The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify (...) many concepts in contemporary mathematics, and thus that both first- and higher-order logics are needed to fully reflect current work. Throughout, the emphasis is on discussing the associated philosophical and historical issues and the implications they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic comparable to that provided in a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in the field today. (shrink)

Embodied cognition is a recent development in psychology that practitioners often present as a superseding standard cognitive science. In this outstanding introduction, Lawrence Shapiro sets out the central themes and debates surrounding embodied cognition, explaining and assessing the work of many of the key figures in the field, including Lawrence Barsalou, Daniel Casasanto, Andy Clark, Alva Noë, and Michael Spivey. Beginning with an outline of the theoretical and methodological commitments of standard cognitive science, Shapiro then examines philosophical and (...) empirical arguments surrounding the traditional perspective, setting the stage for a detailed examination of the embodied alternative. He introduces topics such as dynamical systems theory, ecological psychology, robotics, and connectionism, before addressing core issues in philosophy of mind such as mental representation and extended cognition. This second edition has been updated and revised throughout and includes new chapters that both expand on earlier topics and that introduce new material on embodied concepts, preference formation, and emotion. Including helpful chapter summaries and annotated further reading at the end of each chapter, Embodied Cognition, Second Editionis essential reading for all students of philosophy of mind, psychology, and cognitive science. ition has been updated and revised throughout and includes new chapters that both expand on earlier topics and that introduce new material on embodied concepts, preference formation, and emotion. Including helpful chapter summaries and annotated further reading at the end of each chapter, Embodied Cognition, Second Editionis essential reading for all students of philosophy of mind, psychology, and cognitive science. (shrink)

A line of research within embodied cognition seeks to show that an organism’s body is a determinant of its conceptual capacities. Comparison of this claim of body determinism to linguistic determinism bears interesting results. Just as Slobin’s (1996) idea of thinking for speaking challenges the main thesis of linguistic determinism, so too the possibility of thinking for acting raises difficulties for the proponent of body determinism. However, recent studies suggest that the body may, after all, have a determining role in (...) cognitive processes of sentence comprehension. (shrink)

Deflationists about truth seek to undermine debates about the nature of truth by arguing that the truth predicate is merely a device that allows us to express a certain kind of generality. I argue that a parallel approach is available in the case of logical consequence. Just as deflationism about truth offers an alternative to accounts of truth's nature in terms of correspondence or justification, deflationism about consequence promises an alternative to model-theoretic or proof-theoretic accounts of consequence's nature. I then (...) argue, against considerations put forward by Field and Beall, that Curry's paradox no more rules out deflationism about consequence than the liar paradox rules out deflationism about truth. (shrink)

Democracy and justice are often mutually antagonistic ideas, but in this innovative book Ian Shapiro shows how and why they should be pursued together. Justice must be sought democratically if it is to garner legitimacy in the modern world, he claims, and democracy must be justice-promoting if it is to sustain allegiance over time. _Democratic Justice_ meets these criteria, offering an attractive vision of a practical path to a better future. Wherever power is exercised in human affairs, Shapiro (...) argues, the lack of democracy will be experienced as injustice. The challenge is to democratize social relations so as to diminish injustice, but to do this in ways that are compatible with people’s values and goals. Shapiro shows how this can be done in different phases of the human life cycle, from childhood through the adult worlds of work and domestic life, retirement, old age, and approaching death. He spells out the implications for pressing debates about authority over children, the law of marriage and divorce, population control, governing the firm, basic income guarantees, health insurance, retirement policies, and decisions made by and for the infirm elderly. This refreshing encounter between political philosophy and practical politics will interest all those who aspire to bequeath a more just world to our children than the one we have inherited. (shrink)

There is an interesting logical/semantic issue with some mathematical languages and theories. In the language of (pure) complex analysis, the two square roots of i’ manage to pick out a unique object? This is perhaps the most prominent example of the phenomenon, but there are some others. The issue is related to matters concerning the use of definite descriptions and singular pronouns, such as donkey anaphora and the problem of indistinguishable participants. Taking a cue from some work in linguistics and (...) the philosophy of language, I suggest that i functions like a parameter in natural deduction systems. This may require some rethinking of the role of singular terms, at least in mathematical languages. (shrink)

Graham Priest's In Contradiction (Dordrecht: Martinus Nijhoff Publishers, 1987, chapter 3) contains an argument concerning the intuitive, or ‘naïve’ notion of (arithmetic) proof, or provability. He argues that the intuitively provable arithmetic sentences constitute a recursively enumerable set, which has a Gödel sentence which is itself intuitively provable. The incompleteness theorem does not apply, since the set of provable arithmetic sentences is not consistent. The purpose of this article is to sharpen Priest's argument, avoiding reference to informal notions, consensus, or (...) Church's thesis. We add Priest's dialetheic semantics to ordinary Peano arithmetic PA, to produce a recursively axiomatized formal system PA★ that contains its own truth predicate. Whether one is a dialetheist or not, PA★ is a legitimate, rigorously defined formal system, and one can explore its proof‐theoretic properties. The system is inconsistent (but presumably non‐trivial), and it proves its own Gödel sentence as well as its own soundness. Although this much is perhaps welcome to the dialetheist, it has some untoward consequences. There are purely arithmetic (indeed, Π0) sentences that are both provable and refutable in PA★. So if the dialetheist maintains that PA★ is sound, then he must hold that there are true contradictions in the most elementary language of arithmetic. Moreover, the thorough dialetheist must hold that there is a number g which both is and is not the code of a derivation of the indicated Gödel sentence of PA★. For the thorough dialetheist, it follows ordinary PA and even Robinson arithmetic are themselves inconsistent theories. I argue that this is a bitter pill for the dialetheist to swallow. (shrink)

I here address Descartes' account of human nature as a union of mind and body by appealing to The Passions of the Soul. I first show that Descartes takes us to be able to reform the naturally instituted associations between bodily and mental states. I go on to argue that Descartes offers a teleological explanation of body-mind associations (those instituted both by nature and by artifice). This explanation sheds light on the ontological status of the union. I suggest that it (...) affords a way of understanding how mind and body form a true unit without compromising Descartes' dualism. (shrink)

The partisan and ideological polarization of American politics since the 1970s appears to have affected pubic opinion in striking ways. The American public has become increasingly partisan and ideological along liberal-conservative lines on a wide range of issues, including even foreign policy. This has raised questions about how rational the public is, in the broad sense of the public's responsiveness to objective conditions. Widespread partisan disagreements over what those conditions arei.e., disagreements about the factssuggest that large proportions of the public (...) may be perceiving the facts incorrectly. The facts in question are important enough that these partisan disagreements may translate into sub-optimal policy preferences and electoral decisions. (shrink)

Over the last few decades Michael Dummett developed a rich program for assessing logic and the meaning of the terms of a language. He is also a major exponent of Frege's version of logicism in the philosophy of mathematics. Over the last decade, Neil Tennant developed an extensive version of logicism in Dummettian terms, and Dummett influenced other contemporary logicists such as Crispin Wright and Bob Hale. The purpose of this paper is to explore the prospects for Fregean logicism within (...) a broadly Dummettian framework. The conclusions are mostly negative: Dummett's views on analyticity and the logical/non-logical boundary leave little room for logicism. Dummett's considerations concerning manifestation and separability lead to a conservative extension requirement: if a sentence S is logically true, then there is a proof of S which uses only the introduction and elimination rules of the logical terms that occur in S. If basic arithmetic propositions are logically true-as the logicist contends-then there is tension between this conservation requirement and the ontological commitments of arithmetic. It follows from Dummett's manifestation requirements that if a sentence S is composed entirely of logical terminology, then there is a formal deductive system D such that S is analytic, or logically true, if and only if S is a theorem of D. There is a deep conflict between this result and the essential incompleteness, or as Dummett puts it, the indefinite extensibility, of arithmetic truth. (shrink)

There are a number of regions-based accounts of space/time, due to Whitehead, Roeper, Menger, Tarski, the present authors, and others. They all follow the Aristotelian theme that continua are not composed of points: each region has a proper part. The purpose of this note is to show how to recapture ‘points’ in such frameworks via Scottish neo-logicist abstraction principles. The results recapitulate some Aristotelian themes. A second agenda is to provide a new arena to help decide what is at stake (...) when adjudicating issues concerning the identity of neo-logicist abstracts — so-called ‘Caesar questions’. (shrink)

A number of authors have recently weighed in on the issue of whether it is coherent to have bound variables that range over absolutely everything. Prima facie, it is difficult, and perhaps impossible, to coherently state the “relativist” position without violating it. For example, the relativist might say, or try to say, that for any quantifier used in a proposition of English, there is something outside of its range. What is the range of this quantifier? Or suppose we ask the (...) relativist if there are some things that cannot appear in the range of any bound variable. The likely response would be along these lines: “No. For each object o, it possible to include o in the range of quantifiers, but one cannot quantify over everything at once.” This sentence contains unrestricted quantifiers, or so it seems, pending some clever move from a relativist. On the other hand, in the context of set theory, the reasoning behind the Burali-Forti paradox strongly suggests that there are well-orderings strictly longer than the collection of all ordinals. And set theorists regularly do transfinite recursions and transfinite reductions along such well-orderings. The relativist simply points out that one can always define new ordinals, and thus expand the range of one’s bound variables. The purpose of this paper is to explore the iterative framework, proposed in Zermelo’s 1930 paper, “Über Grenzzahlen und Mengenbereiche” (“On boundary numbers and domains of sets”), in order to shed light on these issues, and see what is involved in resolving them. (shrink)

We show that, by choosing definitions carefully, a version of Frege's theorem can be proved in intuitionistic logic.

There is a standard objection against purported explanations of how a language L can express the notion of being a true sentence of L. According to this objection, such explanations avoid one paradox (the Liar) only to succumb to another of the same kind. Even if L can contain its own truth predicate, we can identify another notion it cannot express, on pain of contradiction via Liar-like reasoning. This paper seeks to undermine such ‘revenge’ by arguing that it presupposes a (...) dubious assumption about the linguistic expression of concepts. Successful revenge would require that there be a notion other than truth that plays the same role with respect to concept-expression that truth is naturally thought to play before we are confronted with the Liar paradox. (shrink)

ABSTRACT Recruitment into the scientific community is one oft-stated goal of science education—in the post-Sputnik United States, for example—but this obscures the fact that science textbooks are often read by people who will never be scientists. It cannot be presupposed that science textbooks for younger audiences, students in primary and secondary schools, function in this way. For this reason, precollegiate-level science textbooks are sometimes discussed as a subset of literature popularizing science. The high school science classroom and the textbook are (...) forums for exposing the public to science. The role of governments and educational institutions in regulating the consumption of these texts not only determines which books are used; it influences how they are written, read, and deemed authoritative. Therefore such science textbooks should not be seen as (at best) the disjunction of texts-for-training and books-for-popularization. A changing sense of what “textbooks” are compels a different understanding of their use in the history of science. (shrink)

Among the aims of this book are: - The discussion of some important philosophical issues using the precision of mathematics. - The development of formal systems that contain both classical and constructive components. This allows the study of constructivity in otherwise classical contexts and represents the formalization of important intensional aspects of mathematical practice. - The direct formalization of intensional concepts (such as computability) in a mixed constructive/classical context.

The purpose of this article is to delimit what can and cannot be claimed on behalf of second-order logic. The starting point is some of the discussions surrounding my Foundations without Foundationalism: A Case for Secondorder Logic.

Computationalism, the notion that cognition is computation, is a working hypothesis of many AI researchers and Cognitive Scientists. Although it has not been proved, neither has it been disproved. In this paper, I give some refutations to some well-known alleged refutations of computationalism. My arguments have two themes: people are more limited than is often recognized in these debates; computer systems are more complicated than is often recognized in these debates. To underline the latter point, I sketch the design and (...) abilities of a possible embodied computer system. (shrink)

According to Gupta and Belnap, the “extensional behavior” of ‘true’ matches that of a circularly defined predicate. Besides promising to explain semantic paradoxicality, their general theory of circular predicates significantly liberalizes the framework of truth-conditional semantics. The authors’ discussions of the rationale behind that liberalization invoke two distinct senses in which a circular predicate’s semantic behavior is explained by a “revision rule” carrying hypothetical information about its extension. Neither attempted explanation succeeds. Their theory may however be modified to employ a (...) relativized notion of extension. The resulting contextualist semantics for ‘true’ construes circularity as a pragmatic phenomenon. (shrink)

What do we know about war crimes and justice? What are the discursive practices through which the dominant images of war crimes, atrocity and justice are understood? In this wide ranging text, Michael J. Shapiro contrasts the justice-related imagery of the war crimes trial with literary justice: representations in literature, film, and biographical testimony, raising questions about atrocities and justice that juridical proceedings exclude. By engaging with the ambiguities exposed by the artistic and experiential genres, reading them alongside policy (...) and archival documentation and critical theoretical discourses, Shapiros War Crimes, Atrocity, and Justice challenges traditional notions of responsibility in juridical settings. His comparative readings instead encourage a focus on the conditions of possibility for war crimes as they arise from the actions of states, non-state agencies and individuals involved in arms trading, peace keeping, sex trafficking, and law enforcement and adjudication. Theory springs to life as Shapiro draws on examples from legal discourse, literature, media, film, and television, to build a nuanced picture of politics and the problem of justice. It will be of great interest to students of film and media, literature, cultural studies, contemporary philosophy and political science. (shrink)

A typical interpreted formal language has (first‐order) variables that range over a collection of objects, sometimes called a domain‐of‐discourse. The domain is what the formal language is about. A language may also contain second‐order variables that range over properties, sets, or relations on the items in the domain‐of‐discourse, or over functions from the domain to itself. For example, the sentence ‘Alexander has all the qualities of a great leader’ would naturally be rendered with a second‐order variable ranging over qualities. Similarly, (...) the sentence ‘there is a property that holds of all and only the prime numbers’ has a variable ranging over properties of natural numbers. Third‐order variables range over properties of properties, sets of sets, functions from properties to sets, etc. For example, according to some logicist accounts, the number 4 is the property shared by all properties that apply to exactly four objects in the domain. Accordingly, the number 4 is a third‐order item. Fourth‐order variables, and beyond, are characterized similarly. The phrase ‘higher‐order variable’ refers to the variables beyond first‐order. (shrink)

Within Western political philosophy, the rights of groups has often been neglected or addressed in only the narrowest fashion. Focusing solely on whether rights are exercised by individuals or groups misses what lies at the heart of ethnocultural conflict, leaving the crucial question unanswered: can the familiar system of common citizenship rights within liberal democracies sufficiently accommodate the legitimate interests of ethnic citizens? Specifically, how does membership in an ethnic group differ from other groups, such as professional, lifestyle, or advocacy (...) groups? How important is ethnicity to personal identity and self-respect, and does accommodating these interests require more than standard citizenship rights? Crucially, what forms of ethnocultural accommodations are consistent with democratic equality, individual freedom, and political stability? Invoking numerous cases studies and addressing the issue of ethnicity from a range of perspectives, Ethnicity and Group Rights seeks to answer these questions. (shrink)

1. Philosophical background: iteration, ineffability, reflection. There are at least two heuristic motivations for the axioms of standard set theory, by which we mean, as usual, first-order Zermelo–Fraenkel set theory with the axiom of choice (ZFC): the iterative conception and limitation of size (see Boolos, 1989). Each strand provides a rather hospitable environment for the hypothesis that the set-theoretic universe is ineffable, which is our target in this paper, although the motivation is different in each case.

An appreciation of the life and word of Barbara McClintock, with special emphasis on what made her a unique and visionary scientist. The obituary indicates unappreciated aspects of her work on biological sensing and how organisms restructure their genomes in response to challenges.

Across the globe, the domain of the litigator and the judge has radically expanded, making it increasingly difficult for those who study comparative and international politics, public policy and regulation, or the evolution of new modes of governance to avoid encountering a great deal of law and courts. In On Law, Politics, and Judicialization, two of the world's leading political scientists present the best of their research, focusing on how to build and test a social science of law and courts. (...) The opening chapter features Shapiro's classic 'Political Jurisprudence,' and Stone Sweet's 'Judicialization and the Construction of Governance,' pieces that critically redefined research agendas on the politics of law and judging. Subsequent chapters take up diverse themes: the strategic contexts of litigation and judging; the discursive foundations of judicial power; the social logic of precedent and appeal; the networking of legal elites; the lawmaking dynamics of rights adjudication; the success and diffusion of constitutional review; the reciprocal impact of courts and legislatures; the globalization of private law; methods, hypothesis-testing, and prediction in comparative law; and the sources and consequences of the creeping 'judicialization of politics' around the world. Chosen empirical settings include the United States, the GATT-WTO, France and Germany, Imperial China and Islam, the European Union, and the transnational world of the Lex Mercatoria. Written for a broad, scholarly audience, the book is also recommended for use in graduate and advanced undergraduate courses in law and the social sciences. (shrink)

This chapter provides an overview of second-order logic and higher-order logic generally. It provides the basic formal languages, deductive systems, and model-theoretic semantics, including a brief account of George Boolos’s interpretation of second-order languages in terms of the plural construction. It then goes into some of the arguments in favor of second-order logic.