A General Completeness Result in Refinement

2000 ◽  
pp. 201-218 ◽  
Author(s):  
Yoshiki Kinoshita ◽  
John Power
Keyword(s):  
Completeness Result

scholarly journals Justification announcements in discrete time. Part II: Frame definability results

2018 ◽  
Vol 27 (5) ◽  
pp. 671-692
Author(s):  
Grigory K Olkhovikov
Keyword(s):  
Discrete Time ◽  
Time Structure ◽  
Stit Logic ◽  
Completeness Result ◽  
Time Part

Abstract In Part I of this paper we presented a Hilbert-style system $\Sigma _D$ axiomatizing stit logic of justification announcements interpreted over models with discrete time structure. In this part, we prove three frame definability results for $\Sigma _D$ using three different definitions of a frame plus another version of completeness result.

Completeness of category-based equational deduction

1995 ◽  
Vol 5 (1) ◽  
pp. 9-40 ◽  
Author(s):  
Răzvan Diaconescu
Keyword(s):  
Logic Programming ◽  
Model Theory ◽  
Constraint Logic Programming ◽  
Horn Clause ◽  
Abstract Model ◽  
Abstract Model Theory ◽  
Completeness Result ◽  
Semantic Treatment ◽  
Theory Framework ◽  
Deduction Modulo

Equational deduction is generalised within a category-based abstract model theory framework, and proved complete under a hypothesis of quantifier projectivity, using a semantic treatment that regards quantifiers as models rather than variables, and valuations as model morphisms rather than functions. Applications include many- and order-sorted (conditional) equational logics, Horn clause logic, equational deduction modulo a theory, constraint logics, and more, as well as any possible combination among them. In the cases of equational deduction modulo a theory and of constraint logic the completeness result is new. One important consequence is an abstract version of Herbrand's Theorem, which provides an abstract model theoretic foundation for equational and constraint logic programming.

scholarly journals Internal coalgebras in cocomplete categories: Generalizing the Eilenberg–Watts theorem

2020 ◽  
pp. 2150165 ◽  
Author(s):  
Laurent Poinsot ◽  
Hans E. Porst
Keyword(s):  
Left Adjoint ◽  
Adjoint Functors ◽  
Locally Presentable Category ◽  
Completeness Result ◽  
Lawvere Theory

The category of internal coalgebras in a cocomplete category [Formula: see text] with respect to a variety [Formula: see text] is equivalent to the category of left adjoint functors from [Formula: see text] to [Formula: see text]. This can be seen best when considering such coalgebras as finite coproduct preserving functors from [Formula: see text], the dual of the Lawvere theory of [Formula: see text], into [Formula: see text]: coalgebras are restrictions of left adjoints and any such left adjoint is the left Kan extension of a coalgebra along the embedding of [Formula: see text] into [Formula: see text]. Since [Formula: see text]-coalgebras in the variety [Formula: see text] for rings [Formula: see text] and [Formula: see text] are nothing but left [Formula: see text]-, right [Formula: see text]-bimodules, the equivalence above generalizes the Eilenberg–Watts theorem and all its previous generalizations. By generalizing and strengthening Bergman’s completeness result for categories of internal coalgebras in varieties, we also prove that the category of coalgebras in a locally presentable category [Formula: see text] is locally presentable and comonadic over [Formula: see text] and, hence, complete in particular. We show, moreover, that Freyd’s canonical constructions of internal coalgebras in a variety define left adjoint functors. Special instances of the respective right adjoints appear in various algebraic contexts and, in the case where [Formula: see text] is a commutative variety, are coreflectors from the category [Formula: see text] into [Formula: see text].

CONTEXTUAL LOGIC WITH MODALITIES FOR TIME AND SPACE

2008 ◽  
Vol 1 (4) ◽  
pp. 433-458 ◽  
Author(s):  
HAIM GAIFMAN
Keyword(s):  
Natural Language ◽  
Deductive System ◽  
Context Dependence ◽  
Full Detail ◽  
Philosophical Discussion ◽  
Modal Operators ◽  
Completeness Result ◽  
Temporal And Spatial ◽  
Linguistic Usage ◽  
Temporal Modality

We develop a formal apparatus to be used as a tool in analyzing common kinds of context dependence in natural language, and their interaction with temporal and spatial modalities. It is based on context-operators, which act on wffs. The interplay between the various modalities and the context-operators is one of the main targets of the analysis. Statements made by different people at different times in different places, using the same personal temporal and spatial indexicals, can be represented in the system, and can be combined by sentential connectives and be subject to quantification. The use of spatial modality and the suggested treatment of adverbial phrases are new as far as we know. So is a certain variant of temporal modality. In the nontechnical part, consisting of Sections 1 and 2, we discuss the role that formalisms can, in principle, play in the analysis of linguistic usage; this is followed by a philosophical discussion of various kinds of context dependence. The semitechnical part, Section 3, introduces the system's components, the context, and the modal operators, and explains their use via natural language examples. In Section 4 the formal language and its semantics are defined, in full detail. The temporal and spatial sublanguages constitute separate sorts, which interact through the modal operators and the context-operators. A sound deductive system is given and a completeness result is stated, without proof.

A COMPLETENESS RESULT FOR FIXED-POINT ALGEBRAS

1984 ◽  
Vol 30 (32-34) ◽  
pp. 525-532 ◽  
Author(s):  
Franco Montagna
Keyword(s):  
Fixed Point ◽  
Completeness Result

A Preference-Based Approach to Defeasible Deontic Inference

2020 ◽  
Author(s):  
James Delgrande
Keyword(s):  
Propositional Logic ◽  
Polynomial Hierarchy ◽  
Deontic Reasoning ◽  
Desirable Outcome ◽  
Interesting Difference ◽  
Completeness Result ◽  
Syntactic Approach ◽  
Soundness And Completeness

In this paper we present an approach to defeasible deontic inference. Given a set of rules R expressing conditional obligations and a formula A giving contingent information, the goal is to determine the most desirable outcome with respect to this information. Semantically, the rules R induce a partial preorder on the set of models, giving the relative desirability of each model. Then the set of minimal A models characterises the best that can be attained given that A holds. A syntactic approach is also given, in terms of maximal subsets of material counterparts of rules in R, and that yields a formula that expresses the best outcome possible given that A holds. These approaches are shown to coincide, providing an analogue to a soundness and completeness result. Complexity is not unreasonable, being at the second level of the polynomial hierarchy when the underlying logic is propositional logic. The approach yields desirable and intuitive results, including for the various “paradoxes” of deontic reasoning. The approach also highlights an interesting difference in how specificity is dealt with in nonmonotonic and deontic reasoning.

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