User:IssaRice/Computability and logic/Semantic completeness: Difference between revisions
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==Definition== | ==Definition== | ||
I want to make sure all these definitions are saying the same thing, so let me list some from several textbooks so I can explicitly compare. | |||
Smith's definition: a logic is semantically complete iff for any set of wffs <math>\Sigma</math> and any sentence <math>\phi</math>, if <math>\Sigma \models \phi</math> then <math>\Sigma\vdash\phi</math>.<ref>Peter Smith. An Introduction to Godel's Theorems. p. 33.</ref> | Smith's definition: a logic is semantically complete iff for any set of wffs <math>\Sigma</math> and any sentence <math>\phi</math>, if <math>\Sigma \models \phi</math> then <math>\Sigma\vdash\phi</math>.<ref>Peter Smith. An Introduction to Godel's Theorems. p. 33.</ref> | ||
Revision as of 05:41, 21 December 2018
Semantic completeness is sometimes written as: if , then .
Semantic completeness differs from negation completeness.
Definition
I want to make sure all these definitions are saying the same thing, so let me list some from several textbooks so I can explicitly compare.
Smith's definition: a logic is semantically complete iff for any set of wffs and any sentence , if then .[1]
Leary/Kristiansen's definition: A deductive system consisting of logical axioms and a collection of rules of inference is said to be complete iff for every set of nonlogical axioms and every -formula , if , then .[2]