Book contents
- Frontmatter
- Contents
- Preface
- Part I Judgments and Rules
- Part II Statics and Dynamics
- Part III Function Types
- Part IV Finite Data Types
- Part V Infinite Data Types
- Part VI Dynamic Types
- Part VII Variable Types
- Part VIII Subtyping
- Part IX Classes and Methods
- Part X Exceptions and Continuations
- Part XI Types and Propositions
- 30 Constructive Logic
- 31 Classical Logic
- Part XII Symbols
- Part XIII State
- Part XIV Laziness
- Part XV Parallelism
- Part XVI Concurrency
- Part XVII Modularity
- Part XVIII Equational Reasoning
- Part XIX Appendix
- Bibliography
- Index
31 - Classical Logic
from Part XI - Types and Propositions
Published online by Cambridge University Press: 05 February 2013
- Frontmatter
- Contents
- Preface
- Part I Judgments and Rules
- Part II Statics and Dynamics
- Part III Function Types
- Part IV Finite Data Types
- Part V Infinite Data Types
- Part VI Dynamic Types
- Part VII Variable Types
- Part VIII Subtyping
- Part IX Classes and Methods
- Part X Exceptions and Continuations
- Part XI Types and Propositions
- 30 Constructive Logic
- 31 Classical Logic
- Part XII Symbols
- Part XIII State
- Part XIV Laziness
- Part XV Parallelism
- Part XVI Concurrency
- Part XVII Modularity
- Part XVIII Equational Reasoning
- Part XIX Appendix
- Bibliography
- Index
Summary
In constructive logic a proposition is true exactly when it has a proof, a derivation of it from axioms and assumptions, and is false exactly when it has a refutation, a derivation of a contradiction from the assumption that it is true. Constructive logic is a logic of positive evidence. To affirm or deny a proposition requires a proof, either of the proposition itself or of a contradiction, under the assumption that it has a proof. We are not always in a position to affirm or deny a proposition. An open problem is one for which we have neither a proof nor a refutation-so that, constructively speaking, it is neither true nor false.
In contrast, classical logic (the one we learned in school) is a logic of perfect information in which every proposition is either true or false. We may say that classical logic corresponds to “god's view” of the world-there are no open problems; rather, all propositions are either true or false. Put another way, to assert that every proposition is either true or false is to weaken the notion of truth to encompass all that is not false, dually to the constructively (and classically) valid interpretation of falsity as all that is not true. The symmetry between truth and falsity is appealing, but there is a price to pay for this: The meanings of the logical connectives are weaker in the classical case than in the constructive.
- Type
- Chapter
- Information
- Practical Foundations for Programming Languages , pp. 249 - 260Publisher: Cambridge University PressPrint publication year: 2012