The consistency property in lattice valued model theory for infinitary logic

1987 ◽  
Vol 3 (2) ◽  
pp. 103-108 ◽  
Author(s):  
Shen Fuxing
Keyword(s):  
Model Theory ◽  
Infinitary Logic ◽  
Consistency Property

Indiscernibility

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Infinitary Logic ◽  
Identity Of Indiscernibles ◽  
Sheer Number

This chapter explores Leibniz's principle of the Identity of Indiscernibles. Model theory supplies us with the resources to distinguish between many different notions of indiscernibility; we can vary: (a) the primitive ideology (b) the background logic and (c) the grade of discernibility. We use these distinctions to discuss the possibility of singling-out “indiscernibles”. And we then use these to distinctions to explicate Leibniz's famous principle. While model theory allows us to make this principle precise, the sheer number of different precise versions of this principle made available by model theory can serve to mitigate some of the initial excitement of this principle. We round out the chapter with two technical topics: indiscernibility in infinitary logic, and the relation between indiscernibility, orders, and stability.

Some model theory for game logics

1979 ◽  
Vol 44 (2) ◽  
pp. 147-152
Author(s):  
Judy Green
Keyword(s):  
Model Theory ◽  
Existence Theorems ◽  
Semantic Interpretation ◽  
Rule Based ◽  
Game Logic ◽  
Important Method ◽  
Consistency Property ◽  
Consistency Properties ◽  
Model Existence

Consistency properties and their model existence theorems have provided an important method of constructing models for fragments of L∞ω. In [E] Ellentuck extended this construction to Suslin logic. One of his extensions, the Borel consistency property, has its extra rule based not on the semantic interpretation of the extra symbols but rather on a theorem of Sierpinski about the classical operation . In this paper we extend that consistency property to the game logic LG and use it to show how one can extend results about and its countable fragments to LG and certain of its countable fragments. The particular formation of LG which we use will allow in the game quantifier infinite alternation of countable conjunctions and disjunctions as well as infinite alternation of quantifiers. In this way LG can be viewed as an extension of Suslin logic.

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.

scholarly journals Strong convergence in finite model theory

2002 ◽  
Vol 67 (3) ◽  
pp. 1083-1092
Author(s):  
Wafik Boulos Lotfallah
Keyword(s):  
Strong Convergence ◽  
Random Graphs ◽  
Model Theory ◽  
Sample Space ◽  
Finite Model ◽  
Additive Measure ◽  
Infinitary Logic ◽  
Uniform Measure ◽  
The Universe ◽  
New Framework

AbstractIn [9] we introduced a new framework for asymptotic probabilities, in which a σ-additive measure is defined on the sample space of all sequences of finite models, where the universe of , is {1,2,…,n}. In this framework we investigated the strong 0-1 law for sentences, which states that each sentence either holds in eventually almost surely or fails in eventually almost surely.In this paper we define the strong convergence law for formulas, which carries over the ideas of the strong 0-1 law to formulas with free variables, and roughly states that for each formula ϕ(x), the fraction of tuples a in , which satisfy the formula ϕ(x), almost surely has a limit as n tends to infinity.We show that the infinitary logic with finitely many variables has the strong convergence law for formulas for the uniform measure, and further characterize the measures on random graphs for which the strong convergence law holds.

κ-Suslin logic

1978 ◽  
Vol 43 (4) ◽  
pp. 659-666
Author(s):  
Judy Green
Keyword(s):  
Model Theory ◽  
Upper Bound ◽  
Completeness Theorem ◽  
Infinitary Logic ◽  
Complete Axiomatization ◽  
Standard Notation ◽  
Weak Completeness

An analogue of a theorem of Sierpinski about the classical operation () provides the motivation for studying κ-Suslin logic, an extension of Lκ+ω which is closed under a propositional connective based on (). This theorem is used to obtain a complete axiomatization for κ-Suslin logic and an upper bound on the κ-Suslin accessible ordinals (for κ = ω these results are due to Ellentuck [E]). It also yields a weak completeness theorem which we use to generalize a result of Barwise and Kunen [B-K] and show that the least ordinal not H(κ+) recursive is the least ordinal not κ-Suslin accessible.We assume familiarity with lectures 3, 4 and 10 of Keisler's Model theory for infinitary logic [Ke]. We use standard notation and terminology including the following.Lκ+ω is the logic closed under negation, finite quantification, and conjunction and disjunction over sets of formulas of cardinality at most κ. For κ singular, conjunctions and disjunctions over sets of cardinality κ can be replaced by conjunctions and disjunctions over sets of cardinality less than κ so that we can (and will in §2) assume the formation rules of Lκ+ω allow conjunctions and disjunctions only over sets of cardinality strictly less than κ whenever κ is singular.

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