# 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?

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– aedia λ Sep 13 '11 at 22:03
Also related (or neither one nor the other may be that related, or they may both be related... take your pick, just don't cut the red and green wires): How does negation affect the use and understanding of “or” and “and”, Should I use 'or' or 'nor'? – aedia λ Sep 13 '11 at 22:14
In Math, you have to be very specific. – user221287 May 23 '12 at 18:48
This question is the wrong way round - it should be why is the mathematical use incongruous with the common meaning ;) – curiousdannii Feb 20 at 13:32

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?

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You might also want to explore this a bit further: for those conjunctions that do appear to work on a 'lower' propositional level (e.g. English "cum"), how common are these and what possibilities exist among languages as a whole? (I don't know the answers to this: I'm just saying, it seems like an interesting direction to take your exploration in.) – Neil Coffey Sep 13 '11 at 22:08
@Neil, you're right, it might be an interesting exercise, although I'll leave it to someone else! – Codie CodeMonkey Sep 13 '11 at 22:13

"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 a 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?

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"Is that dog a Collie or a German shepherd?" - you can change that from an either/or question to a yes/no question by changing the intonation. Go down at the end and the answer should be "Collie", "German Shepherd" or "neither". Go up at the end and the answer should be "yes" or "no". – nnnnnn Sep 14 '11 at 13:27
“English is not a precise in its meaning as math.” — awesomely wonderful understatement. – Simon White Feb 20 at 12:55

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.

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+1. Short answer: Because they are different languages. – T.E.D. Sep 13 '11 at 21:57
I think something a little bit more complex than that is going on, closer to the issues that @DeepYellow is discussing. – Neil Coffey Sep 13 '11 at 22:05
Neil, DeepYellow said what I said, using more words. – JeffSahol Sep 14 '11 at 13:03

Let's have a look at your first statement in detail.

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.

In formal logic, the logical operator and only takes logical statements as parameters. Here, a logical statement means a statement that can be reduced to a truth value (either true or false).

Since neither nuts nor bolts are logical statements, this is not a good example for this exercise. Let's use a similar (almost identical) statement instead: "they want nuts and bolts". This can now be expanded (in English) to mean "they want nuts and they want bolts". This is perfectly understandable in a hardware shop context.

Since both "they want nuts" and "they want bolts" are logical statements, we can also evaluate "they want nuts and they want bolts" formally. If they want both, then both parts are true and so is the larger statement (this is the default understanding in the hardware store). If they really only wanted one or neither of them, then the larger statement is false. In the hardware store, the default response is something along the lines of make up your mind!

This is similar with the or operator. English has both inclusive-or and exclusive-or. Formal logic simply distinguishes between them formally.

(An example of inclusive-or in English: To sample the banquet, you must either be an invited guest or have contributed to the cause. Surely, a contributor would not be denied the banquet simply because he was also an invited guest - the or is therefore an inclusive-or.)

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

There isn't. Formal logic narrows the semantic range of the English terms.

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When people talk with each other they don't use mathematical terms defined by mathematicians according to their needs or terms of logic. They use "and" and "or" as they have used it since at least 2000 years as in Latin. A mathematical term as "exclusive or" was probably defined a while later.

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