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Metatheory

Stewart Shapiro

in Foundations without Foundationalism: A Case for Second-Order Logic

Published in print:
2000
Published Online:
November 2003
ISBN:
9780198250296
eISBN:
9780191598388
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198250290.003.0004
Subject:
Philosophy, Logic/Philosophy of Mathematics

Although second‐order languages, with standard semantics, are sound for standard semantics, they are not compact, and the Löwenheim–Skolem theorems fail. These are consequences of the fact that the ... More


Degraded Acceptability and Markedness in Syntax, and the Stochastic Interpretation of Optimality Theory

RALF VOGEL

in Gradience in Grammar: Generative Perspectives

Published in print:
2006
Published Online:
January 2010
ISBN:
9780199274796
eISBN:
9780191705861
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199274796.003.0013
Subject:
Linguistics, Syntax and Morphology

This chapter shows that violation profiles can be employed to predict contrasts among expressions in empirical investigations, and that markedness is the grammar-internal correlate of (some) ... More


Tarski on Categoricity and Completeness: An Unpublished Lecture from 1940

Paolo Mancosu

in The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940

Published in print:
2010
Published Online:
May 2011
ISBN:
9780199546534
eISBN:
9780191594939
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199546534.003.0017
Subject:
Philosophy, Logic/Philosophy of Mathematics, Philosophy of Mind

One of the central pieces of evidence used in Chapter 16 is an unpublished lecture by Tarski entitled “On the completeness and categoricity of deductive theories”(1940). This Chapter provides a step ... More


Appendix: “On the Completeness and Categoricity of Deductive Systems” (1940) by Alfred Tarski

Paolo Mancosu

in The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940

Published in print:
2010
Published Online:
May 2011
ISBN:
9780199546534
eISBN:
9780191594939
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199546534.003.0018
Subject:
Philosophy, Logic/Philosophy of Mathematics, Philosophy of Mind

This chapter contains a transcription of “On the completeness and categoricity of deductive theories” (1940), an hitherto unpublished lecture by Alfred Tarski. The lecture is central for an ... More


Philosophy and Model Theory

Tim Button and Sean Walsh

Published in print:
2018
Published Online:
May 2018
ISBN:
9780198790396
eISBN:
9780191863424
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198790396.001.0001
Subject:
Philosophy, Logic/Philosophy of Mathematics

Model theory is used in every theoretical branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. ... More


Internal categoricity and the sets

Tim Button and Sean Walsh

in Philosophy and Model Theory

Published in print:
2018
Published Online:
May 2018
ISBN:
9780198790396
eISBN:
9780191863424
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198790396.003.0011
Subject:
Philosophy, Logic/Philosophy of Mathematics

As the previous chapter discussed the internalist perspective on the categoricity of arithmetic, this chapter presents the internalist perspective on sets. In particular, we show both how to ... More


Nietzsche’s Critical Target: Morality

Simon Robertson

in Nietzsche and Contemporary Ethics

Published in print:
2020
Published Online:
July 2020
ISBN:
9780198722212
eISBN:
9780191789069
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198722212.003.0002
Subject:
Philosophy, Moral Philosophy

This chapter characterizes ‘morality’, the object of Nietzsche’s critique. It locates the discussion within a wider interpretive issue, commonly called the Scope Problem, that bears on the coherence ... More


Error Theory and Naturalism

Simon Robertson

in Nietzsche and Contemporary Ethics

Published in print:
2020
Published Online:
July 2020
ISBN:
9780198722212
eISBN:
9780191789069
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198722212.003.0003
Subject:
Philosophy, Moral Philosophy

This is the first of two chapters examining Nietzsche’s attacks on morality’s foundational presuppositions. Presenting him as an error theorist about morality and its categoricity, the chapter ... More


First-order theories

Shawn Hedman

in A First Course in Logic: An Introduction to Model Theory, Proof Theory, Computability, and Complexity

Published in print:
2004
Published Online:
November 2020
ISBN:
9780198529804
eISBN:
9780191916656
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198529804.003.0009
Subject:
Computer Science, Computer Architecture and Logic Design

We continue our study of Model Theory. This is the branch of logic concerned with the interplay between sentences of a formal language and mathematical structures. Primarily, Model Theory studies ... More


Categoricity and the natural numbers

Tim Button and Sean Walsh

in Philosophy and Model Theory

Published in print:
2018
Published Online:
May 2018
ISBN:
9780198790396
eISBN:
9780191863424
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198790396.003.0007
Subject:
Philosophy, Logic/Philosophy of Mathematics

This chapter focuses on modelists who want to pin down the isomorphism type of the natural numbers. This aim immediately runs into two technical barriers: the Compactness Theorem and the ... More


Categoricity and the sets

Tim Button and Sean Walsh

in Philosophy and Model Theory

Published in print:
2018
Published Online:
May 2018
ISBN:
9780198790396
eISBN:
9780191863424
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198790396.003.0008
Subject:
Philosophy, Logic/Philosophy of Mathematics

In this chapter, the focus shifts from numbers to sets. Again, no first-order set theory can hope to get anywhere near categoricity, but Zermelo famously proved the quasi-categoricity of second-order ... More


On the Use of Classical Logic in Set Theory

Ian Rumfitt

in The Boundary Stones of Thought: An Essay in the Philosophy of Logic

Published in print:
2015
Published Online:
June 2015
ISBN:
9780198733638
eISBN:
9780191798016
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198733638.003.0009
Subject:
Philosophy, Philosophy of Mind

Some philosophers have argued that special problems attend the use of classical logic in set theory; this chapter focuses on arguments to this effect due to William Tait. It is conceded to Tait that ... More


Mathematical Determinacy

Jared Warren

in Shadows of Syntax: Revitalizing Logical and Mathematical Conventionalism

Published in print:
2020
Published Online:
December 2020
ISBN:
9780190086152
eISBN:
9780190086183
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780190086152.003.0010
Subject:
Philosophy, Logic/Philosophy of Mathematics

This chapter addresses the second major challenge for the extension of conventionalism from logic to mathematics: the richness of mathematical truth. The chapter begins by distinguishing ... More


Unrestricted Logical Inferentialism

Jared Warren

in Shadows of Syntax: Revitalizing Logical and Mathematical Conventionalism

Published in print:
2020
Published Online:
December 2020
ISBN:
9780190086152
eISBN:
9780190086183
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780190086152.003.0003
Subject:
Philosophy, Logic/Philosophy of Mathematics

This chapter develops and defends an unrestricted inferentialist theory of the meanings of logical constants. Unlike restricted inferentialism, unrestricted inferentialism puts no constraints on ... More


Internal categoricity and the natural numbers

Tim Button and Sean Walsh

in Philosophy and Model Theory

Published in print:
2018
Published Online:
May 2018
ISBN:
9780198790396
eISBN:
9780191863424
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198790396.003.0010
Subject:
Philosophy, Logic/Philosophy of Mathematics

The simple conclusion of the preceding chapters is that moderate modelism fails. But this leaves us with a choice between abandoning moderation and abandoning modelism. The aim of this chapter, and ... More


Wilfrid Hodges A short history of model theory

Tim Button and Sean Walsh

in Philosophy and Model Theory

Published in print:
2018
Published Online:
May 2018
ISBN:
9780198790396
eISBN:
9780191863424
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198790396.003.0018
Subject:
Philosophy, Logic/Philosophy of Mathematics

We give a historical account of mathematical model theory, from its origins to the early twenty-first century. We record how early work in model theory grew from attempts to formalise and systematise ... More


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