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

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

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]

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	Semantic completeness is sometimes written as: if <math>T \models \phi</math>, then <math>T \vdash \phi</math>.		Semantic completeness is sometimes written as: if <math>\Sigma \models \phi</math>, then <math>\Sigma \vdash \phi</math>.

	Semantic completeness differs from [[../Negation completeness\|negation completeness]].		Semantic completeness differs from [[../Negation completeness\|negation completeness]].