T5.6 6 of 6

5.6b

Strategy and Logical Equivalence

We have one final concept for which we need a deriations test:

Two sentences are logically equivalent if and only if it is not possible for one to be true and the other false.

We have seen in the exercises that this takes two derivations: For example, we showed that EQ was a derived rule by taking 'P>(Q>R)' as premise of one derivation and derived '(P&Q)>R'. Then we did things in reverse. So:

To prove two sentences P and Q logically equivalent by doing derivations, do two derivations. The first derivation takes P as its one premise and derives Q. The second derivation takes Q as its one premise and derives P. So, you do your first derivation in "one direction" then back the other way.

.pdf Wrap for 5.6 and some problems...

Notice that each of these problems are set up so that* both derivations can be done on one page*. When you finish with one direction, go to the word "new" and erase the question marks, give the second half any name or no name, then do the other direction.