What does AND/OR mean?

Can someone clear my thinking up on logical AND. I am sure I am missing something fundamental here.

I have two sets, say [cats] AND [black].

If I want to ask for someone to select one cat AND one black item from the sets, I cannot ask select cat AND black. Isn’t that asking for something that is a cat and also black, where the sets overlap.

‘OR’ is easy. If I ask for cat OR black that would be something that is either a cat or black or both. No problems with that one.

So how do you differentiate between instances?
Similarly, if I said I want Fred AND Joe to do a job, this means I want them both to do the job, not something that is Fred and Joe. There is no AND because the [Fred] set and [Joe] set do not overlap.

If I want either Fred or Joe both to do a job, I can say: “I want Fred AND/OR Joe do the job.” Is AND/OR meaningless? What I am really asking is I want (one OR zero instance of the Fred set) OR (one OR zero instance of the Joe set) AND not zero instances from the Fred set AND not zero instances from the Joe set.

Seems like there is an AND which refers to instances and one that refers to the whole set. ??
Does AND/OR mean [AND and OR] or [AND or OR]…or does it mean [AND and OR] and [AND or OR] OR [AND and OR] or [AND or OR] and [AND and OR] and [AND or OR] OR …

NOTE: To head of any lame quips, I am talking logic here, not on how to communicate with real people.

That was confusing as hell, and I’m not sure if this is what you want but…

And/or means “and or or.”

“I want Fred and/or Joe to do the job” means “I want Fred and Joe to do the job, or Fred to do the job, or Joe to do the job.”

And/or doesn’t apply to the cat/black situation you’ve described. You want one cat and one black.

It’s an everyday conversational expression, not one using formal boolean logic; ‘and/or’ in everyday terms means the same as logical OR (one or the other or both), whereas ‘or’ in everyday terms often means something closer to XOR ( one or the other, but not both).

To many people, “OR” connotes exclusivity. One or the other, but not both. AND/OR means one, or the other, or both.

Or in other words, AFAIAW, AND/OR doesn’t exist in formal logic.

I might be missing something here, but I think you are stuck because you are, ultimately, trying to join to non-joinable ideas. I would expect that to be two separate statements.

SELECT one CAT. SELECT one BLACK.

You only join them if there is a logical connection (and as described, there isn’t, if I am reading your example correctly).

Well…that’s basically true. Asking for one CAT or one BLACK doesn’t guarentee anything; you might wind up with one or two items. But yes, OR works the way you describe.

In what context? Periods work well. Semi-colons work well in several programming languages…

That’s an OR condition. I don’t care which does it, as long as Fred OR Joe do it. Or means “AND/OR” as you’ve described it here. If one of the two must do it, then it is an exclusive or.

AND/OR is a little fuzzy. I would expect AND to be a boolean AND (both must be true) and OR to be a boolean OR (one or the other to be true), with an exclusive OR being denoted in some other manner - say, XOR (one or the other must be true, but not both).

I’m not sure what “logic” you are talking about. The logic I am using above is that found in the programming languages with which I am familiar. Symbolic logic might be slightly different, and I am not terribly familiar with formal set theory, so my comments might not apply at all. If you could clarify your questions I might be able to help a little more.

On preview…man, I have to type faster.

The expression “and/or” is necessary because in common usage, “or” is used as the equivalent to what computer programmers call “xor” or “exclusive or.” When somebody says “You can have a cookie or a candy bar,” they mean one or the other – exclusive or, where one or the other is true but not both. Of course, doing an or operation like (cookie OR candyBar) on a computer is true if either cookie or candyBar is true, OR if both of them are true. (At least in Pascal and C and BASIC and the other languages I used to use. It’s probably not true for all programming languages.) My friends and I used to comment on that little programmatical / linguistic disparity all the time in high school.

And darn all of you people who posted the same thing while I was writing. <goes off to sob quietly in the corner for a bit>

If your’e talking about Logical AND/OR as opposed to normal communication, then the following rules apply.

X AND Y means that both X and Y must be true.
In other words, if X represents “Object is cat” and Y represents “Object is black”, then only a black cat will satisfy the comparison.

X OR Y means that either X or Y (or both) must be true.
In this case, a black chair, a white cat, or a black cat would all be true.

Another logical operator is XOR (exclusive OR), which means that only one of the conditions can be true. With an XOR, the black chair and the white cat would be true, but the black cat would be false.

And darn me for not reading the entire OP. There’s a logical AND/OR? I thought that was covered by OR. I’ll just go back to crying in the corner.

Me too. :smack:

I think you will run into problems if you try to reconcile English with formal logic. I used to think about this a lot. You simply have to accept that English is not logical. Consider the following two statements.

I did not eat all of the fries.
I did not eat some of the fries.

Swap a universal quantifier with an existential one without changing any other words in the sentence and you wind up with exactly the same meaning (but with different emphasis).

Part of the problem is that, in English, quantifiers and propositional connectives can appear inside statements that are inseparable in first-order logic. Consider the following statement.

Last semester, I did not take math and English.

Does that mean “I did not math, and I did not take English” or “I did not take both math and English, but I might have taken one of them”? If the former, how is that different from “I did not take math or English”. What would sticking “and/or” in there mean?

That is why I don’t like “and/or”. Once you have a negation, it only confuses the issue.

To really answer this, you need the concept of a choice function, which (roughly speaking) picks an arbitrary element from a non-empty set.

Let C be the set of cats, B the set of black things, and f the choice function. What you’re looking for is a pair (f(C), f(B)).

Asking for black cat is given by f(B [symbol]Ç[/symbol] C), and asking for something that’s either black or a cat (and possibly a black cat) is given by f(B [symbol]È[/symbol] C).

Asking for something that’s either black or a cat but not both is more complicated. In symbols, it’s f((C [symbol]Ç[/symbol] [symbol]Ø[/symbol]B) [symbol]È[/symbol] ([symbol]Ø[/symbol]C [symbol]Ç[/symbol] B))–there is no standard for the XOR operator.

Thanks for the replies.

So,
Conversational ‘or’ is formal logical XOR
Conversational ‘and/or’ is formal logical OR
And ‘and’ in conversation is different to formal boolean logic AND. The former meaning both A+B the latter meaning that that is A and also B.
What does conversational ‘and’ equate to in formal logic?

How does one interpret legislation - using formal logic or everyday expressions?

I read the second sentence as equivalent to “I did not eat any of the fries.”, which clearly has a different meaning from the first. But your interpretation is reasonable, so that just highlights the confusion.

A better example: “I ate lunch and got gas at Joe’s truck stop.” is logically equivalent to “I got gas and ate lunch at Joe’s truck stop.”, but the two sentences are clearly different in some important respect.

Ahhaaa. I suppose my question was asking two things.

  1. Difference between formal logic and conversational.
  2. Choice in sets versus (?equality?) in sets.

I have no idea what you’re getting at here. Can you explain a little further?

The OP is confusing set theory with boolean logic.

Boolean logic is when you say

A and B

If I know the truth value of A and the truth value of B I can determine the truth value of the whole statement. Statements can get complex as you add OR and other logical operations.

Set theory talks about elements of sets. The propositional calculus (IIRC) allows you to make statements like, “There exists an A such that A is an element of the set of all cats AND A is an element of the set of all black things.” This statement is evaluated to be true or false. You also have statements like, “If A is an element of the set of all cats, then A is an element of the set of all black things.” which can also be evaluated for truth value.

. . .then you would say “Select A such that A is an element of the set of call cats and select B such that B is an element of the set of all black things.” What you want to do with A and B is yer business. :wink:

I assume here “at Joe’s truck stop” does not distribute across the “and”. In that case, I do not see this important respect you are talking about. However, if “at Joe’s truck stop” does not distribute, but is attached to “got gas” in the first sentence and “ate lunch” in the second, then the two sentences clearly have different meanings.

I think the problem is that in English, propositional connectives can apply to subjects, verbs, objects, and clauses. (Notice if I replace “and” with “or” in the previous sentence, the meaning is essentially the same.) But in formal logic, connectives can only apply to clauses (i.e., propositions). However, in everyday conversation, restricting connectives to clauses would be extremely tedious. So most people stick connectives in the middle of clauses, and rely on context to pull them out and decide whether a “DeMorgan’s Law” is necessary while pulling.

Oops, that you be distributes.