On the Logical Form of Concessive Conditionals

Abstract

This paper outlines an account of concessive conditionals that rests on two main ideas. One is that the logical form of a sentence as used in a given context is determined by the content expressed by the sentence in that context. The other is that a coherent distinction can be drawn between a reading of 'if' according to which a conditional is true when its consequent holds on the supposition that its antecedent holds, and a stronger reading according to which a conditional is true when its antecedent supports its consequent. As we will suggest, the logical form of concessive conditionals can be elucidated by relying on this distinction.

An Axiomatic System for Concessive Conditionals

“…2 Crupi and Iacona [3]. 3 Crupi and Iacona [2] spells out the notion of incompatibility and explains its relation to the intuitive understanding of support.…”

“…Crupi and Iacona [3] have argued that the three differences just illustrated, together with other observations about concessive conditionals that are equally plausible, pose a serious explanatory challenge, as the extant theories of concessive conditionals do not provide a satisfactory explanation of all the relevant facts. According to their analysis, which is intended to provide such explanation, a concessive conditional is a sentence of the form (p > q) ∧ (¬p q), where > stands for the suppositional conditional in the Stalnaker-Lewis interpretation, and stands for the evidential conditional in their own interpretation.…”

“…This stronger condition, called Chrysippus Test, ensures that a suitably defined relation of incompatibility holds between p and ¬q. 3 Since the two conditions that constitute the Chrysippus Test are respectively the Ramsey Test for p > q and the Ramsey Test for ¬q > ¬p, p q is definable as (p > q) ∧ (¬q > ¬p). Therefore, the analysis of p → q suggested by Crupi and Iacona can equally be phrased as follows: (p > q) ∧ (¬p > q) ∧ (¬q > p).…”

“…More precisely, we will use a method that has been 1 Stalnaker [12], Lewis [7], Crupi and Iacona [2]. 2 Crupi and Iacona [3]. 3 Crupi and Iacona [2] spells out the notion of incompatibility and explains its relation to the intuitive understanding of support.…”

“…first developed by Raidl in a general manner and then employed by Raidl, Iacona, and Crupi for a similar system for evidential conditionals. 4 The structure of the paper is as follows. Section 2 introduces two languages, L > and L → , which differ only in that the former includes > in addition to the usual sentential connectives while the latter includes →.…”