
95Logicism and the Problem of Infinity: The Number of Numbers: ArticlesPhilosophia Mathematica 19 (2): 167212. 2011.Simpletype theory is widely regarded as inadequate to capture the metaphysics of mathematics. The problem, however, is not that some kinds of structure cannot be studied within simpletype theory. Even structures that violate simpletypes are isomorphic to structures that can be studied in simpletype theory. In disputes over the logicist foundations of mathematics, the central issue concerns the problem that simpletype theory fails to assure an infinity of natural numbers as objects. This pap…Read more

91Zermelo and Russell's Paradox: Is There a Universal set?Philosophia Mathematica 21 (2): 180199. 2013.Zermelo once wrote that he had anticipated Russell's contradiction of the set of all sets that are not members of themselves. Is this sufficient for having anticipated Russell's Paradox — the paradox that revealed the untenability of the logical notion of a set as an extension? This paper argues that it is not sufficient and offers criteria that are necessary and sufficient for having discovered Russell's Paradox. It is shown that there is ample evidence that Russell satisfied the criteria and t…Read more

85Wittgenstein's notes on logic – Michael Potter (review)Philosophical Quarterly 60 (240): 645648. 2010.No Abstract

64Cocchiarella’s Formal Ontology and the Paradoxes of HyperintensionalityAxiomathes 19 (2): 115142. 2009.This is a critical discussion of Nino B. Cocchiarella’s book “Formal Ontology and Conceptual Realism.” It focuses on paradoxes of hyperintensionality that may arise in formal systems of intensional logic.

63A new interpretation of russell's multiplerelation theory of judgmentHistory and Philosophy of Logic 12 (1): 3769. 1991.This paper offers an interpretation of Russell's multiplerelation theory of judgment which characterizes it as direct application of the 1905 theory of definite descriptions. The paper maintains that it was by regarding propositional symbols (when occurring as subordinate clauses) as disguised descriptions of complexes, that Russell generated the philosophical explanation of the hierarchy of orders and the ramified theory of types of _Principia mathematica (1910). The interpretation provides a …Read more

61The definability of the set of natural numbers in the 1925 principia mathematicaJournal of Philosophical Logic 25 (6). 1996.In his new introduction to the 1925 second edition of Principia Mathematica, Russell maintained that by adopting Wittgenstein's idea that a logically perfect language should be extensional mathematical induction could be rectified for finite cardinals without the axiom of reducibility. In an Appendix B, Russell set forth a proof. Godel caught a defect in the proof at *89.16, so that the matter of rectification remained open. Myhill later arrived at a negative result: Principia with extensionalit…Read more

54Frege’s Cardinals as ConceptcorrelatesErkenntnis 65 (2): 207243. 2006.In his "Grundgesetze", Frege hints that prior to his theory that cardinal numbers are objects he had an "almost completed" manuscript on cardinals. Taking this early theory to have been an account of cardinals as secondlevel functions, this paper works out the significance of the fact that Frege's cardinal numbers is a theory of conceptcorrelates. Frege held that, where n > 2, there is a one—one correlation between each nlevel function and an n—1 level function, and a one—one correlation betw…Read more

50Frege's Cardinals Do Not Always Obey Hume's PrincipleHistory and Philosophy of Logic 38 (2): 127153. 2017.Hume's Principle, dear to neoLogicists, maintains that equinumerosity is both necessary and sufficient for sameness of cardinal number. All the same, Whitehead demonstrated in Principia Mathematica's logic of relations that Cantor's powerclass theorem entails that Hume's Principle admits of exceptions. Of course, Hume's Principle concerns cardinals and in Principia's ‘noclasses’ theory cardinals are not objects in Frege's sense. But this paper shows that the result applies as well to the theo…Read more

45Russell’s Hidden Substitutional TheoryOxford University Press. 1998.This book explores an important central thread that unifies Russell's thoughts on logic in two works previously considered at odds with each other, the Principles of Mathematics and the later Principia Mathematica. This thread is Russell's doctrine that logic is an absolutely general science and that any calculus for it must embrace wholly unrestricted variables. The heart of Landini's book is a careful analysis of Russell's largely unpublished "substitutional" theory. On Landini's showing, the …Read more

39The persistence of counterexample: Reexamining the debate over Leibniz lawNoûs 25 (1): 4361. 1991.

38Putnam's modeltheoretic argument, natural realism, and the standard conception of theoriesPhilosophical Papers 16 (3): 209233. 1987.No abstract

36Erich H. Reck and Steve Awodey, trans. and ed., Frege's Lectures on Logic: Carnap's Student Notes, 1910–1914. Publications of the Archive of Scientific Philosophy, Hillman Library, University of Pittsburgh. LaSalle, Illinois: Open Court, 2004. Pp. xiv + 170. ISBN 0812695461 (cloth), 0812695534 (paper) (review)Philosophia Mathematica 13 (2): 225227. 2005.

35Russell's Schema, Not Priest's InclosureHistory and Philosophy of Logic 30 (2): 105139. 2009.On investigating a theorem that Russell used in discussing paradoxes of classes, Graham Priest distills a schema and then extends it to form an Inclosure Schema, which he argues is the common structure underlying both classtheoretical paradoxes (such as that of Russell, Cantor, BuraliForti) and the paradoxes of ?definability? (offered by Richard, KönigDixon and Berry). This article shows that Russell's theorem is not Priest's schema and questions the application of Priest's Inclosure Schema t…Read more

35The Ins and Outs of Frege's Way Out †Special thanks to Kevin Klement, Ed Mares, Francesco Orilia, and anonymous referees for helpful comments on this paperPhilosophia Mathematica 14 (1): 125. 2006.Confronted with Russell's Paradox, Frege wrote an appendix to volume II of his _Grundgesetze der Arithmetik_. In it he offered a revision to Basic Law V, and proclaimed with confidence that the major theorems for arithmetic are recoverable. This paper shows that Frege's revised system has been seriously undermined by interpretations that transcribe his system into a predicate logic that is inattentive to important details of his conceptscript. By examining the revised system as a conceptscript…Read more

33Logic in Russell's Principles of MathematicsNotre Dame Journal of Formal Logic 37 (4): 554584. 1996.Unaware of Frege's 1879 Begriffsschrift, Russell's 1903 The Principles of Mathematics set out a calculus for logic whose foundation was the doctrine that any such calculus must adopt only one style of variables–entity (individual) variables. The idea was that logic is a universal and allencompassing science, applying alike to whatever there is–propositions, universals, classes, concrete particulars. Unfortunately, Russell's early calculus has appeared archaic if not completely obscure. This pap…Read more

32Michael Potter Tom Ricketts, eds. The cambridge companion to Frege. Cambridge: Cambridge university press, 2010. Isbn 9780521624794. Pp. XVII+639 (review)Philosophia Mathematica 20 (3): 372387. 2012.

32Words Without Objects: Semantics, Ontology, and Logic for NonSingularityHistory and Philosophy of Logic 30 (2): 204208. 2009.

31Wittgenstein's Apprenticeship with RussellCambridge University Press. 2007.Wittgenstein's Tractatus has generated many interpretations since its publication in 1921, but over the years a consensus has developed concerning its criticisms of Russell's philosophy. In Wittgenstein's Apprenticeship with Russell, Gregory Landini draws extensively from his work on Russell's unpublished manuscripts to show that the consensus characterises Russell with positions he did not hold. Using a careful analysis of Wittgenstein's writings he traces the 'Doctrine of Showing' and the 'fun…Read more

29Russellian Facts About the SlingshotAxiomathes 24 (4): 533547. 2014.The socalled “Slingshot” argument purports to show that an ontology of facts is untenable. In this paper, we address a minimal slingshot restricted to an ontology of physical facts as truthmakers for empirical physical statements. Accepting that logical matters have no bearing on the physical facts that are truthmakers for empirical physical statements and that objects are themselves constituents of such facts, our minimal slingshot argument purportedly shows that any two physical statements …Read more

29Erik C. Banks, The Realistic Empiricism of Mach, James and Russell (review)Hopos: The Journal of the International Society for the History of Philosophy of Science 6 (2): 329333. 2016.

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27Quantification Theory in *8 of Principia Mathematica and the Empty DomainHistory and Philosophy of Logic 26 (1): 4759. 2005.The second printing of Principia Mathematica in 1925 offered Russell an occasion to assess some criticisms of the Principia and make some suggestions for possible improvements. In Appendix A, Russell offered *8 as a new quantification theory to replace *9 of the original text. As Russell explained in the new introduction to the second edition, the system of *8 sets out quantification theory without free variables. Unfortunately, the system has not been well understood. This paper shows that Russ…Read more

27RussellRoutledge. 2010.Landini discusses the second edition of Principia Mathematica, to show Russella (TM)s intellectual relationship with Wittgenstein and Ramsey.

25Decomposition and analysis in frege’sgrundgesetzeHistory and Philosophy of Logic 17 (12): 121139. 1996.Frege seems to hold two incompatible theses:(i) that sentences differing in structure can yet express the same sense; and (ii) that the senses of the meaningful parts of a complex term are determinate parts of the sense of the term. Dummett offered a solution, distinguishing analysis from decomposition. The present paper offers an embellishment of Dummett?s distinction by providing a way of depicting the internal structures of complex senses?determinate structures that yield distinct decompositi…Read more

24Gregory Landini. Zermelo and Russell’s Paradox: Is There a Universal Set?: Correction NoticePhilosophia Mathematica 22 (1): 142142. 2014.

23Review: D. Bostock. Russell’s Logical Atomism (review)Journal for the History of Analytical Philosophy 2 (1). 2013.This is review of D. David Bostock. Russell’s Logical Atomism

22Truth, Predication and a Family of Contingent ParadoxesJournal of Philosophical Logic 48 (1): 113136. 2019.In truth theory one aims at general formal laws governing the attribution of truth to statements. Gupta’s and Belnap’s revisiontheoretic approach provides various wellmotivated theories of truth, in particular T* and T#, which tame the Liar and related paradoxes without a Tarskian hierarchy of languages. In property theory, one similarly aims at general formal laws governing the predication of properties. To avoid Russell’s paradox in this area a recourse to type theory is still popular, as te…Read more

22Logic as a Universal Science: Russell's Early Logicism and Its Philosophical ContextPhilosophical Quarterly 64 (255): 361364. 2014.

22Typos of Principia MathematicaHistory and Philosophy of Logic 34 (4). 2013.Principia Mathematic goes to great lengths to hide its order/type indices and to make it appear as if its incomplete symbols behave as if they are singular terms. But wellhidden as they are, we cannot understand the proofs in Principia unless we bring them into focus. When we do, some rather surprising results emerge ? which is the subject of this paper
Areas of Specialization
Philosophy of Mind 
Logic and Philosophy of Logic 
Philosophy of Mathematics 
Areas of Interest
Philosophy of Mind 
Logic and Philosophy of Logic 
Philosophy of Mathematics 