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Full Idea
Like the Deduction Theorem, one form of Reductio ad Absurdum (If Γ,φ|-[absurdity] then Γ|-¬φ) 'discharges' an assumption. Assume φ and obtain a contradiction, then we know ¬&phi, without assuming φ.
Gist of Idea
The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth
Source
David Bostock (Intermediate Logic [1997], 5.7)
Book Ref
Bostock,David: 'Intermediate Logic' [OUP 1997], p.227
A Reaction
Thus proofs from assumption either arrive at conditional truths, or at truths that are true irrespective of what was initially assumed.
13615 | 'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ [Bostock] |
13616 | The Deduction Theorem greatly simplifies the search for proof [Bostock] |
13620 | Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem [Bostock] |
13621 | The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth [Bostock] |
13690 | Proof by induction 'on the length of the formula' deconstructs a formula into its accepted atoms [Sider] |
13691 | Induction has a 'base case', then an 'inductive hypothesis', and then the 'inductive step' [Sider] |
10703 | Supposing axioms (rather than accepting them) give truths, but they are conditional [Potter] |