Andrew Bacon

Central to certain versions of logical atomism are claims to the effect that every proposition is a truth-functional combination of elementary propositions. Assuming that propositions form a Boolean algebra, we consider a number of natural formal regimentations of informal claims in this vicinity, and show that they are equivalent. For a number of reasons, such as the need to accommodate quantifiers, logical atomists might consider only complete Boolean algebras, and take into account infinite truth-functional combinations. We show that in such a variant setting, some of the regimentations come apart, and explore how they relate to each other. We also discuss how they relate to the claim that propositions form a double powerset algebra, which has been proposed by a number of authors as a way of capturing the central logical atomist idea.

This paper argues that the theory of structured propositions is not undermined by the Russell-Myhill paradox. I develop a theory of structured propositions in which the Russell-Myhill paradox doesn't arise: the theory does not involve ramification or compromises to the underlying logic, but rather rejects common assumptions, encoded in the notation of the $\lambda$-calculus, about what properties and relations can be built. I argue that the structuralist had independent reasons to reject these underlying assumptions. The theory is given both a diagrammatic representation, and a logical representation in a novel language. In the latter half of the paper I turn to some technical questions concerning the treatment of quantification, and demonstrate various equivalences between the diagrammatic and logical representations, and a fragment of the $\lambda$-calculus.

A number of arguments purport to show that vague properties determine sharp boundaries at higher orders. That is, although we may countenance vagueness concerning the location of boundaries for vague predicates, every predicate can instead be associated with precise knowable cut-off points deriving from precision in their higher order boundaries. I argue that this conclusion is indeed paradoxical, and identify the assumption responsible for the paradox as the Brouwerian principle B for vagueness: that if p then it's determinate that it's not determinate that not p. Other paradoxes which do not appear to turn on B turn instead on some subtle issues concerning the relation between assertion, belief and higher order vagueness. In this paper a theory of assertion, knowledge and logic is outlined which allows one to avoid any kind of higher order precision. A class of realistic models containing counterexamples to B and a number of weakenings of B are introduced and its logic is shown to support vagueness at every order.

In this paper the logic of broad necessity is explored. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. It is shown, moreover, that it is possible to give a reductive analysis of this necessity in extensional language. This relates more generally to a conjecture that it is not possible to define intensional connectives from extensional notions. This conjecture is formulated precisely in higher-order logic, and concrete cases in which it fails are examined. The paper ends with a discussion of the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. Some philosophical reasons to think that the logic of broad necessity does not include the S5 principle are given.

This is the first comprehensive textbook on higher order logic that is written specifically to introduce the subject matter to graduate students in philosophy. The book covers both the formal aspects of higher-order languages -- their model theory and proof theory, the theory of λ-abstraction and its generalizations -- and their philosophical applications, especially to the topics of modality and propositional granularity. The book has a strong focus on non-extensional higher-order logics, making it more appropriate for foundational metaphysics than other introductions to the subject from computer science, mathematics, and linguistics. A Philosophical Introduction to Higher Order Logics assumes only that readers have a basic knowledge of first-order logic. With an emphasis on exercises, it can be used as a textbook though is also ideal for self-study. Author Andrew Bacon organizes the book's 18 chapters around four main parts: I. Typed Language II. Higher Order Languages III. General Higher-Order Languages IV. Higher-Order Model Theory In addition, two appendices cover the Curry-Howard isomorphism and its applications for modeling propositional structure. Each chapter includes exercises that move from easier to more difficult, strategically placed throughout the chapter, and concludes with an annotated suggested reading list providing graduate students with most valuable additional resources. Key Features Is the first comprehensive introduction to higher-order logic as a grounding for addressing problems in metaphysics Introduces the basic formal tools that are needed to theorize in, and model, higher order languages Offers an abundance of: - Simple exercises throughout the book, serving as comprehension checks on basic concepts and definitions - More difficult exercises designed to facilitate long-term learning Contains annotated sections on further reading, pointing the reader to related literature, learning resources, and historical context.

Higher-order logic augments first-order logic with devices that let us generalize into grammatical positions other than that of a singular term. Some recent metaphysicians have advocated for using these devices to raise and answer questions that bear on many traditional issues in philosophy. In contrast to these 'higher-order metaphysicians', traditional metaphysics has often focused on parallel, but importantly different, questions concerning special sorts of abstract objects: propositions, properties and relations. The answers to the higher-order and the property-theoretic questions may coincide sometimes but will often come apart. I argue that when they do, the higher-order questions are closer to the metaphysical action and so it would be better for these debates to proceed in higher-order terms.

We develop a theory of necessity operators within a version of higher-order logic that is neutral about how fine-grained reality is. The theory is axiomatized in terms of the primitive of *being a necessity*, and we show how the central notions in the philosophy of modality can be recovered from it. Various questions are formulated and settled within the framework, including questions about the ordering of necessities under strength, the existence of broadest necessities satisfying various logical conditions, and questions about their logical behaviour. We also wield the framework to probe the conditions under which a logicist account of necessities is possible, in which the theory is completely reducible to logic.

This three-part chapter explores a higher-order logic we call ‘Classicism’, which extends a minimal classical higher-order logic with further axioms which guarantee that provable coextensiveness is sufficient for identity. The first part presents several different ways of axiomatizing this theory and makes the case for its naturalness. The second part discusses two kinds of extensions of Classicism: some which take the view in the direction of coarseness of grain (whose endpoint is the maximally coarse-grained view that coextensiveness is sufficient for identity), and some which take the view in the direction of fineness of grain (whose endpoint is the maximally fine-grained theory containing all distinctness claims compatible with Classicism). The third part introduces some techniques for constructing models of Classicism, and uses them to prove the consistency of many of the extensions of Classicism introduced in the second part.

Prior to Kripke’s seminal work on the semantics of modal logic, McKinsey offered an alternative interpretation of the necessity operator, inspired by the Bolzano–Tarski notion of logical truth. According to this interpretation, ‘it is necessary that A’ is true just in case every sentence with the same logical form as A is true. In our paper, we investigate this interpretation of the modal operator, resolving some technical questions, and relating it to the logical interpretation of modality and some views in modal metaphysics. In particular, we present an hitherto unpublished solution to problems 41 and 42 from Friedman’s 102 problems, which uses a different method of proof from the solution presented in the paper of Tadeusz Prucnal.

I consider two puzzles in which an agent undergoes a sequence of decision problems. In both cases it is possible to respond rationally to any given problem yet it is impossible to respond rationally to every problem in the sequence, even though the choices are independent. In particular, although it might be a requirement of rationality that one must respond in a certain way at each point in the sequence, it seems it cannot be a requirement to respond as such at every point for that would be to require the impossible.

In explaining the notion of a fundamental property or relation, metaphysicians will often draw an analogy with languages. The fundamental properties and relations stand to reality as the primitive predicates and relations stand to a language: the smallest set of vocabulary God would need in order to write the “book of the world.” This paper attempts to make good on this metaphor. To that end, a modality is introduced that, put informally, stands to propositions as logical truth stands to sentences. The resulting theory, formulated in higher-order logic, also vindicates the Humean idea that fundamental properties and relations are freely recombinable and a variant of the structural idea that propositions can be decomposed into their fundamental constituents via logical operations. Indeed, it is seen that, although these ideas are seemingly distinct, they are not independent, and fall out of a natural and general theory about the granularity of reality.

This paper formulates some paradoxes of inductive knowledge. Two responses in particular are explored: According to the first sort of theory, one is able to know in advance that certain observations will not be made unless a law exists. According to the other, this sort of knowledge is not available until after the observations have been made. Certain natural assumptions, such as the idea that the observations are just as informative as each other, the idea that they are independent, and that they increase your knowledge monotonically (among others) are given precise formulations. Some surprising consequences of these assumptions are drawn, and their ramifications for the two theories examined. Finally, a simple model of inductive knowledge is offered, and independently derived from other principles concerning the interaction of knowledge and counterfactuals.

We explore the view that Frege's puzzle is a source of straightforward counterexamples to Leibniz's law. Taking this seriously requires us to revise the classical logic of quantifiers and identity; we work out the options, in the context of higher-order logic. The logics we arrive at provide the resources for a straightforward semantics of attitude reports that is consistent with the Millian thesis that the meaning of a name is just the thing it stands for. We provide models to show that some of these logics are non-degenerate.

According to orthodoxy the study of vagueness belongs to the domain of the philosophy of language. On that view, to solve the paradoxes of vagueness we need to investigate the nature of words like ‘heap’ and ‘bald’. This book criticizes linguistic explanations of the state of ignorance we find ourselves in when confronted with borderline cases and develops, within the framework of classical logic, a theory of propositional vagueness in its stead. The view places the study of vagueness squarely in epistemological terms, situating it within a theory of rational propositional attitudes. Once one has accepted vague propositions, a number of questions about their role in thought become conspicuous. Can one’s total evidence be vague? What sort of support does vague evidence lend to precise matters and conversely? Can rational people agree about the precise whilst disagreeing about the vague? Is it rational to care intrinsically about vague matters? Can one’s attitudes towards vague propositions be relevant in decision making? The book develops a set of positions on these matters, and exploits them in expounding a novel theory of vagueness in which vagueness is defined in terms of its role in thought. The resulting view is applied to a number of problems in the philosophy of vagueness.