This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. David Elohim examines the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...) propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal and hyperintensional cognitivism and modal and hyperintensional expressivism. Elohim develops a novel, topic-sensitive truthmaker semantics for dynamic epistemic logic, and develops a novel, dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for epistemic (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between Elohim's hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. Elohim countenances a hyperintensional semantics for novel epistemic abstractionist modalities. Elohim suggests, too, that higher observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the truthmakers thereof. This yields the first hyperintensional category theory in the literature. Elohim invents a new mathematical trick in which first-order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals in the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Elohim provides a counter-example to epistemic closure for logical deduction. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

This influential 1888 publication explained the real numbers, and their construction and properties, from first principles.

David Wallace argues that we should take quantum theory seriously as an account of what the world is like--which means accepting the idea that the universe is constantly branching into new universes. He presents an accessible but rigorous account of the 'Everett interpretation', the best way to make coherent sense of quantum physics.

Gödel's two incompleteness theorems are among the most important results in modern logic, and have deep implications for various issues. They concern the limits of provability in formal axiomatic theories. The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. According to the second incompleteness theorem, such a formal system cannot (...) prove that the system itself is consistent (assuming it is indeed consistent). These results have had a great impact on the philosophy of mathematics and logic. There have been attempts to apply the results also in other areas of philosophy such as the philosophy of mind, but these attempted applications are more controversial. The present entry surveys the two incompleteness theorems and various issues surrounding them. (shrink)

1 Cognitive Architectures 2 Multilayer Perceptrons 3 Relations between Variables 4 Structured Representations 5 Individuals 6 Where does the Machinery of Symbol Manipulation Come From? 7 Conclusions.

This work is a step toward the development of a logic for types and computation that includes not only the usual spaces of mathematics and constructions, but also spaces from logic and domain theory. Using realizability, we investigate a configuration of three toposes that we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we first study axiomatically, and then by deriving a modal calculus as its internal logic. The resulting (...) framework is intended as a setting for the logical and categorical study of relative computability. (shrink)

In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant (...) and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Ω-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing. (shrink)

There are four broad grounds upon which the intelligibility of quantification over absolutely everything has been questioned—one based upon the existence of semantic indeterminacy, another on the relativity of ontology to a conceptual scheme, a third upon the necessity of sortal restriction, and the last upon the possibility of indefinite extendibility. The argument from semantic indeterminacy derives from general philosophical considerations concerning our understanding of language. For the Skolem–Lowenheim Theorem appears to show that an understanding of quanti- fication over absolutely (...) everything (assuming a suitably infinite domain) is semantically indistinguishable from the understanding of quantification over something less than absolutely everything; the same first-order sentences are true and even the same first-order conditions will be satisfied by objects from the narrower domain. From this it is then argued that the two kinds of understanding are indistinguishable tout court and that nothing could count as having the one kind of understanding as opposed to the other. (shrink)

Following Lewis, it is widely held that branching worlds differ in important ways from diverging worlds. There is, however, a simple and natural semantics under which ordinary sentences uttered in branching worlds have much the same truth values as they conventionally have in diverging worlds. Under this semantics, whether branching or diverging, speakers cannot say in advance which branch or world is theirs. They are uncertain as to the outcome. This same semantics ensures the truth of utterances typically made about (...) quantum mechanical contingencies, including statements of uncertainty, if the Everett interpretation of quantum mechanics is true. The ‘incoherence problem’ of the Everett interpretation, that it can give no meaning to the notion of uncertainty, is thereby solved. IntroductionMetaphysicsPersonal fissionBranching worldsPhysicsObjections. (shrink)

I hear the patter of little feet around the house, I expect Bruce. What I expect is a cat, a particular cat. If I heard such a patter in another house, I might expect a cat but no particular cat. What I expect then seems to be a Meinongian incomplete cat. I expect winter, expect stormy weather, expect to shovel snow, expect fatigue---a season, a phenomenon, an activity, a state. I expect that someday mankind will inhabit at least five planets. (...) This time what I expect is a state of affairs. If we let surface grammar be our guide, the objects of expectation seem quite a miscellany. The same goes for belief, since expectation is one kind of belief. The same goes for desire: I could want Bruce, want a cat but no particular cat, want winter, want stormy weather, want to shovel snow, want fatigue, or want that someday mankind will inhabit at least five planets. The same goes for other attitudes to the extent that they consist partly of beliefs or desires or lacks thereof. But the seeming diversity of objects might be an illusion. Perhaps the objects of attitudes are uniform in category, and it is our ways of speaking elliptically about these uniform objects that are diverse. That indeed is our consensus. We mostly think that the attitudes uniformly have propositions as their objects. That is why we speak habitually of "propositional attitudes.". (shrink)

This chapter explains the basic framework of truthmaker or 'exact' semantics, an approach to semantics that has recently received a growing amount of interest, and discusses a number of different applications within philosophy and linguistics. The idea of truthmaking is the idea of something on the side of the world ‐ a fact, perhaps, or a state of affairs ‐ verifying, or making true, something on the side of language or thought ‐ a statement, perhaps, or a proposition. The chapter (...) focuses on the ultimate truthmakers, on what in the world ultimately makes something true, and the question of how the truthmakers make the statements of our language true is of no great concern. It also focuses on the immediate truthmakers, and the question of how they make the statements of the language true will be of greatest concern. (shrink)

The book is an attempt at explaining to the nation the ideas of Frege's Grundlagen. It is wordy and trite, a paradigm case of a redundant piece of writing. The reader is advised to steer clear of it.

The Continuum Problem has inspired set theorists and philosophers since the days of Cantorian set theory. In the last 15 years, W. Hugh Woodin, a leading set theorist, has not only taken it upon himself to engage in this question, he has also changed his mind about the answer. This paper illustrates Woodin’s solutions to the problem, starting in Sect. 3 with his 1999–2004 argument that Cantor’s hypothesis about the continuum was incorrect. From 2010 onwards, Woodin presents a very different (...) argument, an argument that Cantor’s hypothesis is in fact true. This argument is still incomplete, but according to Woodin, some of the philosophical issues surrounding the Continuum Problem have been reduced to precise mathematical questions, questions that are, unlike Cantor’s hypothesis, solvable from our current theory of sets. (shrink)

Turing computation over a non-linguistic domain presupposes a notation for the domain. Accordingly, computability theory studies notations for various non-linguistic domains. It illuminates how different ways of representing a domain support different finite mechanical procedures over that domain. Formal definitions and theorems yield a principled classification of notations based upon their computational properties. To understand computability theory, we must recognize that representation is a key target of mathematical inquiry. We must also recognize that computability theory is an intensional enterprise: it (...) studies entities as represented in certain ways, rather than entities detached from any means of representing them. (shrink)

Constructor theory seeks to express all fundamental scientific theories in terms of a dichotomy between possible and impossible physical transformations–those that can be caused to happen and those that cannot. This is a departure from the prevailing conception of fundamental physics which is to predict what will happen from initial conditions and laws of motion. Several converging motivations for expecting constructor theory to be a fundamental branch of physics are discussed. Some principles of the theory are suggested and its potential (...) for solving various problems and achieving various unifications is explored. These include providing a theory of information underlying classical and quantum information; generalising the theory of computation to include all physical transformations; unifying formal statements of conservation laws with the stronger operational ones (such as the ruling-out of perpetual motion machines); expressing the principles of testability and of the computability of nature (currently deemed methodological and metaphysical respectively) as laws of physics; allowing exact statements of emergent laws (such as the second law of thermodynamics); and expressing certain apparently anthropocentric attributes such as knowledge in physical terms. (shrink)

We look at recent accounts of the indefinite extensibility of the concept set and compare them with a certain linguistic model of indefinite extensibility. We suggest that the linguistic model has much to recommend over alternative accounts of indefinite extensibility, and we defend it against three prima facie objections.

We discuss the work of Paul Cohen in set theory and its influence, especially the background, discovery, development of forcing.

Kurt Gödel with his work on the constructible universeLestablished the relative consistency of the Axiom of Choice and the Continuum Hypothesis. More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of set theoretic constructions (...) and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about Gödel's work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence. Beyond the surface of things we delve deeper into the mathematics. What emerges are the roots and anticipations in work of Russell and Hilbert, and most prominently the sustained motif of truth as formalizable in the “next higher system”. We especially work at bringing out how transforming Gödel's work was for set theory. It is difficult now to see what conceptual and technical distance Gödel had to cover and how dramatic his re-orientation of set theory was. (shrink)

Treating the existential quantification ∃ν i as a diamond $\diamond_i$ and the identity ν i = ν j as a constant δ ij , we study restricted versions of first order logic as if they were modal formalisms. This approach is closely related to algebraic logic, as the Kripke frames of our system have the type of the atom structures of cylindric algebras; the full cylindric set algebras are the complex algebras of the intended multidimensional frames called cubes. The main (...) contribution of the paper is a characterization of these cube frames for the finite-dimensional case and, as a consequence of the special form of this characterization, a completeness theorem for this class. These results lead to finite, though unorthodox, derivation systems for several related formalisms, e.g. for the valid n-variable first order formulas, for type-free valid formulas and for the equational theory of representable cylindric algebras. The result for type-free valid formulas indicates a positive solution to Problem 4.16 of Henkin, Monk and Tarski [16]. (shrink)

I develop a basic theory of content within the framework of truthmaker semantics and, in the second part, consider some of the applications to subject matter, common content, logical subtraction and ground.