Shantila's Inside Logic #18

"Neither .. Nor"

Suppose you are neither going to be on time for your sister's birthday nor are you going to miss the party altogether. Let T represent "You are on time for the party" and M represent "You miss the party altogether." Earlier we represented such sentences as ~T&~M, that is, you aren't going to be on time and you aren't going to miss it altogether.

We also can express the same idea by saying ~(TvM), that is, it is not true that either you will be on time or you will miss it altogether.

Here is another example. Neither Joe nor Sally will go with you. That is, ~J&~S. We can express the same thing by saying that it is not true that either Sally or Joe will go, ~(JvS).

Here's another example of this.. Neither the Yankees nor the Mets will win the World Series this year. That means the Yankees will not win and the Mets will not win, ~Y&~M. But equally it means that it is not the case that either the Yankees or the Mets will win, ~(YvM).

In general "neither P nor Q" can be represented as either ~P&~Q or as ~(PvQ). For this reason we might expect to be able to prove both of the following sequents.

114 ~P&~Q } ~(PvQ)

115 ~(PvQ) } ~P&~Q

In fact we can prove them! (See exercise 19.1 below.)

We also can prove

116 } ~(PvQ) <> (~P&~Q)

which is to say that the two sentences are logically equivalent. They do mean the same thing.

* Practice

18.1 Use our Rules to construct proofs for those two sequents.

114 ~P&~Q } ~(PvQ)

115 ~(PvQ) } ~P&~Q (Hint: to get going see your proof of sequent 113.)

116 } ~(PvQ) <> (~P&~Q) (Hint: Admittedly a bit redundant since you just proved 114 and 115.)

18.2 Symbolize each of these logical truths and construct derivations to show that they are valid.

(a) If not both Boston and Colorado will win, then Boston will not win if Colorado does. (Use these letters: B, C)

(b) If neither the Yankees nor the Mets will win, then the Yankees will not win. (Y, M) (Notice that one can interpet the "neither.. nor" in the antecedent in two different ways and that one can construct proofs for both interpretations. Please do so.)

(c) If the Yankees will not win and the Mets will not win, then it is not true that either the Mets or the Yankees will win.

18.3 Symbolize these arguments and construct derivations to show that they are valid. (from Hurley)

(a) Either animals are mere mechanisms or they feel pain. If either animals feel pain or they have souls, then they have a right not to be subjected to needless pain and humans have a duty to avoid inflicting needless pain on them. Animals are not mere mechanisms. Therefore humans have a duty to avoid inflicting needless pain on animals. (Use these letters: M, F, S, R, D)

(b) If TV viewers send money to evangelist Joe Schmoe and the money does not go to starving children in Africa, then either the money goes to starving children in Africa or Joe Schmoe will continue to live in opulence. TV viewers do send money to evangelist Joe Schmoe. Also the money does not go to starving children in Africa. So Joe Schmoe will continue to live in opulence. (T, A, O)

(c) If topaz is harder than quartz, then it will scratch quartz and also feldspar. Topaz is harder than quartz and it is also harder than calcite. So topaz will scratch quartz or it will scratch corundum. (T: Topaz is harder than quartz. Q: Topaz will scratch quartz. F: Topaz will scratch feldspar. C: Topaz is harder than calcite. O: Topaz will scratch corundum)