display all the ideas for this combination of texts
6 ideas
| 13623 | The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem' [Bostock] |
| Full Idea: The syntactic turnstile |- φ means 'There is a proof of φ' (in the system currently being considered). Another way of saying the same thing is 'φ is a theorem'. | |
| From: David Bostock (Intermediate Logic [1997], 5.1) |
| 13347 | Validity is a conclusion following for premises, even if there is no proof [Bostock] |
| 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. | |
| From: David Bostock (Intermediate Logic [1997], 1.2) | |
| 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. |
| 13348 | It seems more natural to express |= as 'therefore', rather than 'entails' [Bostock] |
| 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'. | |
| From: David Bostock (Intermediate Logic [1997], 1.3) | |
| 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 |-. |
| 13349 | Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [Bostock] |
| Full Idea: If we write Γ |= φ, with one formula to the right, then the turnstile abbreviates 'entails'. For a sequent of the form Γ |= it can be read as 'is inconsistent'. For |= φ we read it as 'valid'. | |
| From: David Bostock (Intermediate Logic [1997], 1.3) |
| 13614 | MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [Bostock] |
| Full Idea: The Rule of Detachment is a version of Modus Ponens, and says 'If |=φ and |=φ→ψ then |=ψ'. This has no assumptions. Modus Ponens is the more general rule that 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ'. | |
| From: David Bostock (Intermediate Logic [1997], 5.3) | |
| A reaction: Modus Ponens is actually designed for use in proof based on assumptions (which isn't always the case). In Detachment the formulae are just valid, without dependence on assumptions to support them. |
| 13617 | MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock] |
| Full Idea: Modus Ponens is equivalent to the converse of the Deduction Theorem, namely 'If Γ |- φ→ψ then Γ,φ|-ψ'. | |
| From: David Bostock (Intermediate Logic [1997], 5.3) | |
| A reaction: See 13615 for details of the Deduction Theorem. See 13614 for Modus Ponens. |