5.3.s. Summary

1

The truth conditions of the conditional recall the definition of implication. Indeed, an implication φ ⇒ ψ will hold if and only if the conditional φ → ψ is a tautology. We can apply similar ideas to conditionals that are conclusions from factual premises by considering a notion of relative implication, implication depending on factual information. This idea appears in our law for the conditional as a conclusion. An entailment Γ ⇒ φ → ψ holds when Γ, φ ⇒ ψ—i.e., when ψ is implied by φ given the further premises Γ. The first of these entailments is a conditionalization of the second, and the second asserts the validity of a hypothetical argument. So an argument with a conditional conclusion is valid if and only if the hypothetical argument it conditionalizes is also valid. The derivation rule implementing this idea is Conditional Proof (CP).

2

The detachment principles for the conditional include the well-known modus ponendo ponens (usually called modus ponens), which is implemented as a rule Modus Ponendo Ponens (MPP), and a second detachment principle modus tollendo tollens (usually called modus tollens), which is implemented as a rule Modus Tollendo Tollens (MTT). Modus ponens in particular can be understood as the use of a conditional as an inference ticket licensing transitions from its antecedent to its consequent.

Glen Helman 28 Aug 2008