scholarly journals On an Application of Intermediate Logics

1960 ◽  
Vol 16 ◽  
pp. 119-133
Author(s):  
Toshio Umezawa
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Logical System ◽  
Intermediate Logics ◽  
Individual Variables ◽  
Classical Predicate

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.

Undecidability of modal and intermediate first-order logics with two individual variables

1993 ◽  
Vol 58 (3) ◽  
pp. 800-823 ◽  
Author(s):  
D. M. Gabbay ◽  
V. B. Shehtman
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Predicate Calculus ◽  
Single Individual ◽  
First Order ◽  
Short Summary ◽  
Binary Predicate ◽  
Individual Variables ◽  
Image Position ◽  
Classical Predicate

The interest in fragments of predicate logics is motivated by the well-known fact that full classical predicate calculus is undecidable (cf. Church [1936]). So it is desirable to find decidable fragments which are in some sense “maximal”, i.e., which become undecidable if they are “slightly” extended. Or, alternatively, we can look for “minimal” undecidable fragments and try to identify the vague boundary between decidability and undecidability. A great deal of work in this area concerning mainly classical logic has been done since the thirties. We will not give a complete review of decidability and undecidability results in classical logic, referring the reader to existing monographs (cf. Suranyi [1959], Lewis [1979], and Dreben, Goldfarb [1979]). A short summary can also be found in the well-known book Church [1956]. Let us recall only several facts. Herein we will consider only logics without functional symbols, constants, and equality.(C1) The fragment of the classical logic with only monadic predicate letters is decidable (cf. Behmann [1922]).(C2) The fragment of the classical logic with a single binary predicate letter is undecidable. (This is a consequence of Gödel [1933].)(C3) The fragment of the classical logic with a single individual variable is decidable; in fact it is equivalent to Lewis S5 (cf. Wajsberg [1933]).(C4) The fragment of the classical logic with two individual variables is decidable (Segerberg [1973] contains a proof using modal logic; Scott [1962] and Mortimer [1975] give traditional proofs.)(C5) The fragment of the classical logic with three individual variables and binary predicate letters is undecidable (cf. Surańyi [1943]). In fact this paper considers formulas of the following typeφ,ψ being quantifier-free and the set of binary predicate letters which can appear in φ or ψ being fixed and finite.

scholarly journals LOGIC FOR THE THEORY OF CONCEPTS

ARHE ◽  
2021 ◽  
Vol 27 (34) ◽  
pp. 85-102
Author(s):  
JOVANA KOSTIĆ
Keyword(s):  
Classical Logic ◽  
Predicate Calculus ◽  
Logical System ◽  
Logical Basis ◽  
Logical Foundation ◽  
Starting Point ◽  
Partial Predicates ◽  
Classical Predicate

In this paper, we follow Gödel’s remarks on an envisioned theory of concepts to determine which properties should a logical basis of such a theory have. The discussion is organized around the question of suitability of the classical predicate calculus for this role. Some reasons to think that classical logic is not an appropriate basis for the theory of concepts, will be presented. We consider, based on these reasons, which alternative logical system could fare better as a logical foundation of, in Gödel’s opinion, the most important theory in logic yet to be developed. This paper should, in particular, motivate the study of partial predicates in a certain system of three-valued logic, as a promising starting point for the foundation of the theory of concepts.

Completeness of global intuitionistic set theory

1997 ◽  
Vol 62 (2) ◽  
pp. 506-528 ◽  
Author(s):  
Satoko Titani
Keyword(s):  
Set Theory ◽  
Intuitionistic Logic ◽  
Classical Logic ◽  
Mathematical Object ◽  
The Other ◽  
Logical System ◽  
Logical Operators ◽  
Logical Symbol ◽  
Image Position ◽  
Intuitionistic Set Theory

Gentzen's sequential system LJ of intuitionistic logic has two symbols of implication. One is the logical symbol → and the other is the metalogical symbol ⇒ in sequentsConsidering the logical system LJ as a mathematical object, we understand that the logical symbols ∧, ∨, →, ¬, ∀, ∃ are operators on formulas, and ⇒ is a relation. That is, φ ⇒ Ψ is a metalogical sentence which is true or false, on the understanding that our metalogic is a classical logic. In other words, we discuss the logical system LJ in the classical set theory ZFC, in which φ ⇒ Ψ is a sentence.The aim of this paper is to formulate an intuitionistic set theory together with its metatheory. In Takeuti and Titani [6], we formulated an intuitionistic set theory together with its metatheory based on intuitionistic logic. In this paper we postulate that the metatheory is based on classical logic.Let Ω be a cHa. Ω can be a truth value set of a model of LJ. Then the logical symbols ∧, ∨, →, ¬, ∀x, ∃x are interpreted as operators on Ω, and the sentence φ ⇒ Ψ is interpreted as 1 (true) or 0 (false). This means that the metalogical symbol ⇒ also can be expressed as a logical operators such that φ ⇒ Ψ is interpreted as 1 or 0.

scholarly journals Prawda i fałsz wypowiedzi literackiej

2020 ◽  
Vol 15 (6) ◽  
pp. 72-91
Author(s):  
Roman Magryś ◽  
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Specific Model ◽  
Logical System ◽  
Theoretical Assumptions ◽  
Point Of Reference

The article “Truth and Falsity of Literary Statements” investigates the issue of logical values, and consequently communication related status of sentences in indicative mood occurring in works of literature. The problem is discussed with reference to theoretical assumptions of classical logic, intuitionistic logic, and other possible intensional logics as well as phenomenological concepts proposed by Roman Ingarden. In this context it is suggested that intensional systems, mainly intuitionistic logic be adequately applied to identify logical value of literary sentences. As a result, it is assumed that the logical value of literary sentences depends on the specific logical system selected; according to the standards of intuitionistic logic, literary statements are true, likely or false. In this context it seems necessary to revise Roman Ingarden’s phenomenological assumption that sentences in indicative mood in a work of fiction do not have objective point of reference. It is suggested that such sentences be recognised as false, and therefore indicative of the group of their intentional meanings as a specific model of reality which can be deemed true or false.

scholarly journals On Correspondence between Selective CPS Transformation and Selective Double Negation Translation

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 385
Author(s):  
Hyeonseung Im
Keyword(s):  
Programming Languages ◽  
Intuitionistic Logic ◽  
Reference Point ◽  
Classical Logic ◽  
Proof System ◽  
Double Negation

A double negation translation (DNT) embeds classical logic into intuitionistic logic. Such translations correspond to continuation passing style (CPS) transformations in programming languages via the Curry-Howard isomorphism. A selective CPS transformation uses a type and effect system to selectively translate only nontrivial expressions possibly with computational effects into CPS functions. In this paper, we review the conventional call-by-value (CBV) CPS transformation and its corresponding DNT, and provide a logical account of a CBV selective CPS transformation by defining a selective DNT via the Curry-Howard isomorphism. By using an annotated proof system derived from the corresponding type and effect system, our selective DNT translates classical proofs into equivalent intuitionistic proofs, which are smaller than those obtained by the usual DNTs. We believe that our work can serve as a reference point for further study on the Curry-Howard isomorphism between CPS transformations and DNTs.

scholarly journals On density of truth of the intuitionistic logic in one variable

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Zofia Kostrzycka
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Propositional Logic ◽  
Propositional Variable ◽  
Intuitionistic Propositional Logic ◽  
International Audience

International audience In this paper we focus on the intuitionistic propositional logic with one propositional variable. More precisely we consider the standard fragment $\{ \to ,\vee ,\bot \}$ of this logic and compute the proportion of tautologies among all formulas. It turns out that this proportion is different from the analog one in the classical logic case.

scholarly journals A Certain Kind of Formal Theories

1965 ◽  
Vol 25 ◽  
pp. 59-86 ◽  
Author(s):  
Katuzi Ono
Keyword(s):  
Classical Logic ◽  
Axiom System ◽  
Logical System ◽  
First Order ◽  
Formal Theories

A common feature of formal theories is that each theory has its own system of axioms described in terms of some symbols for its primitive notions together with logical symbols. Each of these theories is developed by deduction from its axiom system in a certain logical system which is usually the classical logic of the first order.

An embedding of classical logic in S4

1970 ◽  
Vol 35 (4) ◽  
pp. 529-534 ◽  
Author(s):  
Melvin Fitting
Keyword(s):  
Intuitionistic Logic ◽  
Model Theory ◽  
Classical Logic ◽  
First Order ◽  
Barcan Formula ◽  
Complete Sequences

There are well-known embeddings of intuitionistic logic into S4 and of classical logic into S5. In this paper we give a related embedding of (first order) classical logic directly into (first order) S4, with or without the Barcan formula. If one reads the necessity operator of S4 as ‘provable’, the translation may be roughly stated as: truth may be replaced by provable consistency. A proper statement will be found below. The proof is based ultimately on the notion of complete sequences used in Cohen's technique of forcing [1], and is given in terms of Kripke's model theory [3], [4].

On epistemic and ontological interpretations of intuitionistic and paraconsistent paradigms

2020 ◽  
Author(s):  
Walter Carnielli ◽  
Abilio Rodrigues
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Point Of View ◽  
Constructive Proof ◽  
Technical Point ◽  
Preservation Of Evidence ◽  
Epistemic Approach ◽  
Epistemic Interpretation ◽  
Excluded Middle

Abstract From the technical point of view, philosophically neutral, the duality between a paraconsistent and a paracomplete logic (for example intuitionistic logic) lies in the fact that explosion does not hold in the former and excluded middle does not hold in the latter. From the point of view of the motivations for rejecting explosion and excluded middle, this duality can be interpreted either ontologically or epistemically. An ontological interpretation of intuitionistic logic is Brouwer’s idealism; of paraconsistency is dialetheism. The epistemic interpretation of intuitionistic logic is in terms of preservation of constructive proof; of paraconsistency is in terms of preservation of evidence. In this paper, we explain and defend the epistemic approach to paraconsistency. We argue that it is more plausible than dialetheism and allows a peaceful and fruitful coexistence with classical logic.

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