Book contents
- Frontmatter
- Contents
- Preface to the Second Edition
- Preface
- Introduction: What Is Modal Logic?
- 1 The System K: A Foundation for Modal Logic
- 2 Extensions of K
- 3 Basic Concepts of Intensional Semantics
- 4 Trees for K
- 5 The Accessibility Relation
- 6 Trees for Extensions of K
- 7 Converting Trees to Proofs
- 8 Adequacy of Propositional Modal Logics
- 9 Completeness Using Canonical Models
- 10 Axioms and Their Corresponding Conditions on R
- 11 Relationships between the Modal Logics
- 12 Systems for Quantified Modal Logic
- 13 Semantics for Quantified Modal Logics
- 14 Trees for Quantified Modal Logic
- 15 The Adequacy of Quantified Modal Logics
- 16 Completeness of Quantified Modal Logics Using Trees
- 17 Completeness Using Canonical Models
- 18 Descriptions
- 19 Lambda Abstraction
- 20 Conditionals
- Answers to Selected Exercises
- Bibliography of Works Cited
- Index
16 - Completeness of Quantified Modal Logics Using Trees
Published online by Cambridge University Press: 05 June 2014
- Frontmatter
- Contents
- Preface to the Second Edition
- Preface
- Introduction: What Is Modal Logic?
- 1 The System K: A Foundation for Modal Logic
- 2 Extensions of K
- 3 Basic Concepts of Intensional Semantics
- 4 Trees for K
- 5 The Accessibility Relation
- 6 Trees for Extensions of K
- 7 Converting Trees to Proofs
- 8 Adequacy of Propositional Modal Logics
- 9 Completeness Using Canonical Models
- 10 Axioms and Their Corresponding Conditions on R
- 11 Relationships between the Modal Logics
- 12 Systems for Quantified Modal Logic
- 13 Semantics for Quantified Modal Logics
- 14 Trees for Quantified Modal Logic
- 15 The Adequacy of Quantified Modal Logics
- 16 Completeness of Quantified Modal Logics Using Trees
- 17 Completeness Using Canonical Models
- 18 Descriptions
- 19 Lambda Abstraction
- 20 Conditionals
- Answers to Selected Exercises
- Bibliography of Works Cited
- Index
Summary
The completeness of quantified modal logics can be shown with the tree method by modifying the strategy used in propositional modal logic. Section 8.4 explains how to use trees to demonstrate the completeness of propositional modal logics S that result from adding one or more of the following axioms to K: (D), (M), (4), (B), (5), (CD). In this chapter, the tree method will be extended to quantified modal logics based on the same propositional modal logics. The reader may want to review Sections 8.3 and 8.4 now, since details there will be central to this discussion. The fundamental idea is to show that every S-valid argument is provable in S in two stages. Assuming that H / C is S-valid, use the Tree Model Theorem (of Section 8.3) to prove that the S-tree for H / C closes. Then use the method for converting closed S-trees into proofs to construct a proof in S of H / C from the closed S-tree. This will show that any S-valid argument has a proof in S, which is, of course, what the completeness of S amounts to.
The Quantified Tree Model Theorem
In order to demonstrate completeness for quantified modal logics, a quantified version of the Tree Model Theorem will be developed here. This will also be useful in showing the correctness of trees for the quantified systems. Proofs of the appropriate tree model theorems are complicated by the fact that there are so many different systems to be considered in quantified modal logic. Several different interpretations of the quantifiers have been developed. However, the transfer theorems proven in Sections 15.7 and 15.8 show how results for all these systems can be obtained from a proof of completeness for truth value models (tS-models). So a demonstration of the Tree Model Theorem will be given for tS-validity.
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- Modal Logic for Philosophers , pp. 354 - 362Publisher: Cambridge University PressPrint publication year: 2013