Normalized natural deduction systems for some relevant logics I: The logic DW

Abstract

Fitch-style natural deduction was first introduced into relevant logic by Anderson in [1960], for the sentential logic E of entailment and its quantincational extension EQ. This was extended by Anderson and Belnap to the sentential relevant logics R and T and some of their fragments in [ENT1], and further extended to a wide range of sentential and quantified relevant logics by Brady in [1984]. This was done by putting conditions on the elimination rules, →E, ~E, ⋁E and ∃E, pertaining to the set of dependent hypotheses for formulae used in the application of the rule. Each of these rules were subjected to the same condition, this condition varying from logic to logic. These conditions, together with the set of natural deduction rules, precisely determine the particular relevant logic. Generally, this is a simpler representation of a relevant logic than the standard Routley-Meyer semantics, with its existential modelling conditions stated in terms of an otherwise arbitrary 3-place relation R, which is defined over a possibly infinite set of worlds. Readers are urged to refer to Brady [1984], if unfamiliar with the above natural deduction systems, but we will introduce in §2 a modified version in full detail.Natural deduction for classical logic was invented by Jaskowski and Gentzen, but it was Prawitz in [1965] who normalized natural deduction, streamlining its rules so as to allow a subformula property to be proved. (This key property ensures that each formula in the proof of a theorem is indeed a subformula of that theorem.)

The nature of information: a relevant approach

“…Brady (2006) has developed a natural deduction system for relevant logics that allows the derivation of the distribution rules in a non-ad hoc way. He does so by introducing a structural connective into his system and he uses a notion of "streams of proof".…”

Relevance and Conjunction

“…The logic of our paper does not include distribution. For a system that allows the derivation of distribution without a distribution rule, see Brady (2006). Brady's system still includes negation rules that are not strictly speaking introduction or elimination rules, and so does not satisfy Dummett's harmony constraint.…”

“…Thus, it looks as though our interpretation of the proof theory does not justify the^I rule in its full form. 5 In Mares (2004), I argue for the introduction rule on the basis of the intuitiveness of the inference derived in the above argument, that is, from s j= A ! B and s j= A !…”

A natural deduction and its corresponding sequent calculus for positive contraction - less relevant logic

“…In this paper we set up a natural deduction calculus for the logic RW • + , which is positive contraction-less relevant system R (see [1], p. 341), with co-tenability •. Several natural deduction calculi are known for relevant logics, including RW • + , however, all of them, use a kind of relevance numerals in order to keep track of the use of hypotheses (see Anderson and Belnap [1], Dunn [6], [7], Brady [3], Urquhart [18], and Meyer and McRobbie [12]). This is needed to unable the inference of irrelevant formulae.…”

“…(We should note that Kron formulated sequent calculus for a contraction-less relevant logic based on intensional sequences only, where the inference of the distributive law is enabled by the explicit distribution rule; however, Kron's logic is different from RW , see [11]. ) Brady sets up (normalizing Fitch-style) natural deduction calculi, for a wide range of relevant logics, including RW • + in [3]. In order to remove the need for the special distribution rule, Brady allows, like Dunn, several formulae, separated by commas, to be listed on a line of a derivation tree.…”