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5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=

[fitting with the truth of some formulae]

15 ideas
Deduction is true when the premises facts necessarily make the conclusion fact true [Peirce]
X follows from sentences K iff every model of K also models X [Tarski]
Logical consequence is when in any model in which the premises are true, the conclusion is true [Tarski ,by Beall/Restall]
Logical consequence: true premises give true conclusions under all interpretations [Tarski ,by Hodges,W]
Validity is a conclusion following for premises, even if there is no proof [Bostock]
Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [Bostock]
It seems more natural to express |= as 'therefore', rather than 'entails' [Bostock]
|= in model-theory means 'logical consequence' - it holds in all models [Hodges,W]
Semantic consequence is ineffective in second-order logic [Shapiro]
If a logic is incomplete, its semantic consequence relation is not effective [Shapiro]
Γ |= φ for sentences if φ is true when all of Γ is true [Zalabardo]
Γ |= φ if φ is true when all of Γ is true, for all structures and interpretations [Zalabardo]
Logical consequence is either necessary truth preservation, or preservation based on interpretation [Beall/Restall]
A sentence follows from others if they always model it [Beall/Restall]
'Roses are red; therefore, roses are colored' seems truth-preserving, but not valid in a system [Koslicki]