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31Peter Schroeder-Heister on Proof-Theoretic Semantics (edited book)Springer. 2024.This open access book is a superb collection of some fifteen chapters inspired by Schroeder-Heister's groundbreaking work, written by leading experts in the field, plus an extensive autobiography and comments on the various contributions by Schroeder-Heister himself. For several decades, Peter Schroeder-Heister has been a central figure in proof-theoretic semantics, a field of study situated at the interface of logic, theoretical computer science, natural-language semantics, and the philosophy o…Read more
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164Critical Remarks on Frege’s Conception of Logic by Patricia Blanchette (review)Journal for the History of Analytical Philosophy 3 (7). 2015.All contributions included in the present issue were originally presented at an ‘Author Meets Critics’ session organised by Richard Zach at the Pacific Meeting of the American Philosophical Association in San Diego in the Spring of 2014.
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109Tractarian First-Order Logic: Identity and the N-OperatorReview of Symbolic Logic 5 (4): 538-573. 2012.In theTractatus, Wittgenstein advocates two major notational innovations in logic. First, identity is to be expressed by identity of the sign only, not by a sign for identity. Secondly, only one logical operator, called “N” by Wittgenstein, should be employed in the construction of compound formulas. We show that, despite claims to the contrary in the literature, both of these proposals can be realized, severally and jointly, in expressively complete systems of first-order logic. Building on ear…Read more
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34On Equivalence Relations Between Interpreted Languages, with an Application to Modal and First-Order LanguageErkenntnis 88 (1): 193-213. 2021.I examine notions of equivalence between logics (understood as languages interpreted model-theoretically) and develop two new ones that invoke not only the algebraic but also the string-theoretic structure of the underlying language. As an application, I show how to construe modal operator languages as what might be called typographical notational variants of _bona fide_ first-order languages.
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24The Quest for Frege's NachlassIn Michael Beaney & Erich Reck (eds.), Gottlob Frege: Critical Assessments of Leading Philosophers, Vol. I, Routledge. pp. 54-67. 2005.
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Auf der Suche nach Freges NachlaßIn Gottfried Gabriel & Uwe Dathe (eds.), Gottlob Frege - Werk und Wirkung, Mentis. pp. 267-282. 2000.
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74Are quantifiers intensional operators?Inquiry: An Interdisciplinary Journal of Philosophy 64 (5-6): 511-532. 2021.ABSTRACT In this paper, I ask whether quantifiers are intensional operators, with variable assignments playing the role of indices. Certain formulations of extensional type theory suggest an affirmative answer, but the most satisfactory among them suffer from a contamination of their semantic ontology with syntactic material. I lay out ‘Fregean’ versions of extensional type theory that are free from syntactic contamination and suggest a negative answer to our question.
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138The proper treatment of variables in predicate logicLinguistics and Philosophy 41 (2): 209-249. 2018.In §93 of The Principles of Mathematics, Bertrand Russell observes that “the variable is a very complicated logical entity, by no means easy to analyze correctly”. This assessment is borne out by the fact that even now we have no fully satisfactory understanding of the role of variables in a compositional semantics for first-order logic. In standard Tarskian semantics, variables are treated as meaning-bearing entities; moreover, they serve as the basic building blocks of all meanings, which are …Read more
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68Classical and Intuitionistic Models of ArithmeticNotre Dame Journal of Formal Logic 37 (3): 452-461. 1996.Given a classical theory T, a Kripke model K for the language L of T is called T-normal or locally PA just in case the classical L-structure attached to each node of K is a classical model of T. Van Dalen, Mulder, Krabbe, and Visser showed that Kripke models of Heyting Arithmetic (HA) over finite frames are locally PA, and that Kripke models of HA over frames ordered like the natural numbers contain infinitely many PA-nodes. We show that Kripke models of the latter sort are in fact PA-normal. Th…Read more
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39On the relations between Heinrich Scholz and Jan ŁukasiewiczHistory and Philosophy of Logic 28 (1): 67-81. 2007.The aim of the present study is (1) to show, on the basis of a number of unpublished documents, how Heinrich Scholz supported his Warsaw colleague Jan Łukasiewicz, the Polish logician, during World War II, and (2) to discuss the efforts he made in order to enable Jan Łukasiewicz and his wife Regina to move from Warsaw to Münster under life-threatening circumstances. In the first section, we explain how Scholz provided financial help to Łukasiewicz, and we also adduce evidence of the risks incurr…Read more
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71Subjunctivity and cross-world predicationPhilosophical Studies 159 (1): 107-122. 2012.The main goal of this paper is to present and compare two approaches to formalizing cross-world comparisons like John might have been taller than he is in quantified modal logics. One is the standard method employing degrees and graded positives, according to which the example just given is to be paraphrased as something like The height that John has is such that he might have had a height greater than it, which is amenable to familiar formalization strategies with respect to quantified modal lo…Read more
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260How to Live Without Identity—And WhyAustralasian Journal of Philosophy 90 (4). 2012.Identity, we're told, is the binary relation that every object bears to itself, and to itself only. But how can a relation be binary if it never relates two objects? This puzzled Russell and led Wittgenstein to declare that identity is not a relation between objects. The now standard view is that Wittgenstein's position is untenable, and that worries regarding the relational status of identity are the result of confusion. I argue that the rejection of identity as a binary relation is perfectly t…Read more
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103Consistent fragments of grundgesetze and the existence of non-logical objectsSynthese 121 (3): 309-328. 1999.In this paper, I consider two curious subsystems ofFrege's Grundgesetze der Arithmetik: Richard Heck's predicative fragment H, consisting of schema V together with predicative second-order comprehension (in a language containing a syntactical abstraction operator), and a theory T in monadic second-order logic, consisting of axiom V and 1 1-comprehension (in a language containing anabstraction function). I provide a consistency proof for the latter theory, thereby refuting a version of a conjectu…Read more
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21On the Consistency of the Δ1 1-CA Fragment of Frege's GrundgesetzeJournal of Philosophical Logic 31 (4): 301-311. 2002.It is well known that Frege's system in the Grundgesetze der Arithmetik is formally inconsistent. Frege's instantiation rule for the second-order universal quantifier makes his system, except for minor differences, full (i.e., with unrestricted comprehension) second-order logic, augmented by an abstraction operator that abides to Frege's basic law V. A few years ago, Richard Heck proved the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicat…Read more
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126Wittgensteinian Predicate LogicNotre Dame Journal of Formal Logic 45 (1): 1-11. 2004.We investigate a rst-order predicate logic based on Wittgenstein's suggestion to express identity of object by identity of sign, and difference of objects by difference of signs. Hintikka has shown that predicate logic can indeed be set up in such a way; we show that it can be done nicely. More specically, we provide a perspicuous cut-free sequent calculus, as well as a Hilbert-type calculus, for Wittgensteinian predicate logic and prove soundness and completeness theorems.
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17Fragments of [mathematical formula] based on [mathematical formula]-inductionArchive for Mathematical Logic 36 (1). 1997.
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The Association for Symbolic Logic publishes analytical reviews of selected books and articles in the field of symbolic logic. The reviews were published in The Journal of Symbolic Logic from the founding of the Journal in 1936 until the end of 1999. The Association moved the reviews to this Bulletin, beginning in 2000. The Reviews Section is edited by Steve Awodey (Managing Editor). John Baldwin, John (review)Bulletin of Symbolic Logic 15 (2). 2009.
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18College, 124 Raymond avenue, poughkeepsie, ny 12604, usa. In a review, a reference “jsl xliii 148,” for example, refers either to the publication reviewed on page 148 of volume 43 of the journal, or to the review itself (which contains full bibliographical information for the reviewed publication). Analogously, a reference “bsl VII 376” refers to the review beginning on page 376 in volume 7 of this bulletin, or (review)Bulletin of Symbolic Logic 16 (3). 2010.
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72Still in the Mood: The Versatility of Subjunctive Markers in Modal LogicTopoi 38 (2): 361-377. 2019.We investigate and compare two major approaches to enhancing the expressive capacities of modal languages, namely the addition of subjunctive markers on the one hand, and the addition of scope-bearing actuality operators, on the other. It turns out that the subjunctive marker approach is not only every bit as versatile as the actuality operator approach, but that it in fact outperforms its rival in the context of cross-world predication.
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88Identity and quantificationPhilosophical Studies 174 (3): 759-770. 2017.It is a philosophical commonplace that quantification involves, invokes, or presupposes, the relation of identity. There seem to be two major sources for this belief: the conviction that identity is implicated in the phenomenon of bound variable recurrence within the scope of a quantifier; memories of Quine’s insistence that quantification requires absolute identity for the values of variables. With respect to, I show that the only extant argument for a dependence of variable recurrence on ident…Read more
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34On the relations between Heinrich Scholz and Jan ŁukasiewiczHistory and Philosophy of Logic 28 (1): 67-81. 2007.The aim of the present study is (1) to show, on the basis of a number of unpublished documents, how Heinrich Scholz supported his Warsaw colleague Jan ?ukasiewicz, the Polish logician, during World War II, and (2) to discuss the efforts he made in order to enable Jan ?ukasiewicz and his wife Regina to move from Warsaw to Münster under life-threatening circumstances. In the first section, we explain how Scholz provided financial help to ?ukasiewicz, and we also adduce evidence of the risks incurr…Read more
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43Fragments of $HA$ based on $\Sigma_1$ -inductionArchive for Mathematical Logic 37 (1): 37-49. 1997.In the first part of this paper we investigate the intuitionistic version $iI\!\Sigma_1$ of $I\!\Sigma_1$ (in the language of $PRA$ ), using Kleene's recursive realizability techniques. Our treatment closely parallels the usual one for $HA$ and establishes a number of nice properties for $iI\!\Sigma_1$ , e.g. existence of primitive recursive choice functions (this is established by different means also in [D94]). We then sharpen an unpublished theorem of Visser's to the effect that quantifier al…Read more
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43World travelling and mood swingsIn Benedikt Löwe, Thoralf Räsch & Wolfgang Malzkorn (eds.), Foundations of the Formal Sciences II, Kluwer Academic Publishers. 2003.It is not quite as easy to see that there is in fact no formula of this modal language having the same truth conditions (in terms of S5 Kripke semantics) as (1). This was rst conjectured by Allen Hazen2 and later proved by Harold Hodes3. We present a simple direct proof of this result and discuss some consequences for the logical analysis of ordinary modal discourse.
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60On Ramsey's 'Silly Delusion' Regarding Tractatus 5.53In Giuseppe Primiero & Shahid Rahman (eds.), Acts of Knowledge: History, Philosophy and Logic, College Publications. 2009.We investigate a variant of the variable convention proposed at Tractatus 5.53ff for the purpose of eliminating the identity sign from logical notation. The variant in question is what Hintikka has called the strongly exclusive interpretation of the variables, and turns out to be what Ramsey initially (and erroneously) took to be Wittgenstein's intended method. We provide a tableau calculus for this identity-free logic, together with soundness and completeness proofs, as well as a proof of mutua…Read more
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132Frege’s permutation argument revisitedSynthese 147 (1): 43-61. 2005.In Section 10 of Grundgesetze, Volume I, Frege advances a mathematical argument (known as the permutation argument), by means of which he intends to show that an arbitrary value-range may be identified with the True, and any other one with the False, without contradicting any stipulations previously introduced (we shall call this claim the identifiability thesis, following Schroeder-Heister (1987)). As far as we are aware, there is no consensus in the literature as to (i) the proper inte…Read more
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76Gingerbread Nuts and Pebbles: Frege and the Neo-Kantians–Two Recently Discovered DocumentsBritish Journal for the History of Philosophy 21 (3). 2013.(2012). Gingerbread Nuts and Pebbles: Frege and the Neo-Kantians – Two Recently Discovered Documents. British Journal for the History of Philosophy. ???aop.label???. doi: 10.1080/09608788.2012.692665
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24College, 124 Raymond avenue, poughkeepsie, ny 12604, usa. In a review, a reference “jsl xliii 148,” for example, refers either to the publication reviewed on page 148 of volume 43 of the journal, or to the review itself (which contains full bibliographical information for the reviewed publication). Analogously, a reference “bsl VII 376” refers to the review beginning on page 376 in volume 7 of this bulletin, or (review)Bulletin of Symbolic Logic 17 (1). 2011.
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