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AN ANALOGUE OF BETH'S THEOREM IN NORMAL EXTENSIONS OF THE MODAL LOGIC K4

D.M. Gabbay and L. Maksimova

in Interpolation and Definability: Modal and Intuitionistic Logics

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198511748
eISBN:
9780191705779
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198511748.003.0011
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

Beth's theorems state that in the classical predicate logic, any implicitly definable predicate is explicitly definable. This chapter proves an analogue of this theorem for all extensions of K4. It ... More


INTERPOLATION IN GOAL- DIRECTED PROOF SYSTEMS

D.M. Gabbay and L. Maksimova

in Interpolation and Definability: Modal and Intuitionistic Logics

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198511748
eISBN:
9780191705779
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198511748.003.0016
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter studies the interpolation properties for implicational fragment of a variety of substructural, strict modal, and intuitionistic and intermediate logics. The methodology is ... More


Opening Questions, Following Rules

Paul Bloomfield

in Metaethics after Moore

Published in print:
2006
Published Online:
May 2010
ISBN:
9780199269914
eISBN:
9780191710032
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199269914.003.0009
Subject:
Philosophy, Moral Philosophy, General

This chapter begins by noting that the 20th century beneficiary of the open question argument has been (rather ironically) the class of non-realist views, including non-cognitivism and expressivism. ... More


Inessential features and expressive power of descriptive metalanguages

Geoffrey K. Pullum and Hans‐Jörg Tiede

in Features: Perspectives on a Key Notion in Linguistics

Published in print:
2010
Published Online:
September 2010
ISBN:
9780199577743
eISBN:
9780191722844
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199577743.003.0011
Subject:
Linguistics, Semantics and Pragmatics, Syntax and Morphology

Linguists feel intuitively that genuine features, those reflecting properties of natural language, are distinguishable from spurious ones – mere tricks exploiting feature machinery. Unfortunately, no ... More


Truth and Merit

Charles Travis

in Objectivity and the Parochial

Published in print:
2010
Published Online:
January 2011
ISBN:
9780199596218
eISBN:
9780191595783
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199596218.003.0009
Subject:
Philosophy, Aesthetics

This chapter compares Frege and Austin on truth. One might think that these thinkers held opposing views on truth. But, at least for the most part, there is no reason to think this. It is better, it ... More


Formalization: Expressibility, Definability, Compositionality

Stanley Peters

in Quantifiers in Language and Logic

Published in print:
2008
Published Online:
October 2011
ISBN:
9780199291267
eISBN:
9780191700590
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199291267.003.0013
Subject:
Philosophy, Philosophy of Language, Logic/Philosophy of Mathematics

This chapter discusses logical equivalence across languages in terms of the concept of a lexical mapping and addresses the issue of compositionality. Compositionality is perhaps the first extra ... More


Definability and Undefinability in Logical Languages: Tools for the Monadic Case

Stanley Peters

in Quantifiers in Language and Logic

Published in print:
2008
Published Online:
October 2011
ISBN:
9780199291267
eISBN:
9780191700590
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199291267.003.0014
Subject:
Philosophy, Philosophy of Language, Logic/Philosophy of Mathematics

To prove a positive definability result, one usually has to exhibit the defining sentence, and it helps if this sentence belongs to a well-defined language. To prove a negative undefinability result, ... More


Applications to Monadic Definability

Stanley Peters

in Quantifiers in Language and Logic

Published in print:
2008
Published Online:
October 2011
ISBN:
9780199291267
eISBN:
9780191700590
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199291267.003.0015
Subject:
Philosophy, Philosophy of Language, Logic/Philosophy of Mathematics

This chapter applies the Ehrenfeucht-Fraïssé (EF) technique to present a variety of results about the expressive power of certain monadic quantifiers. After repeating the general method, ... More


Advanced 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:
10.1093/0198250290.003.0006
Subject:
Philosophy, Logic/Philosophy of Mathematics

After setting out some issues concerning the meta‐language, this chapter presents a number of results about second‐order languages with standard semantics. Since the membership relation can be ... More


Curves

Ehud Hrushovski and François Loeser

in Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

Published in print:
2016
Published Online:
October 2017
ISBN:
9780691161686
eISBN:
9781400881222
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161686.003.0007
Subject:
Mathematics, Geometry / Topology

This chapter proves the iso-definability of unit vector C when C is a curve using Riemann-Roch. Recall that a pro-definable set is called iso-definable if it is isomorphic, as a pro-definable set, to ... More


Theories of presheaf type: general criteria

Olivia Caramello

in Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic 'bridges'

Published in print:
2017
Published Online:
March 2018
ISBN:
9780198758914
eISBN:
9780191818752
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198758914.003.0008
Subject:
Mathematics, Geometry / Topology, Pure Mathematics

This chapter carries out a systematic investigation of the class of geometric theories of presheaf type (i.e. classified by a presheaf topos), by using in particular the results on flat functors ... More


Paradoxes of Definability, Russell’s Paradox, the Liar

Keith Simmons

in Semantic Singularities: Paradoxes of Reference, Predication, and Truth

Published in print:
2018
Published Online:
July 2018
ISBN:
9780198791546
eISBN:
9780191852923
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198791546.003.0005
Subject:
Philosophy, Philosophy of Language, Logic/Philosophy of Mathematics

Chapter 5 moves beyond the simple paradoxes discussed in Chapters 2-4. The chapter applies the singularity approach to the traditional paradoxes of definability (or denotation), associated with ... 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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