Indicative conditionals: probabilities and relevance

Abstract

We propose a new account of indicative conditionals, giving acceptability and logical closure conditions for them. We start from Adams’ Thesis: the claim that the acceptability of a simple indicative equals the corresponding conditional probability. The Thesis is widely endorsed, but arguably false and refuted by empirical research. To fix it, we submit, we need a relevance constraint: we accept a simple conditional $$\varphi \rightarrow \psi$$ φ → ψ to the extent that (i) the conditional probability $$\mathrm{p}(\psi |\varphi )$$ p ( ψ | φ ) is high, provided that (ii) $$\varphi$$ φ is relevant for $$\psi$$ ψ . How (i) should work is well-understood. It is (ii) that holds the key to improve our understanding of conditionals. Our account has (i) a probabilistic component, using Popper functions; (ii) a relevance component, given via an algebraic structure of topics or subject matters. We present a probabilistic logic for simple indicatives, and argue that its (in)validities are both theoretically desirable and in line with empirical results on how people reason with conditionals.

Williamson’s Abductive Case for the Material Conditional Account

“…According to this proposal, a conditional always expresses a conditional assertion: it asserts the consequent on the condition that the antecedent is true; else, nothing is asserted. 7 Various authors have noted that, on this analysis, (EQ) almost immediately follows. Where A = {v ∈ W :…”

“…6 Huitink [32] argued for such a three-valued semantics, if we want to give a uniform treatment of if-clauses to provide a compositional analysis of embedded conditionals like "Harry usually drinks Butterbeer, if he is happy." 7 The latter part, "else, nothing is asserted," is highly controversial. Stalnaker and Jeffrey [34] famously worked out an alternative proposal where…”

“…19 For work on the logic of the inferential conditional, see Crupi and Iacona [11,12], Raidl et al [66], and Iacona [33]. 20 Related proposals are to be found in Rott [69,70], Oaksford and Chater [57][58][59][60][61], van Rooij and Schulz [85], and Berto and Özgün [7]. Oaksford and Chater as well as van Rooij and Schulz analyze the connection between a conditional's component parts in terms of causality.…”

Relevance through topical unconnectedness

“…[5, p. 722] The heart of [5]'s argument is that a theory of entailment must be able to account for the difference between a valid argument and an enthymeme which they identify rather vaguely as a valid argument had it not been for a required premise which happens to be true, necessarily true, or logically true. To exemplify their view, they showed forth a two-premised argument in which the minor premise is true, yet where the argument without this premise is not valid: 9 (M) All bodies moving in elliptic orbits are subject to the law of gravitation; (m) Comets move in elliptic orbits; (C) Therefore comets are subject to the law of gravitation.…”

“…The metaphysical side to topics will not concern us in this paper, nor will I attempt a general discussion of theories of topics. 16 I will follow the recent and topically related [9] in regarding the topic of a formula as the fusion of the topics of its atomic subformulas. If A and B, then, are about nonoverlapping topics, then so will A and any subformula C of B.…”