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Full Idea
Our deductive practices seem to presuppose the Cut Law.
Gist of Idea
Normal deduction presupposes the Cut Law
Source
Ian Rumfitt (The Boundary Stones of Thought [2015], 2.3)
Book Ref
Rumfitt,Ian: 'The Boundary Stones of Thought' [OUP 2015], p.43
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.
| 11148 | Deduction is when we suppose one thing, and another necessarily follows [Aristotle] |
| 18277 | If q implies p, that is justified by q and p, not by some 'laws' of inference [Wittgenstein] |
| 13623 | The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem' [Bostock] |
| 13722 | A 'theorem' is an axiom, or the last line of a legitimate proof [Sider] |
| 22279 | Frege's sign |--- meant judgements, but the modern |- turnstile means inference, with intecedents [Potter] |
| 10752 | Γ |- S says S can be deduced from Γ; Γ |= S says a good model for Γ makes S true [Rossberg] |
| 18808 | Normal deduction presupposes the Cut Law [Rumfitt] |