Book contents
- Frontmatter
- Contents
- Preface
- Part I Background
- 1 Introduction
- 2 The program and its roots
- 3 Introduction and Elimination conditions in a general setting
- 4 The Belnap program
- Part II Implication relations
- Part III The logical operators
- 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
3 - Introduction and Elimination conditions in a general setting
Published online by Cambridge University Press: 05 May 2010
- Frontmatter
- Contents
- Preface
- Part I Background
- 1 Introduction
- 2 The program and its roots
- 3 Introduction and Elimination conditions in a general setting
- 4 The Belnap program
- Part II Implication relations
- Part III The logical operators
- 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
Although the particular Introduction and Elimination rules for each connective are easily understood, the overall characterization is not easy to comprehend, nor is the general connection that each has to the other. Gentzen thought (1) that the Introduction rules provided a “definition” of the connectives, (2) that they also provided a sense of those connectives, (3) that it should be possible to display the E-inferences as unique functions of their corresponding I-inferences, and (4) that “in the final analysis” the Elimination rules were consequences of the corresponding Introduction rules. What he said about the character of each type of rule and their connections with one another is very much worth quoting in full:
To every logical symbol &, ∨, ∀, ∃, ⊃, ¬, belongs precisely one inference figure which “introduces” the symbol – as the terminal symbol of a formula – and one which ‘eliminates’ it. The fact that the inference figures &-E and ∨-I each have two forms constitutes a trivial, purely external deviation and is of no interest. The introductions represent, as it were, the ‘definitions’ of the symbols concerned, and the eliminations are no more, in the final analysis, than the consequences of these definitions. This fact may be expressed as follows: in eliminating a symbol we are dealing ‘only in the sense afforded it by the introduction of that symbol’. […]
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- A Structuralist Theory of Logic , pp. 19 - 25Publisher: Cambridge University PressPrint publication year: 1992