My statement “a rectangle is a square” is the hypothesis of the following conditional statement.
(1) If a rectangle is a square, then it is regular.
(1) is true; all rectangles that are squares are regular. The subject of the hypothesis of (1) is “a rectangle.” (1) is intended to be a general statement and thus to apply to every rectangle. So, the referent of the subject is any rectangle.
The inverse of (1) is the conditional statement that has the negation of the hypothesis of (1) as the hypothesis and the negation of the conclusion of (1) as the conclusion. The inverse of (1) is the following conditional statement.
(2) If a rectangle is not a square, then it is not regular.
(2) is true; no rectangles that are not squares are regular. The subject of the hypothesis of (2) is “a rectangle.” (2) is intended to be a general statement and thus to apply to every rectangle. So, the referent of the subject is any rectangle.
To summarize, “a rectangle” in the statement “a rectangle is a square” and the statement’s negation “a rectangle is not a square” refers to any rectangle. That may seem counterintuitive, but as I have just shown, it’s true.
As quoted above, fuse mentioned logical equivalence. I have made an interesting observation regarding logical equivalence and my argument. That observation is that if the statement “some rectangles are squares” is logically equivalent to the statement “at least one rectangle is a square,” then the statement “some rectangles are squares” is logically equivalent to the statement “a rectangle is a square.” The argument for this equivalence follows.
Argument. It is given that the statement “some rectangles are squares” is logically equivalent to the statement “at least one rectangle is a square.”
As shown by the argument I offered to Mad Man P earlier in this thread, at viewtopic.php?p=2695612#p2695612, if some rectangles are squares, then a rectangle is a square.
To show the converse, assume a rectangle is a square. Then at least one rectangle is a square. Since, by the given information, that is logically equivalent to “some rectangles are squares,” some rectangles are squares. This concludes the argument.
Another interesting observation I have made regards the statement “a rectangle is a square.” Conventionally, the statement “a rectangle is a square” is taken to be sometimes but not always true. However, since the statement “a rectangle is a square” is implied from the always true statement “some rectangles are squares,” the statement “a rectangle is a square” is also always true. That result is in contradiction with the conventional view. I do not, yet perhaps, see any proper way out of this contradiction. I know there is more than one sense of each of the words always, sometimes, and never, but the statement “some rectangles are squares” is always true in one of the broadest senses of always, and perhaps in the broadest sense of always. One of the broadest senses of always in which the statement “some rectangles are squares” is always true is logical possibility. Another may be conceptual possibility, which may be the broadest sense of always. The contradiction displayed earlier in this paragraph lies on an alternate route to trivialism. A contradiction regarding the statement “a rectangle is not a square” can be similarly exhibited.
In my argument, the statements “a rectangle is a square” and “a rectangle is not a square” may “reference two different, particular rectangles.” It may seem that the statements do, but as I showed through an example in an earlier post in this thread, the post located at viewtopic.php?p=2695639#p2695639, there is more than one sense of rectangle sameness. As the following example demonstrates, there is more than one sense of rectangle difference.
Example. Rectangle MNOP is represented through a computer program by points and lines on a computer screen. The computer program allows the location of at least two of the four vertices of rectangle MNOP to be moved simultaneously, with respect to a coordinate grid. 5 minutes pass. Three of the vertices of rectangle MNOP are then simultaneously moved to new locations on the coordinate grid that none of the four vertices were initially at. All four vertices of the rectangle keep their respective names before the transformation, during all of the transformation, and after the transformation. In one sense of rectangle difference, rectangle MNOP of after the transformation is different from rectangle MNOP of before the transformation. With respect to the coordinate grid, not all vertices of rectangle MNOP of after the transformation are in the same locations as the same named vertices of rectangle MNOP of before the transformation. In another sense of rectangle difference, rectangle MNOP of after the transformation is not different from rectangle MNOP of before the transformation. Although the locations of three vertices of rectangle MNOP have changed with respect to the coordinate grid, the rectangle is considered to be the same rectangle MNOP that existed before the change. So, there is more than one sense of rectangle difference. This concludes the example.
The statement “a rectangle is a square and a rectangle is not a square” is sufficient to conclude that in some sense of rectangle difference, two rectangles that are different from each other are being referred to. However, since there are multiple senses of rectangle difference, the statement “a rectangle is a square and a rectangle is not a square” may refer simultaneously to the following two listed items.
- In one sense of rectangle difference, “two different, particular rectangles.”
- In another sense of rectangle difference, exactly one rectangle.
I know that previously in this thread, I used the noun phrase “one quadrilateral that has four right angles” to describe the noun phrase “a rectangle.” I would prefer to use the noun phrase “one parallelogram that has four right angles” to describe the noun phrase “a rectangle.” Both descriptions of “a rectangle” are correct, but my preferred description agrees more with the definitions of rectangle in the two geometry textbooks I think I most respect. Those definitions of rectangle are located in Geometry (2004) by Ron Larson, Laurie Boswell, and Lee Stiff, at p. 347, and in Larson Geometry (2012) by Larson, Boswell, Timothy D. Kanold, and Stiff, at p. 527. My preferred description uses a more specific term, which I think has been suggested to me by multiple sources, including the aforementioned Geometry (2004), to be a virtuous property for definitions. For the reasons provided, my preferred description is better.
Every rectangle is one parallelogram that has four right angles. Each rectangle is one parallelogram that has four right angles.