Book contents
- Frontmatter
- Contents
- 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 Relations 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
- Answers to Selected Exercises
- Bibliography of Works Cited
- Index
13 - Semantics for Quantified Modal Logics
Published online by Cambridge University Press: 09 January 2010
- Frontmatter
- Contents
- 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 Relations 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
- Answers to Selected Exercises
- Bibliography of Works Cited
- Index
Summary
There are a number of different approaches one can take to giving the semantics for the quantifiers. The simplest method uses truth value semantics with the substitution interpretation of the quantifiers (Leblanc, 1976). Although the substitution interpretation can be criticized, it provides an excellent starting point for understanding the alternatives since it avoids a number of annoying technical complications. For students who prefer to learn the adequacy proofs in easy stages, it is best to master the reasoning for the substitution interpretation first. This will provide a core understanding of the basic strategies, which may be embellished (if one wishes) to accommodate more complex treatments of quantification.
Truth Value Semantics with the Substitution Interpretation
The substitution interpretation is based on the idea that a universal sentence ∀xAx is true exactly when each of its instances Aa, Ab, Ac, ‥, is true. For classical logic, ∀xAx is T if and only if Ac is T for each constant c of the language. In the case of free logic, the truth condition states that ∀xAx is T if and only if Ac is T for all constants that refer to a real object. Since the sentence Ec indicates that c refers to a real object, the free logic truth condition should say that Ac is T for all those constants c such that Ec is also true.
Semantics for quantified modal logic can be defined by incorporating these ideas into the definition of a model for propositional modal logic.
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- Chapter
- Information
- Modal Logic for Philosophers , pp. 265 - 302Publisher: Cambridge University PressPrint publication year: 2006