I disagree. The principle of explosion always applies to all statements, regardless of their language and style.
I disagree. Any two contradictory statements imply “any arbitrarily chosen conclusion.” How formally statements are expressed is never relevant.
I agree. However, while the rectangle in the quoted statement “a rectangle is a square” may not be the same rectangle in the quoted statement “a rectangle is not a square,” the quoted noun phrase “a rectangle” has the same meaning in both statements. That common meaning is “one quadrilateral that has four right angles.”
While high school geometry textbooks here in the United States define the notion of rectangle congruence, they interestingly do not make clear the notion of rectangle equality. This may be because there are multiple senses of sameness. For example, two rectangles that share no vertices are congruent. The rectangles are equal in one sense of equal, but unequal in another sense of equal. This concludes the example. With respect to all I know, it is epistemically possible that all rectangles are the same one rectangle. However, there seems to be a strong sense of rectangle sameness in which not all rectangles are the same one rectangle. See twitter.com/paulemok/status/970468311828975616.
Bringing the notion of multiple senses of sameness forward, while your statement (5) may be the same statement as the quoted statement “a rectangle is a square and a rectangle is not a square” in one sense of statement sameness, the statements are not the same in another sense of statement sameness. From a syntactic standpoint, the statements are different. Also, from a semantic standpoint, the statements are different; your statement (5) contains the additional, and possibly false, quoted information that “the first rectangle is named S and the second rectangle is named N,” which is information that the quoted statement “a rectangle is a square and a rectangle is not a square” does not contain. Your statement (5) requires its first mentioned rectangle to be named S and its second mentioned rectangle to be named N. Although those rectangles may be able to have those names, those rectangles might not have those names.
Your statement (5) simply is not the statement I am interested in.
Actually, I think using rectangles and squares is better. Using them makes the argument more absolute. There may be a possible world in which the quoted statement “all mammals are cats and all cats are mammals” is true.
I disagree. “A rectangle is a square” is not logically equivalent to “every rectangle is a square.” The former statement is sometimes but not always true, but the latter statement is never true. Also, “a rectangle is a square” is not logically equivalent to “any rectangle is a square.” The first statement is sometimes but not always true, but the second statement is never true.
I am not speaking generally. What you could have said is that “a cat is always a mammal” is true, but “a mammal is always a cat” is false.
My second statement does mean “a particular rectangle is a square.” However, “referring to any other rectangle and saying that it is not a square” may “produce a contradiction,” because the rectangles do not need to be the same rectangle. Both rectangles are appropriately represented by the quoted noun phrase “a rectangle.”
No, I’m not saying your quoted statement. “There is at least one rectangle that is a square” is not logically equivalent to “a rectangle is a square.” The former statement is always true, but the latter statement is sometimes but not always true. Similarly, “there is at least one rectangle that is not a square” is not logically equivalent to “a rectangle is not a square.” The former statement is always true, but the latter statement is sometimes but not always true. Overall, “there is at least one rectangle that is a square and there is at least one rectangle that is not a square” is not logically equivalent to “a rectangle is a square and a rectangle is not a square.” The former statement is always true, but the latter statement is sometimes but not always true. To better understand how the latter statement is not always true, consider squares ABCD and EFGH that share no vertices. Since all squares are rectangles, square ABCD is a rectangle. Thus, a rectangle is a square. Since all squares are rectangles, square EFGH is a rectangle. Thus, a rectangle is a square. The negation of the previous statement, “a rectangle is a square,” is the false quoted statement “a rectangle is not a square.” The conjunction of the true quoted statement “a rectangle is a square” that was derived from the fact that square ABCD is a rectangle, and the false quoted statement “a rectangle is not a square” that was previously mentioned, is the false quoted statement “a rectangle is a square and a rectangle is not a square.” So, “a rectangle is a square and a rectangle is not a square” is not always true.