[...]Basic logic — connectives — AND and OR « Gowers's Weblog[...]…

“the set of cats and dogs”, I’d intuitively think of as containing both cats and dogs whereas “the set of animals that are cats and dogs” I would say is empty.

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Clearly, “not( P and Q)” will be true whenever “P and Q” is false. But, “P and Q” is false precisely when “not P or not Q” is true, with “or” being understood to be inclusive.

Then I can see what you mean. “And” and “or” go together nicely, so long as “or” is inclusive.

1. “P and Q” is true if and only if P is true and Q is true.

The above definition of “and” is relatively intuitive as it is consistent with everyday usage.

2. Given 1, when would a statement such as “not (P and Q)” be true?

Clearly, “not( P and Q)” will be true whenever “P and Q” is false. But, “P and Q” is false precisely when “P or Q” is true with or being understood to be inclusive.

Thus, the above argument indicates that the inclusive or is needed to make ensure that the truth values of “not (P and Q)” is consistent with the truth values of “P and Q”.

Hopefully the above makes sense and is of use to someone.

Similar remarks apply to other connectives. If I hear somebody say, “Put your hand up if you would like to go to the seaside,” and I have no wish whatsoever to go the seaside, it would be considered misleading of me to put my hand up: that particular “if” is understood as “if and only if”.

I’m sure you’re familiar with such examples, but in case anyone else is reading this comment who isn’t, the notion of Gricean implicature is quite entertaining, and relevant to this.

[But if all you know is that “P or Q” is true, then your task is more difficult. This time what you have to do is split your argument into cases. If you are given that “P or Q” is true, and you want to deduce R, then you must show that you can deduce R if you assume P (and whatever else you know) and that you can also deduce R if you assume Q (and whatever else you know). That way, since at least one of P and Q is guaranteed to be true, R must be true. So in a proof you might write something like this: “First let us assume that P. Then … blah blah … so R follows. Now let us assume that Q. Then … blah blah … and again R follows. Therefore the result is proved.”]

Is this where “[proof by cases](http://en.wikipedia.org/wiki/Proof_by_exhaustion)” from?

Thanks!

In my opinion, it isn’t so surprising that “OR” and “AND” distribute over each other. Two very simple ways to interpret “OR” and “AND” are to think of “OR” as the maximum (minimum), and “AND” as the minimum (maximum) on an ordered set of two natural numbers, integers, rational numbers, or real numbers {a, b}. Via either of those interpretations distribution of both connectives over each other readily follows. The greater surprise, in my opinion, comes as that multiplication distributes over addition in arithmetic.

I’ve decided to add the word “don’t” so as not to confuse people. Of course, that makes your response confusing, but I hope that this reply will deal with that.

Just one point: you say `Clearly we mean these as names of statements, or we wouldn’t be able to say P\wedgeQ.’ But I think the example works against the principle you are trying to draw: the wedge symbol is written between statements, not names of statements, to create a further statement. There are complications here depending on whether `P’ and `Q’ are supposed to be meta-variables in our metalanguage or, as I’d taken them, dummy sentence letters in the object language, but clearly `the Axiom of Choice and Zorn’s Lemma’ isn’t a statement, but just a conjunction of names, whereas what we want is something like `1 + 1 = 2 and 2 + 2 = 4′ where we have on each side of the connective a statement, not a name.

(Some people, e.g. Machover, have two sorts of arrows, one (single-shafted) to abbreviate `if . . . then’ and one (double-shafted) in the metalanguage for `implies’.)

Anyway, I am looking forward to the rest of the posts — thanks again for doing this.

The difficulty I have is that (i) we often write things like and (ii) we read the symbol “” as “implies”. I see that the Wikipedia article is quite careful about this and says that the symbol means if … then.

Another thing is that we happily write when and are unknown statements. Clearly we don’t mean these as names of statements, or we wouldn’t be able to say And yet most people (I think) read as “P implies Q”.

Anyhow, thanks for the comment. As you suspected, pedantic objections are welcome. I now have to think how to strike the right balance between getting everything correct and being intelligible. What I really care about is that people should be able to do things like negating implications, so I don’t want to scare people off with subtle distinctions. But I’d much rather be accurate than inaccurate. I’ve got some rewriting to do before I put up my next post …

The point is that in almost any real-life situation where one uses “or”, the “and” is ruled out for (a term of art in linguistics) pragmatic reasons. For example, “Frank is a cat or a dog” feels exclusive, but only because we know that in real life “Frank is a cat and a dog” is not possible.

Statements like “London is in England or Paris is in France.” are rejected as ill-formed by most people not because they violate the meaning of “or”, but because they violate the hearer’s expectation that the speaker will have made the most specific (i.e. mathematically strongest) statement possible–here, “London is in England and Paris is in France.” The speaker’s failure to do so upsets the hearer, but again this is a perceived deficiency of the speaker, not of the statement itself.

Thanks — I’ve clarified that now.