Classical Model Existence Theorem in Subclassical Predicate Logics. II

2017 ◽  
pp. 197-212
Author(s):  
Jui-Lin Lee
Keyword(s):  
Existence Theorem ◽  
Classical Model ◽  
Model Existence ◽  
Model Existence Theorem

Model existence theorem in algorithmic logic with non-deterministic programs1

1980 ◽  
Vol 3 (2) ◽  
pp. 157-170
Author(s):  
Grażyna Mirkowska
Keyword(s):  
Existence Theorem ◽  
Algorithmic Logic ◽  
Model Existence ◽  
Model Existence Theorem

The paper is a continuation of the considerations connected with non-deterministic algorithmic logic. We will formulate a Hilbert style axiomatization basing on the analogous one defined for algorithmic logic. The main result is the theorem asserting that every consistent non-deterministic algorithmic theory possesses a model.

Model existence theorems for modal and intuitionistic logics

1973 ◽  
Vol 38 (4) ◽  
pp. 613-627 ◽  
Author(s):  
Melvin Fitting
Keyword(s):  
Existence Theorem ◽  
Intuitionistic Logic ◽  
Classical Logic ◽  
Existence Theorems ◽  
Compactness Theorem ◽  
Infinitary Logic ◽  
Consistency Property ◽  
Model Existence ◽  
Model Existence Theorem ◽  
Intuitionistic Logics

In classical logic a collection of sets of statements (or equivalently, a property of sets of statements) is called a consistency property if it meets certain simple closure conditions (a definition is given in §2). The simplest example of a consistency property is the collection of all consistent sets in some formal system for classical logic. The Model Existence Theorem then says that any member of a consistency property is satisfiable in a countable domain. From this theorem many basic results of classical logic follow rather simply: completeness theorems, the compactness theorem, the Lowenheim-Skolem theorem, and the Craig interpolation lemma among others. The central position of the theorem in classical logic is obvious. For the infinitary logic the Model Existence Theorem is even more basic as the compactness theorem is not available; [8] is largely based on it.In this paper we define appropriate notions of consistency properties for the first-order modal logics S4, T and K (without the Barcan formula) and for intuitionistic logic. Indeed we define two versions for intuitionistic logic, one deriving from the work of Gentzen, one from Beth; both have their uses. Model Existence Theorems are proved, from which the usual known basic results follow. We remark that Craig interpolation lemmas have been proved model theoretically for these logics by Gabbay ([5], [6]) using ultraproducts. The existence of both ultra-product and consistency property proofs of the same result is a common phenomena in classical and infinitary logic. We also present extremely simple tableau proof systems for S4, T, K and intuitionistic logics, systems whose completeness is an easy consequence of the Model Existence Theorems.

Realization using the model existence theorem

2013 ◽  
Vol 26 (1) ◽  
pp. 213-234 ◽  
Author(s):  
Melvin Fitting
Keyword(s):  
Existence Theorem ◽  
Model Existence ◽  
Model Existence Theorem

Model existence theorem in superrelevant predicate logics

1987 ◽  
Vol 26 (1) ◽  
pp. 111-121
Author(s):  
Mirosław Szatkowski
Keyword(s):  
Existence Theorem ◽  
Model Existence ◽  
Model Existence Theorem

Fragments of first order logic, I: universal Horn logic

1977 ◽  
Vol 42 (2) ◽  
pp. 221-237 ◽  
Author(s):  
George F. McNulty
Keyword(s):  
Existence Theorem ◽  
Predicate Logic ◽  
Logical Consequence ◽  
Horn Logic ◽  
First Order ◽  
Consistency Properties ◽  
Model Existence ◽  
Model Existence Theorem ◽  
Sharp Version ◽  
Horn Sentences

Let L be any finitary language. By restricting our attention to the universal Horn sentences of L and appealing to a semantical notion of logical consequence, we can formulate the universal Horn logic of L. The present paper provides some theorems about universal Horn logic that serve to distinguish it from the full first order predicate logic. Universal Horn equivalence between structures is characterized in two ways, one resembling Kochen's ultralimit theorem. A sharp version of Beth's definability theorem is established for universal Horn logic by means of a reduced product construction. The notion of a consistency property is relativized to universal Horn logic and the corresponding model existence theorem is proven. Using the model existence theorem another proof of the definability result is presented. The relativized consistency properties also suggest a syntactical notion of proof that lies entirely within the universal Horn logic. Finally, a decision problem in universal Horn logic is discussed. It is shown that the set of universal Horn sentences preserved under the formation of homomorphic images (or direct factors) is not recursive, provided the language has at least two unary function symbols or at least one function symbol of rank more than one.This paper begins with a discussion of how algebraic relations between structures can be used to obtain fragments of a given logic. Only two such fragments seem to be under current investigation: equational logic and universal Horn logic. Other fragments which seem interesting are pointed out.

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