Just to illustrate these points, let’s look at the connectives in propositional logic. It’s not hard to find English language uses of “and”, “or” and “if-then” that are NOT well described by the logical connectives of propositional logic.
Compare the following two sentences:
(1) “Tom and William ate.”
(2) “Tom and William met.”
In both cases we have a plural subject consisting of conjoined noun phrases and a verb.
The statement in (1) can be decomposed into the conjunction of statements
“Tom ate and William ate.”
However, it would be odd to translate (2) as
“Tom met and William met.”
We can still rewrite (2) as a conjunction of statements, but it’s not the usual conjunction:
“Tom met William and William met Tom.”
Note that this conjunction actually contains a redundancy — “Tom met William” means exactly the same thing as “William met Tom”. One could argue that a simple statement is all that is required to fully capture the meaning (e.g. the first conjunct).
But if so, then we could argue that (2) isn’t a logical conjunction at all.
Philosophers are pretty good at coming up with examples like this. They illustrate that not all English sentences of the form “A and B” are easily described by the logical conjunction as defined in propositional logic, and some resist all our efforts to symbolize them using the logical connective.
Not every statement with an “or” in it is a logical disjunction. Consider for instance this pair of sentences:
(1) “Either Tom wants a cat or Tom wants a dog.”
(2) “Tom wants a cat or a dog.”
Does (1) mean the same thing as (2)?
The problem is that the meaning of (2) is ambiguous, and only one of these meanings is a logical disjunction.
(2) could mean “Tom wants a dog or a cat; but I don’t know which.”
or
(2) could mean “Tom wants a dog or a cat; he doesn’t care which.”
If it’s the former case (Tom wants one or the other, but I don’t know which), then it’s a logical disjunction, since if it turns out that Tom wants a dog, then the statement is true; and if it turns out that Tom wants a cat, the statement is also true.
But if it’s the latter case (Tom wants either one, he doesn’t care which), then it’s not a logical disjunction. Why? Because if this statement is true, it doesn’t guarantee either that Tom wants a dog or that Tom wants a cat. And that would violate the rules for the logical disjunction.
Just to give another example, if I say “I hope Timmy or Tommy takes out the garbage, I don’t care who”, one can’t validly infer that “I hope Timmy takes out the garbage”, or that “I hope Tommy takes out the garbage”. Neither of these statements follow deductively.
What we’re highlighting here is that ordinary language connectives don’t behave in familiar ways when they fall within the scope of certain psychological verbs (“wants”, “cares”, etc.).
Again, there are other examples you can give which illustrate that the English use of “or” does not always translate as the logical disjunction.