Logic 4

Assume, for the sake of making a comprehensible logic example, that all antelopes are brown.

It might not be obvious at first, but a formulation such as "If your pet is an antelope, then it is brown," can be exchanged for other formulations such as, "It is not the case that (your pet is an antelope and it is not brown,)" and, "Either it is not an antelope or it is brown."

A	B	A ⇒ B	￢(A & ￢B)	(￢A) V B
T	T	T	T	T
T	F	F	F	F
F	T	T	T	T
F	F	T	T	T

Sometimes people can get a little confused about whether two sentences like the ones just used for our example really mean the same thing. Making the truth table for each of them will tell whether they are true and/or false under the same conditions.

How about something like, "Either your brother is a professor at Cambridge University or he is a member of the Cambridge Board of Trustees." (Maybe somebody is making this judgment because he has seen the brother in a meeting that was closed to everybody but the faculty and the trustees of that university.) If it can somehow be shown that the brother is not a Cambridge professor and is not on their board of trustees, then the speaker who was making claims about "your brother" was wrong.

Sometimes this kind of substitution may be useful when somebody doesn't understand something you have said. Perhaps you said, "It's very dangerous around here. If I give you the ten thousand American dollars, then I will also give you my lucky four-leafed clover."  M ⇒ C. Later you decided not to give this person any money, but you also were concerned for his safety so you gave him your four-leafed clover anyway. Then he got angry, claiming that you had given him the lucky four-leaf clover but not the money. To which you could say, "What I meant was that I would not both give you the money and not give you the four-leaf clover. I didn't promise that I would not both give you the four-leaf clover and not give you the money."  ￢(M & ￢C)

C	M	C ⇒ M	(￢C) V M
T	T	T	T
T	F	F	F
F	T	T	T
F	F	T	T

Suppose I say, " If I give you $100 then I'll give you an extra $100"0.
The one case when I am being a liar is if I give you $100 but do not give you $1000.

Alternate ways of making the same promise:

"I will not give you $100 and/or I'll give you $1000."
Suppose I then do give you $100 and also give you $1000.
(The first part wasn't done according to the first half of my statemenet, but the second part is right, and that is enough to keep me from being a liar.)

"I will not give you $100 and/or I'll give you $1000."
Suppose I do give you $100 and that is all I give you.
 (You got $100 that I said I wouldn't give you, but not the $1000 that you were obligated to be given, so I broke my promise to you.) (That's F ⋁ F, and so the whole sentence is false.)

"I will not give you $100 and/or I'll give you $1000."
Suppose that I don't give you the $100, but I do give you the $1000.
(I told the truth on the first part and also the truth on the second part. So I'm not a liar.)

"I will not give you $100 and/or I'll give your $1000."
Suppose that I do not give you either the hundred dollars or the thousand dollars.
(I kept my promise on the first part, and that is enough to keep me from being a liar.)

Here is another case where using "and" might help. Often people confuse logical or "⋁" with the exclusive or (either x or else y) often used in everyday speech. So the logical or could be explained by finding the combination of "and" statements that creates the same truth table.

A	B	A ⋁ B	￢(￢A & ￢B)
T	T	T	T
T	F	T	T
F	T	T	T
F	F	F	F

If you claim that, e.g., "She is intelligent or she is beautiful," somebody might object, "You can't say that! Why can't somebody be both intelligent and beautiful." You could then explain, "All I meant was that it is not the case that she is both not intelligent and also not beautiful. She can be one or the other or both."

What if you claim that she is intelligent if and only if she is beautiful? It does make you a liar if it turns out that somebody is neither intelligent nor beautiful? No. You didn't say anything about people who weren't intelligent or weren't beautiful. "↔" symbolizes "if and only if."

		I ↔ B	￢(￢I & ￢B) V (I & B)	(I & B) V ((￢I) & (￢B))
T	T	T	t __ T__ t	t       T            f
T	F	F	f __ F__ f	f       F            f
F	T	F	f __ F__ f	f       F            f
F	F	T	f __ T__ t	f       T            t

In the table above I've used lower-case letters to keep track of the components that are part of a longer statement, and I've used upper-case letters to indicate the truth or falsity of the longer statements.
 
Suppose somebody says, "Either it is not the case that she is not intelligent and she is not beautiful, or else it is the case that she is both  intelligent and she is beautiful." ￢(￢I & ￢B) V (I & B)

If it turns out that she is not beautiful and still intelligent, or beautiful and yet not intelligent, then that does make the person who said that she has to be both or neither  into a liar, or at least somebody who doesn't know what he or she is talking about? It certainly makes the statement an empirical generalization that happens not to be true. The speaker might not be deliberately lying, but he or she is still telling an untruth.

(I & B) ⊻ ((￢I) & (￢B)) 
"⊻" is "exclusive or," which is better to use here since we can't claim something to have both some characteristic and to not have it, e.g., hot and not-hot, at the same time.

Suppose somebody says,  "She is intelligent and she is beautiful, or else she is both not intelligent and she is not beautiful." Is it clear that this is just another way of claiming that there are no mixed cases, e.g., nobody who is intelligent and not beautiful, or not intelligent and yet beautiful?

The way that humans work out their knowledge of the world is to propose hypotheses on the basis of whatever knowledge they may have been able to gather, and then to use those hypotheses to guide their behavior until such time as they discover evidence that disproves the hypothesis. Somebody who believes this business about beauty and intelligence always being found together will probably act on that belief and could work out the logical consequences based on that belief, but experience might show that the original belief was wrong. That's a good thing to learn because that is how human beings sort out bad ideas from good ideas. That is how humans have gone from having superstitious ideas or ideas that just sound like they have to be right to having ideas that reliably correspond to what they will find in the real world.

Once there was an old lady herbalist who could cure people of a certain kind of heart disease. The people of that community had a traditional belief, "If somebody is a witch, then that person can cure heart disease." They had another belief, "If  somebody is a witch, then he or she deserves to be burned at the stake." An important man in the community got heart disease. The regular doctors in the community could not treat him successfully. So he went to the old lady. She showed him how to brew tea from some dried flowers, told him how often to drink a cup of this tea, and sold him a bag of dried flowers. The big shot was cured. Then he said to the judge, "This old lady cured me of my heart disease. She is a witch! She must be burned at the stake." If you were her lawyer, how would you use logic to defend this lady herbalist?