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Single Idea 10694

[filed under theme 5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |= ]

Full Idea

Tarski's 1936 definition of logical consequence is that in any model in which the premises are true, the conclusion is true too (so that no model can make the conclusion false).

Gist of Idea

Logical consequence is when in any model in which the premises are true, the conclusion is true

Source

report of Alfred Tarski (works [1936]) by JC Beall / G Restall - Logical Consequence 3

Book Ref

'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.6

A Reaction

So the general idea is that a logical consequence is distinguished by being unstoppable. Sounds good. But then we have monotonic and non-monotonic logics, which (I'm guessing) embody different notions of consequence.

The 16 ideas with the same theme [fitting with the truth of some formulae]:

19237 Deduction is true when the premises facts necessarily make the conclusion fact true [Peirce]
13344 X follows from sentences K iff every model of K also models X [Tarski]
10694 Logical consequence is when in any model in which the premises are true, the conclusion is true [Tarski, by Beall/Restall]
10479 Logical consequence: true premises give true conclusions under all interpretations [Tarski, by Hodges,W]
13347 Validity is a conclusion following for premises, even if there is no proof [Bostock]
13348 It seems more natural to express |= as 'therefore', rather than 'entails' [Bostock]
13349 Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [Bostock]
10477 |= in model-theory means 'logical consequence' - it holds in all models [Hodges,W]
13626 Semantic consequence is ineffective in second-order logic [Shapiro]
13637 If a logic is incomplete, its semantic consequence relation is not effective [Shapiro]
10893 Γ |= φ for sentences if φ is true when all of Γ is true [Zalabardo]
10899 Γ |= φ if φ is true when all of Γ is true, for all structures and interpretations [Zalabardo]
21611 Formal semantics defines validity as truth preserved in every model [Williamson]
10695 Logical consequence is either necessary truth preservation, or preservation based on interpretation [Beall/Restall]
13240 A sentence follows from others if they always model it [Beall/Restall]
14506 'Roses are red; therefore, roses are colored' seems truth-preserving, but not valid in a system [Koslicki]