Search results for 'Lionel Stefan Shapiro' (try it on Scholar)

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Profile: Lionel Shapiro (University of Connecticut)
  1. Lionel Stefan Shapiro (1994). 'Coordinative Definition' and Reichenbach's Semantic Framework: A Reassessment. Erkenntnis 41 (3):287 - 323.score: 870.0
    Reichenbach's Philosophy of Space and Time (1928) avoids most of the logical positivist pitfalls it is generally held to exemplify, notably both conventionalism and verificationism. To see why, we must appreciate that Reichenbach's interest lies in how mathematical structures can be used to describe reality, not in how words like 'distance' acquire meaning. Examination of his proposed "coordinative definition" of congruence shows that Reichenbach advocates a reductionist analysis of the relations figuring in physical geometry (contrary to common readings that attribute (...)
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  2. Lionel Stefan Shapiro (2001). “The Transition From Sensibility to Reason in Regressu”: Indeterminism in Kant's Reflexionen. Kant-Studien 92 (1):3-12.score: 870.0
    In a remarkable series of Critical-period Reflexionen (5611-4, 5616-9), Kant sketches a defense of the possibility of freedom that differs radically from his usual compatibilism by incorporating an indeterministic account of the phenomena. Anticipating Łukasiewicz, Kant reconciles universal causal determination with an open future by positing a lower temporal bound for the infinite regress of prior determining causes issuing in a contingent action. However, Kant concedes that on this account the unity of experience "cannot fully obtain in the case of (...)
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  3. Lionel Shapiro (2011). Deflating Logical Consequence. Philosophical Quarterly 61 (243):320-342.score: 240.0
    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 (...)
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  4. Lionel Shapiro (2004). Brandom on the Normativity of Meaning. Philosophy and Phenomenological Research 68 (1):141-60.score: 240.0
    Brandom's "inferentialism"—his theory that contentfulness consists in being governed by inferential norms—proves dubiously compatible with his own deflationary approach to intentional objectivity. This is because a deflationist argument, adapted from the case of truth to that of correct inference, undermines the criterion of adequacy Brandom employs in motivating inferentialism. Once that constraint is abandoned, moreover, the very constitutive-explanatory availability of Brandom's inferential norms becomes suspect. Yet Brandom intertwines inferentialism with a separate explanatory project, one that in explaining the pragmatic significance (...)
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  5. Lionel Shapiro (1999). Toward 'Perfect Collections of Properties': Locke on the Constitution of Substantial Sorts. Canadian Journal of Philosophy 29 (4):551-593.score: 240.0
    Locke's claims about the "inadequacy" of substance-ideas can only be understood once it is recognized that the "sort" represented by such an idea is not wholly determined by the idea's descriptive content. The key to his compromise between classificatory conventionalism and essentialism is his injunction to "perfect" the abstract ideas that serve as "nominal essences." This injunction promotes the pursuit of collections of perceptible qualities that approach ever closer to singling out things that possess some shared explanatory-level constitution. It is (...)
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  6. Lionel Shapiro (2012). Objective Being and “Ofness” in Descartes. Philosophy and Phenomenological Research 84 (2):378-418.score: 240.0
    It is generally assumed that Descartes invokes “objective being in the intellect” in order to explain or describe an idea’s status as being “of something.” I argue that this assumption is mistaken. As emerges in his discussion of “materially false ideas” in the Fourth Replies, Descartes recognizes two senses of ‘idea of’. One, a theoretical sense, is itself introduced in terms of objective being. Hence Descartes can’t be introducing objective being to explain or describe “ofness” understood in this sense. Descartes (...)
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  7. Lionel Shapiro (2010). Review of Robert Brandom, Between Saying and Doing. [REVIEW] Australasian Journal of Philosophy 88 (2):367-71.score: 240.0
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  8. Lionel Shapiro (2011). Expressibility and the Liar's Revenge. Australasian Journal of Philosophy 89 (2):297-314.score: 240.0
    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 (...)
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  9. Lionel Shapiro (2010). Two Kinds of Intentionality in Locke. Pacific Philosophical Quarterly 91 (4):554-586.score: 240.0
    Ideas play at least two roles in Locke's theory of the understanding. They are constituents of ‘propositions,’ and some of them ‘represent’ the qualities and sorts of surrounding bodies. I argue that each role involves a distinct kind of intentional directedness. The same idea will in general be an ‘idea of’ two different objects, in different senses of the expression. Identifying Locke's scheme of twofold ‘ofness’ reveals a common structure to his accounts of simple ideas and complex ideas of substances. (...)
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  10. Lionel Shapiro (2013). Intentional Relations and the Sideways‐on View: On McDowell's Critique of Sellars. European Journal of Philosophy 21 (2):300-319.score: 240.0
    : McDowell opposes the view that the intentionality of language and thought remains mysterious unless it can be understood ‘from outside the conceptual order’. While he thinks the demand for such a ‘sideways-on’ understanding can be the result of ‘scientistic prejudice’, he points to Sellars's thought as exhibiting a different source: a distortion of our perspective ‘from within the conceptual order’. The distortion involves a failure on Sellars's part to see how descriptions from within the conceptual order can present expressions (...)
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  11. Lionel Shapiro (2008). Naïve Truth-Conditions and Meaning. Philosophical Quarterly 58 (231):265–277.score: 240.0
    Critics of attempts to explain meaning in terms of truth-conditions have tended to charge their opponents with misconceptions regarding truth. I shall argue that the 'naïve' version of the truth-conditional theory which best accounts for its resilience fails for a different and more basic reason, namely, circularity arising from the contingency of meaning. One reason why this problem has been overlooked is a tendency (noted by Dummett in a different connection) to assimilate the naïve truth-conditional theory to an idealized verificationism.
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  12. Lionel Shapiro (2014). Sellars on the Function of Semantic Vocabulary. British Journal for the History of Philosophy 22 (4):792-811.score: 240.0
    This paper examines two explanations Sellars gives, at successive stages of his career, of how semantic vocabulary (paradigmatically ?means that???? and ?is true if and only if????) lets us relate linguistic expressions to extra-linguistic reality. Despite their differences, both explanations reveal a distinctive pragmatist approach. According to Sellars, we do not use semantic vocabulary to describe language-world relations. Rather, our taking language to relate to the world is implicit in the moves (inferential or non-inferential) licensed by our semantic assertions. I (...)
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  13. Julien Murzi & Lionel Shapiro (forthcoming). Validity and Truth-Preservation. In D. Achourioti, H. Galinon & J. Martinez (eds.), Unifying the Philosophy of Truth Springer. Springer.score: 240.0
    The revisionary approach to semantic paradox is commonly thought to have a somewhat uncomfortable corollary, viz. that, on pain of triviality, we cannot affirm that all valid arguments preserve truth (Beall2007, Beall2009, Field2008, Field2009). We show that the standard arguments for this conclusion all break down once (i) the structural rule of contraction is restricted and (ii) how the premises can be aggregated---so that they can be said to jointly entail a given conclusion---is appropriately understood. In addition, we briefly rehearse (...)
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  14. Lionel Shapiro (2013). Intentionality Bifurcated: A Lesson From Early Modern Philosophy? In Martin Lenz & Anik Waldow (eds.), Contemporary Perspectives on Early Modern Philosophy: Nature and Norms in Thought. Springer.score: 240.0
    This paper examines the pressures leading two very different Early Modern philosophers, Descartes and Locke, to invoke two ways in which thought is directed at objects. According to both philosophers, I argue, the same idea can simultaneously count as “of” two different objects—in two different senses of the phrase ‘idea of’. One kind of intentional directedness is invoked in answering the question What is it to think that thus-and-so? The other kind is invoked in answering the question What accounts for (...)
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  15. Lionel Shapiro (2006). The Rationale Behind Revision-Rule Semantics. Philosophical Studies 129 (3):477 - 515.score: 240.0
    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 (...)
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  16. Lionel Shapiro (forthcoming). Naive Structure, Contraction, and Paradox. Topoi:1-13.score: 240.0
    Rejecting structural contraction has been pro- posed as a strategy for escaping semantic paradoxes. The challenge for its advocates has been to make intuitive sense of how contraction might fail. I offer a way of doing so, based on a "naive" interpretation of the relation between structure and logical vocabulary in a sequent proof system. The naive interpretation of structure motivates the most common way of blaming Curry-style paradoxes on illicit contraction. By contrast, the naive interpretation will not as easily (...)
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  17. Lionel Shapiro (forthcoming). Linguistic Function and Content: Reflections on Price's Pragmatism. Philosophical Quarterly.score: 240.0
    Huw Price proposes a strategy for dissolving ontological puzzles through a pragmatist account of our conceptual activity. Here I consider the proper place for conceptual content in Price’s pragmatism. Price himself rules out any explanatory role for content, just as he rules out any explanatory role for representational notions such as reference and truth. I argue that the cases are disanalogous and that he offers no good reasons for avoiding explanatory appeal to content. Furthermore, I argue that doing so is (...)
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  18. Lionel Shapiro (2013). Validity Curry Strengthened. Thought: A Journal of Philosophy 2 (1):100-107.score: 240.0
    Several authors have argued that a version of Curry's paradox involving validity motivates rejecting the structural rule of contraction. This paper criticizes two recently suggested alternative responses to “validity Curry.” There are three salient stages in a validity Curry derivation. Rejecting contraction blocks the first, while the alternative responses focus on the second and third. I show that a distinguishing feature of validity Curry, as contrasted with more familiar forms of Curry's paradox, is that paradox arises already at the first (...)
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  19. Lionel Shapiro (2014). Assertoric Force Perspectivalism: Relativism Without Relative Truth. Ergo 1 (6).score: 240.0
    According to relativist accounts of discourse about, e.g., epistemic possibility and matters of taste, the truth of propositions must be relativized to nonstandard parameters. This paper argues that the central thrust of such accounts should be understood independently of relative truth, in terms of a perspectival account of assertoric force. My point of departure is a stripped-down version of Brandom’s analysis of the normative structure of discursive practice. By generalizing that structure, I make room for an analogue of the “assessment (...)
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  20. Ann-Louise Shapiro (1997). How Real is the Reality in Documentary Film?Jill Godmilow, in Conversation with Ann-Louise Shapiro. History and Theory 36 (4):80–101.score: 180.0
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  21. Stewart Shapiro & Patrick Greenough (2005). Stewart Shapiro. Context, Conversation, and so-Called 'Higher-Order Vagueness'. Aristotelian Society Supplementary Volume 79 (1):147–165.score: 180.0
    After a brief account of the problem of higher-order vagueness, and its seeming intractability, I explore what comes of the issue on a linguistic, contextualist account of vagueness. On the view in question, predicates like ‘borderline red’ and ‘determinately red’ are, or at least can be, vague, but they are different in kind from ‘red’. In particular, ‘borderline red’ and ‘determinately red’ are not colours. These predicates have linguistic components, and invoke notions like ‘competent user of the language’. On my (...)
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  22. Lawrence Shapiro & Kevin Ryan (2012). Krytyczna Historia Ucieleśniania Jako Paragydmatu Badawczego Nauk o Poznaniu:(Lawrence Shapiro, Embodied Cognitive)/Kevin Ryan. Avant 3 (1):386 - 389.score: 180.0
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  23. Thomas W. Polger & Lawrence Shapiro (2008). Understanding the Dimensions of Realization. Journal of Philosophy 105 (4):213-222.score: 60.0
    Carl Gillett has defended what he calls the “dimensioned” view of the realization relation, which he contrasts with the traditional “flat” view of realization (2003, 2007; see also Gillett 2002). Intuitively, the dimensioned approach characterizes realization in terms of composition whereas the flat approach views realization in terms of occupiers of functional roles. Elsewhere we have argued that the general view of realization and multiple realization that Gillett advances is not able to discharge the theoretical duties of those relations ( (...) 2004, unpublished manuscript; Polger 2004, 2007, forthcoming). Here we focus on an internal objection to Gillett’s account and then raise some broader reasons to reject it. (shrink)
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  24. Stewart Shapiro (2000). Thinking About Mathematics: The Philosophy of Mathematics. Oxford University Press.score: 60.0
    This unique book by Stewart 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 (...)
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  25. Kevin Scharp & Stewart Shapiro (2012). On Richard's When Truth Gives Out. [REVIEW] Philosophical Studies 160 (3):455-463.score: 60.0
    On Richard’s When Truth Gives Out Content Type Journal Article Pages 1-9 DOI 10.1007/s11098-011-9796-0 Authors Kevin Scharp, Department of Philosophy, The Ohio State University, 350 University Hall, 230 North Oval Mall, Columbus, OH 43210, USA Stewart Shapiro, Department of Philosophy, The Ohio State University, 350 University Hall, 230 North Oval Mall, Columbus, OH 43210, USA Journal Philosophical Studies Online ISSN 1573-0883 Print ISSN 0031-8116.
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  26. Stewart Shapiro (1997). Philosophy of Mathematics: Structure and Ontology. Oxford University Press.score: 60.0
    Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable (...)
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  27. Lawrence A. Shapiro (2004). The Mind Incarnate. MIT Press.score: 60.0
    Shapiro tests these hypotheses against two rivals, the mental constraint thesis and the embodied mind thesis. Collecting evidence from a variety of sources (e.g., neuroscience, evolutionary theory, and embodied cognition) he concludes that the multiple realizability thesis, accepted by most philosophers as a virtual truism, is much less obvious than commonly assumed, and that there is even stronger reason to give up the separability thesis. In contrast to views of mind that tempt us to see the mind as simply (...)
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  28. Stewart Shapiro (2006/2008). Vagueness in Context. Oxford University Press.score: 60.0
    Stewart Shapiro's ambition in Vagueness in Context is to develop a comprehensive account of the meaning, function, and logic of vague terms in an idealized version of a natural language like English. It is a commonplace that the extensions of vague terms vary according to their context: a person can be tall with respect to male accountants and not tall (even short) with respect to professional basketball players. The key feature of Shapiro's account is that the extensions of (...)
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  29. Lawrence Shapiro (2008). Understand the Dimensions of Realization. Journal of Philosophy 105 (4):213-222.score: 60.0
    Carl Gillett has defended what he calls the “dimensioned” view of the realization relation, which he contrasts with the traditional “flat” view of realization (2003, 2007; see also Gillett 2002). Intuitively, the dimensioned approach characterizes realization in terms of composition whereas the flat approach views realization in terms of occupiers of functional roles. Elsewhere we have argued that the general view of realization and multiple realization that Gillett advances is not able to discharge the theoretical duties of those relations ( (...) 2004, unpublished manuscript; Polger 2004, 2007, forthcoming). Here we focus on an internal objection to Gillett’s account and then raise some broader reasons to reject it. (shrink)
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  30. Debbie Shapiro (2006). Your Body Speaks Your Mind: Decoding the Emotional, Psychological, and Spiritual Messages That Underlie Illness. Sounds True.score: 60.0
    In Your Body Speaks Your Mind, renowned teacher and best-selling author Deb Shapiro shows you how mastering the language of your symptoms can actually increase ...
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  31. David A. Shapiro (2002). Philosophy in the Schools Project. Questions: Philosophy for Young People 2:8-8.score: 60.0
    In the pursuit of a quality and well-rounded education with philosophy, Shapiro conducts an introductory lesson to students and teachers alike in order to develop deeper, more philosophical questions from their students. Academically, the article expands detail on tutoring in philosophy, analytical practices, and metaphysical activities.
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  32. Michael J. Shapiro (1999). Cinematic Political Thought: Narrating Race, Nation, and Gender. New York University Press.score: 60.0
    In Cinematic Political Thought , Michael J. Shapiro investigates aspects of contemporary politics and articulates a critical philosophical perspective with politically disposed treatments of contemporary cinema. Reading such films as Hoop Dreams, Lone Star, Father of the Bride II and To Live and Die in LA through the lens of Deleuze, Derrida, Foucault and Lyotard, Shapiro demonstrates what it can mean to think the political both in terms of cinema studies and in wider aesthetic and social contexts. Cinematic (...)
     
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  33. Harvey Shapiro (2012). Educational Theory and Jewish Studies in Conversation: From Volozhin to Buczacz. Lexington Books.score: 60.0
    Educational Theory and Jewish Studies in Conversation: From Volozhin to Buczacz, by Harvey Shapiro, PhD, brings together two different fields of study—modern Jewish studies and contemporary educational theory—to provide new theoretical ...
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  34. Joan Poliner Shapiro (2001). Ethical Leadership and Decision Making in Education: Applying Theoretical Perspectives to Complex Dilemmas. L. Erlbaum Associates.score: 30.0
    The authors developed this textbook in response to an increasing interest in ethics, and a growing number of courses on this topic that are now being offered in educational leadership programs. It is designed to fill a gap in instructional materials for teaching the ethics component of the knowledge base that has been established for the profession. The text has several purposes: First, it demonstrates the application of different ethical paradigms (the ethics of justice, care, critique, and the profession) through (...)
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  35. Scott J. Shapiro (2007). The "Hart-Dworkin" Debate : A Short Guide for the Perplexed. In Arthur Ripstein (ed.), Ronald Dworkin. Cambridge University Press. 22--49.score: 30.0
    For the past four decades, Anglo-American legal philosophy has been preoccupied – some might say obsessed – with something called the “Hart-Dworkin” debate. Since the appearance in 1967 of “The Model of Rules I,” Ronald Dworkin’s seminal critique of H.L.A. Hart’s theory of legal positivism, countless books and articles have been written either defending Hart against Dworkin’s objections or defending Dworkin against Hart’s defenders. My purpose in this essay is not to declare an ultimate victor; rather it is to identify (...)
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  36. Stuart C. Shapiro & William J. Rapaport (1991). Models and Minds. In Robert E. Cummins & John L. Pollock (eds.), Philosophy and AI. Cambridge: MIT Press. 215--259.score: 30.0
    Cognitive agents, whether human or computer, that engage in natural-language discourse and that have beliefs about the beliefs of other cognitive agents must be able to represent objects the way they believe them to be and the way they believe others believe them to be. They must be able to represent other cognitive agents both as objects of beliefs and as agents of beliefs. They must be able to represent their own beliefs, and they must be able to represent beliefs (...)
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  37. Stewart Shapiro (2011). Epistemology of Mathematics: What Are the Questions? What Count as Answers? Philosophical Quarterly 61 (242):130-150.score: 30.0
    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 (...)
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  38. Lawrence A. Shapiro & Elliott Sober (forthcoming). Epiphenomenalism - the Do's and the Don'ts. In G. Wolters & Peter K. Machamer (eds.), Studies in Causality: Historical and Contemporary. University of Pittsburgh Press.score: 30.0
    When philosophers defend epiphenomenalist doctrines, they often do so by way of a priori arguments. Here we suggest an empirical approach that is modeled on August Weismann.
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  39. Stewart Shapiro (2009). We Hold These Truths to Be Self-Evident: But What Do We Mean by That? Review of Symbolic Logic 2 (1):175-207.score: 30.0
    At the beginning of Die Grundlagen der Arithmetik (§2) [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 (...)
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  40. Scott J. Shapiro (2009). Was Inclusive Legal Positivism Founded on a Mistake? Ratio Juris 22 (3):326-338.score: 30.0
    In this paper, I present a new argument against inclusive legal positivism. As I show, any theory which permits morality to be a condition on legality cannot account for a core feature of legal activity, namely, that it is an activity of social planning. If the aim of a legal institution is to guide the conduct of the community through plans, it would be self-defeating if the existence of these plans could only be determined through deliberation on the merits. I (...)
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  41. Larry Shapiro (2007). The Embodied Cognition Research Programme. Philosophy Compass 2 (2):338–346.score: 30.0
  42. Scott J. Shapiro, What is the Rule of Recognition (and Does It Exist)?score: 30.0
    One of the principal lessons of The Concept of Law is that legal systems are not only comprised of rules, but founded on them as well. As Hart painstakingly showed, we cannot account for the way in which we talk and think about the law - that is, as an institution which persists over time despite turnover of officials, imposes duties and confers powers, enjoys supremacy over other kinds of practices, resolves doubts and disagreements about what is to be done (...)
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  43. Lawrence A. Shapiro (2010). Embodied Cognition. Routledge.score: 30.0
    Introduction: toward an understanding of embodied cognition -- Standard cognitive science -- Challenging standard cognitive science -- Conceptions of embodiment -- Embodied cognition: the conceptualization hypothesis -- Embodied cognition: the replacement hypothesis -- Embodied cognition: the constitution hypothesis -- Concluding thoughts.
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  44. Stewart Shapiro (2007). The Objectivity of Mathematics. Synthese 156 (2):337 - 381.score: 30.0
    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.
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  45. Stewart Shapiro (2010). So Truth is Safe From Paradox: Now What? [REVIEW] Philosophical Studies 147 (3):445 - 455.score: 30.0
    The article is part of a symposium on Hartry Field’s “Saving truth from paradox”. The book is one of the most significant intellectual achievements of the past decades, but it is not clear what, exactly, it accomplishes. I explore some alternatives, relating the developed view to the intuitive, pre-theoretic notion of truth.
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  46. Jack Arnold & Stewart Shapiro (2007). Where in the (World Wide) Web of Belief is the Law of Non-Contradiction? Noûs 41 (2):276–297.score: 30.0
    It is sometimes said that there are two, competing versions of W. V. O. Quine’s unrelenting empiricism, perhaps divided according to temporal periods of his career. According to one, logic is exempt from, or lies outside the scope of, the attack on the analytic-synthetic distinction. This logic-friendly Quine holds that logical truths and, presumably, logical inferences are analytic in the traditional sense. Logical truths are knowable a priori, and, importantly, they are incorrigible, and so immune from revision. The other, radical (...)
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  47. Stewart Shapiro (1998). Proof and Truth: Through Thick and Thin. Journal of Philosophy 95 (10):493-521.score: 30.0
  48. Lawrence A. Shapiro (2000). Multiple Realizations. Journal of Philosophy 97 (12):635-654.score: 30.0
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  49. Stewart Shapiro (1999). Do Not Claim Too Much: Second-Order Logic and First-Order Logic. Philosophia Mathematica 7 (1):42-64.score: 30.0
    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.
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  50. Stewart Shapiro (2008). Identity, Indiscernibility, and Ante Rem Structuralism: The Tale of I and –I. Philosophia Mathematica 16 (3):285-309.score: 30.0
    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 (...)
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