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The book provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. The notion of abstraction in question is that related to the use of abstraction principles in neo-logicism. The most familiar abstraction principle in this context is Hume’s Principle. Hume’s Principle says that two concepts have the same number if and only if the objects falling under each one of them can be put in one–one correspondence. Chapter 1 shows that abstraction principles were quite widespread in the mathematical practice that preceded Frege’s discussion of them. The second chapter provides the first contextual analysis of Frege’s discussion of abstraction principles in section 64 of the Grundlagen; the second part investigates the foundational reflection on abstraction principles in the Peanosets not by using school and Russell. Chapter 3 discusses a novel approach to measuring the size of infinite sets known as the theory of numerosities. This theory assigns numerosities to infinite sets not by using one–one correspondence but by preserving the part–whole principle, namely the principle according to which if a set A is strictly included in a set B, then the numerosity of A is strictly smaller than the numerosity of B. Mancosu shows how this new development leads to deep mathematical, historical, and philosophical problems. Chapter 4 brings the previous strands together by offering some surprising novel perspectives on neo-logicism.

The collection contains an extensive introduction and 16 original papers on the philosophical and mathematical aspects of Abstractionism—a position in the philosophy of mathematics which is a development of Frege’s original Logicism. The collection is structured as follows: After an extensive editors’ introduction to the topic of abstractionism, part II contains five contributions that deal with semantics and metaontology of Abstractionism, as well as the so-called Caesar Problem. Part III collects four contributions that discuss abstractionist epistemology, focusing on the idea of implicit definitions and non-evidential warrants (entitlements) to account for a priori mathematical knowledge. Four papers in part IV concern the mathematics of Abstractionism, in particular the issue of impredicativity, the Bad Company objection, and the question of abstractionist set theory. The last section contains three contributions that discuss Frege’s application constraint within an abstractionist setting.

The book contains innovative contributions to the history and the philosophy of logic and mathematics in the first half of the twentieth century. It is divided into five main sections: history of logic (from Russell to Tarski); foundational issues (Hilbert’s program, constructivity, Wittgenstein, Gödel); mathematics and phenomenology (Weyl, Becker, Mahnke); nominalism (Quine, Tarski); semantics (Tarski, Carnap, Neurath). The treatment exploits extensively untapped archival sources thereby making available a wealth of new material that deepens in significant ways our understanding of the above-mentioned areas. At the same time, the book is a contribution to recent debates on, among other things, the prospects for a successful nominalist reconstruction of mathematics, the nature of finitist intuition, the viability of alternative definitions of logical consequence, and the extent to which phenomenology can hope to account for the exact sciences.

What has been called ‘the unreasonable effectiveness of mathematics’ sets a challenge for philosophers. Some have responded to that challenge by arguing that mathematics is essentially anthropocentric in character whereas others have pointed to the range of structures that mathematics offers. Here a middle way is offered that focuses on the moves that have to be made in both the mathematics and the relevant physics in order to bring the two into appropriate relation. This relation can be captured via the inferential conception of the applicability of mathematics which is formulated in terms of immersion inference and interpretation. In particular the roles of idealizations and of surplus structure in science and mathematics respectively are brought to the fore and captured via an approach to models and theories that emphasizes the partiality of the available information: the partial structures approach. The discussion as a whole is grounded in a number of case studies drawn from the history of quantum physics and extended to contest recent claims that the explanatory role of certain mathematical structures in scientific practice supports a realist attitude towards them. The overall conclusion is that the effectiveness of mathematics does not seem unreasonable at all once close attention is paid to how it is actually applied in practice.

Medieval logic consists of theories and practices clustered around a core of rules and axioms; the logic contains widely known principles which can be derived from the core. These are used by Aristotle to prove conversion principles, and reduce some syllogisms to others: Exposition (“existential instantiation”), Expository Syllogism (“existential generalization”) and Reductio. Medieval logicians expand on Aristotle’s notation, and this brings new logical principles, such as quantifier interchanges. Theorists use the flexible word order to Latin to let surface order of expressions determine semantic scope. They also make assumptions about existential import that need dealing with. More important, medieval logic is formulated within a natural language, Latin, so there is no logical form except for grammatical form. We look at what is most distinctive of late medieval logic, the useful theory of modes of personal supposition. We examine special terms used to accommodate sentences containing three or more main quantified phrases. Logicians theorize about the logical behavior of pronouns with antecedents; without such pronouns the system of logic is weak; with them added it is expressively as rich as contemporary predicate logic. Also touched on are tense and modality, and intentional contexts, and artificial signs used to alter scope.

Ascriptions of mental states to oneself and others give rise to many interesting logical and semantic problems. This problem presents an original account of mental state ascriptions that are made using intensional transitive verbs such as ‘want’, ‘seek’, ‘imaginer’, and ‘worship’. This book offers a theory of how such verbs work that draws on ideas from natural language semantics, philosophy of language, and aesthetics.

This book presents an expanded edition of the author's exploration of the nature and limits of thought. Embracing contradiction and challenging traditional logic, the book engages with issues across philosophical borders, from the historical to the modern, from Eastern to Western, and from the continental to the analytic. This edition of the text includes new chapters on European and Indian philosophy, and reflections on responses to the previous edition of the book.

This book proposes a theory of analogical arguments, with special focus on analogies in mathematics and science. The core principle of the theory is that a good analogical argument must articulate a clear relationship capable of generalization. This idea leads to a set of distinct models for the critical analysis of prominent forms of analogical argument, corresponding to different logical, causal and probabilistic relationships that occur in scientific reasoning. The same principle allows us to relate analogical reasoning to broad norms and values of scientific practice, such as symmetry and unification. Elaborating this principle, the book raises questions and proposes answers regarding (1) criteria for evaluating analogical arguments, (2) the philosophical justification for analogical reasoning, and (3) the place of scientific analogies in the context of theoretical confirmation.

This book provides a presentation of views on the relations between metaphysics and logic from Aristotle through twentieth century philosophers who contributed to the return of metaphysics in the analytic tradition. The collection combines interest in logic and its history with interest in analytical metaphysics and the history of metaphysical thought. The focus is on metaphysica generalis, or the systematic study of the most general categories of being. The volume aims at historical coverage of certain influential figures and themes. As the tradition is very rich, some choices between important philosophers and topics cannot be avoided. The volume seeks for a balance between different periods; still, early modern, modern and twentieth century metaphysics are more extensively studied than the pre-modern tradition. Thinkers discussed include Aristotle, Avicenna, Thomas Aquinas, Duns Scotus, William Ockham, Gottfried Wilhelm Leibniz, Immanuel Kant, Georg Wilhelm Friedrich Hegel, Bernard Bolzano, Charles Sanders Peirce, Georg Cantor, Gottlob Frege, Alexius Meinong, Edmund Husserl, Bertrand Russell, G. E. Moore, C. I. Lewis, Martin Heidegger, Ludwig Wittgenstein, Rudolf Carnap, Willard Van Orman Quine, Wilfrid Sellars, Peter F. Strawson, Ruth Barcan Marcus, David Armstrong, Saul Kripke, and David Lewis. Not all of these have a chapter of their own, however, for some figure only in connection with other thinkers and specific themes related with their work. The individual chapters seek to cover more than one philosopher's thought and also to take notice of other periods in the history than what is their main focus.

This account of rational belief revision explains how a rational agent ought to proceed when adopting a new belief — a difficult matter if the new belief contradicts the agent’s old beliefs. Belief systems are modeled as finite dependency networks. So one can attend not only to what the agent believes, but also to the variety of reasons the agent has for so believing. The computational complexity of the revision problem is characterized. Algorithms for belief revision are formulated, and implemented in Prolog. The implementation tests well on a range of simple belief‐revision problems that pose a variety of challenges for any account of belief‐revision. The notion of ‘minimal mutilation’ of a belief system is explicated precisely. The proposed revision methods are invariant across different global justificatory structures (foundationalist, coherentist, etc.). They respect the intuition that, when revising one's beliefs, one should not hold on to any belief that has lost all its former justifications. The limitation to finite dependency networks is shown not to compromise theoretical generality. This account affords a novel way to argue that there is an inviolable core of logical principles. These principles, which form the system of Core Logic, cannot be given up, on pain of not being able to carry out the reasoning involved in rationally revising beliefs. The book ends by comparing and contrasting the new account with some major representatives of earlier alternative approaches, from the fields of formal epistemology, artificial intelligence and mathematical logic.

A theory of collective rationality identifies collective acts that are evaluable for rationality and formulates principles for their evaluation. This book argues that a group's act is evaluable for rationality if it is the product of acts its members freely and fully control. It also argues that such an act is collectively rational if the acts of the group's members are rational. Efficiency is a goal of collective rationality, but not a requirement, except in cases where conditions are ideal for joint action and agents have rationally prepared for coordination. A theory of collective rationality also yields principles concerning solutions to games. One principle requires that a solution constitute an equilibrium among the incentives of the agents in the game. In a cooperative game some agents are coalitions of individuals, and it may be impossible for all agents to pursue all incentives. Because rationality is attainable, the appropriate equilibrium‐standard for cooperative games requires pursuit of an incentive only if it provides a sufficient reason to act. The book's theory of collective rationality supports an attainable equilibrium‐standard for solutions to cooperative games and shows that its realization follows from individuals' rational acts. This book's theory of collective rationality contributes to philosophical projects such as contractarian ethics and to practical projects such as the design of social institutions.

Composition is the relation between a whole and its parts—the parts are said to compose the whole; the whole is composed of the parts. But is a whole anything distinct from its parts taken collectively? It is often said that ‘a whole is nothing over and above its parts’; but what might we mean by that? Could it be that a single whole just is its many parts? This collection of essays is the first of its kind to focus on the relationship between composition and identity. These twelve original articles—written by internationally renowned scholars and rising stars in the field—argue for and against the controversial doctrine that composition is identity. An editor’s introduction sets out the formal, mereological, and philosophical groundwork to bring readers to the forefront of the debate.

Our conception of logical space is the set of distinctions we use to navigate the world. This book defends the idea that one’s conception of logical space is shaped by one’s acceptance or rejection of ‘just is’-statements: statements like ‘to be composed of water just is to be composed of H2O’, or ‘for the number of the dinosaurs to be Zero just is for there to be no dinosaurs’. The resulting picture is used to articulate a conception of metaphysical possibility that does not depend on a reduction of the modal to the non-modal, and to develop a trivialist philosophy of mathematics, according to which the truths of pure mathematics have trivial truth-conditions.

Core Logic has unusual philosophical, proof-theoretic, metalogical, computational, and revision-theoretic virtues. It is an elegant kernel lying deep within Classical Logic, a canon for constructive and relevant deduction furnishing faithful formalizations of informal constructive mathematical proofs. Its classicized extension provides likewise for non-constructive mathematical reasoning. Confining one’s search to core proofs affords automated reasoners great gains in efficiency. All logico-semantical paradoxes involve only core reasoning. Core proofs are in normal form, and relevant in a highly exigent ‘vocabulary-sharing’ sense never attained before. Essential advances on the traditional Gentzenian treatment are that core natural deductions are isomorphic to their corresponding sequent proofs, and make do without the structural rules of Cut and Thinning. This ensures relevance of premises to conclusions of proofs, without loss of logical completeness. Every core proof converts any verifications of its premises into a verification of its conclusion. Core Logic makes one reassess the dogma of ‘unrestricted’ transitivity of deduction, because any core ‘restriction’ of transitivity ensures a more than compensatory payoff of epistemic gain: A core proof of A from X and one of B from {A}∪Y effectively determine a proof of B or of absurdity from some subset of X∪Y. The primitive introduction and elimination rules governing the logical operators in Core Logic are subtly different from Gentzen’s. They are obtained by smoothly extrapolating protean rules for determining truth values of sentences under interpretations. Core rules are inviolable: One needs all of them in order to revise beliefs rationally in light of new evidence.

In this book, the semantics of counterfactuals is studied by drawing on contexts of epistemic uncertainty. Probability plays a twofold role: the concept is used both to describe how counterfactuals are evaluated and to determine an objective goal guiding this evaluation process. It is shown in detail which problems one faces in trying to reconcile the behaviour of counterfactuals in contexts of uncertainty with a truth-conditional semantics for counterfactuals. To face the resulting difficulties, the prospects of adapting the restrictor view to counterfactuals is discussed. In response to problems with this approach, the book develops the idea of accounting for our ways of thinking in counterfactual terms by construing counterfactual thought as being concerned with arbitrary worlds at which the hypothetical assumption is satisfied. This account is subsequently applied to various pertinent issues in the debate about counterfactuals.

Vagueness is a familiar but deeply puzzling aspect of the relation between language and the world. It is highly controversial what the nature of vagueness is; a feature of the way we represent reality in language, or rather a feature of reality itself? Assuming standard logical principles, Sorites' arguments suggest that vague terms are either inconsistent or have a sharp boundary. The account we give of such paradoxes plays a pivotal role for our understanding of natural languages. If our reasoning involves any vague concepts, is it safe from contradiction? Do vague concepts really lack any sharp boundary? If not, why are we reluctant to accept the existence of any sharp boundary for them? And what rules of inference can we validly apply, if we reason in vague terms? This book presents the latest work towards a clearer understanding of these old puzzles about the nature and logic of vagueness. The collection offers a stimulating series of original chapters on these and related issues by some of the world's leading experts.

Mathematics depends on proofs, and proofs have to begin somewhere, from some fundamental assumptions. Chapter I traces the historical rise of pure mathematics and the development of set theory, eventually axiomatic set theory, to play this foundational role for contemporary classical mathematics. Here the Euclidean ideal of postulates that are simply obvious or self-evident can't be the whole story, which raises two basic questions: what are the proper methods for defending set-theoretic axioms? And, why are these the proper methods? Chapter II introduces the meta-philosophical perspective, called Second Philosophy, from which the inquiry into these questions will take place, and identifies straightforward mathematical answers to the first question. Addressing the second requires engagement with the troublesome ontological and epistemological issues that have dogged the philosophy of mathematics from its beginnings. Chapters III and IV describe and explore two apparently conflicting stands on these issues—called Thin Realism and Arealism—not so much to recommend either one, but with an eye to suggesting that the question of which is correct has less bite than it might appear. In the end, the hope is to shift attention away from these elusive matters of truth and existence, and to direct it toward the distinctive type of mathematical objectivity emphasized in the opening section of Chapter V. The concluding sections of chapter V return, at last, to the question of set-theoretic method and draw some concrete morals for the project of defending the axioms.