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].

Elementary intuitionistic theories

1973 ◽  
Vol 38 (1) ◽  
pp. 102-134 ◽  
Author(s):  
C. Smorynski
Keyword(s):  
Intuitionistic Logic ◽  
Model Theory ◽  
General Result ◽  
Classical Logic ◽  
Decision Problem ◽  
Quantifier Elimination ◽  
Kripke Model ◽  
Logic Model ◽  
Intuitionistic Theory ◽  
Classical Extension

The present paper concerns itself primarily with the decision problem for formal elementary intuitionistic theories and the method is primarily model-theoretic. The chief tool is the Kripke model for which the reader may find sufficient background in Fitting's book Intuitionistic logic model theory and forcing (North-Holland, Amsterdam, 1969). Our notation is basically that of Fitting, the differences being to favor more standard notations in various places.The author owes a great debt to many people and would particularly like to thank S. Feferman, D. Gabbay, W. Howard, G. Kreisel, G. Mints, and R. Statman for their valuable assistance.The method of elimination of quantifiers, which has long since proven its use in classical logic, has also been applied to intuitionistic theories (i) to demonstrate decidability ([9], [15], [17]), (ii) to prove the coincidence of an intuitionistic theory with its classical extension ([9], [17]), and (iii), as we will see below, to establish relations between an intuitionistic theory and its classical extension. The most general of these results is to be obtained from the method of Lifshits' quantifier elimination for the intuitionistic theory of decidable equality.Since the details of Lifshits' proof have not been published, and since the proof yields a more general result than that stated in his abstract [15], we include the proof and several corollaries.

ON THE WAY TO A WIDER MODEL THEORY: COMPLETENESS THEOREMS FOR FIRST-ORDER LOGICS OF FORMAL INCONSISTENCY

2014 ◽  
Vol 7 (3) ◽  
pp. 548-578 ◽  
Author(s):  
WALTER CARNIELLI ◽  
MARCELO E. CONIGLIO ◽  
RODRIGO PODIACKI ◽  
TARCÍSIO RODRIGUES
Keyword(s):  
Model Theory ◽  
Classical Logic ◽  
Large Family ◽  
Paraconsistent Logics ◽  
Traditional Logic ◽  
First Order ◽  
Basic Model ◽  
Technical Details ◽  
Completeness Theorems ◽  
Refined Version

AbstractThis paper investigates the question of characterizing first-orderLFIs (logics of formal inconsistency) by means of two-valued semantics.LFIs are powerful paraconsistent logics that encode classical logic and permit a finer distinction between contradictions and inconsistencies, with a deep involvement in philosophical and foundational questions. Although focused on just one particular case, namely, the quantified logicQmbC, the method proposed here is completely general for this kind of logics, and can be easily extended to a large family of quantified paraconsistent logics, supplying a sound and complete semantical interpretation for such logics. However, certain subtleties involving term substitution and replacement, that are hidden in classical structures, have to be taken into account when one ventures into the realm of nonclassical reasoning. This paper shows how such difficulties can be overcome, and offers detailed proofs showing that a smooth treatment of semantical characterization can be given to all such logics. Although the paper is well-endowed in technical details and results, it has a significant philosophical aside: it shows how slight extensions of classical methods can be used to construct the basic model theory of logics that are weaker than traditional logic due to the absence of certain rules present in classical logic. Several such logics, however, as in the case of theLFIs treated here, are notorious for their wealth of models precisely because they do not make indiscriminate use of certain rules; these models thus require new methods. In the case of this paper, by just appealing to a refined version of the Principle of Explosion, or Pseudo-Scotus, some new constructions and crafty solutions to certain nonobvious subtleties are proposed. The result is that a richer extension of model theory can be inaugurated, with interest not only for paraconsistency, but hopefully to other enlargements of traditional logic.

scholarly journals Intuitionistic completeness for first order classical logic

1999 ◽  
Vol 64 (1) ◽  
pp. 304-312 ◽  
Author(s):  
Stefano Berardi
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
First Order ◽  
The Past

AbstractIn the past sixty years or so, a real forest of intuitionistic models for classical theories has grown. In this paper we will compare intuitionistic models of first order classical theories according to relevant issues, like completeness (w.r.t. first order classical provability), consistency, and relationship between a connective and its interpretation in a model. We briefly consider also intuitionistic models for classical ω-logic.All results included here, but a part of the proposition (a) below, are new. This work is, ideally, a continuation of a paper by McCarty, who considered intuitionistic completeness mostly for first order intuitionistic logic.

Some results in modal model theory

1974 ◽  
Vol 39 (3) ◽  
pp. 496-508 ◽  
Author(s):  
Michael Mortimer
Keyword(s):  
Modal Logic ◽  
Model Theory ◽  
Classical Logic ◽  
Classical Model ◽  
Completeness Theorem ◽  
Strong Completeness ◽  
Barcan Formula ◽  
Weak Completeness ◽  
Modal Model ◽  
Omitting Types

This paper is concerned with extending some basic results from classical model theory to modal logic.In §1, we define the majority of terms used in the paper, and explain our notation. A full catalogue would be excessive, and we cite [3] and [7] as general references.Many papers on modal logic that have appeared are concerned with (i) introducing a new modal logic, and (ii) proving a weak completeness theorem for it. Theorem 1, in §2, in many cases allows us to conclude immediately that a strong completeness theorem holds for such a logic in languages of arbitrary cardinality. In particular, this is true of S4 with the Barcan formula.In §3 we strengthen Theorem 1 for a number of modal logics to deal with the satisfaction of several sets of sentences, and so obtain a realizing types theorem. Finally, an omitting types theorem, generalizing the result for classical logic (see [5]) is proved in §4.Several consequences of Theorem 1 are already to be found in the literature. [2] gives a proof of strong completeness in languages of arbitrary cardinality of various logics without the Barcan formula, and [8] for some logics in countable languages with it. In the latter case, the result for uncountable languages is cited, without proof, in [1], and there credited to Montague. Our proof was found independently.

scholarly journals Dialogues for proof search

10.29007/5t86 ◽  
2018 ◽  
Author(s):  
Jesse Alama
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Order Logic ◽  
Theorem Prover ◽  
First Order Logic ◽  
Dialogue Games ◽  
First Order ◽  
Proof Search ◽  
Automated Theorem Prover

Dialogue games are a two-player semantics for a variety of logics, including intuitionistic and classical logic. Dialogues can be viewed as a kind of analytic calculus not unlike tableaux. Can dialogue games be an effective foundation for proof search in intuitionistic logic (both first-order and propositional)? We announce Kuno, an automated theorem prover for intuitionistic first-order logic based on dialogue games.

Boolean-valued structures

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Rules ◽  
Mathematical Concepts ◽  
Semantic Framework ◽  
First Order ◽  
Standard Models ◽  
Boolean Valued Model ◽  
Semantic Values

Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin down the semantic frameworks of language. It transpires that there are not just non-standard models, but non-standard ways of doing model theory itself. In more detail: whilst we normally outline a two-valued semantics which makes sentences True or False in a model, the inference rules for first-order logic are compatible with a four-valued semantics; or a semantics with countably many values; or what-have-you. The appropriate level of generality here is that of a Boolean-valued model, which we introduce. And the plurality of possible semantic values gives rise to perhaps the ‘deepest’ level of indeterminacy questions: How can humans pin down the semantic framework for their languages? We consider three different ways for inferentialists to respond to this question.

scholarly journals On the correspondence between nested calculi and semantic systems for intuitionistic logics

2020 ◽  
Author(s):  
Tim Lyon
Keyword(s):  
Intuitionistic Logic ◽  
Structural Rules ◽  
First Order ◽  
Constant Domains ◽  
The Relationship ◽  
Intuitionistic Logics

Abstract This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is shown that Fitting’s nested calculi naturally arise from their corresponding labelled calculi—for each of the aforementioned logics—via the elimination of structural rules in labelled derivations. The translational correspondence between the two types of systems is leveraged to show that the nested calculi inherit proof-theoretic properties from their associated labelled calculi, such as completeness, invertibility of rules and cut admissibility. Since labelled calculi are easily obtained via a logic’s semantics, the method presented in this paper can be seen as one whereby refined versions of labelled calculi (containing nested calculi as fragments) with favourable properties are derived directly from a logic’s semantics.

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.

Weakly atomic-compact relational structures

1971 ◽  
Vol 36 (1) ◽  
pp. 129-140 ◽  
Author(s):  
G. Fuhrken ◽  
W. Taylor
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Compact Abelian Group ◽  
Finite Subset ◽  
Relational Structure ◽  
Similarity Type ◽  
First Order ◽  
Relational Structures ◽  
Order Language ◽  
Weakly Atomic

A relational structure is called weakly atomic-compact if and only if every set Σ of atomic formulas (taken from the first-order language of the similarity type of augmented by a possibly uncountable set of additional variables as “unknowns”) is satisfiable in whenever every finite subset of Σ is so satisfiable. This notion (as well as some related ones which will be mentioned in §4) was introduced by J. Mycielski as a generalization to model theory of I. Kaplansky's notion of an algebraically compact Abelian group (cf. [5], [7], [1], [8]).

Model Theory of Epimorphisms

1974 ◽  
Vol 17 (4) ◽  
pp. 471-477 ◽  
Author(s):  
Paul D. Bacsich
Keyword(s):  
Model Theory ◽  
Order Theory ◽  
First Order ◽  
First Order Theory

Given a first-order theory T, welet be the category of models of T and homomorphisms between them. We shall show that a morphism A→B of is an epimorphism if and only if every element of B is definable from elements of A in a certain precise manner (see Theorem 1), and from this derive the best possible Cowell- power Theorem for .

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