August 27 - 29, 2020
The probability conditional and inferentialism
Psychological research on reasoning has recently advanced from an earlier approach based on binary classical logic to Bayesian, or probabilistic, theories. Our version of the new developments is grounded in what has been called "the Equation", P(if p then q) = P(q|p). We will call a conditional that satisfies the Equation a probability conditional. There is a large body of theoretical and empirical work in support of the Equation, but it has recently been challenged by research on inferentialist accounts of conditionals. These accounts have used counterintuitive missing-link conditionals to argue for the importance of an inferential relation between p and q, similar to an inference ticket, which ensures that P(q|p) > P(q|not-p). We will briefly outline our probabilistic approach to conditionals and alternative proposals from inferentialism. We will then discuss logical, conceptual, and empirical arguments against some inferentialist positions, and propose an alternative account of missing-link conditionals, which draws on pragmatics and on the distinction between singular and general conditionals.
Conditionals, Indeterminacy, Probability and Truth
Many conditionals whose antecedents turn out to be false are not merely uncertain but indeterminate. Nevertheless, I argue, they may be assessed probabilistically. For instance, given the proportions, it’s 90% likely that if I had picked a red ball it would have a black spot; but typically, there is no fact of the matter about which red ball I would have picked, if I had picked a red ball. Thus, probabilities apply even when there is no fact of the matter. I relate these ideas to a different kind of indeterminacy, the case of vagueness. I also relate this to work by Richard Bradley (Phil. Review 2012), where he argues that conditionals (a) are assessed by conditional probability, (b) are not (ordinary) propositions, but a more complex semantic entity, and (c) can nevertheless be given truth conditions such that the probability of their truth is the conditional probability of consequent given antecedent.
Aristotle's logical thinking probabilized: from qualitative to quantitative reasoning and back
Aristotle's probably most elaborated logical thinking handed down to us is presented in his Analytica Priora, which contains the famous syllogisms. Syllogisms are deductive argument forms about categorical propositions. Categorical propositions express whether some or all members of one category (denoted by the subject term S) are included in another category (denoted by the predicate term P). They are qualitative in the sense that they either form an affirmation (e.g., Some S is P) or a denial (e.g., No S is P). In my talk, I present a probabilistic semantics for Aristotelian syllogisms which allows for quantitative reasoning (Pfeifer & Sanfilippo, submitted). By suitable interpretations of categorical propositions in terms of conditional probability and respective concepts of validity, all traditionally valid syllogisms can be rationally reconstructed within coherence-based probability logic. I explain how this quantitative framework also allows for going back to qualitative reasoning, for example, via the notions of defaults and negated defaults. I also show how Aristotle's intuition---that conditionals with contradictory antecedents and consequents should be false (which also matches people's intuitions; Pfeifer, 2012)---can be framed within coherence-based probability logic. Among other instances of contradictions and conditionals, Aristotle's intuition has been investigated within connexive logic in recent years. While connexive logic is qualitative, I show how basic connexive principles can be interpreted quantitatively. Finally, I discuss how interactions among qualitative and quantitative approaches foster our understanding of conditionals.
Compound and iterated conditionals in the setting of coherence.
In our approach, the probability of a natural language conditional, P(if A then C), is interpreted by the subjective probability of the conditional event C|A. The conditional event is, in de Finetti’s sense, a three-valued logical entity. Its probability coincides with the conditional probability of C given A, i.e. P(C|A). An objection to identifying P(if A then C) with P(C|A) was that it appeared unclear how to form compounds and iterations of conditionals by preserving the usual probabilistic properties and by avoiding Lewis' Triviality.
In this talk, I illustrate how to overcome this objection with a probabilistic analysis, based on coherence, of these compounds and iterations.
Specifically, based on Gilio and Sanfilippo (2014, 2019, working paper), I discuss logical operations among conditional events in a coherence-based approach, where the compound and iterated conditionals are conditional random quantities which go beyond the three-valued logic of conditional events. I show that in this "algebra" of conditionals De Morgan’s Law and the usual probabilistic properties are preserved and Lewis' Triviality results are avoided. I also give a characterization of p-consistency and p-entailment, with applications to several inference rules in probabilistic nonmonotonic reasoning. Then, I illustrate two methods which allow to turn not p-valid inference rules into p-valid ones.
Moreover, in analogy to the Deduction Theorem, I show that a conditional event E|H is p-entailed by a p-consistent family of two conditional events A|H and B|K if and only if the prevision of the iterated conditional (E|H)|((A|H)&(B|K)) is constant and equal to 1 (Gilio, Pfeifer, Sanfilippo, 2020). I also explain how to formalize latent information by iterated conditionals. I conclude my talk with some remarks on the supposed “independence” of two conditionals, and I interpret this property as uncorrelation between two random quantities (Sanfilippo, Gilio, Over, and Pfeifer 2020).
Conditionals and the Hierarchy of Causal Queries
In a series of experiments, we test the hypothesis that causal relations require multiple conceptual dimensions (prediction, intervention, and counterfactual dependence), which are differentially encoded in indicative and counterfactual conditionals. Our results indicate that the acceptance of indicative and counterfactual conditionals can become dissociated (Experiments 4, 5), and that the acceptance of both is needed for accepting a causal relation between two co-occurring events (Experiment 2). The implications that these findings have for the hypothesis above and recent debates at the intersection of the psychology of reasoning and causal judgment are discussed.
Gebäude PT, Zi. 4.3.6
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