*Hartry Field*

- Published in print:
- 2008
- Published Online:
- May 2008
- ISBN:
- 9780199230747
- eISBN:
- 9780191710933
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230747.003.0004
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

This chapter is an exposition of the strong Kleene version of Kripke's fixed point semantics, and the theories of truth that can be obtained from it. Emphasis is put on the distinction between ...
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This chapter is an exposition of the strong Kleene version of Kripke's fixed point semantics, and the theories of truth that can be obtained from it. Emphasis is put on the distinction between ‘external’ and ‘internal’ theories of the fixed points: the external theories keep classical logic, posit truth value gaps, and deny the existence of truth value gluts; whereas the internal theories are non-classical and neither assert nor deny the existence of gaps or of gluts. In fact, ‘gap’ and ‘glut’ are equivalent within these theories. The advantages of the internal theories over the external are stressed: primarily, internal theories make True(p) intersubstitutable with p, whereas external theories don't. At the same time, the internal Kripke theories are seriously lacking in expressive power.Less

This chapter is an exposition of the strong Kleene version of Kripke's fixed point semantics, and the theories of truth that can be obtained from it. Emphasis is put on the distinction between ‘external’ and ‘internal’ theories of the fixed points: the external theories keep classical logic, posit truth value gaps, and deny the existence of truth value gluts; whereas the internal theories are non-classical and neither assert nor deny the existence of gaps or of gluts. In fact, ‘gap’ and ‘glut’ are equivalent within these theories. The advantages of the internal theories over the external are stressed: primarily, internal theories make True(p) intersubstitutable with p, whereas external theories don't. At the same time, the internal Kripke theories are seriously lacking in expressive power.

*Bernhard Mühlherr, Holger P. Petersson, and Richard M. Weiss*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691166902
- eISBN:
- 9781400874019
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691166902.001.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This book begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. It then puts forward an algebraic solution into a geometric context by developing a ...
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This book begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. It then puts forward an algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or “form” of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a “residually pseudo-split” building. The book concludes with a display of the Tits indices associated with each of these exceptional forms. This is the third and final volume of a trilogy that began with The Structure of Spherical Buildings and The Structure of Affine Buildings.Less

This book begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. It then puts forward an algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or “form” of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a “residually pseudo-split” building. The book concludes with a display of the Tits indices associated with each of these exceptional forms. This is the third and final volume of a trilogy that began with *The Structure of Spherical Buildings* and *The Structure of Affine Buildings*.