Why is the common meaning of logical terms ('and', 'or') incongruous from that in math?

If someone wrote that they want "nuts and bolts", they would get a bunch of hardware they could attach things with. If this was software or math, they would only receive nothing, because things are (generally) nuts or bolts.

If someone asked for "vanilla or chocolate", they might be given one or the other; "exclusive or" in the math.

Why is there this mix-up of logical operators between normal language and math?

Solution 1:

In mathematics, the logical operators refer to propositions (or statements) regarding a particular object (which may be another statement or proposition). So you are asking for an object x with properties "x is a nut" AND "x is a bolt". Since I know of no such objects, the result is the empty set. Why is it done that way in mathematics? I suppose the answer is that it's useful to mathematicians and logicians in discussing logic.

In English however, and and or often refer to sets of things rather than a proposition about a particular thing. In mathematical language the customer wants a set A, such that A is the union of a non-empty subset of the set of bolts AND a non-empty subset of a set of nuts. Aren't you glad hardware store customers don't speak like that?

Solution 2:

"And" and "or" often have exactly the same meaning in English as they do in math, when they are used in the appropriate construct, however unlike math, they also have other uses and meanings.

For example:

Do you have a passport and an airplane ticket?

Clearly "and" is used the same as in math.

If you have a Red Carpet Club card or a first class ticket, you may board the plane first.

Here again, "or" is used in the same manner as in math. However, in English it is also used in other ways too, such as the examples you cite; English is not as precise in its meaning as math.

There is one particular ambiguity here worth mentioning.

Is that dog a Collie or a German Shepard?

Here there is a peculiar ambiguity. If the dog is a Labrador, the answer would be "no", but if it is a Collie, the answer would most likely be "a Collie"; to answer "yes" in this case would be considered pedantic. Math is, however, pedantic in the extreme.

This is odd because the question means something different depending on the answer. If it is a Labrador the question means "Does the breed of dog occur on this list?", if it is a Collie the question means "Please select the breed of that dog from the following list..."

Which is very odd, don't you think?

Solution 3:

The logical operators' names are borrowed from English, which has different meanings for these words than the precise meanings required in logic/mathematics. It is only a mix-up if you get confused about context, in other words.