User:IssaRice/Computability and logic/Semantic completeness: Difference between revisions

Revision as of 05:29, 21 December 2018

Semantic completeness is sometimes written as: if $T ⊨ ϕ$ , then $T ⊢ ϕ$ .

Semantic completeness differs from negation completeness.

Definition

Smith's definition: a logic is semantically complete iff for any set of wffs $Σ$ and any sentence $ϕ$ , if $Σ ⊨ ϕ$ then $Σ ⊢ ϕ$ .^[1]

Leary/Kristiansen's definition: A deductive system consisting of logical axioms $Λ$ and a collection of rules of inference is said to be complete iff for every set of nonlogical axioms $Σ$ and every $L$ -formula $ϕ$ , if $Σ ⊨ ϕ$ , then $Σ ⊢ ϕ$ .^[2]

References

↑ Peter Smith. An Introduction to Godel's Theorems. p. 33.
↑ Leary; Kristiansen. A Friendly Introduction to Mathematical Logic. p. 74.

[1] Peter Smith. An Introduction to Godel's Theorems. p. 33.

[2] Leary; Kristiansen. A Friendly Introduction to Mathematical Logic. p. 74.

[1]

[2]

@@ Line 5: / Line 5: @@
 ==Definition==
-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>
+Leary/Kristiansen's definition: A deductive system consisting of logical axioms <math>\Lambda</math> and a collection of rules of inference is said to be complete iff for every set of nonlogical axioms <math>\Sigma</math> and every <math>\mathcal L</math>-formula <math>\phi</math>, if <math>\Sigma \models \phi</math>, then <math>\Sigma \vdash \phi</math>.<ref>Leary; Kristiansen. A Friendly Introduction to Mathematical Logic. p. 74.</ref>
 ==References==
 <references/>