scholarly journals Logical Consequence and First-Order Soundness and Completeness: A Bottom Up Approach

2011 ◽  
Vol 52 (1) ◽  
pp. 75-93 ◽  
Author(s):  
Eli Dresner
Keyword(s):  
Logical Consequence ◽  
Bottom Up ◽  
First Order ◽  
Soundness And Completeness

scholarly journals Elementary-base cirquent calculus II: Choice quantifiers

2020 ◽  
Author(s):  
Giorgi Japaridze
Keyword(s):  
Proof Theory ◽  
Predicate Calculus ◽  
First Order ◽  
Complete Axiomatization ◽  
Calculus I ◽  
Soundness And Completeness ◽  
Classical Predicate

Abstract Cirquent calculus is a novel proof theory permitting component-sharing between logical expressions. Using it, the predecessor article ‘Elementary-base cirquent calculus I: Parallel and choice connectives’ built the sound and complete axiomatization $\textbf{CL16}$ of a propositional fragment of computability logic. The atoms of the language of $\textbf{CL16}$ represent elementary, i.e. moveless, games and the logical vocabulary consists of negation, parallel connectives and choice connectives. The present paper constructs the first-order version $\textbf{CL17}$ of $\textbf{CL16}$, also enjoying soundness and completeness. The language of $\textbf{CL17}$ augments that of $\textbf{CL16}$ by including choice quantifiers. Unlike classical predicate calculus, $\textbf{CL17}$ turns out to be decidable.

scholarly journals EVERYONE KNOWS THAT SOMEONE KNOWS: QUANTIFIERS OVER EPISTEMIC AGENTS

2019 ◽  
Vol 12 (2) ◽  
pp. 255-270 ◽  
Author(s):  
PAVEL NAUMOV ◽  
JIA TAO
Keyword(s):  
Modal Logic ◽  
Epistemic Logic ◽  
Possible Worlds ◽  
Completeness Theorem ◽  
Kripke Semantics ◽  
Possible Worlds Semantics ◽  
Distributed Knowledge ◽  
First Order ◽  
Soundness And Completeness ◽  
First Order Modal Logic

AbstractModal logic S5 is commonly viewed as an epistemic logic that captures the most basic properties of knowledge. Kripke proved a completeness theorem for the first-order modal logic S5 with respect to a possible worlds semantics. A multiagent version of the propositional S5 as well as a version of the propositional S5 that describes properties of distributed knowledge in multiagent systems has also been previously studied. This article proposes a version of S5-like epistemic logic of distributed knowledge with quantifiers ranging over the set of agents, and proves its soundness and completeness with respect to a Kripke semantics.

scholarly journals A Condensed Goal-Independent Fixpoint Semantics Modeling the Small-Step Behavior of Rewriting

10.29007/cpt8 ◽  
2018 ◽  
Author(s):  
Marco Comini ◽  
Luca Torella
Keyword(s):  
Term Rewriting ◽  
Small Step ◽  
Top Down ◽  
Term Rewriting Systems ◽  
Bottom Up ◽  
Rewriting Systems ◽  
Term Rewriting System ◽  
Soundness And Completeness ◽  
Full Class ◽  
Minimal Information

In this paper we present a novel condensed narrowing-like semantics that contains the minimal information which is needed to describe compositionally all possible rewritings of a term rewriting system. We provide its goal-dependent top-down definition and, more importantly, an equivalent goal-independent bottom-up fixpoint characterization.We prove soundness and completeness w.r.t. the small-step behavior of rewriting for the full class of term rewriting systems.

Truthmakers and Consequence

2017 ◽  
Author(s):  
Neil Tennant
Keyword(s):  
Winning Strategy ◽  
Logical Consequence ◽  
Game Plan ◽  
Soundness And Completeness ◽  
Truthmaker Semantics ◽  
Definition Of ◽  
Point Of Departure

We compare Tarski’s notion of logical consequence (preservation of truth) with that of Prawitz (transformability of warrants for assertion). The latter is our point of departure for a definition of consequence in terms of the transformability of truthmakers (verifications) relative to all models. A sentence’s Tarskian truth-in-M coincides with its having an M-relative truthmaker. An M-relative truthmaker serves as a winning strategy or game plan for player T in the ‘material game’ played on that sentence against the background of the model M. We enter conjectures about soundness and completeness of Classical Core Logic with respect to the notion of consequence that results when the domain is required to be decidable. We consider whether the truthmaker semantics threatens a slide to realism. We work with examples of core proofs whose premises are given M-relative truthmakers; and show how these can be systematically transformed into a truthmaker for the proof’s conclusion.

Logical consequence relations in logics of monotone predicates and logics of antitone predicates

2017 ◽  
pp. 021-029
Author(s):  
O.S. Shkilniak ◽  
Keyword(s):  
Logical Consequence ◽  
Consequence Relations ◽  
Consequence Relation ◽  
First Order ◽  
Different Types ◽  
Composition Algebras

Logical consequence is one of the fundamental concepts in logic. In this paper we study logical consequence relations for program-oriented logical formalisms: pure first-order composition nominative logics of quasiary predicates. In our research we are giving special attention to different types of logical consequence relations in various semantics of logics of monotone predicates and logics of antitone predicates. For pure first-order logics of quasiary predicates we specify composition algebras of predicates, languages, interpretation classes (sematics) and logical consequence relations. We obtain the pairwise distinct relations: irrefutability consequence P |= IR , consequence on truth P |= T , consequence on falsity P |= F, strong consequence P |= TF in P-sеmantics of partial singlevalued predicates and strong consequence R |= TF in R-sеmantics of partial multi-valued predicates. Of the total of 20 of defined logical consequence relations in logics of monotone predicates and of antitone predicates, the following ones are pairwise distinct: PE |= IR, PE |= T, PE |= F, PE |= TF, RM |= T, RM |= F, RM |= TF. A number of examples showing the differences between various types of logical consequence relations is given. We summarize the results concerning the existence of a particular logical consequence relation for certain sets of formulas in a table and determine interrelations between different types of logical consequence relations.

A Logical Theory of Truth-Makers and Falsity-Makers

2018 ◽  
Author(s):  
Neil Tennant
Keyword(s):  
Logical Consequence ◽  
Suitable Method ◽  
Elimination Rule ◽  
Logical Theory ◽  
Relative Truth ◽  
Logical Operator ◽  
First Order ◽  
Introduction Rule ◽  
Infinite Case ◽  
Ways Of Being

We explicate the different ways that a first-order sentence can be true (resp., false) in a model M, as formal objects, called (M-relative) truth-makers (resp., falsity-makers). M-relative truth-makers and falsity-makers are co-inductively definable, by appeal to the “atomic facts” in M, and to certain rules of verification and of falsification, collectively called rules of evaluation. Each logical operator has a rule of verification, much like an introduction rule; and a rule of falsification, much like an elimination rule. Applications of the rules (∀) and (∃) involve infinite furcation when the domain of M is infinite. But even in the infinite case, truth-makers and falsity-makers are tree-like objects whose branches are at most finitely long. A sentence φ is true (resp., false) in a model M (in the sense of Tarski) if and only if there existsπ such that π is an M-relative truth-maker (resp., falsity-maker) for φ. With “ways of being true” explicated as these logical truthmakers, one can re-conceive logical consequence between given premises and a conclusion. It obtains just in case there is a suitable method for transforming M-relative truthmakers for the premises into an M-relative truthmaker for the conclusion, whatever the model M may be.

scholarly journals A Logic for Vagueness

2010 ◽  
Vol 8 ◽  
Author(s):  
John Slaney
Keyword(s):  
Natural Deduction ◽  
Kripke Semantics ◽  
Substructural Logic ◽  
Relational Semantics ◽  
Deduction System ◽  
First Order ◽  
Natural Deduction System ◽  
Soundness And Completeness ◽  
Propositional System

This paper presents F, substructural logic designed to treat vagueness. Weaker than Lukasiewicz’s infinitely valued logic, it is presented first in a natural deduction system, then given a Kripke semantics in the manner of Routley and Meyer's ternary relational semantics for R and related systems, but in this case, the points are motivated as degrees to which the truth could be stretched. Soundness and completeness are proved, not only for the propositional system, but also for its extension with first-order quantifiers. The first-order models allow not only objects with vague properties, but also objects whose very existence is a matter of degree.

scholarly journals CIRCULARITY IN SOUNDNESS AND COMPLETENESS

2014 ◽  
Vol 20 (1) ◽  
pp. 24-38
Author(s):  
RICHARD KAYE
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Foundational Research ◽  
Soundness And Completeness

AbstractWe raise an issue of circularity in the argument for the completeness of first-order logic. An analysis of the problem sheds light on the development of mathematics, and suggests other possible directions for foundational research.

scholarly journals Tractability of Theory Patching

1998 ◽  
Vol 8 ◽  
pp. 39-65 ◽  
Author(s):  
S. Argamon-Engelson ◽  
M. Koppel
Keyword(s):  
Domain Theory ◽  
Theory Revision ◽  
First Order ◽  
Soundness And Completeness ◽  
Training Examples ◽  
Order Domain ◽  
Domain Concept

In this paper we consider the problem of `theory patching', in which we are given a domain theory, some of whose components are indicated to be possibly flawed, and a set of labeled training examples for the domain concept. The theory patching problem is to revise only the indicated components of the theory, such that the resulting theory correctly classifies all the training examples. Theory patching is thus a type of theory revision in which revisions are made to individual components of the theory. Our concern in this paper is to determine for which classes of logical domain theories the theory patching problem is tractable. We consider both propositional and first-order domain theories, and show that the theory patching problem is equivalent to that of determining what information contained in a theory is `stable' regardless of what revisions might be performed to the theory. We show that determining stability is tractable if the input theory satisfies two conditions: that revisions to each theory component have monotonic effects on the classification of examples, and that theory components act independently in the classification of examples in the theory. We also show how the concepts introduced can be used to determine the soundness and completeness of particular theory patching algorithms.

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