-
1A Consistent Higher‐Order Theory Without a (Higher‐Order) ModelMathematical Logic Quarterly 35 (5): 385-386. 2006.
-
2Permutation Models in the Sense of Rieger‐BernaysMathematical Logic Quarterly 33 (3): 201-210. 2006.
-
2Permutations and stratified formulae a preservation theoremMathematical Logic Quarterly 36 (5): 385-388. 2006.
-
36Synonymy Questions Concerning the Quine SystemsJournal of Symbolic Logic 90 (4): 1779-1795. 2025.There are a variety of (“alternative”) axiomatic set theories available to mathematicians. It is worth asking how “alternative” they really are. Might they be no more than rephrasings of the theory (ZFC) that we already have? Here we give an account of the status of the Quine systems in this regard. Some are merely ZF in wolves’ clothing; some are genuine wolves.
-
50Internal Automorphisms and Antimorphisms of Models of NfJournal of Symbolic Logic 90 (4): 1796-1800. 2025.It is shown that every model of NF admits a permutation model containing an internal automorphism.
-
53Permutation Models in the Sense of Rieger-BernaysZeitschrift fur mathematische Logik und Grundlagen der Mathematik 33 (3): 201-210. 1987.
-
37Permutation Models in the Sense of Rieger‐BernaysMathematical Logic Quarterly 33 (3): 201-210. 1987.
-
48Reasoning About Theoretical EntitiesWorld Scientific. 2003.As such this book fills a void in the philosophical literature and presents a challenge to every would-be (anti-)reductionist.
-
143The status of the axiom of choice in set theory with a universal setJournal of Symbolic Logic 50 (3): 701-707. 1985.
-
174Term models for weak set theories with a universal setJournal of Symbolic Logic 52 (2): 374-387. 1987.
-
133Ramsey’s theorem and König’s LemmaArchive for Mathematical Logic 46 (1): 37-42. 2007.We consider the relation between versions of Ramsey’s Theorem and König’s Infinity Lemma, in the absence of the axiom of choice
-
195Further consistency and independence results in NF obtained by the permutation methodJournal of Symbolic Logic 48 (2): 236-238. 1983.
-
158Non-well-foundedness of well-orderable power setsJournal of Symbolic Logic 68 (3): 879-884. 2003.Tarski [5] showed that for any set X, its set w(X) of well-orderable subsets has cardinality strictly greater than that of X, even in the absence of the axiom of choice. We construct a Fraenkel-Mostowski model in which there is an infinite strictly descending sequence under the relation |w (X)| = |Y|. This contrasts with the corresponding situation for power sets, where use of Hartogs' ℵ-function easily establishes that there can be no infinite descending sequence under the relation |P(X)| = |Y|
-
80Normal subgroups of infinite symmetric groups, with an application to stratified set theoryJournal of Symbolic Logic 74 (1): 17-26. 2009.
-
83Erdös-Rado without ChoiceJournal of Symbolic Logic 72 (3). 2007.A version of the Erdös-Rado theorem on partitions of the unordered n-tuples from uncountable sets is proved, without using the axiom of choice. The case with exponent 1 is just the Sierpinski-Hartogs' result that $\aleph (\alpha)\leq 2^{2^{2^{\alpha}}}$
-
117Sharvy’s Lucy and Benjamin PuzzleStudia Logica 90 (2): 249-256. 2008.Sharvy’s puzzle concerns a situation in which common knowledge of two parties is obtained by repeated observation each of the other, no fixed point being reached in finite time. Can a fixed point be reached?
-
197End-extensions preserving power setJournal of Symbolic Logic 56 (1): 323-328. 1991.We consider the quantifier hierarchy of Takahashi [1972] and show how it gives rise to reflection theorems for some large cardinals in ZF, a new natural subtheory of Zermelo's set theory, a potentially useful new reduction of the consistency problem for Quine's NF, and a sharpening of another reduction of this problem due to Boffa.
-
131Yablo's paradox and the omitting types theorem for propositional languagesLogique Et Analyse 54 (215): 323-326. 2011.
-
95NF at (nearly) 75Logique Et Analyse 53 (212): 483-491. 2010.The consistency question for Quine's NF is still open. This is despite consistency having been established for systems which apparently resemble it very closely. The peculiar difficulties attending the consistency problem for NF are discussed. © 2011 Elsevier B.V., All rights reserved.
-
22Deterministic and Nondeterministic Strategies for Hintikka games in First-order and Branching-quantifier logicLogique Et Analyse 195 265--9. 2006.Applications of game-theoretic semantics à la Hintikka can be extended from Lower Predicate Calculus to languages with branching quantifiers. When one does this, issues which in the LPC could be swept under the carpet suddenly cause unwelcome subtleties. It turns out that which formulae of the branching quantifier logic one accounts true comes to depend on whether one requires that the winning strategies for Team Eloïse in the Hintikka game be deterministic (or allows them to be nondeterministic…Read more
-
76Permutations and Wellfoundedness: The True Meaning of the Bizarre Arithmetic of Quine's NFJournal of Symbolic Logic 71 (1). 2006.It is shown that, according to NF, many of the assertions of ordinal arithmetic involving the T-function which is peculiar to NF turn out to be equivalent to the truth-in-certain-permutation-models of assertions which have perfectly sensible ZF-style meanings, such as: the existence of wellfounded sets of great size or rank, or the nonexistence of small counterexamples to the wellfoundedness of ∈. Everything here holds also for NFU if the permutations are taken to fix all urelemente
-
A Note On Paradoxes In EthicsThe Baltic International Yearbook of Cognition, Logic and Communication 1. 2005.
-
16The axiom of choice and inference to the best explanationLogique Et Analyse 49 191-197. 2006.An argument often given for adopting the Axiom of Choice as an axiom is that it has a lot of obviously true consequences. This looks like a legitimate application of the practice of Inference to the Best Explanation. However, the standard examples of obvious-truths-following-from-AC all turn out, on closer inspection, to involve a fallacy of equivocation. © 2012 Elsevier B.V., All rights reserved.
-
46Permutations and stratified formulae a preservation theoremMathematical Logic Quarterly 36 (5): 385-388. 1990.
-
124A Note on Freedom from Detachment in the Logic of ParadoxNotre Dame Journal of Formal Logic 54 (1): 15-20. 2013.We shed light on an old problem by showing that the logic LP cannot define a binary connective $\odot$ obeying detachment in the sense that every valuation satisfying $\varphi$ and $(\varphi\odot\psi)$ also satisfies $\psi$, except trivially. We derive this as a corollary of a more general result concerning variable sharing.
-
Cambridge UniversityRetired faculty
-
Cambridge UniversityRetired faculty
Cambridge, United Kingdom of Great Britain and Northern Ireland
Areas of Specialization
| Science, Logic, and Mathematics |
Areas of Interest
| Science, Logic, and Mathematics |