INDIRECT DERIVATIONS

We saw earlier in working with consistency trees that it leads to a contradiction to negate an actually valid conclusion and then conjoin it with true premises. We can make use of the same principle for a derivation by assuming a false conclusion and then deriving an obviously false statement.

PQC, QNRO, RN |- PN
1. PQC
2. QNRO
3. RN \ show PN
.....4. P...... hyp [it is often advisable to set off this "subordinate proof"]
.....5. Q....... 1,4 C elim
.....6. R....... 2,5 O elim
.....7. RRNA.. 3,6 A int
8. PN 4-7..... hyp elim

The key here is showing that you have "proved" something impossible (RRNA), which could happen with consistent premises only if the hypothesized statement PN is false.
Of course, we can use the same idea whenever the premises are inconsistent to begin with, even if the results might seem peculiar.
P, PN |- Q
1. P
2. PN \ show Q
3. QN....... hyp
4. PPNA... 1,2 A int
5. Q......... 3-4 hyp elim

Once we set up a hypothetical situation, we cannot end the derivation until we have eliminated it. For an indirect proof this happens once we have asserted the contradiction. All the interim steps are then taken out of play as well: we are not free to use anything established within a subordinate proof as though it followed solely from the original premises.

An indirect proof can be developed for absolutely any valid argument, but ordinarily such proofs are less efficient than either a direct derivation or a conditional proof, when these are possible.

Go ahead to conditional proofs.

Go back to the starting page.