Modal logic provides an elegant way to understand the notion of potential infinity. This raises the question of what the right modal logic is for reasoning about potential infinity. In this article I identify a choice point in determining the right modal logic: Can a potentially infinite collection ever be expanded in two mutually incompatible ways? If not, then the possible expansions are convergent; if so, then the possible expansions are branching. When possible expansions are convergent, the right modal logic (...) is S4.2, and a mirroring theorem due to Linnebo allows for a natural potentialist interpretation of mathematical discourse. When the possible expansions are branching, the right modal logic is S4. However, the usual box and diamond do not suffice to express everything the potentialist wants to express. I argue that the potentialist also needs an operator expressing that something will eventually happen in every possible expansion. I prove that the result of adding this operator to S4 makes the set of validities Pi-1-1 hard. This result makes it unlikely that there is any natural translation of ordinary mathematical discourse into the potentialist framework in the context of branching possibilities. (shrink)

Modal logic provides an elegant way to understand the notion of potential infinity. This raises the question of what the right modal logic is for reasoning about potential infinity. In this article I identify a choice point in determining the right modal logic: Can a potentially infinite collection ever be expanded in two mutually incompatible ways? If not, then the possible expansions are convergent; if so, then the possible expansions are branching. When possible expansions are convergent, the right modal logic (...) is S4.2, and a mirroring theorem due to Linnebo allows for a natural potentialist interpretation of mathematical discourse. When the possible expansions are branching, the right modal logic is S4. However, the usual box and diamond do not suffice to express everything the potentialist wants to express. I argue that the potentialist also needs an operator expressing that something will eventually happen in every possible expansion. I prove that the result of adding this operator to S4 makes the set of validities Pi-1-1 hard. This result makes it unlikely that there is any natural translation of ordinary mathematical discourse into the potentialist framework in the context of branching possibilities. (shrink)

Modal logic has been used to analyze potential infinity and potentialism more generally. However, the standard analysis breaks down in cases of divergent possibilities, where there are two or more possibilities that can be individually realized but which are jointly incompatible. This paper has three aims. First, using the intuitionistic theory of choice sequences, we motivate the need for a modal analysis of divergent potentialism and explain the challenges this involves. Then, using Beth–Kripke semantics for intuitionistic logic, we overcome those (...) challenges. Finally, we apply our modal analysis of divergent potentialism to make choice sequences comprehensible in classical terms. (shrink)

unified treatment of both (families of) interpretations is based on a revised notion of settledness. The main features of this approach are the following: (i) in branching structures, a world can be represented not by a single course of events, but by a node u in the tree, where u itself is seen as the cluster of courses of events passing through it; (ii) the utterance time is uniquely fixed; (iii) the utterance world is not uniquely fixed; (iv) because of (...) (iii), an utterance-event is associated not to a single context, but to a plurality of contexts, depending on which world we are considering as the utterance world. As a consequence, if a future-tensed sentence φ is uttered at u and its truth (falsehood) is already a settled issue at u itself, then the sentence is true (false) at u; otherwise the sentence is neither true nor false at u and, to get a definite truth value, we must wait until settledness is reached in a different context. Since what is crucial, in both cases, is the reference to a given state of information, such a treatment can be extended to other intriguing uses of the future tense, starting from the epistemic reading. These uses will be topic of a related paper. (shrink)

According to the actualist view, what is essential to the truth conditions of a future-tensed sentence ‘it will be the case that ϕ’ is reference to the unique course of events that will become actual. On the other hand, the modal view has it that the truth conditions of such a sentence require that the truth of ϕ be already “settled” at the time of utterance, where “being settled at time t” is defined by universal quantification over a domain of (...) courses of events, the futures compatible with what has happened up to time t. On the proposal we discuss in this paper, the actualist view and the modal view are seen as two related attitudes that speakers can have when they evaluate future-tensed sentences, and the corresponding interpretations undergo a unified semantic treatment based on a contextual notion of settledness. A central feature of our approach is a dynamic view of contexts of utterance, whereby the world of the utterance is not fixed once and for all, as different worlds, by the passing of time, can play this role in turn. Finally, one major goal of the paper is to show how the unified analysis we propose accounts for a particularly interesting interpretation of future-tensed sentences, often referred to as ‘epistemic reading’. (shrink)