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Single Idea 13691

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

Full Idea

A proof by induction starts with a 'base case', usually that an atomic formula has some property. It then assumes an 'inductive hypothesis', that the property is true up to a certain case. The 'inductive step' then says it will be true for the next case.

Gist of Idea

Induction has a 'base case', then an 'inductive hypothesis', and then the 'inductive step'

Source

Theodore Sider (Logic for Philosophy [2010], 2.7)

Book Ref

Sider,Theodore: 'Logic for Philosophy' [OUP 2010], p.53

A Reaction

[compressed]

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]