The Moral Law is fulfilled iff everything that ought to be the case is the case, and The Good is realised in a possible world w at a time t iff w is deontically accessible from w at t. In this paper, I will introduce a set of temporal modal deontic systems with propositional quantifiers that can be used to prove some interesting theorems about The Moral Law and The Good. First, I will describe a set of systems without any (...) propositional quantifiers. Then, I will show how these systems can be extended by a couple of propositional quantifiers. I will use a kind of TxW semantics to describe the systems semantically and semantic tableaux to describe them syntactically. Every system will include a constant · that stands for The Good. ‘·’ is read as ‘The Good is realised’. All systems that contain the propositional quantifiers will also include a constant '*' that stands for The Moral Law. '*' is read as ‘The Moral Law is fulfilled’. I will prove that all systems (without the propositional quantifiers) are sound and complete with respect to their semantics and that all systems (including the extended systems) are sound with respect to their semantics. It is left as an open question whether or not the extended systems are complete. (shrink)

The present dissertation investigates certain facets of the logical structure of oughts – where “ought” is used as a noun, roughly meaning obligation. I do so by following two lines of inquiry. The first part of the thesis places oughts in the context of practical rationality. The second part of the thesis concerns the inference rules governing arguments about oughts, and specifically the inference rule of Reasoning by Cases. These two lines of inquiry, together, aim to expound upon oughts in (...) rationality and reasoning. The methodology used in this dissertation is the one of philosophical logic, in which logical, qualitative models are developed to support and foster conceptual analysis. The dissertation consists of four main chapters. The first two chapters are devoted to the role of oughts in practical rationality. I focus on the so-called Enkratic principle of rationality, which – in its most general formulation – requires that if an agent believes sincerely and with conviction that she ought to do X, then she intends to X. I develop a logical framework to investigate the (static and dynamic) relation between those oughts believed by the agent and her intentions. It is shown that, under certain minimal assumptions, the Enkratic principle of rationality is a principle of limited validity. The following two chapters of the dissertation constitute a study of the classical inference rule of Reasoning by Cases, which – in its simplest form – moves from the premises “A or B”, “if A then C” and “if B then C” to the conclusion “C”. Recent literature has called the validity of Reasoning by Cases into question, with the most influential counterexample being the so-called Miners’ Puzzle – an instance of Reasoning by Cases where “C” involves oughts. I provide a unifying explanation of why the Miners’ Puzzle emerges. It is shown that, within specific boundaries, Reasoning by Cases is a valid inference rule. (shrink)

This paper focuses on the Enkratic principle of rationality, according to which rationality requires that if an agent sincerely and with conviction believes she ought to X, then X-ing is a goal in her plan. We analyze the logical structure of Enkrasia and its implications for deontic logic. To do so, we elaborate on the distinction between basic and derived oughts, and provide a multi-modal neighborhood logic with three characteristic operators: a non-normal operator for basic oughts, a non-normal operator for (...) goals in plans, and a normal operator for derived oughts. We prove two completeness theorems for the resulting logic, and provide a dynamic extension of the logic by means of product updates. We illustrate how this setting informs deontic logic by considering issues related to the filtering of inconsistent oughts, the restricted validity of deontic closure, and the stability of oughts and goals under dynamics. (shrink)

This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...) the semantics, such as inductive skepticism, the logic of discovery, Duhem’s problem, the articulation of theories by auxiliary hypotheses, the role of serendipity in scientific knowledge, Fitch’s paradox, deductive closure of knowability, whether one can know inductively that one knows inductively, whether one can know inductively that one does not know inductively, and whether expert instruction can spread common inductive knowledge—as opposed to mere, true belief—through a community of gullible pupils. (shrink)

The purpose of this paper is to describe a set of temporal alethic–deontic systems, i.e. systems that include temporal, alethic and deontic operators. All in all we will consider 2,147,483,648 systems. All systems are described both semantically and proof theoretically. We use a kind of possible world semantics, inspired by the so-called T x W semantics, to characterize our systems semantically and semantic tableaux to characterize them proof theoretically. We also show that all systems are sound and complete with respect (...) to their semantics. (shrink)

This paper deals with structures ${\langle{\bf T}, I\rangle}$ in which T is a tree and I is a function assigning each moment a partition of the set of histories passing through it. The function I is called indistinguishability and generalizes the notion of undividedness. Belnap’s choices are particular indistinguishability functions. Structures ${\langle{\bf T}, I\rangle}$ provide a semantics for a language ${\mathcal{L}}$ with tense and modal operators. The first part of the paper investigates the set-theoretical properties of the set of indistinguishability (...) classes, which has a tree structure. The significant relations between this tree and T are established within a general theory of trees. The aim of second part is testing the expressive power of the language ${\mathcal{L}}$ . The natural environment for this kind of investigations is Belnap’s seeing to it that (stit). It will be proved that the hybrid extension of ${\mathcal{L}}$ (with a simultaneity operator) is suitable for expressing stit concepts in a purely temporal language. (shrink)

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)