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I added a disambiguation note to conjunction, since most of the links to that page actually wanted something else. Then I changed those links to something else: logical conjunction (not yet extant).
An Internet and dictionary search suggests that there is no analogous danger for disjunction (also not yet extant).
Now they are extant.
Hmm, I forgot about that. Are you sure that’s the right choice of page names? We could also use [[conjunction]]
for the logical notion (it would match with disjunction, for one thing) and call the double-category notion something like [[conjoint pair]]
.
I’m not at all sure about the double category notion, which I am not familiar with.
But it is true that ‘conjunction’ has a variety of meanings in natural language that ‘disjunction’ has not, so logic has a much stronger claim to being the default metaphorical mathematical meaning of the latter than the former.
Compare Wikipedia:
I’ve added inference rules to logical conjunction and disjunction.
I could definitely go with and and or. Perhaps there should be a hatnote at conjunction however.
Actually, I don’t really like and and or because they’re not names; I mean, grammatically ‘and’ and ‘or’ are not nouns. (Of course, I agree with having them as redirects, which is why I made them redirects.)
The question for me is what conjunction by itself should mean. By the way, there is a hatnote there already.
Hey Toby, tangentially related question: do you have a mnemonic to remember which one is the meet and which one is the join in a lattice? Meanwhile, those are awful names, and whoever thought them up should be ashamed.
Where two things meet is what they have in common: intersection. To join two things is to take them together to form larger whole: union.
The terms seem perfectly acceptable to me, and make a lot of sense when you consider Venn diagrams.
Where two things meet is what they have in common: intersection. To join two things is to take them together to form larger whole: union.
But where two things join is also what they have in common. So the only difference between the two words is that you cannot ‘meet’ two things together.
For me, it helps that I had earlier seen ‘meet’ used as a verb, for a binary relation between subsets rather than for a binary operation on them (or on elements of any lattice): two subsets meet if their intersection is inhabited. So their intersection (their meet) is the set of all points where they meet.
I also find it a little disconcerting that ‘meet’ and ‘join’ are used as nouns at all. (Technically, they can be nouns in everyday English, but not with relevant meanings.) I would rather have said ‘meeting’ and ‘joining’, or something like that. (No one tries to speak of the ‘intersect’ of two subsets, do they?)
But I’m used to it now.
Under the mnemonic I was trying to offer, I think it would be hard to construe “meet” as a union. So it’s not just that one verb is transitive and the other isn’t.
(Sorry if I cut in front of you, Toby. But IMO this isn’t that complicated.)
But where two things join is also what they have in common.
That doesn’t make sense to me; “where two things join” doesn’t sound like English to me. Do you mean “where two things are joined together”? To me “join” means to take two things and put them together to make a bigger thing.
Under the mnemonic I was trying to offer, I think it would be hard to construe “meet” as a union.
Right, I accepted that difference. (I wasn’t trying to make a point about transitive and intransitive verbs.) But I don’t accept the other difference; it’s very easy to construe ‘join’ as an intersection, using exactly the same phrasing. And while you cannot ‘meet’ two things together, it is possible for two things to meet to form a union, even though one thing can only join a union that already exists. So while that half of your mnemonic stands with your phrasing, it is not far from something ambiguous.
“where two things join” doesn’t sound like English to me.
To quote http://www.thefreedictionary.com/join:
To come together so as to form a connection: ‘where the two bones join’.
One might just as well say ‘meet’ here.
The two bones join at the joint. The verb ‘join’ is related both to ‘joint’ and to ‘junction’, which can be a synonym for ‘intersection’. (That word, ‘junction’, also appears in ‘conjunction’, which literally means a joining together; it appears with a negative prefix in ‘disjunction’, which literally means a non-joining). In fact, here is the only noun meaning (after 13 verb meanings) from the same reference:
A joint; a junction.
So if the word ‘join’ is used as a noun in ordinary language, it must be an intersection (while ‘meet’ must be a sporting event), which is why I would prefer to use good nouns like ‘meeting’ and ‘joining’ (in which ‘meet’ and ‘join’ are verbs).
I’ll add that The Free Dictionary (which, by the way, I chose only because it came first in a Google search) twice defines ‘meet’ as a synonym of ‘join’ and once conversely.
I certainly agree that the usual meaning of these words in lattice theory is better than the reverse. Overall, reading through the meanings of ‘join’ and ‘meet’, the former tend to have a sense of permanence and irreversible change that the latter do not, and it is a change of becoming a union. More often than not, when two things join, they become a larger whole that is greater than either of its parts; but when two things meet, they retain their individual identities and the meeting is only a part of each.
Ultimately, both of these words are based on spatiotemporal metaphors that don’t fully translate to the purely spatial metaphors that the lattice-theoretic terms capture. Rather than just sitting there like circles in a Venn diagram, $A$ and $B$, originally disjoint, move together to have a inhabited intersection; if they stay together, then their identities may be lost within the union, and they have joined, while if they separate afterwards and become disjoint once more, then they have merely met. So ‘join’ for union and ‘meet’ for intersection are better than the reverse, but neither is really what the words mean.
Actually, ordinary language has a lot of conflating of intersection and union. Some of this is due to contravariance. I teach my algebra students to turn
$x^2 + 12 = 7x$into
$x = 3 \;\text{or}\; x = 4 ,$but many of them want to write
$x = 3,\; x = 4$instead. The mistake is reinforced when the textbook writes the answer as a set,
$\{ 3, 4 \}$(having implictly turned the problem into one of simplifying $\{x | x^2 + 12 = 7x\}$), which is pronounced ‘the set of $3$ and $4$’.
So $\{3\}$ is the set of $3$, $\{4\}$ is the set of $4$, and $\{3,4\} = \{3\} \cup \{4\}$ is the set of $3$ and $4$. More generally, $A \cup B$ consists of the elements of $A$ and the elements of $B$. What is really going on here, as I analyse it, is that we are defining $A \cup B$ by stating the axiom
$(x \in A \Rightarrow x \in A \cup B) \wedge (y \in B \Rightarrow y \in A \cup B)$and then interpreting the definition inductively, so that $A \cup B$ is the smallest thing that satisfies this axiom. It is the contravariance of $\Rightarrow$ (in its first argument) that makes this condition equivalent to
$(x \in A \vee x \in B) \Rightarrow x \in A \cup B ,$and adding the inductive interpretation in directly finally makes
$(x \in A \vee x \in B) \Leftrightarrow x \in A \cup B .$This last form, which has ‘or’, is the only complete definition, but the first form, which has ‘and’, is how people often think about it.
While you just made some good points, Toby, the discussion is becoming a little tiresome to me I’m afraid, and I cling to the feeling that it’s really not hard to keep the meanings straight, and it shouldn’t be made to look harder than it is. (The bit about spatio-temporal metaphors would, I’m sure, apply to many terms used in math.)
Instead, here’s something amusing I noticed yesterday: the symbols $\wedge$ and $\vee$ are exactly wrong! If lesser elements are drawn below greater elements as in a Hasse diagram of a lattice, then the lower vertices of the symbol $\wedge$ ought to remind one of two elements $a$ and $b$, and the top vertex of an upper bound of $a$ and $b$.
That may be worse than ’join’ and ’meet’, but it’s just one of those things that one gets used to (cf. Toby’s last line in #12), and moves on.
@Todd: Draw your lattice upside-down.
Yeah, I know that Harry. Thanks.
Who says you can’t convey sarcasm through the internet? =p Anyway, I tend to draw hasse diagrams inverted, because it’s a clockwise turn from the horizontal notation $a\leq b$. Drawing a Hasse diagram rooted at $a$ requires you to turn counterclockwise and start at the bottom of the page.
I’m sorry that this is tiresome for you, Todd; I find this sort of thing very interesting. I’m not trying to be critical.
I agree that it should not be hard to keep these straight, but still it is something that takes getting used to. So I can see where Harry is coming from. Aside from thinking about the meaning of particular words, that’s my point.
Toby, I also love word origins and word usages and linguistics. Maybe I’m tired of this particular discussion because I don’t see that there’s that much to get used to. In other words, I understand your points, but by now the topic of the aptness (or not) of “join” and “meet” seems just a bit too trivial to spend any more time on. That’s just my opinion of course, and I applaud your empathetic responses to Harry. :-)
The more mathematical portions of #15 are interesting to me, and I’ve registered similar observations over the years.
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