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Single Idea 13621

[filed under theme 5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions ]

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.

The 7 ideas with the same theme [proofs which add assumptions to axioms and rules]:

13616 The Deduction Theorem greatly simplifies the search for proof [Bostock]
13615 'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ [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]