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Single Idea 13347

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

Full Idea

The classical definition of validity counts an argument as valid if and only if the conclusion does in fact follow from the premises, whether or not the argument contains any demonstration of this fact.

Gist of Idea

Validity is a conclusion following for premises, even if there is no proof

Source

David Bostock (Intermediate Logic [1997], 1.2)

Book Ref

Bostock,David: 'Intermediate Logic' [OUP 1997], p.5

A Reaction

Hence validity is given by |= rather than by |-. A common example is 'it is red so it is coloured', which seems true but beyond proof. In the absence of formal proof, you wonder whether validity is merely a psychological notion.

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]