So far we have learned to work with direct derivations, which cannot be used in all situations, and with indirect derivations, which can be used in all situations. Now we are going to work with another type of derivation that relies on setting up and then eliminating (discharging) a hypothesis. This is the conditional or hypothetical proof, which applies only when we are attempting to derive a conditional statement or a statement that could be rewritten as such.
Go to my page on conditional derivations and try the practice exercises that are there.
Conditional derivations are especially useful in establishing that certain wff's are tautologies (or theorems) , as in the example below for what is often listed as the hypothetical syllogism. You'll note there are no assumptions to start with. You'll also note that all we are doing in identifying the rules used is replacing the PLN signal with the corresponding connective.
|- ((P->Q) & (Q->R)) -> (P -> R)
1. (P->Q) & (Q->R)
Hyp we always begin by posing the antecedent
as true
2
P
Hyp we are doing the same thing again
with the next antecedent
3.
P->Q
1 & elim
4.
Q
2,3 -> elim
5.
Q->R
1 & elim
6.
R
4,5 -> elim
7. P->R
2-6 hyp elim
8. ((P->Q) & (Q->R)) -> (P -> R)
1-7 hyp elim
Working with conditional derivations is an
important part of doing efficient proofs. Special examples of how
to apply them come up in converting expressions with quantifiers to what
is called a prenex form. Look over my page on these prenex
forms and note how often we would use a conditional or hypothetical
proof even though we could also have used an indirect one (and note also
the times we have no choice about using an indirect proof).