Grafted frames and S1 -completeness

1999 ◽  
Vol 64 (3) ◽  
pp. 1324-1338 ◽  
Author(s):  
Beihai Zhou
Keyword(s):  
Modal Logic ◽  
Relevance Logic ◽  
Value Assignment ◽  
Combined Logics ◽  
Modal Model ◽  
Truth Value ◽  
Modal Frame ◽  
Relevance Models ◽  
New System ◽  
Relevance Model

AbstractA grafted frame is a new kind of frame which combines a modal frame and some relevance frames. A grafted model consists of a grafted frame and a truth-value assignment. In this paper, the grafted frame and the grafted model are constructed and used to show the completeness of S1. The implications of S1-completeness are discussed.A grafted frame does not combine two kinds of frames simply by putting relations defined in the components together. That is, the resulting grafted frame is not in the form of <W,R,R′>, or more generally, in the form of <W, R, R′,R″>,…>, which consists of a non-empty set with several relations defined on it.1 Rather, it resembles the construction of fibering proposed by D. M. Gabbay and M. Finger (see [4] and [3]). On a grafted frame, some modal worlds, which belong to the initial modal frame, are attached by some relevance frames.However, these two semantics have important differences. Consider the combined semantics involving semantics of relevance logic and modal logic. A fibred model and a grafted model proposed in this paper differ in the following respects. First, a fibred model is constructed from a class of modal models and a class of relevance models. A grafted model consists of a grafted frame and a truth-value assignment, where the grafted frame is constructed from a modal frame and some relevance frames, and the assignment is a union of a modal truth-value assignment VM and some relevance truth-value assignments VR. VM (VR) defined in this paper is not the same as the assignment contained in a modal (relevance) model. Second, in a fibred model each relevance world is associated (or fibred) with a modal model and each modal world with a relevance model.2 To be the grafted frame on which a grafted model is based, it is enough to have some modal worlds attached by some relevance frames. Moreover, no relevance world is associated with a modal frame in the grafted frame. Third, fibred models are intended to provide an appropriate semantics to combined logics. Grafted frames and grafted models are inspired to characterize S1, which, containing only one modality □, is not a combined logic. It is shown in this paper that S1 is sort of a meta-logic of the intersection of S0.4 and F, where S0.4, a new system proposed in this paper, is in turn a meta-logic of the relevance logic.

Georg Henrik von Wright. Form and content in logic. A revised reprint of XV 58(2), 199(2), 280(2). Logical studies. International library of psychology, philosophy and scientific method. The Humanities Press, Inc., New York, and Routledge & Kegan Paul Ltd., London, 1957, pp. 1–21. - Georg Henrik von Wright. On the idea of logical truth (I). A revised reprint of XV 58(1), 199(1), 280(1). Logical studies. International library of psychology, philosophy and scientific method. The Humanities Press, Inc., New York, and Routledge & Kegan Paul Ltd., London, 1957, pp. 22–43. - Georg Henrik von Wright. On double quantification. A revised reprint of XVII 201. Logical studies. International library of psychology, philosophy and scientific method. The Humanities Press, Inc., New York, and Routledge & Kegan Paul Ltd., London, 1957, pp. 44–57. - Georg Henrik von Wright. Deontic logic. A revised reprint of XVII 140. Logical studies. International library of psychology, philosophy and scientific method. The Humanities Press, Inc., New York, and Routledge & Kegan Paul Ltd., London, 1957, pp. 58–74. - Georg Henrik von Wright. Interpretations of modal logic. A revised reprint of XVIII 176. Logical studies. International library of psychology, philosophy and scientific method. The Humanities Press, Inc., New York, and Routledge & Kegan Paul Ltd., London, 1957, pp. 75–88. - Georg Henrik von Wright. A new system of modal logic. A revised version of XIX 66. Logical studies. International library of psychology, philosophy and scientific method. The Humanities Press, Inc., New York, and Routledge & Kegan Paul Ltd., London, 1957, pp. 89–126. - Georg Henrik von Wright. On conditionals. Logical studies. International library of psychology, philosophy and scientific method. The Humanities Press, Inc., New York, and Routledge & Kegan Paul Ltd., London, 1957, pp. 127–165. - Georg Henrik von Wright. The concept of entailment. Logical studies. International library of psychology, philosophy and scientific method. The Humanities Press, Inc., New York, and Routledge & Kegan Paul Ltd., London, 1957, pp. 166–191.

1970 ◽  
Vol 35 (3) ◽  
pp. 460-462
Author(s):  
Timothy Smiley
Keyword(s):  
New York ◽  
Modal Logic ◽  
Scientific Method ◽  
Deontic Logic ◽  
Logical Truth ◽  
Double Quantification ◽  
New System

A completeness theorem for symmetric product phase spaces

2004 ◽  
Vol 69 (2) ◽  
pp. 340-370 ◽  
Author(s):  
Thomas Ehrhard
Keyword(s):  
Linear Logic ◽  
Denotational Semantics ◽  
Completeness Theorem ◽  
Symmetric Product ◽  
Product Phase ◽  
Phase Models ◽  
Truth Value ◽  
Strict Extension ◽  
New System

Abstract.In a previous work with Antonio Bucciarelli, we introduced indexed linear logic as a tool for studying and enlarging the denotational semantics of linear logic. In particular, we showed how to define new denotational models of linear logic using symmetric product phase models (truth-value models) of indexed linear logic. We present here a strict extension of indexed linear logic for which symmetric product phase spaces provide a complete semantics. We study the connection between this new system and indexed linear logic.

Modal Frame Classes Revisited

1993 ◽  
Vol 18 (2-4) ◽  
pp. 307-317
Author(s):  
Johan Van Benthem
Keyword(s):  
Modal Logic ◽  
Alternative Model ◽  
Algebraic Proof ◽  
Versus Model ◽  
Theoretic Argument ◽  
Modal Frame ◽  
General Perspective

We re-analyze the original algebraic proof of the Goldblatt-Thomason theorem characterizing modally definable frame classes, providing an alternative model-theoretic argument. The analysis also provides a more general perspective on the use of algebraic versus model-theoretic methods in Modal Logic.

A New System of Modal Logic

2017 ◽  
pp. 89-126 ◽  
Author(s):  
Georg Henrik von Wright
Keyword(s):  
Modal Logic ◽  
New System

Imperative Wh-Interrogatives: Pragmatic Analysis and Empirical Evidence from German

2014 ◽  
Vol 6 (2) ◽  
pp. 263-305
Author(s):  
Sonja Müller
Keyword(s):  
Empirical Evidence ◽  
Pragmatic Approach ◽  
Value Assignment ◽  
Semantic Account ◽  
Illocutionary Acts ◽  
Pragmatic Analysis ◽  
Truth Value ◽  
Interrogative Clauses ◽  
Verbal Mood

This paper is concerned with wh-interrogative clauses in which the imperative verbal mood occurs. According to some authors (cf. in particular Lohnstein, 2000; Chiba, 2009), such constructions are ill-formed. Lohnstein offers a semantic account (based on his theory of sentence mood) ruling out imperative (wh-)interrogatives by arguing for the impossibility of partitioning propositions which are not open for truth value assignment. Chiba argues for the ungrammaticality of such structures by assuming that sentence types (as defined by Sadock & Zwicky, 1985) cannot be combined. This paper will show that unacceptable as well as acceptable imperative wh-interrogatives exist and that, therefore, both approaches make the wrong predictions. A pragmatic approach is developed which argues for the realization of incompatible illocutionary acts. This account makes it possible to derive the unacceptability of certain imperative wh-interrogatives and to allow the acceptable cases which can be observed. An advantage of referring to illocutionary acts is that it becomes possible to speak about various subtypes of the erothetic illocution which can be proven to play a role in distinguishing between acceptable and unacceptable imperative wh-interrogatives. The results of controlled acceptability judgements confirm this assumption.

scholarly journals Meta-inferences and Supervaluationism

2021 ◽  
Author(s):  
Luca Incurvati ◽  
Julian J. Schlöder
Keyword(s):  
Modal Logic ◽  
Deductive Reasoning ◽  
Higher Order ◽  
Proof System ◽  
Inference Rules ◽  
Consequence Relation ◽  
Normal Modal Logic ◽  
First Order ◽  
Truth Value ◽  
Crucial Ingredient

AbstractMany classically valid meta-inferences fail in a standard supervaluationist framework. This allegedly prevents supervaluationism from offering an account of good deductive reasoning. We provide a proof system for supervaluationist logic which includes supervaluationistically acceptable versions of the classical meta-inferences. The proof system emerges naturally by thinking of truth as licensing assertion, falsity as licensing negative assertion and lack of truth-value as licensing rejection and weak assertion. Moreover, the proof system respects well-known criteria for the admissibility of inference rules. Thus, supervaluationists can provide an account of good deductive reasoning. Our proof system moreover brings to light how one can revise the standard supervaluationist framework to make room for higher-order vagueness. We prove that the resulting logic is sound and complete with respect to the consequence relation that preserves truth in a model of the non-normal modal logic NT. Finally, we extend our approach to a first-order setting and show that supervaluationism can treat vagueness in the same way at every order. The failure of conditional proof and other meta-inferences is a crucial ingredient in this treatment and hence should be embraced, not lamented.

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