First-degree entailments and information.

Abstract

This paper presents two interpretations of the first-degree (FD) entailments of propositional logic that are based directly on the notion of inclusion of information.* It is proved in Section 2 that one of these interpretations exactly characterizes the tautological entailments of [2], while the other exactly characterizes the valid arguments of classical truth-functional logic. In Section 3, following a line of reasoning suggested in part by consideration of these interpretations, it is argued that the claim that relevance logic better captures our intuitions about entailment than classical logic is false. Section 1 presents natural-deduction formulations of both classical and relevant FD entailments that are used in subsequent proofs. Two systems of natural deduction rulesI take -, v, and & as primitive connectives and assume that sentential letters are specified. Wffs are as usual. I let A, B, . . ., F (with or without numerical subscripts) range over wffs and let M and TV range over finite nonempty sets of wffs.Given any finite nonempty set of wffs M, an infinite set of wffs XM is defined recursively as follows:1. If A eM 9 then ,4 e X M . 2. A & B e X M iff A e X M and B e X M . 3. If A e X M or B e X M , then A v B e X M .

Informational Semantics as a Third Alternative?

“…This formal point reinforces our informal remark that as far as (C) is concerned, there is no difference between explaining consequence in terms of content-nonexpansion (i.e. the requirement that CONTðAjCÞ ¼ ;) and explaining it in terms of truth-preservation (see Hanson (1980) for a similar diagnosis). Similarly, as suggested in Beall et al (2011, 3.2), when we identify the first two indices in the ternary relation (and make suitable modifications to ensure the proper behaviour of negation), (R) can be used to individuate intuitionist information because s then ranges over constructions.…”

Outline of a Theory of Strongly Semantic Information

“…For a positive account, in terms of the set of all statedescriptions entailed by σ, see Hanson [1980]. Both accounts equally lead to the formulation of BCP.…”

Outline of a Theory of Strongly Semantic Information

“…8 Popper [1935] was one of the first to suggest that the amount of semantic information in a statement can be analysed in terms of the number of alternative possibilities excluded by that statement, see also Popper [1962]. For a positive account, in terms of the set of all statedescriptions entailed by σ, see Hanson [1980]. Both accounts equally lead to the formulation of BCP.…”