Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- 1 GL and other systems of propositional modal logic
- 2 Peano arithmetic
- 3 The box as Bew(x)
- 4 Semantics for GL and other modal logics
- 5 Completeness and decidability of GL and K, K4, T, B, S4, and S5
- 6 Canonical models
- 7 On GL
- 8 The fixed point theorem
- 9 The arithmetical completeness theorems for GL and GLS
- 10 Trees for GL
- 11 An incomplete system of modal logic
- 12 An S4-preserving proof-theoretical treatment of modality
- 13 Modal logic within set theory
- 14 Modal logic within analysis
- 15 The joint provability logic of consistency and ω-consistency
- 16 On GLB: The fixed point theorem, letterless sentences, and analysis
- 17 Quantified provability logic
- 18 Quantified provability logic with one one-place predicate letter
- Notes
- Bibliography
- Index
- Notation and symbols
7 - On GL
Published online by Cambridge University Press: 08 January 2010
- Frontmatter
- Contents
- Preface
- Introduction
- 1 GL and other systems of propositional modal logic
- 2 Peano arithmetic
- 3 The box as Bew(x)
- 4 Semantics for GL and other modal logics
- 5 Completeness and decidability of GL and K, K4, T, B, S4, and S5
- 6 Canonical models
- 7 On GL
- 8 The fixed point theorem
- 9 The arithmetical completeness theorems for GL and GLS
- 10 Trees for GL
- 11 An incomplete system of modal logic
- 12 An S4-preserving proof-theoretical treatment of modality
- 13 Modal logic within set theory
- 14 Modal logic within analysis
- 15 The joint provability logic of consistency and ω-consistency
- 16 On GLB: The fixed point theorem, letterless sentences, and analysis
- 17 Quantified provability logic
- 18 Quantified provability logic with one one-place predicate letter
- Notes
- Bibliography
- Index
- Notation and symbols
Summary
We here present a number of results about the system GL. Some of these will be of direct interest for the study of provability in PA; others are simply independently interesting (we hope), and these occur toward the end of the chapter. The discussion here of letterless sentences and the notions of rank and trace will be particularly important in the next chapter, where we take up the fixed point theorem, certainly one of the most striking applications of modal logic ever made.
We begin with one of the oldest results of the subject of provability logic, the normal form theorem for letterless sentences. Recall that a modal sentence is called letterless if it contains no sentence letters, equivalently if it is a member of the smallest class containing ⊥ and containing (A→B) and □A whenever it contains A and B.
As ever, □0A = A and □i+1A = □□iA.
We shall say that a letterless sentence C is in normal form if it is a truth-functional combination of sentences of the form □i⊥.
The normal form theorem for letterless sentences
If B is a letterless sentence, there is a letterless sentence C in normal form such that GL⊢B↔C.
Proof. It clearly suffices to show how to construct a letterless sentence in normal form equivalent to □C from a letterless sentence C in normal form.
- Type
- Chapter
- Information
- The Logic of Provability , pp. 92 - 103Publisher: Cambridge University PressPrint publication year: 1994