for a number of modal logics, and in [21] for many-valued predicate calculi.

LAR: A logic of algorithmic reasoning

Reasoning about programs involves some logical framework which seems to go beyond classical predicate logic. LAR is an extension of predicate logic by additional concepts which are to formalize our "natural" reasoning about algorithms. Semantically, this extension introduces an underlying "time scale" on which formulas are considered and "time shifting" connectives. Besides a full model-theoretic treatment, a consistent and complete formal system for LAR is given. The pure logical system can serve as a basis for various theories. As an example, a theory of "while program schemes" is developed which contains Hoare's correctness proof system.

Noncompactness in propositional modal logic

We have come to believe that propositional modal logic (with the usual relational semantics) must be understood as a rather strong fragment of classical second-order predicate logic. (The interpretation of propositional modal logic in second-order predicate logic is well known; see e.g. [2, §1].) “Strong” refers of course to the expressive power of the languages, not to the deductive power of formal systems. By “rather strong” we mean sufficiently strong that theorems about first-order logic which fail for second-order logic usually fail even for propositional modal logic. Some evidence for this belief is contained in [2] and [3]. In the former is exhibited a finitely axiomatized consistent tense logic having no relational models, and the latter presents a finitely axiomatized modal logic between T and S4, such that □p → □2p is valid in all relational models of the logic but is not a thesis of the logic. The result of [2] is strong evidence that bimodal logic is essentially second-order, but that of [3] does not eliminate the possibility that unimodal logic only appears to be incomplete because we have not adopted sufficiently powerful rules of inference. In the present paper we present stronger evidence of the essentially second-order nature of unimodal logic.

Toshio Umezawa. On logics intermediate between intuitionistic and classical predicate logic. The journal of symbolic logic, vol. 24 no. 2 (for 1959, pub. 1960), pp. 141–153.

Toshio Umezawa. On logics intermediate between intuitionistic and classical predicate logic. The journal of symbolic logic, vol. 24 no. 2 (for 1959, pub. 1960), pp. 141–153.

A Study on Formal Deductions in the Primitive Logic

Main purpose of the present paper is to study formal deductions described along the line of my former work [1]. In the present paper, I restrict myself to the primitive logic. To extend this method to other logics such as the lower classical predicate logic or the intuitionistic predicaste logic, my way of practical discription has to undergo a certain extent of modification.

Taboo Versus Axiom

Some important formal systems are really developable from a finite number of axioms in the lower classical predicate logic LK or in the intuitionistic predicate logic LJ. Any system of this kind can be developed in LK (or in LJ) from the single conjunction of all the axioms of the system.

A Formalism for Primitive Logic and Mechanical Proof-Checking

The universal character of the primitive logic LO in the sense that popular logics such as the lower classical predicate logic LK, the intuitionistic predicate logic LJ, Johansson’s minimal predicate logic LM, etc. can be faithfully interpreted in LO is very remarkable even from the view point of mechanical proof-checking. Since LO is very simple, deductions in LO could be mechanized in a simple form if a suitable formalism for LO is found out. Main purpose of this paper is to introduce a practical formalism for LO, practical in the sense that it is suitable at least for mechanical proof-checking business.

On Universal Character of the Primitive Logic

The PRIMITIVE LOGIC LO introduced in my former work is a logic having only two logical constants IMPLICATION → and UNIVERSAL QUANTIFICATION ( ) with their usual inference rules which are admitted even in the INTUITIONISTIC PREDICATE LOGIC LJ. LO is really a very simple logic, maybe the simplest possible logic as one can imagine, but it is very important because of its universal character. In fact, popular logics such as the LOWER CLASSICAL PREDICATE LOGIC LK, the INTUITIONISTIC PREDICATE LOGIC LJ, the MINIMAL PREDICATE .LOGIC LM, etc. can be faithfully interpreted in it. Speaking frankly, I am further expecting that all the important logics would be interpreted faithfully in it and would disclose their intrinsic characteristics by being interpreted in it. Main purpose of this paper is to show the universal character of the primitive logic LO by pointing out that a series of typical logics are faithfully interpretable in LO.

On logics intermediate between intuitionistic and classical predicate logic

In [1] and [2] I investigated logics intermediate between intuitionistic and classical propositional logic. In the present paper I shall study inclusion and non-inclusion between certain intermediate predicate logics. All the logics considered result from intuitionistic predicate logic by addition of classically valid axiom schemes.