Capturing the relationship between conditionals and conditional probability with a trivalent semantics

De Finettian Logics of Indicative Conditionals Part I: Trivalent Semantics and Validity

“…Second, de Finettian conditionals keep the epistemic notion of assertability and the semantic notion of truth separate, while allowing for a fruitful interaction: degrees of assertability can be defined directly in terms of the truth conditions. For a probability function Pr on a propositional language, and assuming X is a Boolean sentence or a simple conditional, 7 we define the degree of assertability to be: [75]). Trivalent semantics accommodates the familiar norm of asserting what is probably true by extending it to cases where the antecedent might be undefined.…”

“…In motivating this interpretation, Reichenbach refers explicitly to the Bohr-Heisenberg interpretation of quantum mechanics 3. Related proposals include, in the philosophical tradition, Jeffrey[45], Belnap[5], Dummett[27], Manor[56], Farrell[35], McDermott[57], Huitink[41], Rothschild[75] and Kapsner[46]. In the more mathematical tradition, they include work on conditionals objects by Schay[76], Calabrese[10], Goodman et al[39], Dubois and Prade[26].…”

Conditionals and inferential connections: toward a new semantics

“…In the three-value view, the probability of a conditional "If φ, ψ" can be equated with the ratio of the probability that it is true to the probability that it has a (determinate) truth value, which is the conditional probability of ψ given φ (Rothschild, 2014). Thus the three-value view could alternatively be paired with Adams' probabilistic notion of validity, according to which high-enough probabilities for the premises guarantee high probability for the conclusion (Adams, 1965(Adams, , 1975.…”

“…We are assuming classical probability theory here. By extending that theory with a primitive notion of conditional event,Gilio et al (2017) obtain an account of conditionals in which Pr(If ψ, χ | φ) is defined, in spite of the fact that the conditional is still not a proposition in the classical sense (see alsoGilio & Sanfilippo, 2013, 2014. How much this is going to help non-propositionalists depends on the extent to which Gilio and coauthors, or others, will be able to clarify the metaphysical status of conditional events, which so far they have not given us much guidance on.5 Weirich (in press) presents a number of other a priori reasons for rejecting non-propositionalism.…”

Yablo’s semantic machinery

“…In my view this is not because trivalence itself is suspect, but rather because it might be difficult to use trivalence too much in explaining different linguistic phenomena. It turns out that many uses of trivalence found in the literature are mutually incompatible[Soames, 1989, Rothschild, 2014, Spector, 2015, Križ, 2015.7 Of course, the divisibility of the content is itself a matter of considerable debate in epistemology. However, at the least, we can all accept that the factive entailment (that p is true) can be true without John bearing any epistemic relation to p. Any further division would be rejected by those who followWilliamson [2000] in thinking that knowledge is unanalysable.8 This strategy can be generalized beyond 'know': the king of France is bald' both asserts the existence of a king of France and says that he is bald, 'John stopped smoking' both says that John used to smoke and that he doesn't now.…”