In a previous post we mentioned the ambiguity of or in day-to-day speech. There is another construction that also exhibits this kind of ambiguity, but while the dual meaning of or is legendary and well documented, the one I’m writing about now is far less known. As a matter of fact, I have never seen it mentioned in any of the many logic books I’ve read.
The language spoken in Classical Logic Land doesn’t have any ambiguities in it. It is forbidden by law. The connective or has got only the inclusive meaning, and if an exclusive meaning is intended, a different word has to be used, for example xor. So, in this peculiar country the menus in restaurants read: “with the meal you can have a drink xor dessert” (the pronunciation of xor is open to debate; some people propose using one of the click consonants found in Bantu languages for it).
One of the connectives shared with everyday languages is called implication, notated with an arrow (A → B) and pronounced thus: If A then B. The implication is considered to be false when A is true and B false, and true in the other cases.
Among the rigid laws of this country there is one that says that the implication A → B is equivalent to (not A) or B. Let’s think a bit about it: for this to be false (not A) has to be false (i.e. A has to be true) and B has to be false, just what we said it was needed for the implication to be false.
Now, what happens if we replace the or by xor and perform the transformation in the other direction? We get a construction that in logicians lingo is known as if and only if, usually notated with a two pointed arrow (A ↔ B) and sometimes abbreviated as A iff B.
Ok, stop it! This is so unbelievably booooring! Yes, indeed. Lucky that I had just finished with our instructive journey through Logic Land, and it is safe now to go back to good old plain English. And where is the ambiguity you promised us at the beginning of the article? Well, I claim that sometimes when people hears if this then that they interpret it like: this iff that. Typical example: I’m standing with an umbrella, and say to my friend: if it rains then I take my umbrella with me, and my friend replies but it’s not raining. No, it’s not, but I never said that I take my umbrella if and only if it rains.
Q.E.D. (quod erat demonstrandum).