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
9 - Implications from implications
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
There are examples of implication relations that are not quite “from scratch.” One kind of example uses the particular structure that the elements have in order to define an implication relation over them. Thus, there are implication relations that can be defined over sets, taking into account that it is sets over which the relation is to be defined. Another example uses theories or Tarskian systems in order to define an implication relation. A third example studies certain implication relations over individuals (or their names) and uses the fact about these individuals that they have parts that can enter into a whole-part relation. A fourth example defines an implication relation over interrogatives, exploiting the fact that these interrogatives are of a type that have “direct answers” (Chapter 23), and there is an interesting implication relation that can be defined for the integers when they are encoded in binary notation in the programming language called BASIC (see Appendix A). These examples are very different from the simple bisection relations. The latter are genuinely “topic-neutral” in that they are not sensitive to whatever structure the elements may have. These examples exploit the particular nature of the elements in the implication structure.
The second kind of example we shall describe involves the construction of implications on a set, by using an implication that is already in place. The most theoretically interesting of this type is the notion of a dual implication relation, although a second example, component implication, has some theoretical interest as well.
- Type
- Chapter
- Information
- A Structuralist Theory of Logic , pp. 53 - 69Publisher: Cambridge University PressPrint publication year: 1992