Philosophy Index

Deduction theorem

Deduction theorem is a basic rule of inference in logic by which a conditional is formed out of an implication. The deduction theorem is also known as the conditional introduction rule of inference (or →I), and explains that if "P implies Q", then "if P then Q" is also true.

Addition may be formally presented as follows:

infers β) infers α → β

Creating a conditional

When constructing a proof of some formula, one may have to invoke the deduction theorem by means of an assumption. To do this, an assumption for conditional proof (ACP) α is created to show that if α is true, then β is true.

RTP α → β
  α ACP
RTP Contradiction
  β
  α → β →I

In this setup, the subproof (by which α is assumed and β is demonstrated as following from α) satisfies the implication, which can be used to form a conditional.