T3.5: 9 of 9

Short-cut Tables

Finally, notice that the Logic Café will let you get away with short-cut truth tables! At least some of the time. When can you do a short-cut table? An incomplete table is acceptable exactly when it includes enough rows to prove your case.

For instance, to prove that a set of sentences is consistent, all you need is one row in which all sentences of the set are true. An example of this occurred in the exercise you just finished: the following table was enough for exercise 3!

3.5ex IV 3:
 A B C [A > B] , [B > C] , [A > C] T T T T T F T F T T F F F T T F T F F F T F F F

All you need to do here is the first row – that row proves that all three sentences can be true together. So, the set is logically consistent.

If you notice that you can fully prove your case with such a "short-cut" table, go ahead and submit it. If you are unsure, you may submit it to see if you've done enough, or (especially on an exam!) you may just go ahead and do the whole table. This latter will never hurt!

Now, think about testing for other properties. Will you ever be able to do a short-cut table to show and argument is valid? invalid? a pair of sentences are equivalent in SL? inequivalent in SL? Think about these questions by consulting the definitions for the terms involved – again, if a single row proves your point, then that's all you need to do.