*Stewart Shapiro*

- Published in print:
- 2000
- Published Online:
- November 2003
- ISBN:
- 9780198250296
- eISBN:
- 9780191598388
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198250290.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

A language is second‐order, or higher‐order, if it has bound variables that range over properties or sets of the items in the range of the ordinary, first‐order variables. This book presents a formal ...
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A language is second‐order, or higher‐order, if it has bound variables that range over properties or sets of the items in the range of the ordinary, first‐order variables. This book presents a formal development of second‐ and higher‐order logic and an extended argument that higher‐order systems have an important role to play in the philosophy and foundations of mathematics. The development includes the languages, deductive systems, and model‐theoretic semantics for higher‐order languages, and the basic and advanced results in its meta‐theory: completeness, compactness, and the Löwenheim–Skolem theorems for Henkin semantics, and the failure of those results for standard semantics. Argues that second‐order theories and formalizations, with standard semantics, provide better models of important aspects of mathematics than their first‐order counterparts. Despite the fact that Quine is the main opponent of second‐order logic (arguing that second‐order logic is set‐theory in disguise), the present argument is broadly Quinean, proposing that there is no sharp line dividing mathematics from logic, especially the logic of mathematics. Also surveys the historical development in logic, tracing the emergence of first‐order logic as the de facto standard among logicians and philosophers. The connection between formal deduction and reasoning is related to Wittgensteinian issues concerning rule‐following. The book closes with an examination of several alternatives to second‐order logic: first‐order set theory, infinitary languages, and systems that are, in a sense, intermediate between first order and second order.Less

A language is second‐order, or higher‐order, if it has bound variables that range over properties or sets of the items in the range of the ordinary, first‐order variables. This book presents a formal development of second‐ and higher‐order logic and an extended argument that higher‐order systems have an important role to play in the philosophy and foundations of mathematics. The development includes the languages, deductive systems, and model‐theoretic semantics for higher‐order languages, and the basic and advanced results in its meta‐theory: completeness, compactness, and the Löwenheim–Skolem theorems for Henkin semantics, and the failure of those results for standard semantics. Argues that second‐order theories and formalizations, with standard semantics, provide better models of important aspects of mathematics than their first‐order counterparts. Despite the fact that Quine is the main opponent of second‐order logic (arguing that second‐order logic is set‐theory in disguise), the present argument is broadly Quinean, proposing that there is no sharp line dividing mathematics from logic, especially the logic of mathematics. Also surveys the historical development in logic, tracing the emergence of first‐order logic as the *de facto* standard among logicians and philosophers. The connection between formal deduction and reasoning is related to Wittgensteinian issues concerning rule‐following. The book closes with an examination of several alternatives to second‐order logic: first‐order set theory, infinitary languages, and systems that are, in a sense, intermediate between first order and second order.

*Charles S. Chihara*

- Published in print:
- 1991
- Published Online:
- November 2003
- ISBN:
- 9780198239758
- eISBN:
- 9780191597190
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198239750.001.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

A continuation of the study of mathematical existence begun in Ontology and the Vicious‐Circle Principle (published in 1973); in the present work, Quine's indispensability argument is rebutted by the ...
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A continuation of the study of mathematical existence begun in Ontology and the Vicious‐Circle Principle (published in 1973); in the present work, Quine's indispensability argument is rebutted by the development of a new nominalistic version of mathematics (the Constructibility Theory) that is specified as an axiomatized theory formalized in a many‐sorted first‐order language. What is new in the present work is its abandonment of the predicative restrictions of the earlier work and its much greater attention to the applications of mathematics in science and everyday life. The book also contains detailed discussions of rival views (Mathematical Structuralism, Field's Instrumentalism, Burgess's Moderate Realism, Maddy's Set Theoretical Realism, and Kitcher's Ideal Agent account of mathematics), in which many comparisons with the Constructibility Theory are made.Less

A continuation of the study of mathematical existence begun in *Ontology and the Vicious‐Circle Principle* (published in 1973); in the present work, Quine's indispensability argument is rebutted by the development of a new nominalistic version of mathematics (the Constructibility Theory) that is specified as an axiomatized theory formalized in a many‐sorted first‐order language. What is new in the present work is its abandonment of the predicative restrictions of the earlier work and its much greater attention to the applications of mathematics in science and everyday life. The book also contains detailed discussions of rival views (Mathematical Structuralism, Field's Instrumentalism, Burgess's Moderate Realism, Maddy's Set Theoretical Realism, and Kitcher's Ideal Agent account of mathematics), in which many comparisons with the Constructibility Theory are made.

*Charles S. Chihara*

- Published in print:
- 1991
- Published Online:
- November 2003
- ISBN:
- 9780198239758
- eISBN:
- 9780191597190
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198239750.003.0007
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

The first of six chapters in which rival views are critically evaluated and compared with the Constructibility view described in earlier chapters. The views considered here (forms of ‘Structuralism’) ...
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The first of six chapters in which rival views are critically evaluated and compared with the Constructibility view described in earlier chapters. The views considered here (forms of ‘Structuralism’) are those of Stewart Shapiro and Michael Resnik. A number of difficulties with these two views are detailed and it is explained how the Constructibility Theory is not troubled by the problems that Structuralism was explicitly developed to resolve.Less

The first of six chapters in which rival views are critically evaluated and compared with the Constructibility view described in earlier chapters. The views considered here (forms of ‘Structuralism’) are those of Stewart Shapiro and Michael Resnik. A number of difficulties with these two views are detailed and it is explained how the Constructibility Theory is not troubled by the problems that Structuralism was explicitly developed to resolve.

*Stewart Shapiro*

- Published in print:
- 2000
- Published Online:
- November 2003
- ISBN:
- 9780199241279
- eISBN:
- 9780191597107
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199241279.003.0014
- Subject:
- Philosophy, Metaphysics/Epistemology

It seems that if a thinker in an argument arrives at an empirical conclusion, then some of the belief‐formation or reasoning principles she employs must be a priori if the reasoning is to be ...
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It seems that if a thinker in an argument arrives at an empirical conclusion, then some of the belief‐formation or reasoning principles she employs must be a priori if the reasoning is to be knowledgeable. Stewart Shapiro accepts this claim, and investigates the way in which the basic principles of logic must have an a priori status if the process of empirical confirmation of propositions reasoning that involves such principles of logic is to make sense.Less

It seems that if a thinker in an argument arrives at an empirical conclusion, then some of the belief‐formation or reasoning principles she employs must be a priori if the reasoning is to be knowledgeable. Stewart Shapiro accepts this claim, and investigates the way in which the basic principles of logic must have an a priori status if the process of empirical confirmation of propositions reasoning that involves such principles of logic is to make sense.