Axiomatizability by a schema

1968 ◽  
Vol 32 (4) ◽  
pp. 473-479 ◽  
Author(s):  
Robert L. Vaught
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Relation Symbol ◽  
First Order ◽  
Free Variables

A theory T is axiomatizable by a schema if there is a formula Γ, involving symbols of T plus a new relation symbol R, such that the set of all (universal closures of) instances of Γ in T is a set of axioms for T. (It is understood that, if R has n places, an instance of Γ in T is obtained by properly substituting for R in Γ a formula of T which has n selected free variables and is allowed to have any number of other free variables as parameters.) Obviously, the notion is unchanged if finitely many Γ's, each involving several new R's, are allowed instead. All theories we consider are assumed to be theories in the first-order logic with equality (as in [8]), to have finitely many nonlogical symbols, and to be recursively axiomatizable.

An Interpolation Theorem

2000 ◽  
Vol 6 (4) ◽  
pp. 447-462 ◽  
Author(s):  
Martin Otto
Keyword(s):  
Interpolation Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Elementary Model ◽  
Relation Symbol ◽  
First Order ◽  
Theoretic Proof ◽  
Universal Quantification ◽  
The Given ◽  
Interpolation Result

AbstractLyndon's Interpolation Theorem asserts that for any valid implication between two purely relational sentences of first-order logic, there is an interpolant in which each relation symbol appears positively (negatively) only if it appears positively (negatively) in both the antecedent and the succedent of the given implication. We prove a similar, more general interpolation result with the additional requirement that, for some fixed tuple of unary predicates U, all formulae under consideration have all quantifiers explicitly relativised to one of the U. Under this stipulation, existential (universal) quantification over U contributes a positive (negative) occurrence of U.It is shown how this single new interpolation theorem, obtained by a canonical and rather elementary model theoretic proof, unifies a number of related results: the classical characterisation theorems concerning extensions (substructures) with those concerning monotonicity, as well as a many-sorted interpolation theorem focusing on positive vs. negative occurrences of predicates and on existentially vs. universally quantified sorts.

The degree of the set of sentences of predicate provability logic that are true under every interpretation

1987 ◽  
Vol 52 (1) ◽  
pp. 165-171 ◽  
Author(s):  
George Boolos ◽  
Vann McGee
Keyword(s):  
Predicate Symbol ◽  
Order Logic ◽  
First Order Logic ◽  
Provability Logic ◽  
First Order ◽  
Free Variables ◽  
Unary Operator

The formalism of P(redicate) P(rovability) L(ogic) is the result of adjoining the unary operator □ to first-order logic without identity, constants, or function symbols. The term “provability” indicates that □ is to be “read” as “it is provable in P(eano) A(rithmetic) that…” and that the formulae of predicate provability logic are to be interpreted via formulae of PA as follows.Pr(x), alias Bew(x), is the standard provability predicate of PA. For any formula F of PA, Pr[F] is the formula of PA that expresses the PA-provability of F “of” the values of the variables free in F, i.e., it is the formula of PA with the same free variables as F that expresses the PA-provability of the result of substituting for each variable free in F the numeral for the value of that variable. For the details of the construction of Pr[F], the reader may consult [B2, p. 42]. If F is a sentence of PA, then Pr[F] = Pr(‘F’), the sentence that expresses the PA-provability of F.Let υ1, υ2,… be an enumeration of the variables of PA. An interpretation * of a formula ϕ of PPL is a function which assigns to each predicate symbol P of ϕ a formula P* of the language of arithmetic whose free variables are the first n variables of PA, where n is the degree of P.

On amalgamations of languages with Magidor-Malitz quantifiers

1979 ◽  
Vol 44 (4) ◽  
pp. 549-558
Author(s):  
Carl F. Morgenstern
Keyword(s):  
Universal Algebra ◽  
Model Theory ◽  
Natural Extension ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Free Variables ◽  
Countably Compact

In this paper we indicate how compact languages containing the Magidor-Malitz quantifiers Qκn in different cardinalities can be amalgamated to yield more expressive, compact languages.The language Lκ<ω, originally introduced by Magidor and Malitz [9], is a natural extension of the language L(Q) introduced by Mostowski and investigated by Fuhrken [6], [7], Keisler [8] and Vaught [13]. Intuitively, Lκ<ω is first-order logic together with quantifiers Qκn (n ∈ ω) binding n free variables which express “there is a set X of cardinality κ such than any n distinct elements of X satisfy …”, or in other words, iff the relation on determined by φ contains an n-cube of cardinality κ. With these languages one can express a variety of combinatorial statements of the type considered by Erdös and his colleagues, as well as concepts in universal algebra which are beyond the scope of first-order logic. The model theory of Lκ<ω has been further developed by Badger [1], Magidor and Malitz [10] and Shelah [12].We refer to a language as being < κ compact if, given any set of sentences Σ of the language, if Σ is finitely satisfiable and ∣Σ∣ < κ, then Σ has a model. The phrase countably compact is used in place of <ℵ1 compact.

scholarly journals The Triguarded Fragment with Transitivity

10.29007/z359 ◽  
2020 ◽  
Author(s):  
Emanuel Kieronski ◽  
Adam Malinowski
Keyword(s):  
Decision Procedure ◽  
Order Logic ◽  
First Order Logic ◽  
Satisfiability Problem ◽  
First Order ◽  
Free Variables ◽  
Guarded Fragment

The triguarded fragment of first-order logic is an extension of the guarded fragment in which quantification for subformulas with at most two free variables need not be guarded. Thus, it unifies two prominent decidable logics: the guarded fragment and the two-variable fragment. Its satisfiability problem is known to be undecidable in the presence of equality, but becomes decidable when equality is forbidden. We consider an extension of the tri- guarded fragment without equality by transitive relations, allowing them to be used only as guards. We show that the satisfiability problem for the obtained formalism is decidable and 2-ExpTime-complete, that is, it is of the same complexity as for the analogous exten- sion of the classical guarded fragment. In fact, in our satisfiability test we use a decision procedure for the latter as a subroutine. We also show how our approach, consisting in exploiting some existing results on guarded logics, can be used to reprove some known facts, as well as to derive some other new results on triguarded logics.

Abstraction Logic

2021 ◽  
Author(s):  
Steven Obua
Keyword(s):  
Binding Constants ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Algebraic Semantics ◽  
Variable Binding ◽  
First Order ◽  
Free Variables ◽  
Direct Support ◽  
Second Order Logic

Abstraction Logic is introduced as a foundation for Practical Types and Practal. It combines the simplicity of first-order logic with direct support for variable binding constants called abstractions. It also allows free variables to depend on parameters, which means that first-order axiom schemata can be encoded as simple axioms. Conceptually abstraction logic is situated between first-order logic and second-order logic. It is sound with respect to an intuitive and simple algebraic semantics. Completeness holds for both intuitionistic and classical abstraction logic, and all abstraction logics in between and beyond.

scholarly journals An elementary definability theorem for first order logic

1999 ◽  
Vol 64 (3) ◽  
pp. 1028-1036 ◽  
Author(s):  
C. Butz ◽  
I. Moerdijk
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Free Variables ◽  
Definable Subset ◽  
Boolean Valued Model

In this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language , then, clearly, any definable subset S ⊂ M (i.e., a subset S = {a ∣ M ⊨ φ(a)} defined by some formula φ) is invariant under all automorphisms of M. The same is of course true for subsets of Mn defined by formulas with n free variables.Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely, for any theory T we will construct a Boolean valued model M, in which precisely the T -provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a formula .Our presentation is entirely selfcontained, and only requires familiarity with the most elementary properties of model theory. In particular, we have added a first section in which we review the basic definitions concerning Boolean valued models.

Provability with Finitely Many Variables

2002 ◽  
Vol 8 (3) ◽  
pp. 348-379 ◽  
Author(s):  
Robin Hirsch ◽  
Ian Hodkinson ◽  
Roger D. Maddux
Keyword(s):  
Predicate Logic ◽  
Algebraic Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Alternative Semantics ◽  
Relation Symbol ◽  
First Order ◽  
Binary Predicate ◽  
Standard Set ◽  
Binary Relation Symbol

AbstractFor every finite n ≥ 4 there is a logically valid sentence φn with the following properties: φn contains only 3 variables (each of which occurs many times); φn contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol); φn has a proof in first-order logic with equality that contains exactly n variables, but no proof containing only n − 1 variables. This result was first proved using the machinery of algebraic logic developed in several research monographs and papers. Here we replicate the result and its proof entirely within the realm of (elementary) first-order binary predicate logic with equality. We need the usual syntax, axioms, and rules of inference to show that φn has a proof with only n variables. To show that φn has no proof with only n − 1 variables we use alternative semantics in place of the usual, standard, set-theoretical semantics of first-order logic.

Abstraction Logic

2021 ◽  
Author(s):  
Steven Obua
Keyword(s):  
Binding Constants ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Algebraic Semantics ◽  
Variable Binding ◽  
First Order ◽  
Free Variables ◽  
Direct Support ◽  
Second Order Logic

Abstraction Logic is introduced as a foundation for Practical Types and Practal. It combines the simplicity of first-order logic with direct support for variable binding constants called abstractions. It also allows free variables to depend on parameters, which means that first-order axiom schemata can be encoded as simple axioms. Conceptually abstraction logic is situated between first-order logic and second-order logic. It is sound and complete with respect to an intuitive and simple algebraic semantics.

First-Order Logic Reasoning Support for the Semantic Web

2009 ◽  
Vol 19 (12) ◽  
pp. 3091-3099 ◽  
Author(s):  
Gui-Hong XU ◽  
Jian ZHANG
Keyword(s):  
Semantic Web ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Logic Reasoning

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.

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