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Revamping the Restriction Strategy by
, 2007
"... This study continues the antirealist’s quest for a principled way to avoid Fitch’s paradox. It is proposed that the Cartesian restriction on the antirealist’s knowability principle ‘ϕ, therefore ✸Kϕ ’ should be formulated as a consistency requirement not on the premise ϕ of an application of the r ..."
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This study continues the antirealist’s quest for a principled way to avoid Fitch’s paradox. It is proposed that the Cartesian restriction on the antirealist’s knowability principle ‘ϕ, therefore ✸Kϕ ’ should be formulated as a consistency requirement not on the premise ϕ of an application of the rule, but rather on the set of assumptions on which the relevant occurrence of ϕ depends. It is stressed, by reference to illustrative proofs, how important it is to have proofs in normal form before applying the proposed restriction. A similar restriction is proposed for the converse inference, the socalled Rule of Factiveness ‘✸Kϕ therefore ϕ’. The proposed restriction appears to block another Fitchstyle derivation that uses the KKthesis in order to get around the Cartesian restriction on applications of the knowability principle. ∗ To appear in Joseph Salerno, ed., All Truths are Known: New Essays on the Knowability Paradox, Oxford University Press. This paper would not have been written without the stimulation, encouragement and criticism that I have enjoyed from Joseph Salerno, Salvatore Florio, Christina Moisa, Nicholaos Jones, and Patrick Reeder.
Induction and Indefinite Extensibility: The Gödel Sentence is True, but Did Someone Change the Subject?
"... Over the last few decades Michael Dummett developed a rich program for assessing logic and the meaning of the terms of a language. He is also a major exponent of Frege’s version of logicism in the philosophy of mathematics. Over the last decade, Neil Tennant developed an extensive version of logicis ..."
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Over the last few decades Michael Dummett developed a rich program for assessing logic and the meaning of the terms of a language. He is also a major exponent of Frege’s version of logicism in the philosophy of mathematics. Over the last decade, Neil Tennant developed an extensive version of logicism in Dummettian terms, and Dummett influenced other contemporary logicists such as Crispin Wright and Bob Hale. The purpose of this paper is to explore the prospects for Fregean logicism within a broadly Dummettian framework. The conclusions are mostly negative: Dummett’s views on analyticity and the logical/nonlogical boundary leave little room for logicism. Dummett’s considerations concerning manifestation and separability lead to a conservative extension requirement: if a sentence S is logically true, then there is a proof of S which uses only the introduction and elimination rules of the logical terms that occur in S. If basic arithmetic propositions are logically true—as the logicist contends—then there is tension between this conservation requirement and the ontological commitments
Concepts and Axioms
, 1998
"... The paper discusses the transition from informal concepts to mathematically precise notions; examples are given, and in some detail the case of lawless sequences, a concept of intuitionistic mathematics, is discussed. A final section comments on philosophical discussions concerning intuitionistic lo ..."
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The paper discusses the transition from informal concepts to mathematically precise notions; examples are given, and in some detail the case of lawless sequences, a concept of intuitionistic mathematics, is discussed. A final section comments on philosophical discussions concerning intuitionistic logic in connection with a "theory of meaning". What I have to tell here is not a new story, and it does not contain any really new ideas. The main difference with my earlier discussions of the same topics ([TD88, chapter16],[Tro91]) is in the emphasis. This paper starts with some examples of the transition from informal concepts to mathematically precise notions, followed by a more detailed discussion of one of these examples, the intuitionistic notion of a choice sequence, arguing for the lasting interest of this notion for the philosophy of mathematics. In a final section, I describe my own position relative to some of the philosophical discussions concerning intuitionistic logic in the wr...
A pragmatic framework for intuitionistic modalities: Classical logic and Lax logic.
"... Summary. We reconsider Dalla Pozza and Garola pragmatic interpretation of intuitionistic logic [13] where sentences and proofs formalize assertions and their justifications and revise it so that the costruction is done within an intuitionistic metatheory. We reconsider also the extension of Dalla Po ..."
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Summary. We reconsider Dalla Pozza and Garola pragmatic interpretation of intuitionistic logic [13] where sentences and proofs formalize assertions and their justifications and revise it so that the costruction is done within an intuitionistic metatheory. We reconsider also the extension of Dalla Pozza and Garola’a approach to cointuitionistic logic, seen as a logic of hypotheses [5, 9, 4] and the duality between assertions and hypotheses represented by two negations, the assertive and the hypothetical ones. By adding illocutionary forces of conjecture, defined as a hypothesis that an assertion is justified and of expectation, an assertion that a hypothesis is justified we obtain pragmatic counterparts of the modalities of classical S4, but also a framework for different interpretations of intuitionistic modalities necessity and possibility. We consider two applications: one is typing Parigot’s λµ calculus in a biintuitionistic logic of expectations. The second is an interpretation of Fairtlough and Mendler’s Propositional Lax Logic as an extension of intuitionistic logic with a cointuitionistic operator of empirical possibility. 1 Preface: intuitionistic pragmatics and its extensions.
by
, 2007
"... This study is in two parts. In the first part, various important principles of classical extensional mereology are derived on the basis of a nice axiomatization involving ‘part of ’ and fusion. All results are proved here with full Fregean (and Gentzenian) rigor. They are chosen because they are nee ..."
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This study is in two parts. In the first part, various important principles of classical extensional mereology are derived on the basis of a nice axiomatization involving ‘part of ’ and fusion. All results are proved here with full Fregean (and Gentzenian) rigor. They are chosen because they are needed for the second part. In the second part, this naturaldeduction framework is used in order to regiment David Lewis’s justification of his Division Thesis, which features prominently in his combination of mereology with class theory. The Division Thesis plays a crucial role in Lewis’s informal argument for his Second Thesis in his book Parts of Classes. In order to present Lewis’s argument in rigorous detail, an elegant new principle is offered for the theory that combines class theory and mereology. The new principle is called the Canonical Decomposition Thesis. It secures Lewis’s Division Thesis on the strong construal required in order for his
Inferentialism, Logicism, Harmony, and a Counterpoint by
, 2007
"... Inferentialism is explained as an attempt to provide an account of meaning that is more sensitive (than the tradition of truthconditional theorizing deriving from Tarski and Davidson) to what is learned when one masters meanings. The logically reformist inferentialism of Dummett and Prawitz is cont ..."
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Inferentialism is explained as an attempt to provide an account of meaning that is more sensitive (than the tradition of truthconditional theorizing deriving from Tarski and Davidson) to what is learned when one masters meanings. The logically reformist inferentialism of Dummett and Prawitz is contrasted with the more recent quietist inferentialism of Brandom. Various other issues are highlighted for inferentialism in general, by reference to which different kinds of inferentialism can be characterized. Inferentialism for the logical operators is explained, with special reference to the Principle of Harmony. The statement of that principle in the author’s book Natural Logic is finetuned here in the way obviously required in order to bar an interesting wouldbe counterexample furnished by Crispin Wright, and to stave off any more of the same.