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

[following from a formula in proof-theory]

7 ideas
Deduction is when we suppose one thing, and another necessarily follows [Aristotle]
     Full Idea: A deduction is a discourse in which, certain things having been supposed, something different from the things supposed results of necessity because these things are so.
     From: Aristotle (Prior Analytics [c.328 BCE], 24b18)
     A reaction: Notice that it is modal ('suppose', rather than 'know'), that necessity is involved, which is presumably metaphysical necessity, and that there are assumptions about what would be true, and not just what follows from what.
If q implies p, that is justified by q and p, not by some 'laws' of inference [Wittgenstein]
     Full Idea: If p follows from q, I can make an inference from q to p, deduce p from q. The nature of the inference can be gathered only from the two propositions. They are the only possible justification of the inference. 'Laws of Inference' would be superfluous.
     From: Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921], 5.132)
     A reaction: That seems to imply that each inference is judged on its particulars. But logic aims to be general. There seem to be 'laws' at a more complex level in the logic.
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)
A 'theorem' is an axiom, or the last line of a legitimate proof [Sider]
     Full Idea: A 'theorem' is defined as the last line of a proof in which each line is either an axiom or follows from earlier lines by a rule.
     From: Theodore Sider (Logic for Philosophy [2010], 9.7)
     A reaction: In other words, theorems are the axioms and their implications.
Frege's sign |--- meant judgements, but the modern |- turnstile means inference, with intecedents [Potter]
     Full Idea: Natural deduction systems generally depend on conditional proof, but for Frege everything is asserted unconditionally. The modern turnstile |- is allowed to have antecedents, and hence to represent inference rather than Frege's judgement sign |---.
     From: Michael Potter (The Rise of Analytic Philosophy 1879-1930 [2020], 03 'Axioms')
     A reaction: [compressed] Shockingly, Frege's approach seems more psychological than the modern approach. I would say that the whole point of logic is that it has to be conditional, because the truth of the antecedents is irrelevant.
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.
Normal deduction presupposes the Cut Law [Rumfitt]
     Full Idea: Our deductive practices seem to presuppose the Cut Law.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 2.3)
     A reaction: That is, if you don't believe that deductions can be transitive (and thus form a successful chain of implications), then you don't really believe in deduction. It remains a well known fact that you can live without the Cut Law.