Book contents
- Frontmatter
- Contents
- Preface
- Part I Background
- 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
Appendix C - Component-style logical operators and relevance
Published online by Cambridge University Press: 05 May 2010
- Frontmatter
- Contents
- Preface
- Part I Background
- 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
Consider the logical operators when they are relativized to component implication. Let us call these operators component conjunctions, component hypotheticals, and so forth, when the implication relation is componential.
Component-style logical operators
Basically, the general picture of component-style logical operators is this: The logical operator Oc (the component-style version of the logical operator O) acts upon the elements A, B, C, …, all of which belong to the same component of the implication structure, and assigns those elements of that component that the operator O assigns – if it assigns anything at all. If A, B, C, … do not all belong to the same component, then there is no member of S that Oc assigns to them. In other words, the component-style logical operators stay in the component of the elements that it acts upon; they are not component-hopping. Later we shall see what the situation is for each of the operators when they act on elements of S that may not all be in the same component.
Component-style conjunctions
In our preliminary (unparameterized) description of the conjunction operator, C(A, B) was taken to be the weakest member of the structure that implied A as well as B.
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- Information
- A Structuralist Theory of Logic , pp. 382 - 387Publisher: Cambridge University PressPrint publication year: 1992