The theory of truth tabular connectives, both truth functional and modal

1997 ◽  
Vol 31 (4) ◽  
pp. 593-608 ◽  
Author(s):  
Gerald J. Massey
Keyword(s):  
Modal Logic ◽  
Duality Theorem ◽  
Functional Completeness ◽  
Partial Truth ◽  
Intuitive Concept ◽  
Close Investigation ◽  
General Duality ◽  
Truth Tables

Briefly, the following things are done in this paper. An informal exposition of the significance of partial truth tables in modal logic is presented. The intuitive concept of a truth tabular sentence connective is examined and made precise. From among a vast assortment of truth tabular connectives, the semantically well-behaved ones (called regular connectives) are singled out for close investigation. The familiar concept of functional completeness is generalized to all sets of regular connectives, and the functional completeness (or incompleteness) of selected sets of connectives is established. Some modal analogues of Sheffer's stroke are presented, i.e., single connectives are introduced which serve to define not only the truth functional connectives but all the regular modal connectives as well. The notion of duality is extended to all regular connectives and a general duality theorem is proved. Lastly, simplified proofs are given of several metatheorems about the system S5.

Binary connectives functionally complete by themselves in S5 modal logic

1967 ◽  
Vol 32 (1) ◽  
pp. 91-92 ◽  
Author(s):  
Gerald J. Massey
Keyword(s):  
Modal Logic ◽  
Modal Theory ◽  
Binary Operator ◽  
Complete Set ◽  
Complete Sets ◽  
Image Position ◽  
Open Question ◽  
Truth Tables

This paper answers affirmatively the open question of Massey [1] concerning the existence of binary connectives functionally complete by themselves in two-valued truth tabular logic, i.e. in the modal theory S5. Since {∼, ⊃, ◊} is a functionally complete set of connectives (Massey [1, § 4]), the following definitions show that the binary operator ф, the semantics of which is given below, is functionally complete by itself: It is left to the reader to verify, by means of complete sets of truth tables (see Massey [1, §§ 1 and 3]), that the foregoing definitions are correct.

scholarly journals A general duality theorem for the Monge–Kantorovich transport problem

2012 ◽  
Vol 209 (2) ◽  
pp. 151-167 ◽  
Author(s):  
Mathias Beiglböck ◽  
Christian Léonard ◽  
Walter Schachermayer
Keyword(s):  
Duality Theorem ◽  
Transport Problem ◽  
General Duality

A general duality theorem for marginal problems

1995 ◽  
Vol 101 (3) ◽  
pp. 311-319 ◽  
Author(s):  
D. Ramachandran ◽  
L. R�schendorf
Keyword(s):  
Duality Theorem ◽  
General Duality

scholarly journals The Whole Truth about Partial Truth Tables

2020 ◽  
Vol 10 (02) ◽  
pp. 192-219
Author(s):  
Keith Burgess-Jackson
Keyword(s):  
Partial Truth ◽  
Truth Tables

Modal operators and functional completeness, II

1977 ◽  
Vol 42 (3) ◽  
pp. 391-399 ◽  
Author(s):  
S. K. Thomason
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Kripke Semantics ◽  
Functional Completeness ◽  
Possible World ◽  
States Of Affairs ◽  
Propositional Modal Logic ◽  
Modal Operators ◽  
Fixed Frame ◽  
State Of Affairs

In the Kripke semantics for propositional modal logic, a frame W = (W, ≺) represents a set of “possible worlds” and a relation of “accessibility” between possible worlds. With respect to a fixed frame W, a proposition is represented by a subset of W (regarded as the set of worlds in which the proposition is true), and an n-ary connective (i.e. a way of forming a new proposition from an ordered n-tuple of given propositions) is represented by a function fw: (P(W))n → P(W). Finally a state of affairs (i.e. a consistent specification whether or not each proposition obtains) is represented by an ultrafilter over W. {To avoid possible confusion, the reader should forget that some people prefer the term “states of affairs” for our “possible worlds”.}In a broader sense, an n-ary connective is represented by an n-ary operatorf = {fw∣ W ∈ Fr}, where Fr is the class of all frames and each fw: (P(W))n → P(W). A connective is modal if it corresponds to a formula of propositional modal logic. A connective C is coherent if whether C(P1,…, Pn) is true in a possible world depends only upon which modal combinations of P1,…,Pn are true in that world. (A modal combination of P1,…,Pn is the result of applying a modal connective to P1,…, Pn.) A connective C is strongly coherent if whether C(P1, …, Pn) obtains in a state of affairs depends only upon which modal combinations of P1,…, Pn obtain in that state of affairs.

Modal Logic

1980 ◽  
Author(s):  
Brian F. Chellas
Keyword(s):  
Modal Logic
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