According to the PCCP thesis, the probability of a conditional A → C is the conditional probability P. This claim is undermined by Lewis’ triviality results, which purport to show that apart from trivial cases, PCCP is not true. In the present article we show that the only rational, “Dutch Book-resistant” extension of the agent’s beliefs concerning non-conditional sentences A and C to the conditional A → C is by assuming that P = P. In other cases a diachronic Dutch (...) Book against the agent can be constructed. There is a tension between our findings and Lewis’ results, which needs to be explained. Therefore, we present a probability space which corresponds in a natural way to the diachronic Dutch Book—and which allows the conditional A → C to be interpreted as an event in a mathematically sound way. It also allows to formalize the notion of conditionalizing A → C on ¬C which plays a crucial role in Lewis’ proof. Our conclusion is that Lewis’ proof is circular, so it cannot be considered to be a sound argument against PCCP. (shrink)

One of central problems in the theory of conditionals is the construction of a probability space, where conditionals can be interpreted as events and assigned probabilities. The problem has been given a technical formulation by van Fraassen (23), who also discussed in great detail the solution in the form of Stalnaker Bernoulli spaces. These spaces are very complex – they have the cardinality of the continuum, even if the language is finite. A natural question is, therefore, whether a technically simpler (...) (in particular finite) partial construction can be given. In the paper we provide a new solution to the problem. We show how to construct a finite probability space $$\mathrm {S}^\#=\left(\mathrm\Omega^\#,\mathrm\Sigma^\#,\mathrm P^\#\right)$$ S # = Ω #, Σ #, P # in which simple conditionals and their Boolean combinations can be interpreted. The structure is minimal in terms of cardinality within a certain, naturally defined class of models – an interesting side-effect is an estimate of the number of non-equivalent propositions in the conditional language. We demand that the structure satisfy certain natural assumptions concerning the logic and semantics of conditionals and also that it satisfy PCCP. The construction can be easily iterated, producing interpretations for conditionals of arbitrary complexity. (shrink)

We define a semantics for conditionals in terms of stochastic graphs which gives a straightforward and simple method of evaluating the probabilities of conditionals. It seems to be a good and useful method in the cases already discussed in the literature, and it can easily be extended to cover more complex situations. In particular, it allows us to describe several possible interpretations of the conditional and to formalize some intuitively valid but formally incorrect considerations concerning the probabilities of conditionals under (...) these two interpretations. It also yields a powerful method of handling more complex issues. The stochastic graph semantics provides a satisfactory answer to Lewis’s arguments against the PC = CP principle, and defends important intuitions which connect the notion of probability of a conditional with the notion of conditional probability. It also illustrates the general problem of finding formal explications of philosophically important notions and applying mathematical methods in analyzing philosophical issues. (shrink)

Although mathematical philosophy is flourishing today, it remains subject to criticism, especially from non-analytical philosophers. The main concern is that even if formal tools serve to clarify reasoning, they themselves contribute nothing new or relevant to philosophy. We defend mathematical philosophy against such concerns here by appealing to its metaphysical foundations. Our thesis is that mathematical philosophy can be founded on the phenomenological theory of ideas as developed by Roman Ingarden. From this platonist perspective, the “unreasonable effectiveness of mathematics in (...) philosophy”—to adapt Wigner’s phrase—is analogous to that of mathematical explanations in science. As success-criteria for mathematical philosophy, we propose that it should be correct, responsive, illuminating, promising, relevant, and adequate. (shrink)