# Not both v Both not, again

Mr. Nance,

In Copi’s 14th edition of Introduction to Logic, one problem reads, “Iran and Libya both do not raise the price of oil.” The symbolic translation is ~I • ~L. I thought it might also be translated as ~(I • L). However, using a truth table to check for equivalence, I found the two are NOT equivalent.

Later in the exercise there is a problem that reads, “Either Iran raises the price of oil and Egypt’s food shortage worsens, or it is not the case both that Jordan requests more U.S. aid and that Saudi Arabia buys five hundred more warplanes.” The symbolic translation is (I • E) ∨ ~(J • S). I’m confused by reading “…it is not the case both that Jordan requests more U.S. aid and that Saudi Arabia buys five hundred more warplanes” as ~(J • S). That seems a lot like saying “It is not the case both that Iran and Libya do not raise the price of oil,” which I thought might be translated ~(I • L).

Can you explain how to read this correctly? That is, why are they not logically equivalent? Or did I just mess up royally?

Thanks so much.

You are correct in saying that ~(p • q) is not equivalent to ~p • ~q. How then do we determine the correct form for statements that use “both” and “not”?

Fundamentally, we must use the forms that reflect the meaning of the statements. The form ~(p • q) means “not both p and q”, as in “Tom and Jim are not both from Idaho.” The form ~p • ~q means “both not p and not q” which is equivalent to “neither p nor q”, as in “Tom and Jim are both not from China.”

Practically, the first thing to ask when symbolizing statements like this is, “Which comes first in the statement, the ‘not’ or the ‘both’?” If it is ‘not both’ then it is probably the form ~(p • q). If it is ‘both not’ then is is probably the form ~p • ~q. Let’s apply this to the statements in question.

1. “Iran and Libya both do not raise the price of oil.” This is correctly symbolized ~I • ~L. The meaning is that neither Iran nor Libya raise the price of oil.
2. “It is not the case both that Jordan requests more U.S. aid and that Saudi Arabia buys five hundred more warplanes.” This is correctly symbolized ~(J • S).

You have too many nots in your second to last paragraph, which is confusing the issue. But I trust that my explanation clears things up.

For more on this issue, read this EARLIER POST.

Blessings!

## One thought on “Not both v Both not, again”

1. Anthony Manda says:

I’m working on proving AAA-3 is not valid so I used meat-eaters for major term, predators as minor term and lions as middle term. It appears to be valid. But my book says it should be invalid.