Book contents
- Frontmatter
- Contents
- Preface
- Part I Background
- Part II Implication relations
- Part III The logical operators
- 11 Hypotheticals
- 12 Negations
- 13 Conjunctions
- 14 The disjunction operator
- 15 The logical operators parameterized
- 16 Further features of the operators
- 17 The dual of negation: Classical and nonclassical implication structures
- 18 The distinctness and relative power of the logical operators
- 19 Extensionality
- 20 Quantification
- 21 Identity
- 22 Special structures I: Logical operators on individuals: Mereology reconstituted
- 23 Special structures II: Interrogatives and implication relations
- 24 Completeness
- Part IV The modal operators
- Appendix A An implication relation for the integers in the programming language BASIC
- Appendix B Symmetric sequents as products of implication relations and their duals
- Appendix C Component-style logical operators and relevance
- Notes
- Bibliography
- Index
17 - The dual of negation: Classical and nonclassical implication structures
Published online by Cambridge University Press: 05 May 2010
- Frontmatter
- Contents
- Preface
- Part I Background
- Part II Implication relations
- Part III The logical operators
- 11 Hypotheticals
- 12 Negations
- 13 Conjunctions
- 14 The disjunction operator
- 15 The logical operators parameterized
- 16 Further features of the operators
- 17 The dual of negation: Classical and nonclassical implication structures
- 18 The distinctness and relative power of the logical operators
- 19 Extensionality
- 20 Quantification
- 21 Identity
- 22 Special structures I: Logical operators on individuals: Mereology reconstituted
- 23 Special structures II: Interrogatives and implication relations
- 24 Completeness
- Part IV The modal operators
- Appendix A An implication relation for the integers in the programming language BASIC
- Appendix B Symmetric sequents as products of implication relations and their duals
- Appendix C Component-style logical operators and relevance
- Notes
- Bibliography
- Index
Summary
The dual of the negation operator on a structure is just the negation operator on the dual of that structure. This characterization of the dual of negation yields some simple results, as we noted in Chapter 12: If N̂ is the dual of N, then N̂N̂(A) ⇒ A for all A, but the converse does not hold in all structures; if A ⇒ B, then N̂(B) ⇒ N̂(A) in all structures, but not conversely; even if N(A) exists for some A, N̂(A) need not exist (if the structure is not classical); N̂ is a logical operator. These simple consequences are part of the story about dual negation, but only a part. There is an interesting story about this little-studied operator that deserves telling.
Theorem 17.1.Let I = 〈S, ⇒〉) be an implication structure for which disjunctions of its members always exist. Then D(A, N̂(A)) [that is, A ∨ N̂(A)] is a thesis of I for all A in S.
Proof. For every A in S, A, N̂(A) ⇒̂ B for all B in S, since N̂ is the negation operator on the dual of I. Since disjunctions always exist in the structure, it follows by Theorem 14.8 that B ⇒ D(A, N̂(A)) for all B in S. Consequently, D(A, N̂(A)) is a thesis of I for all A in S.
Since N and N̂ are logical operators on any structure, it is possible to compare N and N̂ with respect to their implication strengths.
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- Information
- A Structuralist Theory of Logic , pp. 142 - 150Publisher: Cambridge University PressPrint publication year: 1992