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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.
Gist of Idea
A 'theorem' is an axiom, or the last line of a legitimate proof
Source
Theodore Sider (Logic for Philosophy [2010], 9.7)
Book Ref
Sider,Theodore: 'Logic for Philosophy' [OUP 2010], p.250
A Reaction
In other words, theorems are the axioms and their implications.
| 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] |