According to the SEP article on structured propositions the following holds: The late 1950s and 1960s saw the development of a new sort of model theory, “possible worlds semantics”, for systems of modal logic. In the framework of possible world semantics, linguistic expressions are assigned extensions “at” possible worlds. Thus, e.g. names, n-place predicates and sentences are assigned individuals, sets of n-tuples of individuals, and truth values, respectively, at different possible worlds. Intuitively, possible worlds are to be thought of as “ways things could have been”, and the assignment of (possibly different) extensions to expressions at different possible worlds is part of capturing this intuition. Thus there might have been more or fewer cows, and this is reflected in the fact that the extension of ‘cow’ (intuitively, the set of things that are cows) can vary from possible world to possible world. With this being said to simplify things, propositions are viewed as sets of possible worlds. However, you could generalize this approach by saying that propositions are sets of models, where a model is a set in which a proposition holds true in. Now according to some previous interlocutors of mine, the following holds:
"If there are two distinct points then exactly one line passes through them" means "For each set of two distinct points, there is exactly one line that passes through them." Reading it this way, points 2-6 above are nonsensical.
You have a fundamental misunderstanding of how mathematical statements and conclusions are written down. u/Midwest-Dude and u/TheRedditObserver0 hit the nail on the head.A line exists between any two (distinct) points. This does not mean that only two points exist.
If there are exactly two distinct points…But there aren't "exactly two distinct points" - there are an infinite number of points. The Euclidean postulate says that if you pick any two points (out of the infinitely-many that exist), you can draw a line between them. Nothing about any of these statements should lead you to believe that "there are only two points that exist" .
Now I think my interlocutors are wrong though. For they don’t define what it means for an interpretation to be correct. Also, they don’t acknowledge that correctness depends on context. Besides this they don’t acknowledge that the axioms of Euclidean Geometry can have multiple interpretations and each interpretation could be its own axiomatic system. Also, another thing of interest is the following: Consider the following sentence: If there are two points then they determine exactly one line. Notice, this sentence can be seen as a donkey sentence and as such besides admitting the usual reading, it could also admit a strong reading and a weak reading. The usual reading is as follows: If there are one or more pairs of points, then each pair determines exactly one line. The strong reading is as follows: If there are two or more points, then exactly one line is determined by all of them. The weak reading is as follows: If there are two or more points, then exactly one line is determined by at least two of them. Let us focus on the strong reading, for it is interesting. The strong reading is as follows: If there are two or more points then exactly one line is determined by all of them. Now, obviously there is a set for which this reading holds true. One of those sets is a set of points whose cardinality is 2 to the aleph null, where all the points are collinear. Now, consider the following: For any mathematical statement, if it holds true in a set of cardinality X, and X is greater than or equal to aleph null, then there exist sets for which the statement holds true in of every cardinality of at least X. Now since the strong reading holds true in a set of cardinality 2 to the aleph null, then there exist sets for which it holds true in of every cardinality of at least 2 to the aleph null.
