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Single Idea 13620

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

Full Idea

By repeated transformations using the Deduction Theorem, any proof from assumptions can be transformed into a fully conditionalized proof, which is then an axiomatic proof.

Gist of Idea

Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem

Source

David Bostock (Intermediate Logic [1997], 5.6)

Book Ref

Bostock,David: 'Intermediate Logic' [OUP 1997], p.223

A Reaction

Since proof using assumptions is perhaps the most standard proof system (e.g. used in Lemmon, for many years the standard book at Oxford University), the Deduction Theorem is crucial for giving it solid foundations.

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]