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
16 - Further features of the operators
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
Theorem 16.1 (expansion).If I = 〈S, ⇒〉 is an implication structure, and A and D are any elements of S, then H(A, B) ⇒ H(C(A, D), B) [that is, (A, → B) ⇒ ((A & D) → B)].
Proof. C(A, D), H(A, B)⇒A, since C(A, D) ⇒ A. By Projection, C(A, D), H(A, B) ⇒ H(A, B). Therefore C(A, D), H(A, B) ⇒ B. Consequently, H(A, B) ⇒ H(C(A, D), B).
Theorem 16.2 (importation).If I = 〈S, ⇒〉 is an implication structure, and A, B, and D are any members of S, then H(A, H(B, D)) ⇒ H(C(A, B), D) [that is, (A → (B → D)) ⇒ ((A & B) → D)].
Proof. C(A, B), H(A, H(B, D)) ⇒ H(B, D), since C(A, B), H(A, H(B, D)) ⇒ A, as well as H(A, H(B, D)). Moreover, since C(A, B) ⇒ B, C(A, B), H(A, H(B, D)) ⇒ D. Consequently, by H2, H(A, H(B, D)) ⇒ H(C(A, B), D).
Theorem 16.3.If I = 〈S, ⇒,〉 is an implication structure, and A and B are any members of S, then D(N(A), B) ⇒ H(A, B) [that is, (¬A ∨ B) ⇒ (A → B)].
Proof. B ⇒ H(A, B), and N(A) ⇒ H(A, B). Consequently, by D1D(N(A), B) ⇒ H(A, B)].
Theorem 16.4.It is not true that in every implication structure, for all A and B, H(A, B) ⇒ D(N(A), B) [that is, (A → B) ⇒ (¬A ∨ B)].
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- Information
- A Structuralist Theory of Logic , pp. 134 - 141Publisher: Cambridge University PressPrint publication year: 1992