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Ideas for 'That Politics may be reduced to a Science', 'First-order Logic, 2nd-order, Completeness' and 'fragments/reports'

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2 ideas

5. Theory of Logic / B. Logical Consequence / 2. Types of Consequence

10753	Logical consequence is intuitively semantic, and captured by model theory [Rossberg]
	Full Idea: Logical consequence is intuitively taken to be a semantic notion, ...and it is therefore the formal semantics, i.e. the model theory, that captures logical consequence.
	From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2)
	A reaction: If you come at the issue from normal speech, this seems right, but if you start thinking about the necessity of logical consequence, that formal rules and proof-theory seem to be the foundation.

5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-

10752	Γ |- S says S can be deduced from Γ; Γ |= S says a good model for Γ makes S true [Rossberg]
	Full Idea: Deductive consequence, written Γ|-S, is loosely read as 'the sentence S can be deduced from the sentences Γ', and semantic consequence Γ|=S says 'all models that make Γ true make S true as well'.
	From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §2)
	A reaction: We might read |= as 'true in the same model as'. What is the relation, though, between the LHS and the RHS? They seem to be mutually related to some model, but not directly to one another.