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)

We define a model for computing probabilities of right-nested conditionals in terms of graphs representing Markov chains. This is an extension of the model for simple conditionals from Wójtowicz and Wójtowicz. The model makes it possible to give a formal yet simple description of different interpretations of right-nested conditionals and to compute their probabilities in a mathematically rigorous way. In this study we focus on the problem of the probabilities of conditionals; we do not discuss questions concerning logical and metalogical (...) issues such as setting up an axiomatic framework, inference rules, defining semantics, proving completeness, soundness etc. Our theory is motivated by the possible-worlds approach ; however, our model is generally more flexible. In the paper we focus on right-nested conditionals, discussing them in detail. The graph model makes it possible to account in a unified way for both shallow and deep interpretations of right-nested conditionals. In particular, we discuss the status of the Import-Export Principle and PCCP. We briefly discuss some methodological constraints on admissible models and analyze our model with respect to them. The study also illustrates the general problem of finding formal explications of philosophically important notions and applying mathematical methods in analyzing philosophical issues. (shrink)

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)

We analyze three interesting arguments from the literature, where ascribing a probability of 1 to a certain right-nested conditional A→(B→C) leads to strong theses concerning conditionals: they serve as counterexamples to important general claims. The first is the classic and much discussed McGee’s counterexample to Modus Ponens from McGee [“A Counterexample to Modus Ponens.” The Journal of Philosophy 82 (9): 462–471]. The second example was given by Santorio [“Trivializing Informational Consequence.” Philosophy and Phenomenological Research 104:297–320] and is intended to undermine (...) the probabilistic version of Modus Ponens. The third example was presented in Cantwell [“Revisiting McGee’s Probabilistic Analysis of Conditionals.” Journal of Philosophical Logic 51 (5): 973–1017] in order to reject the general version of the Ramsey Test (i.e. PCCP) and Modus Ponens. These examples are based on a specific interpretation of right-nested conditional which is not the only possible one. We provide examples demonstrating that alternative interpretations are feasible. As a result, the arguments presented in the three cited works possess limited efficacy, as they are confined solely to the particular interpretation of the conditional. (shrink)