An algebraic semantics of notional entailment logic Cn

Mathematical Structures in Computer Science, 2008

This paper proposes a new relevant logic B + , which is obtained by adding two binary connectives, intensional conjunction and intensional disjunction , to Meyer-Routley minimal positive relevant logic B + , where and are weaker than fusion • and fission +, respectively. We give Kripke-style semantics for B + , with →, and modelled by ternary relations. We prove the soundness and completeness of the proposed semantics. A number of axiomatic extensions of B + , including negation-extensions, are also considered, together with the corresponding semantic conditions required for soundness and completeness to be maintained. † Dunn's general approach is algebraic, where each logical connective is characterised as an operation on distributive lattices, which 'distributes' in each of its places over at least one of ∧ and ∨, leaving ∧ or Y. Gao and J. Cheng 146 • † , and shares with + ‡. Then, additional axioms or rules can be added to make coincide with •, and with +. This qualifies and as weaker versions of intensional conjunction and disjunction, respectively. To give a semantics for B + , we apply Dunn's strategy (Dunn 1990), that is, we use n + 1-placed accessibility relations to model n-placed connectives. The semantics is defined by adapting and extending the traditional relational semantics for relevant logics. There are four ternary relations: R 1 and R 2 for →; S 1 for ; and S 2 for. To construct canonical models, as well as theories, we define dualtheories and antidualtheories such that R 1 , R 2 , S 1 , S 2 are canonically defined as derivatives of operations on theories and anti-dualtheories. The crucial tools for completeness are extensions or reductions of a given theory or anti-dualtheory to a prime theory. Then, by well-known standard techniques, together with our extra definitions, we can establish the soundness and completeness of the proposed semantics for B +. Furthermore, we consider a number of axiomatic extensions of B + (including negation-extensions with negation modelled by the Routley ' * '-operation), together with the corresponding semantic conditions to ensure that soundness and completeness are maintained. 2. The basic system B + 2.1. An axiom system for B + B + is expressed in a language L, which has the two-place connectives →, ∧, ∨, and , parentheses (and), and a stock of propositional variables p, q, r, ... Formulas are defined recursively in the usual manner. We use the following scope conventions: the connectives are ranked , , ∧, ∨, → in order of increasing scope (that is, binds more strongly than , binds more strongly than ∧, and so on), otherwise, association is to the left. A, B, C, ... will be used to range over arbitrary formulas. We begin by giving an axiom system for B + , which is defined in the same way as that of Priest and Sylvan (1992) and Restall (1993) § : Axioms

2015

In the paper I present and discuss a development of Lewis comparative similarity semantics for conditional logic based on a material or algorithmic definition of the similarity relation between possible worlds. Four conditionals are defined: conceptual, physical, actual and contingent implication, the first, second and third corresponding to Lewis ’ VW logic, the last one to Lewis ’ VC logic. I. Syntax The conditional statements in natural languages are usually reduced to two general types: subjunctive and indicative1 (and, perhaps, some special subtype as counterlegals, might-conditionals, even-kind conditionals...). An example of a subjunctive statement is “If the windows were open the temperature would be lower”, which Lewis ’ theory claims to model in his counterfactual logic [6]. An example of an indicative statement is “If it rains the streets are wet”, which belongs to the class of sentences Burks tries to model with his system of causal statements [2]. A brief description of...

Semantics and Pragmatics

It is argued that the notion of classical entailment faces two problems, the second argument projection problem and the P-to-Q problem, which arise because classical entailment is not designed to handle partial functions. It is shown that while the second argument projection problem can be solved either by flattening the syntactic tree or with naïve multi-valued logics, the P-to-Q problem cannot. Both problems are solved by introducing a new notion of entailment that is defined in terms of Strawson entailment (in the sense of von Fintel 1999, 2001).

2020

Generally speaking, there are two categories of semantics theory: model-theoretic approach and proof-theoretic approach. In the first part of this paper, I will briefly analyze some inadequacies related to these two approaches, and promote an alternative relational approach, which bases semantic notions on relations between expressions. A brief discussion in general for this alternative will be provided. In the second part, I will provide a solid mathematical framework to the study of logical meanings, and show its connection with the other two approaches.

ABSTRACT Expressions, words and symbols without reference to something else which could be called their meanings are semantically helpless. But not all expressions and words refer; some even come with ambiguities and equivocations like Golden Mountain, Chimera etc., however, any symbol which does not refer could not properly be called a symbol. So because every symbol necessarily refers to something definite, it is not the case that ambiguities and equivocations would sneak into symbolic expressions. Hence, logic becomes that science which prefers symbolic or artificial or formal language to natural language. Therefore, since “meaning” or semantics is a central focus of logic together with syntax, we attempted in this work to obtain a logical derivation of it in the symbolic language of logic.

1994

Arhe, 2021

In this paper, we follow Gödel's remarks on an envisioned theory of concepts to determine which properties should a logical basis of such a theory have. The discussion is organized around the question of suitability of the classical predicate calculus for this role. Some reasons to think that classical logic is not an appropriate basis for the theory of concepts, will be presented. We consider, based on these reasons, which alternative logical system could fare better as a logical foundation of, in Gödel's opinion, the most important theory in logic yet to be developed. This paper should, in particular, motivate the study of partial predicates in a certain system of three-valued logic, as a promising starting point for the foundation of the theory of concepts.

The most general questions are what modern logic regards the logical role of compositionality, how it works in two-component logical semantics. After showing different versions of the compositionality of natural language we analyze the possible appearances of the principle of compositionality in two-component logical semantics. Finally, some of the most fundamental notions of intensional logical semantics are given in the way of maintaining the priority of compositionality concerning sense.

Our aim is to present the algebra of concepts in two formal languages. First, after introducing a primary relation between concepts, which is subsumption, we shall specify in a language that uses quantifiers, the Boolean algebra of general concepts. Next, we shall note down the same algebra in simplified non-quantifying language, in order to use it as basis for two specific implementations, i.e. to create the Boolean algebras of deontic concepts and axiological concepts.

2017

Introduction 1 1 Generalities on abstract logics and sentential logics 13 2 Abstract logics as models of sentential logics 29 2.1 Models and full models 29 2.2 5-algebras 34 2.3 The lattice of full models over an algebra 38 2.4 Full models and metalogical properties 42 3 Applications to protoalgebraic and algebraizable logics 55 4 Abstract logics as models of Gentzen systems 69 4.1 Gentzen systems and their models 70 4.2 Selfextensional logics with Conjunction 80 4.3 Selfextensional logics having the Deduction Theorem \ • • • 89 5 Applications to particular sentential logics 97 5.1 Some non-protoalgebraic logics 99 5.1.1 CPC AV , the {A, V}-fragment of Classical Logic 99 5.1.2 The logic of lattices 101 5.1.3 Belnap's four-valued logic, and other related logics 102 5.1.4 The implication-less fragment of IPC and its extensions .... 104 5.2 Some Fregean algebraizable logics 105 5.2.1 Alternative Gentzen systems adequate for IPC_ not having the full Deduction Theorem 107 5.3 Some modal logics 108 5.3.1 A logic without a strongly adequate Gentzen system Ill vi Contents 5.4 Other miscellaneous examples. .. Ill 5.4.1 Two relevance logics 112 5.4.2 Sette's paraconsistent logic 113 5.4.3 Tetravalent modal logic 114 5.4.4 Logics related to cardinality restrictions in the Deduction Theorem 115 Bibliography 119