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.

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)

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.

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)

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 arithmetic, if only because many of its methods and concepts originated in India, it has been the tradition to reason less strictly than in geometry, ...

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)

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)

The first complete English translation of a groundbreaking work. An ambitious account of the relation of mathematics to logic. Includes a foreword by Crispin Wright, translators' Introduction, and an appendix on Frege's logic by Roy T. Cook. The German philosopher and mathematician Gottlob Frege (1848-1925) was the father of analytic philosophy and to all intents and purposes the inventor of modern logic. Basic Laws of Arithmetic, originally published in German in two volumes (1893, 1903), is Freges magnum opus. It was (...) to be the pinnacle of Freges lifes work. It represents the final stage of his logicist project the idea that arithmetic and analysis are reducible to logic and contains his mature philosophy of mathematics and logic. The aim of Basic Laws of Arithmetic is to demonstrate the logical nature of mathematical theorems by providing gapless proofs in Frege's formal system using only basic laws of logic, logical inference, and explicit definitions. The work contains a philosophical foreword, an introduction to Frege's logic, a derivation of arithmetic from this logic, a critique of contemporary approaches to the real numbers, and the beginnings of a logicist treatment of real analysis. As is well-known, a letter received from Bertrand Russell shortly before the publication of the second volume made Frege realise that his basic law V, governing the identity of value-ranges, leads into inconsistency. Frege discusses a revision to basic law V written in response to Russells letter in an afterword to volume II. The continuing importance of Basic Laws of Arithmetic lies not only in its bearing on issues in the foundations of mathematics and logic but in its model of philosophical inquiry. Frege's ability to locate the essential questions, his integration of logical and philosophical analysis, and his rigorous approach to criticism and argument in general are vividly in evidence in this, his most ambitious work. Philip Ebert and Marcus Rossberg present the first full English translation of both volumes of Freges major work preserving the original formalism and pagination. The edition contains a foreword by Crispin Wright and an extensive appendix providing an introduction to Frege's formal system by Roy T. Cook. Readership: Scholars and advanced students in philosophy of logic, philosophy of mathematics, and early analytic philosophy. (shrink)

Simon Saunders and David Wallace have proposed an attractive semantics for interpreting linguistic communities embedded in an Everettian multiverse. It provides a charitable interpretation of our ordinary talk about the future, and allows us to retain a principle of bivalence for propositions and to retain the law of excluded middle in the logic of propositions about the future. But difficulties arise when it comes to providing an appropriate account of the metaphysics of macroscopic objects and events. I evaluate various metaphysical (...) frameworks which might be combined with the Saunders–Wallace semantics. I conclude that the most appropriate metaphysics to underwrite the semantics renders Everettian quantum mechanics a theory of non-overlapping worlds. (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)

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.

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)

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)

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)

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

Each article in this book usually discusses more than one German word, together with their English (and often Greek and Latin) equivalents. Some words are grouped together because they fall under a single broad heading (e.g. SPACE AND SPATIALITY), sometimes because two or more similar seeming words are used by Heidegger in different ways and need to be distinguished (e.g. FATE AND DESTINY, or SIGNS AND HINTS). Not surprisingly, in view of Heidegger's belief that every philosophical question raises every other (...) philosophical question, I have often found it difficult to confine the discussion of a topic to a single article. Thus I have often indicated a cross‐reference from one article to another by capitalizing a word that appears in the title of another relevant article. The occurrence of ‘PRESENT‐AT‐HAND’ means: Consult the article whose title contains the expression ‘present‐at‐hand’. The main discussions of difficult words and concepts are also indicated in the General Index. (shrink)

Metaphysicians should pay attention to quantum mechanics. Why? Not because it provides definitive answers to many metaphysical questions-the theory itself is remarkably silent on the nature of the physical world, and the various interpretations of the theory on offer present conflicting ontological pictures. Rather, quantum mechanics is essential to the metaphysician because it reshapes standard metaphysical debates and opens up unforeseen new metaphysical possibilities. Even if quantum mechanics provides few clear answers, there are good reasons to think that any adequate (...) understanding of the quantum world will result in a radical reshaping of our classical world-view in some way or other. Whatever the world is like at the atomic scale, it is almost certainly not the swarm of particles pushed around by forces that is often presupposed. This book guides readers through the theory of quantum mechanics and its implications for metaphysics in a clear and accessible way. The theory and its various interpretations are presented with a minimum of technicality. The consequences of these interpretations for metaphysical debates concerning realism, indeterminacy, causation, determinism, holism, and individuality are explored in detail, stressing the novel form that the debates take given the empirical facts in the quantum domain. While quantum mechanics may not deliver unconditional pronouncements on these issues, the range of possibilities consistent with our knowledge of the empirical world is relatively small-and each possibility is metaphysically revisionary in some way. This book will appeal to researchers, students, and anybody else interested in how science informs our world-view. (shrink)