"Women and children were present" isn't saying the intersection of the women and children were present but that the things were women and children and they all were present.
The rest of the variety of "meanings" of "and" are just inferences on top of this basic meaning. If things occur in sequence, for instance, we can infer causality from what we know about the things conjoined apart from the conjunction.
Crucially, there is more here than just "joining multiple clauses together" (and many of the things conjoined aren't clauses), because "or" can be described this way too, but "and" does not mean the same thing as "or". With "or" you are only asserting that some of the propositions are true.
1) not every clause is a predicate and applying logical operators to a non-predicate makes very little sense to me.
2) I don’t see how ordering, time, and causation can be inferred, if you treat each clause as a predicate.
“I punched Bob and apologized” has a different meaning than “I apologized, and punched Bob.”
The AND as used in logics simply doesn’t embody this usage.
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It was a sloppy wording of mine to say that AND just join clauses together. I meant to say that it joins clauses together, and the relations between the clauses are not necessarily set union. Of course it can’t embody all relations, but more than set unions.
Since words and phrases necessarily occur in sequence, if you are speaking of things can be and typically are in sequence, it is natural to infer that the speaker is presenting them in the sequence of their occurrence. The speaker can do this, the speaker and the hearer both know that the speaker can do this, and it is a common understanding between the speaker and the hearer that the speaker intends the hearer to receive a particular representation of events: this is the purpose of communication. This being so, the speaker presenting events out of their temporal sequence would be perverse. If the speaker knows the hearer will assume she is presenting events in their temporal sequence, she must present them in this sequence or explain that they are not so presented. To do otherwise would be understood by the speaker and hearer as deliberate deception.
So the sequence comes not from the semantics of "and" but the pragmatics of speaking itself. In fact, you don't need "and" at all. If you leave "and" out -- asyndeton -- and list things, they will be understood to be in temporal sequence if they are the sort of thing that occur that way. "I opened the door. I went out."
All of these "meanings of and" can be explained via pragmatics and a very simple meaning of "and". The meaning I provided was intended as a hand-wavy gesture in the direction of a more formal theory.
Totally agree. I think part of the issue is that "natural-language and" behaves differently when you apply it to different parts of speech, and everyone in this thread is thinking of a different part!
That being said, in the example sentences you gave ("I punched Bob and apologized" vs. "I apologized and punched Bob"), it's actually ambiguous whether they have equivalent meaning or not, when we ONLY consider grammar. Using ONLY the words in the sentence, the relationship between "apologized" and "punched" rests on our interpretation of "and"'s grammatical role. Taken in its usual sense as a conjunction, the two verbs are only related in the sense that the same subject performed them.
But I think you're arguing that "and" places the two verbs in sequence (which prevents the sentence from being flipped without changing meaning.) The more I think about this, the more I don't think it follows. To read the verb phrase this way, the reader has to infer a missing conjunctive adverb: ("punched Bob and [then] apologized")... and in order to make that inference, you need context from the sentence.
To make matters more complicated, we _could_ interpret the sentence in a way that doesn't imply the verbs are related: suppose the speaker is apologizing to some third party, and punches Bob for some unrelated reason. If we interpret it this way, the two sentences can be switched without changing the meaning too much.
But all this just goes to show: your core point is correct: natural languages aren't formal languages, and trying to treat them as such is an exercise in frustration.
Yes and it's usually very hard to summarize what words can mean in different contexts, therefore it's usually harder to learn natural languages than programming languages. For example, in a programming language "while you're out, pick up some milk" would be an infinite loop. I think this is why Alexa, Google Assistant, Siri and many more are only finding usage for the simplest of tasks.
The big difference I see is with 'OR' in programing the options are exclusive. Ex: "Are you tired or hungry" The answer is yes if you are tired. Yes if you are hungry, Yes if you are both. No if you are neither. This is natural English. In programming if you are both. The answer is No.(False)
Note that I’m not joining people with AND. I joined abilities.
For instance, to expand on the example.
I have three Pokémons. They can read and write.
I’m not saying that my three Pokémon are the intersection of reading Pokémon and writing Pokémon. There might be other Pokémon who can do just the same.
I’m saying that what they can do is a union of writing and reading.
EDIT: spelling
(Which means saying that “and” is about intersection and not union is nonsense, everything statement about set union can be rephrased to an equivalent statement about intersection, and vice versa.)
For some relation R between elements of the set X and elements of the set Y (we can think of X as being the set of individuals, and Y as being a set of elementary unary predicates), for each x in X, we can refer to A(x) := {y in Y : x R y} , and for each y in Y we can talk about B(y) := {x in X : x R y} . When we say (x_1 R y_1) && (x_1 R y_2), this is equivalent to both (x_1 in B(y_1)) && (x_1 in B(y_2)) i.e. (x_1 in (B(Y_1) intersect(B(y_2)))) , and also to (y_1 in A(x_1)) && (y_2 in A(x_1)) , i.e. ({y_1} \subset A(x_1)) && ({y_2} \subset A(x_1)) i.e. (({y_1} union {y_2}) \subset A(x_1)) .
Personally, if we are talking about a single x_1, and we are saying "and" about a number of different y_1 , y_2 , etc. , or if we are talking about each x in some collection but are saying the same thing about each of them, namely that for each x in the collection, that x R y_1 and that x R y_2 and that x R y_3 , I think it seems more natural to take the first of the two framings described above. That is, I think it seems more natural to say that for each x in the collection, that x \in (B(y_1) intersect B(y_2) intersect B(y_3)), rather than saying that ({y_1} union {y_2} union {y_3}) subset A(x) .
Or, if we give the name W to the collection we are talking about, we could say that "W subset (B(y_1) intersect B(y_2) intersect B(y_3))". This seems to me a fairly natural way to say "all the W , y_1 , y_2, and y_3" ("each of my 3 pokemon can do reading, writing, and arithmetic") .
The other way would end up more like, "({y_1} union {y_2} union {y_3}) subset (the intersection over all x in W, of A(x))".
This works, but seems less natural to me.
Also, especially in the case where you expect X to be substantially larger than Y.
Now, thanks to what I can only describe as the implicit vagaries of English, the bounding (three) might now imply that I have three ... or more Pokémons! To make that sentence precise: I have three Pokémons and they can read and write. Here I am substituting and for a full stop (period.) When I deployed "that" I managed to muddy the waters.
Mapping logical and Boolean operators to grammatical constructs is tricky and should not be taken lightly. Best to consider them separately. AND is not and (except when it is!)
Be careful out there kids.
>> Set intersection: my jeans were creased and dirty
IMHO those are both unions. One describes two capabilities people have while the other describes two conditions some clothing has.
They both become intersection when using the pair of conditions as selection or decision criteria.
It's the difference between an assertion and a question.
Set union: they can read and write’
Set intersection: my jeans were creased and dirty
Sequential: he turned round and walked out
Causal: there was a flash flood and by the next morning the town was under water’
And, there are many more usages.