This volume seeks to further the use of formal methods in clarifying one of the central problems of philosophy: that of our free human agency and its place in our indeterministic world. It celebrates the important contributions made in this area by Nuel Belnap, American logician and philosopher. Philosophically, indeterminism and free action can seem far apart, but in Belnap’s work, they are intimately linked. This book explores their philosophical interconnectedness through a selection of original research papers that build forth (...) on Belnap’s logical and philosophical work. Some contributions take the form of critical discussions of Belnap's published work, some develop points made in his publications in new directions, and others provide additional insights on the topics of indeterminism and free action. In Nuel Belnap’s work on indeterminism and free action, three formal frameworks figure prominently: the simple branching histories framework known as "branching time;" its relativistic spatio-temporal extension, branching space-times; and the “seeing to it that” logic of agency. As those frameworks provide the formal background for the contributed papers, the volume introduction gives an overview of the current state of their development. It also introduces case-intensional first order logic, a general intensional logic offering resources for a first-order extension of the mentioned frameworks and a recent research focus of Belnap’s. The volume also contains an extended biographical interview with Nuel Belnap. (shrink)

Non-bivalent semantics of the future tense assert that propositions regarding future contingents are neither true nor false. One of the most relevant non-bivalent semantics is supervaluationism :264–281, 1970; Thomason, in: Gabbay, Guenthner Handbook of philosophical logic, Springer, Berlin, 1984), which preserves important logical principles. Recently, non-bivalent semantics are under attack from some pragmatics arguments: these semantics would be incompatible with our practices of asserting future contingents and with the probability we ascribe to such assertions :251–271, 2014; Cariani and Santorio in (...) Mind 127:129–165, 2018). The aim of this paper is to defend supervaluationism against this kind of criticism. We argue that, if probability is interpreted as the subjective belief in a proposition and if a semantic contextualism is adopted, supervaluationism makes correct predictions concerning the norms that govern our practices of asserting future contingents. Obviously, other arguments can be proposed against supervaluationism, but they must be of a different kind, such as metaphysical arguments. (shrink)

We care not only about what experiences we have, but when we have them too. However, on the B-theory of time, something’s timing isn’t an intrinsic way for that thing to be or become. Given B-theory, should we be rationally indifferent about the timing per se of an experience? In this paper, I argue that B-theorists can justify time-biased preferences for pains to be past rather than present and for pleasures to be present rather than past. In support of this (...) argument, I appeal to the doctrine of temporal parts or “four-dimensionalism” for short. When held in conjunction with a certain evaluative principle about whose experiences matter, four-dimensionalism reconciles B-theory with some time-biased preferences. (shrink)

Provisional draft, pre-production copy of my book “The Modal Future” (forthcoming with Cambridge University Press).

There is a long-standing disagreement among Branching-Time theorists. Even though they all believe that the branching representation accurately grasps the idea that the future, contrary to the past, is open, they argue whether this representation is compatible with the claim that one among many possible futures is distinguished—the single future that will come to be. This disagreement is paralleled in an argument about the bivalence of future contingents. The single, privileged future is often called the Thin Red Line. I reconstruct (...) the history of the arguments for and against this idea. Then, I propose my own version of the Thin Red Line theory which is immune to the major objections found in the literature. I argue that the semantic disagreement is grounded in distinct metaphysical presuppositions. My solution is expressed in a conceptual framework proposed by John MacFarlane, who distinguishes semantics from postsemantics. I extend his distinction and introduce a new notion of presemantics to elucidate my idea. (shrink)

The purpose of the paper is to rethink the role of actuality in the branching model of possibilities. We investigate the idea that the model should be enriched with an additional factor—the so-called Thin Red Line—which is supposed to represent the single possible course of events that gets actualized in time. We believe that this idea was often misconceived which prompted some unfortunate reactions. On the one hand, it suggested problematic semantic models of future tense and and on the other, (...) it provoked questionable lines of criticism. We reassess the debate and point to potential pitfalls, focusing on the semantic dimension of the Thin Red Line theory. Our agenda transcends the semantics, however. We conclude that semantic considerations do not threaten the Thin Red Line theory and that the proper debate should be carried in the domain of metaphysics. (shrink)

A commonly held idea regarding the nature of time is that the future is open and the past is fixed or closed. This article investigates the notion that there is an asymmetry in openness between the past and the future. The following questions are considered: How exactly is this asymmetry in openness to be understood? What is the relation between an open future and various ontological views about the future? Is an open future a branching future? What is the relation (...) between an open future and the question of whether contingent statements about the future are true or false? Is an open future compatible with a single determinate future? (shrink)

The aim of this paper is to argue that the adoption of an unrestricted principle of bivalence is compatible with a metaphysics that (i) denies that the future is real, (ii) adopts nomological indeterminism, and (iii) exploits a branching structure to provide a semantics for future contingent claims. To this end, we elaborate what we call Flow Fragmentalism, a view inspired by Kit Fine (2005)’s non-standard tense realism, according to which reality is divided up into maximally coherent collections of tensed (...) facts. In this way, we show how to reconcile a genuinely A-theoretic branching-time model with the idea that there is a branch corresponding to the thin red line, that is, the branch that will turn out to be the actual future history of the world. (shrink)

There is a familiar debate between Russell and Strawson concerning bivalence and ‘the present King of France’. According to the Strawsonian view, ‘The present King of France is bald’ is neither true nor false, whereas, on the Russellian view, that proposition is simply false. In this paper, I develop what I take to be a crucial connection between this debate and a different domain where bivalence has been at stake: future contingents. On the familiar ‘Aristotelian’ view, future contingent propositions are (...) neither true nor false. However, I argue that, just as there is a Russellian alternative to the Strawsonian view concerning ‘the present King of France’, according to which the relevant class of propositions all turn out false, so there is a Russellian alternative to the Aristotelian view, according to which future contingents all turn out false, not neither true nor false. The result: contrary to millennia of philosophical tradition, we can be open futurists without denying bivalence. (shrink)

In Facing the Future, Belnap et al. reject bivalence and propose double time reference semantics to give a pragmatic response to the following assertion problem: how can we make sense of assertions about future events made at a time when the outcomes of those events are not yet determined? John MacFarlane employs the same semantics, now bolstered with a relative-truth predicate, to accommodate the following apparently conflicting intuitions regarding the truth-value of an uttered future contingent: at the moment of utterance, (...) if asked to evaluate the truth-value of the asserted future contingent one has the intuition that the assertion is neither true nor false, yet later, at the moment of the predicted event, one has the intuition that the assertion was, already, either true or false. Both MacFarlane and Belnap assume that assertions of future contingents have complete propositional content – the traditional propositional contents that, according to him ‘are the contents of assertions and beliefs’. This assumption is challenged. (shrink)

Open futurism is the indeterministic position according to which the future is ‘open’, i.e., there is now no fact of the matter as to what future contingent events will actually obtain. Many open futurists hold a branching conception of time, in which a variety of possible futures exist. This paper introduces two challenges to branching-time open futurism, which are similar in spirit to a challenge posed by Fine to tense realism. The paper argues that, to address the new challenges, open (...) futurists must adopt an objective, non-perspectival notion of actuality and subscribe to an A-theoretic, dynamic conception of reality. Moreover, given a natural understanding of “actual future”, it is perfectly sensible for open futurists to hold that a unique, objectively actual future exists, contrary to a common assumption in the current debate. The paper also contends that recognising the existence of a unique actual future helps open futurists to avoid potential misconceptions. (shrink)

According to an influential line of thought, from the assumption that indeterminism makes future contingents neither true nor false, one can conclude that assertions of future contingents are never permissible. This conclusion, however, fails to recognize that we ordinarily assert future contingents even when we take the future to be unsettled. Several attempts have been made to solve this puzzle, either by arguing that, albeit truth-valueless, future contingents can be correctly assertable, or by rejecting the claim that future contingents are (...) truth-valueless. The paper examines three of most representative accounts in line with the first attempt, and concludes that none of them succeed in providing a persuasive answer as to why we felicitously assert future contingents. (shrink)

In this paper we develop a novel propositional semantics based on the framework of branching time. The basic idea is to replace the moment-history pairs employed as parameters of truth in the standard Ockhamist semantics by pairs consisting of a moment and a consistent, downward closed set of so-called transitions. Whereas histories represent complete possible courses of events, sets of transitions can represent incomplete parts thereof as well. Each transition captures one of the alternative immediate future possibilities open at a (...) branching point. The transition semantics exploits the structural resources a branching time structure has to offer and provides a fine-grained picture of the interrelation of modality and time. In addition to temporal and modal operators, a so-called stability operator becomes interpretable as a universal quantifier over the possible future extensions of a given transition set. The stability operator allows us to specify how and how far time has to unfold for the truth value of a sentence at a moment to become settled and enables a perspicuous treatment of future contingents. We show that the semantics developed along those lines generalizes and extends extant approaches: both Peirceanism and Ockhamism can be viewed as limiting cases of the transition approach that build on restricted resources only, and on both accounts, stability collapses into truth. (shrink)

Ockhamism implies that future contingents may be true, their historical contingency notwithstanding. It is thus opposed to both the Peircean view according to which all future contingents are false, and Supervaluationist Indeterminism according to which all future contingents are neither true nor false. The paper seeks to defend Ockhamism against two charges: the charge that it cannot meet the requirement that truths be grounded in reality, and the charge that it proves incompatible with objective indeterminism about the future. In each (...) case, the defence draws on the idea that certain truths are truths only courtesy of others and of what makes the latter true. After introduction of the Ockhamist view, its competitors and implications, a suitable definition of grounded truth is being devised that both is faithful to the spirit of the grounding-requirement and allows the Ockhamist to heed that requirement quite comfortably. Then two senses in which the future might be open are being introduced, indeterminacy as failure of predetermination by past and present facts, and indeterminacy as failure of entailment by past and present truths. It is argued that while openness in the former sense, but not in the latter sense, coheres with the Ockhamist view, it is only openness in the former sense that matters for objective indeterminism. (shrink)

The paper defends an Aristotelian notion of indeterminism, as rigorously formulated in the framework of branching space-times (BST) of Belnap (1992), against the model-theoretic characterization of indeterminism that Montague (1962) introduced into the philosophy of science. It delineates BST branching against the background provided by Earman's (2008) distinction between individual vs. ensemble branching. It describes a construction of physically-motivated BST models, in which histories are isomorphic to Minkowski spacetime. Finally it responds to criticism leveled against BST by addressing some semantical (...) questions, a topological issue, and the past/future asymmetry. (shrink)

This paper gives an overview of logico-philosophical issues of time and determinism. After a brief review of historical roots and 20th century developments, three current research areas are discussed: the definition of determinism, space-time indeterminism, and the temporality of individual things and their possibilities.

The metaphor of a branching tree of future possibilities has a number of important philosophical and logical uses. In this paper we trace this metaphor through some of its uses and argue that the metaphor works the same way in physics as in philosophy. We then give an overview of formal systems for branching possibilities, viz., branching time and (briefly) branching space-times. In a next step we describe a number of different notions of possibility, thereby sketching a landscape of possibilities. (...) In the final section of the paper we look at the place of branching-based possibilities in that larger landscape of possibilities. Our main message is that far from being an outlandish metaphysical extravagancy, branching-based possibilities are epistemically as well as metaphysically basic. (shrink)

Branching theories are popular frameworks for modeling objective indeterminism in the form of a future of open possibilities. In such theories, the notion of a history plays a crucial role: it is both a basic ingredient in the axiomatic definition of the framework, and it is used as a parameter of truth in semantics for languages with a future tense. Furthermore, histories—complete possible courses of events—ground the notion of modal consistency: a set of events is modally consistent iff there is (...) a history containing that set. We will explain these roles of histories and highlight some critical aspects having to do with the fact that histories are global and, in a relevant sense, “big” objects. The notion of modal consistency, on the other hand, has both local and global aspects. We ask in how far a local notion of modal consistency can serve as an alternative to the common uses of histories, and work out two recent approaches to alternatives to histories. Combining these approaches, we develop a novel semantics for branching time. (shrink)

Explanations of the genuine openness of the future often appeal to objective indeterminacy. According to the received view, such indeterminacy is indeterminacy of certain future-tensed state of affairs that presently obtain. We shall call this view the weak indeterminate present, to distinguish it from the view we will defend in this paper, which we dub the strong indeterminate present. According to our view, unsettledness of the future is grounded on the present indeterminacy of some present-tensed state of affairs. In order (...) for an indeterminate present-tensed state of affairs to explain the unsettledness of a future-tensed state of affairs, there has to be a connection between the two. We argue that this connection can only be provided if we look at the internal structure of the relevant state of affairs. Finally, we will suggest that the best background theory to explain the connection are the so-called spontaneous collapse models of quantum mechanics. (shrink)

The thin red line ( TRL ) is a theory about the semantics of future-contingents. The central idea is that there is such a thing as the ‘actual future’, even in the presence of indeterminism. It is inspired by a famous solution to the problem of divine foreknowledge associated with William of Ockham, in which the freedom of agents is argued to be compatible with God’s omniscience. In the modern branching time setting, the theory of the TRL is widely regarded (...) to suffer from several fundamental problems. In this paper we propose several new TRL semantics, each with differing degrees of success. This leads up to our final semantics, which is a cross between the TRL and supervaluationism. We discuss the notions of truth, validity and semantic consequence which result from our final semantics, and demonstrate some of its pleasing results. This account, we believe, answers the main objection in the literature, and thus places the TRL on the same level as any other competing semantics for future contingents. (shrink)

Open future is incompatible with realism about possible worlds. Since realistically conceived (concrete or abstract) possible worlds are maximal in the sense that they contain/represent the full history of a possible spacetime, past and future included, if such a world is actual now, the future is fully settled now, which rules out openness. The kind of metaphysical indeterminacy required for open future is incompatible with the kind of maximality which is built into the concept of possible worlds. The paper discusses (...) various modal realist responses and argues that they provide ersatz openness only, or they lead to incoherence, or they render the resulting theory inadequate as a theory of modality. The paper also considers various accounts of the open future, including rejection of bivalence, supervaluationism, and the ‘thin red line’ view (TRL), and claims that a version of (TRL) can avoid the incompatibility problem, but only at the cost of deflating the notion of openness. (shrink)

This paper investigates the logic of Ockhamism, a view according to which future contingents are either true or false. Several attempts have been made to give rigorous shape to this view by defining a suitable formal semantics, but arguably none of them is fully satisfactory. The paper draws attention to some problems that beset such attempts, and suggests that these problems are different symptoms of the same initial confusion, in that they stem from the unjustified assumption that the actual course (...) of events must be represented in the semantics as a distinguished history, the Thin Red Line. (shrink)

This paper articulates in formal terms a crucial distinction concerning future contingents, the distinction between what is true about the future and what is reasonable to believe about the future. Its key idea is that the branching structures that have been used so far to model truth can be employed to define an epistemic property, credibility, which we take to be closely related to knowledge and assertibility, and which is ultimately reducible to probability. As a result, two kinds of claims (...) about future contingents—one concerning truth, the other concerning credibility—can be smoothly handled within a single semantic framework. (shrink)

This paper has a twofold aim. The first is to offer a precise definition of soft fact. Without such definition it is impossible to assess the Ockhamist solution to the problem of divine foreknowledge and human freedom in an accurate way. The second purpose is to identify the costs of such a solution, distinguishing them from some of the other costs usually ascribed to Ockhamism, which Ockhamism does not actually need to pay. In particular, it is argued that Ockhamism is (...) committed to the view that a true future exists and to a form of backward causation. (shrink)

In this article, we take into consideration two semantics of the future tense: linearism, according to which future-tense sentences are interpreted on a single history, and universalism, according to which they are evaluated by universally quantifying on the plurality of future histories that radiate from the present instant. Specifically, we focus on a objection advanced against universalism: if universalism were correct semantics of will, negated future-tense sentences of natural language should have two readings, depending on the scope of negation with (...) respect to the universal quantifier on histories. However, since natural language does not show this difference, one may conclude that there is no universal quantifier in the interpretation of these sentences. We show that this conclusion is premature. First, will has clear scope interactions with indefinite nouns phrases, contrary to what linearism predicts. Second, it is possible to extend the treatment of vague predicates as partial predicates to will: Since partial predicates have no scope interactions with negation, this can account for the scopelessness of will. The partiality of truth conditions is not restricted to will but also pertains to counterfactuals and generics and probably is part of a general tendency to maximize contraries. (shrink)

This paper follows Part I of our essay on case-intensional first-order logic (CIFOL; Belnap and Müller (2013)). We introduce a framework of branching histories to take account of indeterminism. Our system BH-CIFOL adds structure to the cases, which in Part I formed just a set: a case in BH-CIFOL is a moment/history pair, specifying both an element of a partial ordering of moments and one of the total courses of events (extending all the way into the future) that that moment (...) is part of. This framework allows us to define the familiar Ockhamist temporal/modal connectives, most notably for past, future, and settledness. The novelty of our framework becomes visible in our discussion of substances in branching histories, i.e., in its first-order part. That discussion shows how the basic idea of tracing an individual thing from case to case via an absolute property is applicable in a branching histories framework. We stress the importance of keeping apart extensionality and moment-definiteness, and give a formal account of how the specification of natural sortals and natural qualities turns out to be a coordination task in BH-CIFOL. We also provide a detailed answer to Lewis’s well-known argument against branching histories, exposing the fallacy in that argument. (shrink)

This paper follows Part I of our essay on case-intensional first-order logic ). We introduce a framework of branching histories to take account of indeterminism. Our system BH-CIFOL adds structure to the cases, which in Part I formed just a set: a case in BH-CIFOL is a moment/history pair, specifying both an element of a partial ordering of moments and one of the total courses of events that that moment is part of. This framework allows us to define the familiar (...) Ockhamist temporal/modal connectives, most notably for past, future, and settledness. The novelty of our framework becomes visible in our discussion of substances in branching histories, i.e., in its first-order part. That discussion shows how the basic idea of tracing an individual thing from case to case via an absolute property is applicable in a branching histories framework. We stress the importance of keeping apart extensionality and moment-definiteness, and give a formal account of how the specification of natural sortals and natural qualities turns out to be a coordination task in BH-CIFOL. We also provide a detailed answer to Lewis’s well-known argument against branching histories, exposing the fallacy in that argument. (shrink)

In this paper I address issues related to the problem of future contingents and the metaphysical doctrine of fatalism. Two classical responses to the problem of future contingents are the third truth value view and the all-false view. According to the former, future contingents take a third truth value which goes beyond truth and falsity. According to the latter, they are all false. I here illustrate and discuss two ways to respectively argue for those two views. Both ways are similar (...) in spirit and intimately connected with fatalism, in the sense that they engage with the doctrine of fatalism and accept a large part of a standard fatalistic machinery. (shrink)

Over the past few years, the tree model of time has been widely employed to deal with issues concerning the semantics of tensed discourse. The thought that has motivated its adoption is that the most plausible way to make sense of indeterminism is to conceive of future possibilities as branches that depart from a common trunk, constituted by the past and the present. However, the thought still needs to be further articulated and defended, and several important questions remain open, such (...) as the question of how actuality can be understood and formally represented in a branching framework. The present volume is intended to be a 360 degree reflection on the tree model. The contributions is gathers concern the model and its alternatives, both from a semantic and from a metaphysical point of view. ​. (shrink)

A fairly simple theory of the semantics of tense is obtained by combining three claims: (i) for any time t, a present-tense sentence `p' is either true or false at t; (ii) for any time t0 earlier than t, the future-tense sentence `It will be the case that p at t' is true at t0 if `p' is true at t, false otherwise; (iii) for any time t0 later than t, the past-tense sentence `It was the case that p at (...) t' is true at t0 if `p' is true at t, false otherwise. This theory, which has been called the theory of timeless truth, is often dismissed on the basis of its alleged incapacity to comply with indeterminism. Here, instead, it will be suggested that there is no reason to be dismissive. Section 1 provides some elucidations about the theory. Sections 2 and 3 explain how a foregone objection to it can be resisted. Sections 4 and 5 show that there is a plausible sense in which (i)-(iii) are compatible with indeterminism. Sections 6 and 7 dispel some misunderstandings that may lead to think that indeterminism is not vindicated in some important sense other than that outlined. (shrink)