Indicative conditionals: probabilities and relevance

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→ψto the extent that (i) the conditional probability$$\mathrm{p}(\psi |\varphi )$$p(ψ|φ)is high, provided that (ii)$$\varphi$$φis 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.

TRIANGULATING NON-ARCHIMEDEAN PROBABILITY

We relate Popper functions to regular and perfectly additive such non-Archimedean probability functions by means of a representation theorem: every such non-Archimedean probability function is infinitesimally close to some Popper function, and vice versa. We also show that regular and perfectly additive non-Archimedean probability functions can be given a lexicographic representation. Thus Popper functions, a specific kind of non-Archimedean probability functions, and lexicographic probability functions triangulate to the same place: they are in a good sense interchangeable.

A Logic of Comparative Support: Qualitative Conditional Probability Relations Representable by Popper Functions

Revising classical logic—to deal with the paradoxes of self-reference, or vague propositions, for the purposes of scientific theory or of metaphysical anti-realism—requires the revision of probability theory. This chapter reviews the connection between classical logic and classical probability, clarifies nonclassical logic, giving simple examples, explores modifications of probability theory, using formal analogies to the classical setting, and provides two foundational justifications for these ‘nonclassical probabilities’. There follows an examination of extensions of the nonclassical framework: to conditionalization and decision theory in particular, before a final review of open questions and alternative approaches, and an evaluation of current progress.

Probability and Nonclassical Logic

This chapter presents axioms for comparative conditional probability relations. The axioms presented here are more general than usual. Each comparative relation is a weak partial order on pairs of sentences but need not be a complete order relation. The axioms for these comparative relations are probabilistically sound for the broad class of conditional probability functions known as Popper functions. Furthermore, these axioms are probabilistically complete. Arguably, the notion of comparative conditional probability provides a foundation for Bayesian confirmation theory. Bayesian confirmation functions are overly precise probabilistic representations of the more fundamental logic of comparative support. The most important features of evidential support are captured by comparative relationships among argument strengths, realized by the comparative support relations and their logic.