# 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?

• Commented Sep 13, 2011 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'? Commented Sep 13, 2011 at 22:14
• In Math, you have to be very specific.
– user19917
Commented May 23, 2012 at 18:48
• This question is the wrong way round - it should be why is the mathematical use incongruous with the common meaning ;) Commented Feb 20, 2016 at 13:32
• Don't forget XOR and NOR. In contracts, person ''A or B" on a contract means A must sign off, and B must also sign off. To be quite specific, the term "A and or B" is sometimes used. Commented Aug 16, 2019 at 15:19

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?

• 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.) Commented Sep 13, 2011 at 22:08
• @Neil, you're right, it might be an interesting exercise, although I'll leave it to someone else! Commented Sep 13, 2011 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 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?

• "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". Commented Sep 14, 2011 at 13:27
• “English is not a precise in its meaning as math.” — awesomely wonderful understatement. Commented Feb 20, 2016 at 12:55
• And if the dog is half Collie half German Shepherd, the pedantic English answer would be ‘yes’, the expected English answer would be ‘both’, and the expected maths answer would be ‘no’. Commented Aug 16, 2019 at 11:55

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.

• +1. Short answer: Because they are different languages. Commented Sep 13, 2011 at 21:57
• I think something a little bit more complex than that is going on, closer to the issues that @DeepYellow is discussing. Commented Sep 13, 2011 at 22:05
• Neil, DeepYellow said what I said, using more words. Commented Sep 14, 2011 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.

To the best of my knowledge, there is not difference between the meaning of and in regular English and in mathematics or logic. In your example, "I want nuts and bolts," the discrepancy you claim requires an unnatural interpretation of the statement. You would need to interpret it as

I want a collection of objects, each of which is both a nut and a bolt.

Since few if any English speakers would interpret it that way, it's not clear why moving into the realm of logic requires us to adopt this interpretation. A more natural interpretation would be

I want a collection of objects composed of multiple nuts and multiple bolts.

The difference between these two interpretations has nothing to do with our understanding of the word and. And our expectations in the second interpretation are based on an entirely logical interpretation of and. If I bring you nuts and I bring you bolts, you should be pleased. If I bring you only one or the other (or neither) I have not fulfilled your request.

As to the question of or, it is accurate that logic always interprets or inclusively, while natural English interprets it either inclusively or exclusively depending on context. However, it's important to note that formal logic picks an existing interpretation. It doesn't make one up that doesn't occur in natural language.

For purposes of creating formal rules of logic out of natural language, it makes sense to pick one sense or the other. The inclusive interpretation of or has advantages over the exclusive. The truth conditions for exclusive or are a subset of those for inclusive or, which makes it fairly straightforward to construct an exclusive or from and, not, and inclusive or. It is more difficult to create a formal expression for inclusive or if you take exclusive or as the default relationship.

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.

There are two ways to look at this:

1 - Math has a technical language inspired by natural language but forced into narrow meanings to make manipulation easier.

and

2 - In your example, you can use 'and' logically the same way, you're just inferring quantifiers wrong.

First, mathematics uses natural language terms but with stipulated definitions in order to fit some very exact manipulations. 'add' means something vaguely like 'put these two things together'. You can add numbers and you can add ingredients in a recipe: numbers have very precise definitions and add means only one thing there, but everyone knows that you can't just add more ingredients to make a recipe bigger. As to 'and', the OED has a number of entries for the natural language version of 'and', and only one really comes close to the math version.

Second, as to your example, you're using 'and' logically/mathematically in 'nuts and bolts' but you're misapplying the inferred quantifiers. You're inferring that you want a set of things such that each thing is a nut and the thing is a bolt. Because of what we know about nuts and bolts (they have parts of their definitions that are impossible together), the logical 'and' of those features is always false. But that is not what the inference is taken as by anyone who understands English. You should infer that you want a set of things that are nuts and you want a set of things that are bolts.

"I want some nuts and bolts"

is not saying

"I want some things and one individual thing is both a nut and a bolt.

but is really saying

"I want two not trivial collections, some nuts and some bolts".

You'd be happy if they came in one bundle or in two separate bundles, but the logical analysis is that you'd only be happy if both are true, and unhappy if either are untrue. That this can be turned around to say "I want some things that are either a nut or a bolt." is just further logical manipulation.