Concept
Completeness (logic)
Overview
In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete.
The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantic validity. Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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In Kantian language this is called satisfying equally the law of homogeneity and that of specification. It arises from this peculiar nature of scientific completeness, that the aim of science is not greater certainty—for certainty may be possessed in just as high a degree by the most disconnected particular knowledge—but its aim is rather the facilitating of rational knowledge by means of its form, and the…
Arthur Schopenhauer, The World as Will and Idea
States what Completeness is; names Completeness several times.
Such a statement demands accuracy. Completeness is not so much an additional requirement as it is a condition of accuracy. For accuracy depends fundamentally upon relevancy to the determination of what is to be done. Completeness does not mean exhaustiveness _per se_, but adequacy as respects end and its means.
John Dewey, Essays in Experimental Logic
States what Completeness is; names Completeness twice.
But this completeness can be a completeness of principles only, not of intuitions (i.e., concrete atsights or Anschauungen) and of objects. In order however to represent the ideas definitely, reason conceives them after the fashion of the cognition of an object. The cognition is as far as these rules are concerned completely determined, but the object is only an idea invented for the purpose of bringing the…
Immanuel Kant, Kant's Prolegomena to Any Future Metaphysics
Names Completeness several times.
Philosophers who discussed Completeness (logic)
Related ideas
- Formal language
- First-order logic
- Gödel's completeness theorem
- Negation
- Logical connective
- Propositional logic
- Mathematical logic
- Sentence (mathematical logic)
- Validity (logic)
- Peano axioms
- Soundness
- Theorem
- Modal logic
- Parallel postulate
- Metalogic
- Formal system
- Well-formed formula
- Tautology (logic)
- Property (philosophy)
- Peter Norvig
- Artificial Intelligence: A Modern Approach
- Stuart J. Russell
- Gödel's incompleteness theorems
- Consistency
Place in the tradition
Part of First-order logic.