Intermediate Logics Research Articles

Some Results on the Intermediate Logics

In ClOj, we developed the method of Kripke models and gave some applications of it to the study of the intermediate logics. We found that the use of Kripke models is very efficient, since in many cases the algebraic structure of Kripke models reflects well the properties of the logics characterized by them. In dll]], we proved that a certain relation holds between the logics characterized by some Kripke models and the logics having the finite model property. As we stated in the correction at the end of CllH, the original proof contained an error. So, we emphasize here that the following problem remains open: Has any intermediate logic a characteristic Kripke model? In this paper, we will proceed in the same direction as Hi OH anc* Ell]. At present, we have at hand many particular intermediate logics. But we have very little knowledge about the general properties common to many logics. For instance, though many logics having the disjunction property have been known, we don't know what conditions make a logic have the disjunction property. We think that the central aim of the study of intermedate logics is to construct the theory about the general properties of them. The notion of the slice introduced by Hosoi Q4] gave us the first clue to our purpose. We will introduce in §1 other classifications of intermediate logics. In §2, we will characterize them by Kripke models, just as the slice was characterized by the height of Kripke models in [110]. In §3, we will investigate about the disjunction property in connection with these classifications. We assume familiarity with the terminologies and the notions of ^10]. In ^10], we consider a

Axiomatization of Models for Intermediate Logics Constructed with Boolean Models by Piling Up

Axiomatization of Models for Intermediate Logics Constructed with Boolean Models by Piling Up

Applications of trees to intermediate logics

We investigate extensions of Heyting's predicate calculus (HPC). We relate geometric properties of the trees of Kripke models (see [2]) with schemas of HPC and thus obtain completeness theorems for several intermediate logics defined by schemas. Our main results are:(a) ∼(∀x ∼ ∼ϕ(x) Λ ∼∀xϕ(x)) is characterized by all Kripke models with trees T with the property that every point is below an endpoint. (From this we shall deduce Glivenko type theorems for this extension.)(b) The fragment of HPC without ∨ and ∃ is complete for all Kripke models with constant domains.We assume familiarity with Kripke [2]. Our notation is different from his since we want to stress properties of the trees. A Kripke model will be denoted by (Aα, ≤ 0), α ∈ T where (T, ≤, 0) is the tree with the least member 0 ∈ T and Aα, α ∈ T, is the model standing at the node α. The truth value at α of a formula ϕ(a1 … an) under the indicated assignment at α is denoted by [ϕ(a1 … an)]α.A Kripke model is said to be of constant domains if all the models Aα, α ∈ T, have the same domain.

A class of models for intermediate logics.

A class of models for intermediate logics.

Tsutomu Hosoi. On intermediate logics. Journal of the Faculty of Science, University of Tokyo, section I, vol. 14 (1967), pp. 293–312, and vol. 16 (1969), pp. 1–12.

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C. G. McKay. The decidability of certain intermediate propositional logics. The journal of symbolic logic, vol. 33 (1968), pp. 258–264.

C. G. McKay. The decidability of certain intermediate propositional logics. The journal of symbolic logic, vol. 33 (1968), pp. 258–264.

On the Finite Model Property for Kripke Models

This work is a sequel to Q4]. The familiarity with the results and terminology of T4] is presupposed. It is a well-known fact that the intuitionistic prepositional logic (abbreviated as LJ) has not a finite characteristic model. But, Jaskowski proved that there is a monotonic descending sequence of finite models which converges to LJ. Using the notation in Q4], we can restate this result as follows; there are finite (pseudo-Boolean) models Pi(i € /) such that LJ^Cr\ieIPi. Now, let's consider the following problem. Let L be any intermediate logic. Are there finite pseudo-Boolean models Pi(i €E /) such that ZOCAie/ Pi? If this problem is solved affirmatively for a logic Z,, we say L has the finite model property, following Harrop's terminology Ql]. In Ql], it is proved that if a logic L is finitely axiomatizable and has the finite model property then L is decidable. It is an interesting problem whether all intermediate logics are decidable. But we don't know even whether all intermediate logics have the finite model property. In this paper, we show that this problem can be reduced to the following problem presented in \j£}: Has any intermediate logic a characteristic Kripke model ? We extend the result to that for logics between Johansson's minimal logic and the classical prepositional logic, and show that the logics whobe decidability are not known in £5] are decidable. Now, we state our main theorem.

A class of decidable intermediate propositional logics

For the terminology and motivation of this note, my earlier paper [3] should be consulted. Here I make a slight change by requiring the Intuitionist Propositional Logic H to be given in terms of axiom schemata rather than axioms.Definition. An axiom schema F is essentially negative iff each schematic letter appearing in F is negated.Thus the schemata (¬P ∨ ¬¬P), (¬¬(¬P ∨ ¬Q)→(¬P ∨ ¬Q)) are essentially negative, whereas (¬P ∨ P) and (¬¬P→P) are not.Lemma. Let F be an essentially negative axiom schema. Then F yields at most finitely many intuitionistically nonequivalent axioms whose atoms are chosen from a fixed finite set.

Kripke Models and Intermediate Logics

In [10], Kripke gave a definition of the semantics of the intuitionistic logic. Fitting [2] showed that Kripke's models are equivalent to algebraic models (i.e., pseudo-Boolean models) in a certain sense. As a corollary of this result, we can show that any partially ordered set is regarded as a (characteristic) model of a intermediate logic ^ We shall study the relations between intermediate logics and partially ordered sets as models of them, in this paper. We call a partially ordered set, a Kripke model.2^ At present we don't know whether any intermediate logic 'has a Kripke model. But Kripke models have some interesting properties and are useful when we study the models of intermediate logics. In §2, we shall study general properties of Kripke models. In §3, we shall define the height of a Kripke model and show the close connection between the height and the slice, which is introduced in [7]. In §4, we shall give a model of LP» which is the least element in n-ih slice Sn (see [7]).

Toshio Umezawa. On some properties of intermediate logics. Proceedings of the Japan Academy, vol. 35 (1959), pp. 575–577.

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On an Application of Intermediate Logics

In [1] I investigated some logics intermediate between intuitionistic and classical predicate logics. The purpose of this paper is to show the possibility of applying some intermediate logics to mathematics namely, to show that some mathematical theorems which are provable in the classical logic but not provable in the intuitionistic logic are provable in some intermediate logics. Let LZ be the logical system obtained from LJ′ a variant of Gentzen’s LJ [2], by adding as axioms all those sequents which can be obtained from a sequent scheme Z by substitution for propositional, predicate, or individual variables.

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.

On intermediate propositional logics

By intermediate prepositional logics we mean prepositional logics between the intuitionistic and classical logics.K. Gödel [1] proved that there is a set of intermediate prepositional logics which possesses the order type ω. The method enables us to define intermediate logics in terms of axioms and rules of inference. We shall call it the axiomatic method.

On some properties of intermediate logics

On some properties of intermediate logics

Intermediate Logic. By James Welton D.Lit., and A. J. Monahan M.A. Third Edition, revised by E. M. Whetnall Ph.D., B.A. (London: University Tutorial Press, Ltd.1928. Pp. xvi + 508. Price 10s. 6d.)

Intermediate Logic. By James Welton D.Lit., and A. J. Monahan M.A. Third Edition, revised by E. M. Whetnall Ph.D., B.A. (London: University Tutorial Press, Ltd.1928. Pp. xvi + 508. Price 10s. 6d.)