Results for 'Rosalyn Stewart'
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- To Cure Sometimes, To Relieve Often, and To Comfort Always.RosalynStewart &Valerie Gray Hardcastle -2019 -American Journal of Bioethics 19 (12):66-68.Volume 19, Issue 12, December 2019, Page 66-68.
- Medical Decision-Making Capacity: High Stakes, Complex, and Fluid.Valerie Gray Hardcastle &Rosalyn W.Stewart -2013 -American Journal of Bioethics Neuroscience 4 (4):21-22.
- Reduction and embodied cognition : perspectives from medicine and psychiatry.Valerie Gray Hardcastle &Rosalyn W.Stewart -2008 - In Jakob Hohwy & Jesper Kallestrup,Being Reduced: New Essays on Reduction, Explanation, and Causation. New York: Oxford University Press.
- Supporting Irrational Suicide.Valerie Gray Hardcastle &Rosalyn WalkerStewart -2002 -Bioethics 16 (5):425-438.In this essay, we present three case studies which suggest that sometimes we are better off supporting a so–called irrational suicide, and that emotional or psychological distress – even if medically controllable – might justify a suicide. We underscore how complicated these decisions are and how murky a physician's moral role can be. We advocate a more individualized route to end–of–life care, eschewing well–meaning, principled, generalizations in favor of a highly contextualized, patient–centered, approach. We conclude that our Western traditions of (...) promoting reasoned behavior and life themselves may at times be counter–productive. (shrink)
- Varieties of Logic.Stewart Shapiro -2014 - Oxford and New York: Oxford University Press.Logical pluralism is the view that different logics are equally appropriate, or equally correct. Logical relativism is a pluralism according to which validity and logical consequence are relative to something.Stewart Shapiro explores various such views. He argues that the question of meaning shift is itself context-sensitive and interest-relative.
- Thinking about mathematics: the philosophy of mathematics.Stewart Shapiro -2000 - New York: Oxford University Press.This unique book byStewart Shapiro looks at a range of philosophical issues and positions concerning mathematics in four comprehensive sections. Part I describes questions and issues about mathematics that have motivated philosophers since the beginning of intellectual history. Part II is an historical survey, discussing the role of mathematics in the thought of such philosophers as Plato, Aristotle, Kant, and Mill. Part III covers the three major positions held throughout the twentieth century: the idea that mathematics is logic (...) (logicism), the view that the essence of mathematics is the rule-governed manipulation of characters (formalism), and a revisionist philosophy that focuses on the mental activity of mathematics (intuitionism). Finally, Part IV brings the reader up-to-date with a look at contemporary developments within the discipline. This sweeping introductory guide to the philosophy of mathematics makes these fascinating concepts accessible to those with little background in either mathematics or philosophy. (shrink)
- Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro -1997 - Oxford, England: Oxford University Press USA.Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.
- Foundations without foundationalism: a case for second-order logic.Stewart Shapiro -1991 - New York: Oxford University Press.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)
- Proof and Truth.Stewart Shapiro -1998 -Journal of Philosophy 95 (10):493-521.
- Logical Consequence: Models and Modality.Stewart Shapiro -1998 - In Matthias Schirn,The Philosophy of Mathematics Today: Papers From a Conference Held in Munich From June 28 to July 4,1993. Oxford, England: Clarendon Press. pp. 131–156.
- Oxford Handbook of Philosophy of Mathematics and Logic.Stewart Shapiro (ed.) -2005 - Oxford and New York: Oxford University Press.This Oxford Handbook covers the current state of the art in the philosophy of maths and logic in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 newly-commissioned chapters are by established experts in the field and contain both exposition and criticism as well as substantial development of their own positions. Select major positions are represented by two chapters - one supportive and one critical. The book includes a comprehensive (...) bibliography. (shrink)
- Identity, indiscernibility, and Ante Rem structuralism: The tale of I and –I.Stewart Shapiro -2008 -Philosophia Mathematica 16 (3):285-309.Some authors have claimed that ante rem structuralism has problems with structures that have indiscernible places. In response, I argue that there is no requirement that mathematical objects be individuated in a non-trivial way. Metaphysical principles and intuitions to the contrary do not stand up to ordinary mathematical practice, which presupposes an identity relation that, in a sense, cannot be defined. In complex analysis, the two square roots of –1 are indiscernible: anything true of one of them is true of (...) the other. I suggest that i functions like a parameter in natural deduction systems. I gave an early version of this paper at a workshop on structuralism in mathematics and science, held in the Autumn of 2006, at Bristol University. Thanks to the organizers, particularly Hannes Leitgeb, James Ladyman, and Øystein Linnebo, to my commentator Richard Pettigrew, and to the audience there. The paper also benefited considerably from a preliminary session at the Arché Research Centre at the University of St Andrews. I am indebted to my colleagues Craige Roberts, for help with the linguistics literature, and Ben Caplan and Gabriel Uzquiano, for help with the metaphysics. Thanks also to Hannes Leitgeb and Jeffrey Ketland for reading an earlier version of the manuscript and making helpful suggestions. I also benefited from conversations with Richard Heck, John Mayberry, Kevin Scharp, and Jason Stanley. CiteULike Connotea Del.icio.us What's this? (shrink)
- Logical consequence, proof theory, and model theory.Stewart Shapiro -2005 - InOxford Handbook of Philosophy of Mathematics and Logic. Oxford and New York: Oxford University Press. pp. 651--670.This chapter provides broad coverage of the notion of logical consequence, exploring its modal, semantic, and epistemic aspects. It develops the contrast between proof-theoretic notion of consequence, in terms of deduction, and a model-theoretic approach, in terms of truth-conditions. The main purpose is to relate the formal, technical work in logic to the philosophical concepts that underlie reasoning.
- Conservativeness and incompleteness.Stewart Shapiro -1983 -Journal of Philosophy 80 (9):521-531.
- (1 other version)Mathematical structuralism.Stewart Shapiro -1996 -Philosophia Mathematica 4 (2):81-82.STEWART SHAPIRO; Mathematical Structuralism, Philosophia Mathematica, Volume 4, Issue 2, 1 May 1996, Pages 81–82, https://doi.org/10.1093/philmat/4.2.81.
- Mathematics and reality.Stewart Shapiro -1983 -Philosophy of Science 50 (4):523-548.The subject of this paper is the philosophical problem of accounting for the relationship between mathematics and non-mathematical reality. The first section, devoted to the importance of the problem, suggests that many of the reasons for engaging in philosophy at all make an account of the relationship between mathematics and reality a priority, not only in philosophy of mathematics and philosophy of science, but also in general epistemology/metaphysics. This is followed by a (rather brief) survey of the major, traditional philosophies (...) of mathematics indicating how each is prepared to deal with the present problem. It is shown that (the standard formulations of) some views seem to deny outright that there is a relationship between mathematics and any non-mathematical reality; such philosophies are clearly unacceptable. Other views leave the relationship rather mysterious and, thus, are incomplete at best. The final, more speculative section provides the direction of a positive account. A structuralist philosophy of mathematics is outlined and it is proposed that mathematics applies to reality though the discovery of mathematical structures underlying the non-mathematical universe. (shrink)
- (1 other version)Thinking about Mathematics.Stewart Shapiro -2001 - Oxford University Press.
- All Things Indefinitely Extensible.Stewart Shapiro &Crispin Wright -2006 - In Stewart Shapiro & Crispin Wright,All Things Indefinitely Extensible. pp. 255--304.
- Categories, Structures, and the Frege-Hilbert Controversy: The Status of Meta-mathematics.Stewart Shapiro -2005 -Philosophia Mathematica 13 (1):61-77.There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or is it an exception to the (...) slogan that mathematics is the science of structure? (shrink)
- Open Texture and Mathematics.Stewart Shapiro &Craige Roberts -2021 -Notre Dame Journal of Formal Logic 62 (1):173-191.The purpose of this article is to explore the extent to which mathematics is subject to open texture and the extent to which mathematics resists open texture. The resistance is tied to the importance of proof and, in particular, rigor, in mathematics.
- Modality and ontology.Stewart Shapiro -1993 -Mind 102 (407):455-481.
- We hold these truths to be self-evident: But what do we mean by that?: We hold these truths to be self-evident.Stewart Shapiro -2009 -Review of Symbolic Logic 2 (1):175-207.At the beginning of Die Grundlagen der Arithmetik [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”. This, of course, is true, but thinkers differ on why it is that mathematicians prefer proof. And what of propositions for which no proof is possible? What of axioms? This talk explores various notions of self-evidence, and the role they play in various foundational systems, notably those of Frege and Zermelo. I argue that both (...) programs are undermined at a crucial point, namely when self-evidence is supported by holistic and even pragmatic considerations. (shrink)
- Structure and identity.Stewart Shapiro -2006 - In Fraser MacBride,Identity and modality. New York: Oxford University Press. pp. 34--69.According to ante rem structuralism a branch of mathematics, such as arithmetic, is about a structure, or structures, that exist independent of the mathematician, and independent of any systems that exemplify the structure. A structure is a universal of sorts: structure is to exemplified system as property is to object. So ante rem structuralist is a form of ante rem realism concerning universals. Since the appearance of my Philosophy of mathematics: Structure and ontology, a number of criticisms of the idea (...) of ante rem structures have appeared. Some argue that it is impossible to give identity conditions for places in homogeneous ante rem structures, invoking a version of the identity of indiscernibles. Others raise issues concerning the identity and distinctness of places in different structures, such as the the natural number 2 and the real number 2. The purpose of this paper is to take the measure of these objections, and to further articulate ante rem structuralism to take them into account. (shrink)
- An “I” for an I: Singular terms, uniqueness, and reference.Stewart Shapiro -2012 -Review of Symbolic Logic 5 (3):380-415.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)
- Incompleteness and inconsistency.Stewart Shapiro -2002 -Mind 111 (444):817-832.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)
- The Analysis of Knowing: A Decade of Research.Stewart Cohen -1986 -Philosophy and Phenomenological Research 46 (3):523-528.No categories
- Prolegomenon To Any Future Neo‐Logicist Set Theory: Abstraction And Indefinite Extensibility.Stewart Shapiro -2003 -British Journal for the Philosophy of Science 54 (1):59-91.The purpose of this paper is to assess the prospects for a neo‐logicist development of set theory based on a restriction of Frege's Basic Law V, which we call (RV): ∀P∀Q[Ext(P) = Ext(Q) ≡ [(BAD(P) & BAD(Q)) ∨ ∀x(Px ≡ Qx)]] BAD is taken as a primitive property of properties. We explore the features it must have for (RV) to sanction the various strong axioms of Zermelo–Fraenkel set theory. The primary interpretation is where ‘BAD’ is Dummett's ‘indefinitely extensible’.1 Background: what (...) and why?2 Framework3 GOOD candidates, indefinite extensibility4 The framework of (RV) alone, or almost alone5 The axioms6 Brief closing. (shrink)
- Incompleteness, mechanism, and optimism.Stewart Shapiro -1998 -Bulletin of Symbolic Logic 4 (3):273-302.§1. Overview. Philosophers and mathematicians have drawn lots of conclusions from Gödel's incompleteness theorems, and related results from mathematical logic. Languages, minds, and machines figure prominently in the discussion. Gödel's theorems surely tell us something about these important matters. But what?A descriptive title for this paper would be “Gödel, Lucas, Penrose, Turing, Feferman, Dummett, mechanism, optimism, reflection, and indefinite extensibility”. Adding “God and the Devil” would probably be redundant. Despite the breath-taking, whirlwind tour, I have the modest aim of forging (...) connections between different parts of this literature and clearing up some confusions, together with the less modest aim of not introducing any more confusions.I propose to focus on three spheres within the literature on incompleteness. The first, and primary, one concerns arguments that Gödel's theorem refutes the mechanistic thesis that the human mind is, or can be accurately modeled as, a digital computer or a Turing machine. The most famous instance is the much reprinted J. R. Lucas [18]. To summarize, suppose that a mechanist provides plans for a machine,M, and claims that the output ofMconsists of all and only the arithmetic truths that a human, or the totality of human mathematicians, will ever or can ever know. We assume that the output ofMis consistent. (shrink)
- Intuitionism, Pluralism, and Cognitive Command.Stewart Shapiro &William W. Taschek -1996 -Journal of Philosophy 93 (2):74.
- The Limits of Logic: Higher-order Logic and the Löwenheim-Skolem Theorem.Stewart Shapiro -1996 - Routledge.The articles in this volume represent a part of the philosophical literature on higher-order logic and the Skolem paradox. They ask the question what is second-order logic? and examine various interpretations of the Lowenheim-Skolem theorem.
- Ineffability within the limits of abstraction alone.Stewart Shapiro &Gabriel Uzquiano -2016 - In Philip A. Ebert & Marcus Rossberg,Abstractionism: Essays in Philosophy of Mathematics. Oxford, England: Oxford University Press UK.The purpose of this article is to assess the prospects for a Scottish neo-logicist foundation for a set theory. We show how to reformulate a key aspect of our set theory as a neo-logicist abstraction principle. That puts the enterprise on the neo-logicist map, and allows us to assess its prospects, both as a mathematical theory in its own right and in terms of the foundational role that has been advertised for set theory. On the positive side, we show that (...) our abstraction based theory can be modified to yield much of ordinary mathematics, indeed everything needed to recapture all branches of mathematics short of set theory itself. However, our conclusions are mostly negative. The theory will fall far short of the power of ordinary Zermelo-Fraenkel set theory. It is consistent that our set theory has models that are relatively small, smaller than the first cardinal with an uncountable index. More important, there is a strong tension between the idea that the iterative hierarchy is somehow ineffable, or indefinitely extensible, and the neo-logicist theme of capturing mathematical theories with abstraction principles. (shrink)
- Classical Logic.Stewart Shapiro &Teresa Kouri Kissel -2012 - In Ed Zalta,Stanford Encyclopedia of Philosophy. Stanford, CA: Stanford Encyclopedia of Philosophy.Typically, a logic consists of a formal or informal language together with a deductive system and/or a model-theoretic semantics. The language is, or corresponds to, a part of a natural language like English or Greek. The deductive system is to capture, codify, or simply record which inferences are correct for the given language, and the semantics is to capture, codify, or record the meanings, or truth-conditions, or possible truth conditions, for at least part of the language.
- Induction and Indefinite Extensibility: The Gödel Sentence is True, but Did Someone Change the Subject?Stewart Shapiro -1998 -Mind 107 (427):597-624.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)
- The brain circuitry of attention.Stewart Shipp -2004 -Trends in Cognitive Sciences 8 (5):223-230.
- Modesty and the Egalitarian Ethos.S.Stewart Braun -2017 - In Noell Birondo & S. Stewart Braun,Virtue’s Reasons: New Essays on Virtue, Character, and Reasons. New York: Routledge. pp. 168-188.Export citation
Bookmark - Reading modern law: critical methodologies and sovereign formations.Ruth Margaret Buchanan,Stewart J. Motha &Sundhya Pahuja (eds.) -2012 - New York, NY: Routledge.Export citation Bookmark
- The Objectivity of Mathematics.Stewart Shapiro -2007 -Synthese 156 (2):337-381.The purpose of this paper is to apply Crispin Wright’s criteria and various axes of objectivity to mathematics. I test the criteria and the objectivity of mathematics against each other. Along the way, various issues concerning general logic and epistemology are encountered.
- Frege Meets Aristotle: Points as Abstracts.Stewart Shapiro &Geoffrey Hellman -2015 -Philosophia Mathematica:nkv021.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)
- Mechanism, truth, and Penrose's new argument.Stewart Shapiro -2003 -Journal of Philosophical Logic 32 (1):19-42.Sections 3.16 and 3.23 of Roger Penrose's Shadows of the mind (Oxford, Oxford University Press, 1994) contain a subtle and intriguing new argument against mechanism, the thesis that the human mind can be accurately modeled by a Turing machine. The argument, based on the incompleteness theorem, is designed to meet standard objections to the original Lucas-Penrose formulations. The new argument, however, seems to invoke an unrestricted truth predicate (and an unrestricted knowability predicate). If so, its premises are inconsistent. The usual (...) ways of restricting the predicates either invalidate Penrose's reasoning or require presuppositions that the mechanist can reject. (shrink)
- Computability, Notation, and de re Knowledge of Numbers.Stewart Shapiro,Eric Snyder &Richard Samuels -2022 -Philosophies 1 (7):20.Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship between (...) the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects. (shrink)
- Does Logical Pluralism Imply, or Suggest, Truth Pluralism, or Vice Versa?Stewart Shapiro &Michael Lynch -2019 -Synthese 198 (Suppl 20):4925-4936.The answers to the questions in the title depend on the kind of pluralism one is talking about. We will focus here on our own views. The purpose of this article is to trace out some possible connections between these kinds of pluralism. We show how each of them might bear on the other, depending on how certain open questions are resolved.
- Translating Logical Terms.Stewart Shapiro -2019 -Topoi 38 (2):291-303.The is an old question over whether there is a substantial disagreement between advocates of different logics, as they simply attach different meanings to the crucial logical terminology. The purpose of this article is to revisit this old question in light a pluralism/relativism that regards the various logics as equally legitimate, in their own contexts. We thereby address the vexed notion of translation, as it occurs between mathematical theories. We articulate and defend a thesis that the notion of “same meaning” (...) is itself context-sensitive, depending on the purposes of a given conversation. (shrink)
- (1 other version)Computability, Proof, and Open-Texture.Stewart Shapiro -2006 - In Adam Olszewski, Jan Wolenski & Robert Janusz,Church's Thesis After 70 Years. Ontos Verlag. pp. 420-455.
- Epistemology of mathematics: What are the questions? What count as answers?Stewart Shapiro -2011 -Philosophical Quarterly 61 (242):130-150.A paper in this journal by Fraser MacBride, ‘Can Ante Rem Structuralism Solve the Access Problem?’, raises important issues concerning the epistemological goals and burdens of contemporary philosophy of mathematics, and perhaps philosophy of science and other disciplines as well. I use a response to MacBride's paper as a framework for developing a broadly holistic framework for these issues, and I attempt to steer a middle course between reductive foundationalism and extreme naturalistic quietism. For this purpose the notion of entitlement (...) is invoked along the way, suitably modified for the present anti-foundationalist setting. (shrink)
- Structures and Logics: A Case for (a) Relativism.Stewart Shapiro -2014 -Erkenntnis 79 (2):309-329.In this paper, I use the cases of intuitionistic arithmetic with Church’s thesis, intuitionistic analysis, and smooth infinitesimal analysis to argue for a sort of pluralism or relativism about logic. The thesis is that logic is relative to a structure. There are classical structures, intuitionistic structures, and (possibly) paraconsistent structures. Each such structure is a legitimate branch of mathematics, and there does not seem to be an interesting logic that is common to all of them. One main theme of my (...) ante rem structuralism is that any coherent axiomatization describes a structure, or a class of structures. If one weakens the logic, then more axiomatizations become coherent. (shrink)
- Second-Order Logic, Foundations, and Rules.Stewart Shapiro -1990 -Journal of Philosophy 87 (5):234.
- Classical logic II: Higher-order logic.Stewart Shapiro -2001 - In Lou Goble,The Blackwell Guide to Philosophical Logic. Malden, Mass.: Wiley-Blackwell. pp. 33--54.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)
- Varieties of Pluralism and Relativism for Logic.Stewart Shapiro -2010 - In Steven D. Hales,A Companion to Relativism. Malden, MA: Wiley-Blackwell. pp. 526–552.This chapter contains sections titled: Abstract Introduction Defining Terms: Relativism, Pluralism, Tolerance What Is Logic? One Route to Pluralism: Logic ‐ as ‐ Model The Boundary Between Logical and Non ‐ Logical Terminology Vagueness Relativity to Structure References.No categories
- The problem of recognition, erasure, and epistemic injustice in medicine : Harms to Transgender and Gender non-binary patients - why we should be worried.Lauren Freeman &HeatherStewart -2023 - In Paul Giladi & Nicola McMillan,Epistemic injustice and the philosophy of recognition. New York, NY: Routledge Taylor & Francis Group.
- Intuitionistic sets and numbers: small set theory and Heyting arithmetic.Stewart Shapiro,Charles McCarty &Michael Rathjen -2025 -Archive for Mathematical Logic 64 (1).It has long been known that (classical) Peano arithmetic is, in some strong sense, “equivalent” to the variant of (classical) Zermelo–Fraenkel set theory (including choice) in which the axiom of infinity is replaced by its negation. The intended model of the latter is the set of hereditarily finite sets. The connection between the theories is so tight that they may be taken as notational variants of each other. Our purpose here is to develop and establish a constructive version of this. (...) We present an intuitionistic theory of the hereditarily finite sets, and show that it is definitionally equivalent to Heyting Arithmetic, in a sense to be made precise. Our main target theory, the intuitionistic small set theory is remarkably simple, and intuitive. It has just one non-logical primitive, for membership, and three straightforward axioms plus one axiom scheme. We locate our theory within intuitionistic mathematics generally. (shrink)
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