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
11 - Relationships between the Modal Logics
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
Since there are so many different possible systems for modal logic, it is important to determine which system are equivalent, and which ones distinct from others. Figure 11.1 (on the next page) lays out these relationships for some of the best-known modal logics. It names systems by listing their axioms. So, for example, M4B is the system that results from adding (M), (4), and (B) to K. In boldface, we have also indicated traditional names of some systems, namely, S4, B, and S5. When system S appears below and/or to the left of S′ connected by a line, then S′ is an extension of S. This means that every argument provable in S is provable in S′, but S is weaker than S′, that is, not all arguments provable in S′ are provable in S.
Showing Systems Are Equivalent
One striking fact shown in Figure 11.1 is the large number of alternative ways of formulating S5. It is possible to prove these formulations are equivalent by proving the derivability of the official axioms of S5 (namely, (M) and (5)) in each of these systems and vice versa. However, there is an easier way. By the adequacy results given in Chapter 8 (or Chapter 9), we know that for each collection of axioms, there is a corresponding concept of validity. Adequacy guarantees that these notions of provability and validity correspond. So if we can show that two forms of validity are equivalent, then it will follow that the corresponding systems are equivalent. Let us illustrate with an example.
- Type
- Chapter
- Information
- Modal Logic for Philosophers , pp. 219 - 225Publisher: Cambridge University PressPrint publication year: 2013