scholarly journals Completeness for the Classical Antecedent Fragment of Inquisitive First-Order Logic

2021 ◽  
Author(s):  
Gianluca Grilletti
Keyword(s):  
Classical Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Strong Completeness ◽  
New Approach ◽  
First Order ◽  
Logic Equation

AbstractInquisitive first order logic "Equation missing" is an extension of first order classical logic, introducing questions and studying the logical relations between questions and quantifiers. It is not known whether "Equation missing" is recursively axiomatizable, even though an axiomatization has been found for fragments of the logic (Ciardelli, 2016). In this paper we define the $$\mathsf {ClAnt}$$ ClAnt —classical antecedent—fragment, together with an axiomatization and a proof of its strong completeness. This result extends the ones presented in the literature and introduces a new approach to study the axiomatization problem for fragments of the logic.

Visual And Diagrammatic Languages

2011 ◽  
pp. 22-50
Author(s):  
Bernd Meyer ◽  
Paolo Bottoni
Keyword(s):  
Logic Programming ◽  
Programming Language ◽  
Linear Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Visual Languages ◽  
New Approach ◽  
Logic Programming Language ◽  
First Order ◽  
Complex Forms

In this paper we investigate a new approach to formalizing interpretation of and reasoning with visual languages based on linear logic. We argue that an approach based on logic makes it possible to deal with different computational tasks in the usage of visual notations, from parsing and animation to reasoning about diagrams. However, classical first order logic, being monotonic, is not a suitable basis for such an approach. The paper therefore explores linear logic as an alternative. We demonstrate how parsing corresponds to linear proofs and prove the soundness and correctness of this mapping. As our mapping of grammars is into a subset of a linear logic programming language, we also demonstrate how multi-dimensional parsing can be understood as automated linear deduction. We proceed to discuss how the same framework can be used as the foundation of more complex forms of reasoning with and about diagrams.

scholarly journals A proof of completeness for continuous first-order logic

2010 ◽  
Vol 75 (1) ◽  
pp. 168-190 ◽  
Author(s):  
Itaï Ben Yaacov ◽  
Arthur Paul Pedersen
Keyword(s):  
Model Theory ◽  
Completeness Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Complete Theory ◽  
Strong Completeness ◽  
First Order ◽  
Completeness Result ◽  
Metric Structures ◽  
Natural Classes

AbstractContinuous first-order logic has found interest among model theorists who wish to extend the classical analysis of “algebraic” structures (such as fields, group, and graphs) to various natural classes of complete metric structures (such as probability algebras, Hilbert spaces, and Banach spaces). With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result?The primary purpose of this article is to show that a certain, interesting set of axioms does indeed yield a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies anapproximatedform of strong completeness, whereby Σ⊧φ(if and) only if Σ⊢φ∸2−nfor alln < ω. This approximated form of strong completeness asserts that if Σ⊧φ, then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth ofφ.Additionally, we consider a different kind of question traditionally arising in model theory—that of decidability. When is the set of all consequences of a theory (in a countable, recursive language) recursive? Say that a complete theoryTisdecidableif for every sentenceφ, the valueφTis a recursive real, and moreover, uniformly computable fromφ. IfTis incomplete, we say it is decidable if for every sentenceφthe real numberφTois uniformly recursive fromφ, whereφTois the maximal value ofφconsistent withT. As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) axiomatization then it is decidable.

Natural deduction based set theories: a new resolution of the old paradoxes

1986 ◽  
Vol 51 (2) ◽  
pp. 393-411 ◽  
Author(s):  
Paul C. Gilmore
Keyword(s):  
Set Theory ◽  
Classical Logic ◽  
Natural Deduction ◽  
Order Logic ◽  
First Order Logic ◽  
Truth Values ◽  
First Order ◽  
Axiom Scheme ◽  
Universe Of Sets ◽  
Comprehension Axiom

AbstractThe comprehension principle of set theory asserts that a set can be formed from the objects satisfying any given property. The principle leads to immediate contradictions if it is formalized as an axiom scheme within classical first order logic. A resolution of the set paradoxes results if the principle is formalized instead as two rules of deduction in a natural deduction presentation of logic. This presentation of the comprehension principle for sets as semantic rules, instead of as a comprehension axiom scheme, can be viewed as an extension of classical logic, in contrast to the assertion of extra-logical axioms expressing truths about a pre-existing or constructed universe of sets. The paradoxes are disarmed in the extended classical semantics because truth values are only assigned to those sentences that can be grounded in atomic sentences.

scholarly journals WITTGENSTEIN’S ELIMINATION OF IDENTITY FOR QUANTIFIER-FREE LOGIC

2020 ◽  
pp. 1-21
Author(s):  
TIMM LAMPERT ◽  
MARKUS SÄBEL
Keyword(s):  
Classical Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Free Logic ◽  
First Order ◽  
Logical Equivalence

Abstract One of the central logical ideas in Wittgenstein’s Tractatus logico-philosophicus is the elimination of the identity sign in favor of the so-called “exclusive interpretation” of names and quantifiers requiring different names to refer to different objects and (roughly) different variables to take different values. In this paper, we examine a recent development of these ideas in papers by Kai Wehmeier. We diagnose two main problems of Wehmeier’s account, the first concerning the treatment of individual constants, the second concerning so-called “pseudo-propositions” (Scheinsätze) of classical logic such as $a=a$ or $a=b \wedge b=c \rightarrow a=c$ . We argue that overcoming these problems requires two fairly drastic departures from Wehmeier’s account: (1) Not every formula of classical first-order logic will be translatable into a single formula of Wittgenstein’s exclusive notation. Instead, there will often be a multiplicity of possible translations, revealing the original “inclusive” formulas to be ambiguous. (2) Certain formulas of first-order logic such as $a=a$ will not be translatable into Wittgenstein’s notation at all, being thereby revealed as nonsensical pseudo-propositions which should be excluded from a “correct” conceptual notation. We provide translation procedures from inclusive quantifier-free logic into the exclusive notation that take these modifications into account and define a notion of logical equivalence suitable for assessing these translations.

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.

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.

scholarly journals Extensions in graph normal form

2020 ◽  
Author(s):  
Michał Walicki
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
Propositional Logic ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Special Cases

Abstract Graph normal form, introduced earlier for propositional logic, is shown to be a normal form also for first-order logic. It allows to view syntax of theories as digraphs, while their semantics as kernels of these digraphs. Graphs are particularly well suited for studying circularity, and we provide some general means for verifying that circular or apparently circular extensions are conservative. Traditional syntactic means of ensuring conservativity, like definitional extensions or positive occurrences guaranteeing exsitence of fixed points, emerge as special cases.

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