On the model theory of denumerably long formulas with finite strings of quantifiers

1969 ◽  
Vol 34 (3) ◽  
pp. 437-459 ◽  
Author(s):  
M. Makkai
Keyword(s):  
Model Theory ◽  
Predicate Calculus

In this paper we prove infinitary analogues of model-theoretic results known for finitary logic. The infinitary language we deal with is Lω1ω which is roughly described by saying that, in addition to the usual formation rules of the lower predicate calculus with identity, also the formation of the conjunction and disjunction of countably many formulas is allowed.

On the strong semantical completeness of the intuitionistic predicate calculus

1968 ◽  
Vol 33 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Richmond H. Thomason
Keyword(s):  
Model Theory ◽  
Completeness Theorem ◽  
Predicate Calculus ◽  
First Order ◽  
Semantic Tableaux ◽  
Weak Completeness ◽  
Skolem Theorem ◽  
Completeness Theorems

In Kripke [8] the first-order intuitionjstic predicate calculus (without identity) is proved semantically complete with respect to a certain model theory, in the sense that every formula of this calculus is shown to be provable if and only if it is valid. Metatheorems of this sort are frequently called weak completeness theorems—the object of the present paper is to extend Kripke's result to obtain a strong completeness theorem for the intuitionistic predicate calculus of first order; i.e., we will show that a formula A of this calculus can be deduced from a set Γ of formulas if and only if Γ implies A. In notes 3 and 5, below, we will indicate how to account for identity, as well. Our proof of the completeness theorem employs techniques adapted from Henkin [6], and makes no use of semantic tableaux; this proof will also yield a Löwenheim-Skolem theorem for the modeling.

A THREE-VALUED QUANTIFIED ARGUMENT CALCULUS: DOMAIN-FREE MODEL-THEORY, COMPLETENESS, AND EMBEDDING OF FOL

2017 ◽  
Vol 10 (3) ◽  
pp. 549-582 ◽  
Author(s):  
RAN LANZET
Keyword(s):  
Natural Language ◽  
Model Theory ◽  
Predicate Calculus ◽  
Extended Version ◽  
First Order ◽  
Completeness Result ◽  
Free Model ◽  
Simplifying Assumptions ◽  
Syntactic Restrictions ◽  
Order Predicate Calculus

AbstractThis paper presents an extended version of the Quantified Argument Calculus (Quarc). Quarc is a logic comparable to the first-order predicate calculus. It employs several nonstandard syntactic and semantic devices, which bring it closer to natural language in several respects. Most notably, quantifiers in this logic are attached to one-place predicates; the resulting quantified constructions are then allowed to occupy the argument places of predicates. The version presented here is capable of straightforwardly translating natural-language sentences involving defining clauses. A three-valued, model-theoretic semantics for Quarc is presented. Interpretations in this semantics are not equipped with domains of quantification: they are just interpretation functions. This reflects the analysis of natural-language quantification on which Quarc is based. A proof system is presented, and a completeness result is obtained. The logic presented here is capable of straightforward translation of the classical first-order predicate calculus, the translation preserving truth values as well as entailment. The first-order predicate calculus and its devices of quantification can be seen as resulting from Quarc on certain semantic and syntactic restrictions, akin to simplifying assumptions. An analogous, straightforward translation of Quarc into the first-order predicate calculus is impossible.

Projective model completeness

1974 ◽  
Vol 39 (1) ◽  
pp. 117-123 ◽  
Author(s):  
George S. Sacerdote
Keyword(s):  
Model Theory ◽  
Predicate Calculus ◽  
Model Completeness ◽  
New Model ◽  
Projective Model ◽  
New Criteria ◽  
Completeness Theorems

The concept of model completeness has been very useful in model theory. In this article we obtain a new model theoretic tool by “reversing the arrows.” Specifically, model completeness deals with the relations between a model of a theory and its extensions; in this paper we shall be concerned with the relation between a model of a theory and its homomorphic pre-images.This work is based on the intuitive principle that metamathematical theorems about universal sentences of the lower predicate calculus and substructures can be translated in a truth-preserving way to theorems about positive sentences and homomorphic images. From model completeness and the completeness theorems which depend on it, this translation gives new criteria for completeness.Special thanks are due to Professor A. Robinson whose encouragement and suggestions have contributed much to these results.

LePUS

2011 ◽  
pp. 357-372 ◽  
Author(s):  
Epameinondas Gasparis
Keyword(s):  
Mathematical Logic ◽  
Conceptual Framework ◽  
Model Theory ◽  
Design Patterns ◽  
Frame Of Reference ◽  
Predicate Calculus ◽  
Tool Support ◽  
First Order ◽  
Finite Structures ◽  
Order Predicate Calculus

We present LePUS, a formal language for modeling object oriented (O-O) Design patterns. We demonstrate the language’s unique efficacy in producing precise, concise, generic, and appropriately abstract specifications that effectively model the Gang of Four’s Design patterns. Mathematical logic is used as a main frame of reference: LePUS is defined as a subset of first-order predicate calculus and implementations (programs) are modeled as finite structures in model theory. We also demonstrate the conceptual framework in which the verification of implementations against pattern specifications is possible and our ongoing endeavour to develop effective tool support for LePUS.

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