Results 1 
7 of
7
Natural Logicism via the Logic of Orderly Pairing by
, 2008
"... Schumm, Timothy Smiley and Matthias Wille. Comments by two anonymous referees have also led to significant improvements. The aim here is to describe how to complete the constructive logicist program, in the author’s book AntiRealism and Logic, of deriving all the PeanoDedekind postulates for arith ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Schumm, Timothy Smiley and Matthias Wille. Comments by two anonymous referees have also led to significant improvements. The aim here is to describe how to complete the constructive logicist program, in the author’s book AntiRealism and Logic, of deriving all the PeanoDedekind postulates for arithmetic within a theory of natural numbers that also accounts for their applicability in counting finite collections of objects. The axioms still to be derived are those for addition and multiplication. Frege did not derive them in a fully explicit, conceptually illuminating way. Nor has any neoFregean done so. These outstanding axioms need to be derived in a way fully in keeping with the spirit and the letter of Frege’s logicism and his doctrine of definition. To that end this study develops a logic, in the GentzenPrawitz style of natural deduction, for the operation of orderly pairing. The logic is an extension of free firstorder logic with identity. Orderly pairing is treated as a primitive. No notion of set is presupposed, nor any settheoretic notion of membership. The formation of ordered pairs, and the two projection operations yielding their left and right coordinates, form a coeval family of logical notions. The challenge is to furnish them with introduction and elimination rules that capture their exact meanings, and no more. Orderly pairing as a logical primitive is then used in order to introduce addition and multiplication in a conceptually satisfying way within a constructive logicist theory of the natural numbers. Because of its reliance, throughout, on senseconstituting rules of natural deduction, the completed account can be described as ‘natural logicism’. 2 1 Introduction: historical
(to appear in J. Salerno, ed., New Essays on the Knowability Paradox, Oxford: Oxford University Press) Tennant’s Troubles
"... First, some reminiscences. In the years 197380, when I was an undergraduate and then graduate student at Oxford, Michael Dummett’s formidable and creative philosophical presence made his arguments impossible to ignore. In consequence, one pole of discussion was always a form of antirealism. It end ..."
Abstract
 Add to MetaCart
First, some reminiscences. In the years 197380, when I was an undergraduate and then graduate student at Oxford, Michael Dummett’s formidable and creative philosophical presence made his arguments impossible to ignore. In consequence, one pole of discussion was always a form of antirealism. It endorsed something like the replacement of truthconditional semantics by verificationconditional semantics and of classical logic by intuitionistic logic, and the principle that all truths are knowable. It did not endorse the principle that all truths are known. Nor did it mention the now celebrated argument, first published by Frederic Fitch (1963), that if all truths are knowable then all truths are known. Even in 1970s Oxford, intuitionistic antirealism was a strictly minority view, but many others regarded it as a live theoretical option in a way that now seems very distant. As the extreme verificationist commitments of the view have combined with accumulating decades of failure to reply convincingly to criticisms of the arguments in its favour or to carry out the programme of generalizing intuitionistic semantics for 1 mathematics to empirical discourse, even in toy examples, the impression has been
c ○ Peter SchroederHeisterPreface
, 1987
"... and Neil Tennant for helpful comments and discussions on topics related to this work. ..."
Abstract
 Add to MetaCart
and Neil Tennant for helpful comments and discussions on topics related to this work.
Planetary Orbits in a Sun’s
"... Abstract. We present a logically detailed casestudy of explanation and prediction in Newtonian mechanics. The case in question is that of a planet’s elliptical orbit in the Sun’s gravitational field. Care is taken to distinguish the respective contributions of the mathematics that is being applied, ..."
Abstract
 Add to MetaCart
Abstract. We present a logically detailed casestudy of explanation and prediction in Newtonian mechanics. The case in question is that of a planet’s elliptical orbit in the Sun’s gravitational field. Care is taken to distinguish the respective contributions of the mathematics that is being applied, and of the empirical hypotheses that receive a mathematical formulation. This enables one to appreciate how in this case the overall logical structure of scientific explanation and prediction is exactly in accordance with the hypotheticodeductive model.
Inferential Semantics for FirstOrder Logic: Motivating Rules of Inference from Rules of Evaluation
, 2009
"... I am grateful to Tim, as my teacher in logic, for emphasizing that its main notions are relational: ‘ϕ is deducible from ∆ ’ and ‘ϕ is a logical consequence of ∆’. The notions ‘ϕ is a theorem ’ and ‘ϕ is logically true’ are special cases. In the early 1970s Tim gave a formative series of lectures em ..."
Abstract
 Add to MetaCart
I am grateful to Tim, as my teacher in logic, for emphasizing that its main notions are relational: ‘ϕ is deducible from ∆ ’ and ‘ϕ is a logical consequence of ∆’. The notions ‘ϕ is a theorem ’ and ‘ϕ is logically true’ are special cases. In the early 1970s Tim gave a formative series of lectures emphasizing how proofs are to be understood as perfected arguments, in Aristotle’s sense. The present discussion of verification and falsification is fully in the inferentialist spirit of Tim’s emphases. The aim is to render even the notions ‘ϕ is true ’ and ‘ϕ is false ’ as essentially relational and inferential. A sentence’s truthvalue is determined relative to collections of rules of inference that constitute an interpretation. Moreover, truthmakers and falsitymakers are themselves prooflike objects, encoding the inferential process of evaluation involved. The inference rules involved in the determination of truthvalue are almost identical to those involved in securing the transmission of truth from premises to conclusion of a valid argument. We shall see how smoothly one can ‘morph ’ the former into the
THE REVIEW OF SYMBOLIC LOGIC, Page 1 of 30 CUT FOR CORE LOGIC
"... Abstract. The motivation for Core Logic is explained. Its system of proof is set out. It is then shown that, although the system has no Cut rule, its relation of deducibility obeys Cut with epistemic gain. §1. The debate over logical reform. There is much dispute over which logic is the right logic— ..."
Abstract
 Add to MetaCart
Abstract. The motivation for Core Logic is explained. Its system of proof is set out. It is then shown that, although the system has no Cut rule, its relation of deducibility obeys Cut with epistemic gain. §1. The debate over logical reform. There is much dispute over which logic is the right logic—indeed, over whether there could even be such a thing as the right logic, rather than a spectrum of logics variously suited for different applications in different areas. Absolutists about logic regard the use of the definite article as justified; pluralists have their principled doubts. For those who engage in the absolutist debate, those whom we can call the quietists are willing to accept the full canon C of classical logic. Their opponents, whom we can call the reformists—intuitionists and relevantists prominent among them— argue that certain rules of classical logic lack validity, and have no right to be in the canon. Intuitionists, on the one hand, originally drew inspiration for their critique of classical logic from the requirements of constructivity in mathematical proof. According to the intuitionist’s construal of existence, a mathematical existence claim of the form ‘there is a natural number n such that F(n) ’ requires its asserter to be able to provide a justifying instance—a constructively determinable number t for which one can prove (intuitionistically!) that F(t): F(t) ∃xF(x) This means that one may not use the ‘backdoor’, or indirect, reasoning that would be available to a classical mathematician, whereby in order to derive the conclusion that there is a natural number n such that F(n), it would be sufficient simply to assume that no natural number has the property F, and then (classically!) derive an absurdity from that assumption: ¬∃xF(x)
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 ..."
Abstract
 Add to MetaCart
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.