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850Hilbert’s Finitism: Historical, Philosophical, and Metamathematical PerspectivesDissertation, University of California, Berkeley. 2001.In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing---using only so-called finitistic principles---that these formalizations are free of contradictions. ;In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as compl…Read more
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117Carnap’s early metatheory: scope and limitsSynthese 194 (1): 33-65. 2017.In Untersuchungen zur allgemeinen Axiomatik and Abriss der Logistik, Carnap attempted to formulate the metatheory of axiomatic theories within a single, fully interpreted type-theoretic framework and to investigate a number of meta-logical notions in it, such as those of model, consequence, consistency, completeness, and decidability. These attempts were largely unsuccessful, also in his own considered judgment. A detailed assessment of Carnap’s attempt shows, nevertheless, that his approach is …Read more
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282Labeled calculi and finite-valued logicsStudia Logica 61 (1): 7-33. 1998.A general class of labeled sequent calculi is investigated, and necessary and sufficient conditions are given for when such a calculus is sound and complete for a finite -valued logic if the labels are interpreted as sets of truth values. Furthermore, it is shown that any finite -valued logic can be given an axiomatization by such a labeled calculus using arbitrary "systems of signs," i.e., of sets of truth values, as labels. The number of labels needed is logarithmic in the number of truth valu…Read more
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777Natural Deduction for the Sheffer Stroke and Peirce’s Arrow (and any Other Truth-Functional Connective)Journal of Philosophical Logic 45 (2): 183-197. 2015.Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke and Peirce’s arrow. The restriction to a single conclusion in standard systems of natural deduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absu…Read more
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85First-order Gödel logicsAnnals of Pure and Applied Logic 147 (1): 23-47. 2007.First-order Gödel logics are a family of finite- or infinite-valued logics where the sets of truth values V are closed subsets of [0,1] containing both 0 and 1. Different such sets V in general determine different Gödel logics GV (sets of those formulas which evaluate to 1 in every interpretation into V). It is shown that GV is axiomatizable iff V is finite, V is uncountable with 0 isolated in V, or every neighborhood of 0 in V is uncountable. Complete axiomatizations for each of these cases ar…Read more
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Areas of Specialization
Logic and Philosophy of Logic |
Philosophy of Mathematics |
20th Century Philosophy |