The logical structure of linguistic commitment II: Systems of relevant commitment entailment

In ‘The Logical Structure of Linguistic Commitment I” (The Journal of Philovophicul Logic 23 (1994). 369400), we sketch a linguistic theory (inspired by Brandom’s Making it Explicit) which includes an “exprcssivist” account of the implication connective, -+: the role of + is to “make explicit” the inferential proprieties among possible commitments which proprieties determine, in part, the significances of sentences. This motivates reading (A -+ B) as “commitment to A is, in part, commitment to L3”. Our project is to study the logic of 4. LSLC I approximates (A + L?) as “anyone committed to A is committed to B”, ignoring issues of whether A is rekvant to B. The present paper includes considerations of relevance, motivating systems of relevant commitment entailment related to the systems of commitment entailment of LSLC I. We also consider the relevance logics that result from a commitment reading of Fine’s semantics for relevance logics, a reading that Fine suggests. “The Logical Structure of Linguistic Commitment I” (LSLC I) sketches a linguistic theory (inspired by Brandom, 1983, 1985 and 1994) according to which the significance of an expression is, in part, cashed out in terms of its role in the inferential structure of language. This theory emphasizes the importance of asserting as a linguistic act: when a person makes an assertion, she undertakes certain commitments to justify the assertion, and its consequences and if these commitments are appropriately discharged, she secures prima facie entitlement to the assertion. This motivates consideration of an entailment-like connective “+“, where “A + B” is to be read as “commitment to A is, in part, commitment to B”. Given such a connective, to say (correctly) “A + II” is, in part, to make explicit the inferential moves to which the members of the linguistic community are committed, and thereby to shed light on the meaning or significance of the terms occurring in A and B. Our project in LSLC I and in the present paper is to study the logical structure of “+“. LSLC I notes that a claim of the form “A -+ I?” suggests two things: “universality of agent”, i.e. that anyone committed to A is committed to B; and “commitment relevance”, i.e. that commitment to A is relevant to commitment to B. LSLC I ignores considerations of relevance, however, and formulates four logics of non-relevant Journal of Philosophical Logic 25: 425449, 1996. @ 1996 Kluwer Academic Publishers. Printed in the Netherlands. 426 MARK LANCE AND PHILIP KREh4ER commitment entailment. Differences among these logics reflect different intuitions concerning the upshot of embedded commitment claims. The present paper formulates similar logics, this time of relevant commitment entailment. (As in LSLC I, we include conjunction, “&“, to facilitate axiomatization.) Although our main object of formal and philosophical concern is the concept of commitment entailment, we report a pleasing connection with earlier work. Anderson and Belnap’s original investigation of systems of relevant entailment led them to endorse the system E as the correct analysis of relevant entailment and to construct this system by combining, in a certain sense, the modal system S4 and the relevant system R. Later work in the area of relevance logic, however, has tended to diverge more and more from this traditional project. Many (including Lance, 1988a) have argued against Anderson and Belnap’s interpretation of the arrow of R. As for E, those still endorsing the project of constructing an entailment system from R and a modal system have tended to reject E in favor of the closely related RN. Most other researchers in the area have rejected strong relevance systems like E altogether, arguing for and investigating weaker systems with little relation to R. Thus E and the related project of demonstrating its joint relevanccmodal ancestry has been relegated, if not to the trash-heap of history, at least to a provincial sector of the relevance logic commonwealth. The present paper provides a partial rehabilitation of E-,k and R. LSLC I motivated the modal system S4 as one plausible logic of non-relevant commitment entailment, and we show here that adding considerations of relevance to this yields E as one possible formalization of the notion of relevant commitment entailment. Further, the corresponding algebraic semantics offers a natural and important interpretation of both E and R. This rehabilitation is only partial, alas, since our endorsement of the principles of asscrtional commitment underlying S4 (and, hence, E) is tentative at best (and we will diverge even further when negation is added to the language). Nonetheless, this work should cast new light and generate new interest in an old comrade in the struggle against classical logic as the single legitimate entailment system. Note: for reasons made clear by the main results of LSLC I (which are summarized in $1, below), the present paper uses “4” for non-relevant commitment entailment and “--+” for relevant commitment entailment. LOGICAL STRUCTURE OF LINGUKTIC 427 1. SUMMARY OF LSLC I 1.1. Fitch-style Systems LSLC I presents four related Fitch-style natural deduction systems of non-relevant commitment entailment. These systems differ merely in the restrictions placed on the use of modus ponens. In these systems, each line of a proof consists of a formula A, prefixed by a (possibly empty) Commitment prefix cl . -. ck. “cl . . . CkA” is given the intuitive gloss: “personi is committed to person2 being committed to . . . personk being committed to A”. Before we stipulate the rules, note that for any line of a Fitch-style natural deduction proof there is a number of vertical lines to the left of the sentence written at that line. This number is the rank of the line. The rules are as follows: Hyp: A step may be introduced as the hypothesis of a new subproof and each new hypothesis receives a prefix Ci e . . Ck, where k is the rank of the subproof. Rep: A sentence occurring at an earlier line may be repeated, retaining the prefix. Reit: A sentence occurring earlier may be reiterated into hypothetical subproofs, retaining the prefix. CP: From a proof of Ci . . . Cn+l B on hypothesis Ci . . . &+,A to infer Cl . . .C,(A-3B), n 2 0. MP: From Ci . . . C,(A-SB) and Cl . . . C,, e e . &+,A to infer Cl . . ’ G+m B, where n 2 0, and m. is restricted according to the table below. (The following rules for conjunction, “W, are natural and facilitate axiomatization.) &I: From Ci . . s C,A and Cl . . . C,B to infer Cl . . . C,(A&B). &E: From Cl . . . C,(A&B) to infer either Cl . . . C, A or Cr . . . C,, B. LSLC I motivates four ranges for m in MP: m = 1; m = 0 or 1; m 2 1; and m > 0. These four ranges lead to four Fitch-style systems. The four corresponding sets of theorems are the four corresponding non-relevant commitment logics. Table I indicates the names of these systems and logics. Each commitment logic is the strict implication conjunction fragment of a modal logic (where “4” is strict implication), and has a natural axiomatization. Table I also summarizes this information. 428 MARK LANCE AND PHILIP KREWER