In Support of Trivialism

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.

  1. In one sense of rectangle difference, “two different, particular rectangles.”
  2. 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.

Then, by your own admission, your argument is a non sequitur…

Mad Man P:

The basis for your claim is unclear. I’m unsure how your two quotes support what you have claimed.

You cannot perform a deduction without a second premise… you may restate the first premise so long as it is a logical equivalent, but the moment you alter it’s logical implications it’s a non-sequitur.

You did this with the statement “Some rectangles are squares”

  1. Some rectangles are squares
  2. In any set containing more than one there is at least one.
    Conclusion: At least one rectangle is a square.

You have no such deductive argument leading you from “at least one rectangle is a square” to “a rectangle is a square”…

Which means, in so far as it’s logically different, it is a non sequitur.

That is not true. A deduction does not have to involve at least two premises. The following example provides a deduction from exactly one premise.

Example.
Domain: All rectangles.
[i]S/i is the statement “x is a square.”
Statements (Reasons)

  1. x [i]S/i (Premise)
  2. [i]S/i (Existential instantiation from (1))
    This concludes the example.

What you have said is unclear.

Yes, I do. It’s part of the argument I’ve previously offered you, the argument you’ve already quoted twice. I quote the relevant portion.

The example I provided earlier in this reply provides a deduction akin to a deduction of “a rectangle is a square” from “at least one rectangle is a square.” A deduction of “a rectangle is a square” from “at least one rectangle is a square” can be similar to, or even can be itself, an invocation of existential instantiation.

You are misinformed, and what’s worse is that you’re showing confidence in your own misinformation by mounting a defense thereof.
Educating someone who believes to already be educated on a topic has proven to be a futile effort, so I’ll be excusing myself from that nightmarish prospect…
So as a parting note I will clarify my meaning to the best of my ability, for you to do with as you please.

“At least one rectangle is a square” is logically equivalent to any existential instantiation thereof, given that it is an existential claim in the first place.

“At least one rectangle is a square” changed to “at least THIS rectangle is a square” does not denote any logical difference.
Both statements are an affirmation that there exists at least one rectangle that is a square… and nothing more
What worse is when you say “a rectangle is a square” it is no more concretised than when you say “at least one rectangle is a square” so your claim that it’s an instantiation is inaccurate.

If you were to say “Rectangle #21 on page 13 of book 2 is a square” THAT would be an instantiation… provided such a book existed.

In either case, you would not get any contradiction to occur from an instantiation. You would have to show the SAME rectangle that is a square is also not a square for there to be a contradiction.
But you have dodged this point before…

“A rectangle is a square” does not contradict “a rectangle is not a square” unless both statements were referring to the SAME rectangle… or unless both were categorical statements.

Neither of which you have demonstrated to any degree.

The proceeding statement may surprise you. I agree. The same rectangle that is a square is also not a square. As I’ve pointed out in a reply of mine at viewtopic.php?p=2695954#p2695954,

As a result, that any rectangle referred to by the statement “a rectangle is not a square” can be made to be the same rectangle referred to by the statement “a rectangle is a square.” In making the referents the same, a contradiction is produced.

I like how you use the word referring there. Since I am allowed to set the referent of “a rectangle” in each of the two statements to any rectangle, I am allowed to set the referent of “a rectangle” in each of the two statements to the same rectangle and have a contradiction.

Only by electing to ignoring the law of non-contradiction.
In other words, you are not allowed to do this by the laws of logic… so I don’t know who or what you think is “allowing” you to make that move.

Consider for a moment that in this instance violating the law of non-contradiction is elected and not only logically unnecessary it’s very specifically prohibited by that axiom…

All you are demonstrating is that the laws of nature allow you to make nonsensical and contradictory statements… which for anyone who frequents this forum will not be news.

The laws of logic do allow it to be done. Regardless of whether my argument for trivialism is sound, no law of logic is violated.

If trivialism is true, then no law of logic is violated. If trivialism is false, then no law of logic is violated.

Perhaps you should place more emphasis on what I’ve already said in this thread. There’s no need for me to restate what I’ve already said.

Regarding my post at viewtopic.php?p=2695639#p2695639,

Similarly, “any rectangle is not a square” is not logically equivalent to “a rectangle is not a square.” “Any rectangle is not a square” is never true, but “a rectangle is not a square” is sometimes but not always true.

The referents of the subjects of the hypotheses of (1) and (2) rather are one particular rectangle each. That is, the referent of the subject of the hypothesis of (1) is a particular rectangle, and the referent of the subject of the hypothesis of (2) is a particular rectangle.

To correct myself, I disagree; there is no such requirement. Whether the rectangle that is a square is the same rectangle that is not a square is not relevant. As (1) and its inverse (2) demonstrate, the negation of “a rectangle is a square” is not “the rectangle is not a square;” it is “a rectangle is not a square.”

Since my last post, I have posted an image that helps describe my argument for trivialism. The image is located at twitter.com/paulemok/status/975234801409118208.

Paul, here is what is wrong with your argument.

Premise 1: There exists a rectangle that’s a square.

Absolutely true. A 1 x 1 rectangle is a square, for example. There are many others.

Premise 2: There exists a rectangle that’s not a square.

Also true. A 1 x 2 rectangle is not a square, for example. There are many others.

Premise three: P1 and P2 are each other’s negations.

No no no no no. False. That’s your error. If P1 and P2 were each other’s negation, your explosion argument would work. But P1 and P2 are not each other’s negations.

What is the negation of “There exists a rectangle that’s a square?” It is: “There does NOT exist a rectangle that’s a square.” That’s the proper negation. Now your argument fails.

I also wanted to mention that I looked for your post on thephilosophyforum.com/ and I see they removed it. Sorry that happened, their moderation is a little inconsistent over there to say the least. Wish I’d seen the thread. Sometimes one of their moderators gets a bug up their butt and deletes something that annoys them. I’ve had it happen to me. Don’t take it personally.

Anyway, you do see that the negation of “There exists an X that’s a Y” is, “There does NOT exist an X that’s a Y.” Equivalently, the negation can be stated as “All Y’s are NOT X’s.”

It is perfectly possible that there exists an X that’s a Y, and there exists an X that’s not a Y. Those two statements can both be true, and often are. There’s a fish that’s a shark, and there’s some other fish that’s not a shark. Those statements do not negate each other! This is basic predicate logic. You need to read up on the negation of the existential quantifier. Start here. en.wikipedia.org/wiki/Existenti … tification

This is not a premise of my argument. I am aware that “there exists a rectangle that is not a square” is not the negation of “there exists a rectangle that is a square.” Mad Man P thought I was claiming something similar. My reply to Mad Man P is located in my post at viewtopic.php?p=2695639#p2695639.

What may be of interest regarding my argument, however, is whether “a rectangle is not a square” is the negation of “a rectangle is a square.” It is.

It may be said that the negation of “a rectangle is a square” is “the rectangle is not a square.” However, the change from the indefinite article a to the definite article the is not justified through negation alone. The negation of a statement should always be the negation of the same, exact statement. “A rectangle is not a square” is a better negation of “a rectangle is a square” than “the rectangle is not a square” is.

Consider the following conditional statement (1) from my post at viewtopic.php?p=2695954#p2695954.

The contrapositive of (1) is the conditional statement that has the negation of the conclusion of (1) as the hypothesis and the negation of the hypothesis of (1) as the conclusion. The contrapositive of (1) is the following conditional statement.

(3) If a rectangle is not regular, then it is not a square.

(3) is true; no rectangles that are not regular are squares. Notice how, despite the difference in syntax between the subjects, the conclusion of (3), “it is not a square,” is considered to be the negation of the hypothesis of (1), “a rectangle is a square.” In (3), the antecedent of the pronoun it is the rectangle from the hypothesis of (3). Without further consideration, “a rectangle is not a square” is a better negation of “a rectangle is a square” than “it is not a square” is. In constructing the conclusion of (3), however, there is an additional consideration that comes into play than simply negating the hypothesis of (1). Like in (1) where the information that the rectangle of the conclusion of (1) is the same rectangle as the rectangle of the hypothesis of (1) is considered and provided, in (3), the information that the rectangle of the conclusion of (3) is the same rectangle as the rectangle of the hypothesis of (3) is considered and provided.

The purest negation of “a rectangle is a square” does not consider whether the rectangle in the negation is the same rectangle as in the original statement.

As several posters have already noted, you seem to be equivocating the English usage of “a” to sometimes mean:

  • There exists a rectangle that’s a square; and

  • All rectangles are squares.

Since you are making a point of formal logic, can you please say which interpretation of “a rectangle is a square” you mean?

wtf:

It should be clear through my use in this thread that I mean “a rectangle is a square” in a sense more like, although perhaps not exactly like, the former. As I’ve told phyllo in my post at viewtopic.php?p=2695639#p2695639,

There seems to be some sort of contradiction regarding the statement “a rectangle is a square.” The statement both is and is not logically equivalent to the statement “There exists a rectangle that is a square.” “A rectangle is a square” is logically equivalent to “there exists a rectangle that is a square” because a rectangle is a square if and only if there exists a rectangle that is a square. On the other hand, they are not logically equivalent because the former is sometimes but not always true, but the latter is always true.

It’s not a contradiction. It’s an ambiguity in natural language. In predicate logic we remove the ambiguity by use of either an existential quantifier, “There exists …” or a universal quantifier, “For all …”

You are just exploiting and/or being confused by an ambiguity in natural language.

It appears to be a contradiction to me.

You may be right. However, with respect to all I know, just because it’s ambiguous, doesn’t mean it’s not a contradiction. It seems much of my argument revolves around the contradictory nature of the statement “a rectangle is a square.”

I know what formal logic and the two quantifiers can do.

An ambiguity in natural language could imply a legitimate contradiction that infects all languages, natural and unnatural. To disregard the contradiction regarding “a rectangle is a square” would be or be similar to committing the red herring logical fallacy.

Omg you guys have been arguing about rectangles and squares for a page and a half? :open_mouth:

This is a matter of accepted definitions:

Rectangle - a parallelogram having four right angles.

Square - a rectangle having all four sides of equal length.

A square is a rectangle.

A rectangle is not always a square.

No. I just got here. I’m fresh meat.

As you should be able to see from this thread, it’s more than that.

I agree. Your statement is always true.

Again, I agree. Your statement is always true.

The statement “some rectangles are squares” is also always true. But as the argument I’ve given Mad Man P in my post at viewtopic.php?p=2695954#p2695954.

One interesting defense I have noticed against my argument for trivialism would be that the set of all possible worlds in which a rectangle is a square is always disjoint from the set of all possible worlds in which a rectangle is not a square. This defense does not work because it is false. The premises “some rectangles are squares” and “some rectangles are not squares” of my argument are both true in all possible worlds. Thus, the statement “a rectangle is a square” is true in all possible worlds, and the statement “a rectangle is not a square” is true in all possible worlds. So, a contradiction exists in all possible worlds. By the principle of explosion therefore, trivialism is true in all possible worlds.

No, “some” means “at least one”. “Is” means “equal to”.

At least one rectangle = square. That does not imply: All rectangles = squares.

You’re taking liberties with language and conflating/equivocating sets of shapes with specific shapes.

Some rectangles are squares implies at least one specific rectangle is a square and that’s all you can conclude.

Some men are doctors.
Some doctors are tall.
Therefore, some men are tall.

That’s false.

That’s false so every conclusion based on it does not follow.

Square? :confusion-shrug:

Nonsense.