Book contents
- Frontmatter
- Contents
- Preface
- Part I Background
- Part II Implication relations
- 5 The theory of implication relations
- 6 Implications: Variations and emendations
- 7 Familiar implication relations: Deducibility and logical consequence
- 8 Implication relations: Direct and derived
- 9 Implications from implications
- 10 Implication relations and the a priori: A further condition?
- 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
7 - Familiar implication relations: Deducibility and logical consequence
Published online by Cambridge University Press: 05 May 2010
- Frontmatter
- Contents
- Preface
- Part I Background
- Part II Implication relations
- 5 The theory of implication relations
- 6 Implications: Variations and emendations
- 7 Familiar implication relations: Deducibility and logical consequence
- 8 Implication relations: Direct and derived
- 9 Implications from implications
- 10 Implication relations and the a priori: A further condition?
- 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
It would be implausible to think of implication relations as not including the familiar notions of deducibility and logical consequence, and it would be misleading to think that implication relations were limited to cases of just these two types. A concept of (syntactic) deducibility is usually introduced over a well-defined set of sentences, according to which A1, …, An ⊢ B holds if and only if there is a sequence of sentences c1, …, Cm such that Cm is B, and every Ci is either an Aj (or an axiom if we are concerned with axiomatic formulations of the theory) or follows from one or more preceding sentences of the sequence by any member of a finite collection of rules R. It is easy to verify that “⊢” satisfies the condition for being an implication relation.
The semantic concept of consequence is also familiar. Starting with a well-defined set of sentences of, say, the propositional calculus, the notion of an interpretation is defined, and for any A1 …, An and B, we say that B is a logical consequence of A1 …, An that is, A1, …, An ╞ B if and only if every interpretation that assigns “true” to all the Ai's also assigns “true” to B. It is easy to see that “╞” is an implication relation. The details are familiar and need not be reviewed here.
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- Information
- A Structuralist Theory of Logic , pp. 42 - 44Publisher: Cambridge University PressPrint publication year: 1992