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Single Idea 13348

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

Full Idea

In practice we avoid quotation marks and explicitly set-theoretic notation that explaining |= as 'entails' appears to demand. Hence it seems more natural to explain |= as simply representing the word 'therefore'.

Gist of Idea

It seems more natural to express |= as 'therefore', rather than 'entails'

Source

David Bostock (Intermediate Logic [1997], 1.3)

Book Ref

Bostock,David: 'Intermediate Logic' [OUP 1997], p.10

A Reaction

Not sure I quite understand that, but I have trained myself to say 'therefore' for the generic use of |=. In other consequences it seems better to read it as 'semantic consequence', to distinguish it from |-.

Related Idea

Idea 13623 The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem' [Bostock]

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]