Double Negative Arrows

[al_godnessmary]

So I went back and forth on this after reading the "double negative arrow" which I hadn't encountered anywhere else. Then I wondered:

Either A or B is chosen, but not both.

This means A can be chosen if B isn't, or B can be chosen if A isn't, or NEITHER can be chosen. Doesn't this mean that, technically in this situation, A and B are linked by a double negative arrow?

A B

For example, (page 196, explanation for "Either/Or...Drill" #6) "it is either feast or famine."

NO Feast Famine
NO Famine Feast

As stated in the explanation and through commonsense, you can't really have BOTH feast and famine...but what I'm thinking is, technically you can have a midpoint (unless of course for the sake of definition we're calling anything that is not a famine a feast). Which leads me to think that it could be:

Feast Famine

But the only reasonable way to consider this NOT to be the case (that is, the double negative would be INCORRECT for this conditional statement) is to accept that the situation of there being NEITHER feast NOR famine to be impossible.

But...is it?

(#2 of the same drill raises the same question. After much pondering, I tentatively concluded that perhaps that's what is alluded to in the answer key explanation that if this were an election where there can only be a single winner, "a second set of diagrams would apply." Would I be correct in thinking this second set of diagrams would be the double negative arrow? )

Thanks!

[Robert Carroll]

al,

The double-not arrow means that the two things related by that arrow cannot both be selected (or together, or whatever is indicated by the context).

"Either A or B is chosen, but not both" does mean that A B. This is because the second part of the statement does not allow both to be chosen. However, "neither is chosen" is excluded by the first part of the statement - "Either A or B is chosen" means you cannot lack both A and B - at least one must be chosen.

Thus, A B is true, but this double-not arrow is not a complete way of representing what the rule says, because it covers only the second half of the statement. The first half says you have to have at least one of them, so "A or B" is also true. The double-not arrow is true, but not complete. You need to supplement it with that "A or B" statement in order to capture both parts of this rule.

Note that a statement that "A or B is chosen" can also be represented as A B, because the double-not arrow between the negations of A and B means that you can't lack both of them. "A or B" is, I think, a much more clear way to state this than
A B.

In the feast or famine example you're talking about, of course you cannot have both feast and famine. This is a situation where the meanings of the words, and not any information the statement gives you, allows you to infer a double-not arrow: feast famine. However, the statement says that "it is either feast or famine." If we are representing what that statement says, the possibility that NEITHER happens has been excluded by the statement itself - normally, there are situations where there is neither feast nor famine, but the statement is telling us that things are not normal - either a feast situation or a famine situation exists.

What is normally true may be restricted by rules or by statements, and getting down how to represent those restrictions is important in learning when to use the double-not arrow, "either...or...", or conditionals.

If I was told nothing, I could say:

feast famine

That's because these are polar opposites, so they can't both be true, just because of the meanings of the words themselves. I don't need a Logic Games rule or a Logical Reasoning stimulus to tell me the double-not arrow relates these two.

If, however, I am told "it is either feast or famine," I know the double-not arrow, because that's something I know in any case, but the statement excludes the possibility of "it is NEITHER feast NOR famine." If this were stated in a situation where I trust the information (a Must Be True stimulus, or a Logic Games rule), then I am no longer at liberty to consider that "neither feast nor famine" may be true - the "either...or..." excludes that.

A statement that says "A or B is chosen, but not both" says something different from each of the following:

"A or B is chosen"

"A and B are not both chosen"

The original statement is the combination of the two, and needs to show that. The double-not arrow is only one side of the coin, as is the "either...or..."

I hope this helps!

Robert Carroll