I give a probabilistic semantics for modal logic in which modal operators function as quantifiers over Popper functions in probabilistic model sets, thereby generalizing Kripke's semantics for modal logic.

This volume is a collation of original contributions from the key actors of a new trend in the contemporary theory of knowledge and belief, that we call “dynamic epistemology”. It brings the works of these researchers under a single umbrella by highlighting the coherence of their current themes, and by establishing connections between topics that, up until now, have been investigated independently. It also illustrates how the new analytical toolbox unveils questions about the theory of knowledge, belief, preference, action, and (...) rationality, in a number of central axes in dynamic epistemology: temporal, social, probabilistic and even deontic dynamics. (shrink)

Most physics theories are deterministic, with the notable exception of quantum mechanics which, however, comes plagued by the so-called measurement problem. This state of affairs might well be due to the inability of standard mathematics to “speak” of indeterminism, its inability to present us a worldview in which new information is created as time passes. In such a case, scientific determinism would only be an illusion due to the timeless mathematical language scientists use. To investigate this possibility it is necessary (...) to develop an alternative mathematical language that is both powerful enough to allow scientists to compute predictions and compatible with indeterminism and the passage of time. We suggest that intuitionistic mathematics provides such a language and we illustrate it in simple terms. (shrink)

The use of probability theory is widespread in our daily life as well as in scientific theories. In virtually all cases, calculations can be carried out within the framework of classical probability theory. A special exception is given by quantum mechanics, which gives rise to a new probability theory: quantum probability theory. This dissertation deals with the question of how this formalism can be understood from a philosophical and physical perspective. The dissertation is divided into three parts. In the first (...) part, the formalism of quantum probability theory and its relation to quantum mechanics is presented. The second part considers the possibility of reformulating quantum probability theory to embed it within classical probability. Mathematically, this possibility overlaps with the possibility of the existence of hidden variables theories of quantum mechanics: theories that attempt to solve fundamental problems in quantum mechanics by introducing additional physical properties of systems. The conclusion of this part is that, although classical reformulations of quantum probability theory are formally possible, they offer little insight into the formalism of quantum probability theory itself. The third part regards a more direct investigation of quantum probability theory. A reformulation of quantum probability theory is obtained by constructing a quantum logic on the basis of empirical non-probabilistic predictions of quantum mechanics. In this reformulation quantum probability functions are conditional probability functions on an algebra of propositions about measurements and measurement outcomes. (shrink)

Personal probability is now a familiar subject in epistemology, together with such more venerable notions as knowledge and belief. But there are severe strains between probability and belief; if either is taken as the more basic, the other may suffer. After explaining the difficulties of attempts to accommodate both, I shall propose a unified account which takes conditional personal probability as basic. Full belief is therefore a defined, derivative notion. Yet we will still be able to picture opinion as follows: (...) my subjective probability is only a grading of the possibilities left open by my beliefs. My conditional probabilities generally derive from the strongest belief I can maintain when admitting the relevant condition. (shrink)

Provided here is a characterisation of absolute probability functions for intuitionistic (propositional) logic L, i.e. a set of constraints on the unary functions P from the statements of L to the reals, which insures that (i) if a statement A of L is provable in L, then P(A) = 1 for every P, L's axiomatisation being thus sound in the probabilistic sense, and (ii) if P(A) = 1 for every P, then A is provable in L, L's axiomatisation being thus (...) complete in the probabilistic sense. As there are theorems of classical (propositional) logic that are not intuitionistic ones, there are unary probability functions for intuitionistic logic that are not classical ones. Provided here because of this is a means of singling out the classical probability functions from among the intuitionistic ones. (shrink)

I present some counterexamples to Adams's Thesis and explain how they undermine arguments that indicative conditionals cannot be truth-evaluable propositions.

We show how to obtain a probabilistic semantics and calculus for a logic presented by a valuation specification. By identifying general forms of valuation constraints we are able to accommodate a wide class of propositional based logics encompassing multi-valued logics like Łukasiewicz 3-valued logic and the Belnap–Dunn four-valued logic as well as paraconsistent logics like $${\textsf{mbC}}$$ and $${\textsf{LFI1}}$$. The probabilistic calculus is automatically generated from the valuation specification. Although not having explicit probability constructors in the language, the rules of the (...) calculus reflect the valuation constraints in a probabilistic way. Indeed the probability of the premises of each rule coincides with the probability of the conclusions. Moreover, a failed exhaustive attempt of proving a formula in this calculus means non-derivability. Nevertheless when the non-derived formula is consistent then it is possible to extract a satisfying valuation from the failed exhaustive attempt. Soundness and completeness of the calculi are established with respect to the probabilistic semantics consisting of probability spaces also induced by the valuation specification. Furthermore we prove the equivalence between the probabilistic and the valuation semantics. (shrink)