In Support of Trivialism

[browser32]

My name is Paul E. Mokrzecki. In February, I wrote a somewhat similar opening post as this opening post, for a discussion named "An Argument for Trivialism" on another online philosophy forum. Unfortunately, that thread is no longer publicly accessible. The other online philosophy forum is located at https://thephilosophyforum.com/.

In the past year, I've been advancing an argument in favor of trivialism. Previous content regarding the argument, including a depicted variation of the argument, can be located through https://twitter.com/paulemok/status/970444013512871936. The argument may seem like a classic argument that has been defeated in the past. However, I am taken by how strong the argument is. The argument goes as follows.

Some rectangles are squares. So, a rectangle is a square. Some rectangles are not squares. So, a rectangle is not a square. It follows by conjunction introduction that "a rectangle is a square and a rectangle is not a square." But the quoted statement is a contradiction. Therefore, by the principle of explosion, trivialism is true. This concludes the argument.

[phyllo]

Some rectangles are squares. So, a rectangle is a square. Some rectangles are not squares. So, a rectangle is not a square. It follows by conjunction introduction that "a rectangle is a square and a rectangle is not a square." But the quoted statement is a contradiction. Therefore, by the principle of explosion, trivialism is true. This concludes the argument.

How is this different from :

Some mammals are cats. So, a mammal is a cat. Some mammals are not cats. So, a mammal is not a cat. It follows by conjunction introduction that "a mammal is a cat and a mammal is not a cat."

With the same logic flaw.

[fuse]

Previous content regarding the argument, including a depicted variation of the argument, can be located through https://twitter.com/paulemok/status/970444013512871936.

twitter, the perfect place to espouse trivialism
of course, non-trivialism must also be true - why privilege just one side of the trivial coin?

Some rectangles are squares. So, [any] rectangle is a square.

The implied [any] is the mistake in your premises.

First rule of making a good argument: eliminate ambiguity/assumption/hinted implication from your terms.

[browser32]

phyllo:

Your argument involving mammals and cats involves different categories as my argument involving rectangles and squares. Your argument involves the categories mammal and cat, but my argument does not involve those categories. Instead, my argument involves the categories rectangle and square, which are categories your argument does not involve. Both arguments, however, have the same form. The same conclusion that trivialism is true can be reached with your argument as it can be reached with my argument. However, neither of our arguments has a logic flaw. They are both sound arguments.

If you feel the arguments have "the same logic flaw," as you have contended, feel free to point out precisely what it is.

With respect to all I know, it is epistemically possible that the quoted statement "trivialism is true and false simultaneously" is true. So, if trivialism is false, then with respect to all I know, it is epistemically possible that the quoted statement "trivialism is true" is still true. To show that trivialism is false therefore is not enough to convince me that I'm wrong.

fuse:

of course, non-trivialism must also be true

I understand that if trivialism is true, trivialism is also false.

why privilege just one side of the trivial coin?

The truth of trivialism is less evident than is the falsity of trivialism. So, it seems more appropriate to better show the truth of trivialism rather than to better show the falsity of trivialism.

The implied [any] is the mistake in your premises.

There is no implied any in my premises. The statement "a rectangle is a square" is not logically equivalent to the statement "any rectangle is a square." The former statement is sometimes but not always true, while the latter statement is never true. By logically equivalent statements, I am referring to statements that always have the same truth value as each other.

First rule of making a good argument: eliminate ambiguity/assumption/hinted implication from your terms.

Changing or formalizing a statement to make it more suitable for a particular argument may change the properties of the statement. Some of the properties of the original statement may not carry over to the transformed statement. Also, some of the properties of the transformed statement may not have been present in the original statement. It seems that when "making a good argument," formalizing a statement, for example, into first-order logic, isn't always the best option.

[Mad Man P]

Some rectangles are squares. So, a rectangle is a square.

This is a non sequitur...

[browser32]

Mad Man P:

No, it is not a non sequitur. The following argument shows how the statement "a rectangle is a square" follows from the statement "some rectangles are squares."

Argument. It is given that some rectangles are squares. The given statement is conventionally taken to mean that at least one rectangle is a square. Otherwise, since the word rectangles is plural, it is taken to mean that at least two rectangles are squares. In either case, at least one rectangle is a square. Since at least one rectangle is a square, it is also true that one rectangle is a square. The implication of the previous statement is similar to the implication that if at least five peaches fell off of a peach tree today, then it is also true that five peaches fell off of the tree today. Since one rectangle is a square, one rectangle that is a square is a rectangle. So, a rectangle is a square. This concludes the argument.

[Carleas]

Changing or formalizing a statement to make it more suitable for a particular argument may change the properties of the statement.

This point applies with even greater force to your argument. The principle of explosion no longer applies when you are dealing with the imprecision of every day language, as it follows from formal logic. In every day speech, it isn't the case that two contradictory statements imply any arbitrarily chosen conclusion. In every day speech, the same word or locution can be and often are used to mean different things. When stated precisely, there is no contradiction between them, and so no explosion:

"Some rectangles are squares" ->
1) There exists a rectangle that is a square.

"a rectangle is a square" ->
This can be translated in a couple ways:
2a) Rectangle S is a square.
or
2b) If (x is a rectangle) then (x is a square).

Only 2a is true from 1.

"Some rectangles are not squares." ->
3) There exists a rectangle that is not a square.

"a rectangle is not a square." ->
again, this can be translated in more than one way:
4a) Rectangle N is not a square.
or
4b) If (x is a rectangle) then (x is not a square).

only 4a is true from 3.

"a rectangle is a square and a rectangle is not a square"
If this is just 2a & 4a, it's true and non-contradictory:
5) Rectangle S is a square and rectangle N is not a square.

[phyllo]

Your argument involving mammals and cats involves different categories as my argument involving rectangles and squares. Your argument involves the categories mammal and cat, but my argument does not involve those categories. Instead, my argument involves the categories rectangle and square, which are categories your argument does not involve.

A square is a rectangle with particular distinguishing characteristics, just as a cat is a mammal with particular distinguishing characteristics.

I switched from rectangles and squares to mammals and cats so that the logic would be more intuitive and not dependent on mathematical concepts and definitions.

Your first statement : Some rectangles are squares.

Your second statement : So, a rectangle is a square.

Your second statement has merely removed the word "some" from the first statement. That change has altered the meaning to "Every rectangle is a square" or alternatively "Any rectangle is a square" as Fuse pointed out.

In other words, your second statement is a contradiction of the first and it's clearly false.

One can see that more easily by substituting mammals and cats :

1.Some mammals are cats.
2.So, a mammal is a cat.

"A cat is a mammal" is true but "A mammal is a cat" is false.

Another alternate meaning for your second statement is "A particular rectangle is a square", but in that case referring to any other rectangle and saying that it is not a square does not produce a contradiction.

[Mad Man P]

Argument. It is given that some rectangles are squares. The given statement is conventionally taken to mean that at least one rectangle is a square. Otherwise, since the word rectangles is plural, it is taken to mean that at least two rectangles are squares. In either case, at least one rectangle is a square. Since at least one rectangle is a square, it is also true that one rectangle is a square. The implication of the previous statement is similar to the implication that if at least five peaches fell off of a peach tree today, then it is also true that five peaches fell off of the tree today. Since one rectangle is a square, one rectangle that is a square is a rectangle. So, a rectangle is a square. This concludes the argument.

In which case, the statement "a rectangle is a square" is meant in the singular, that is to say a specific rectangle that is a square.
For there to be a contradiction that very same rectangle that is a square must also NOT be a square... otherwise there is no contradiction.

You see all you're saying is "there is at least one rectangle that is a square and there is at least one rectangle that is NOT a square" there is no contradiction in that.

[browser32]

The principle of explosion no longer applies when you are dealing with the imprecision of every day language, as it follows from formal logic.

I disagree. The principle of explosion always applies to all statements, regardless of their language and style.

In every day speech, it isn't the case that two contradictory statements imply any arbitrarily chosen conclusion.

I disagree. Any two contradictory statements imply "any arbitrarily chosen conclusion." How formally statements are expressed is never relevant.

In every day speech, the same word or locution can be and often are used to mean different things.

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 https://twitter.com/paulemok/status/970468311828975616.

5) Rectangle S is a square and rectangle N is not a square.

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.

I switched from rectangles and squares to mammals and cats so that the logic would be more intuitive and not dependent on mathematical concepts and definitions.

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.

That change has altered the meaning to "Every rectangle is a square" or alternatively "Any rectangle is a square" as Fuse pointed out.

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.

"A cat is a mammal" is true but "A mammal is a cat" is false.

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.

Another alternate meaning for your second statement is "A particular rectangle is a square", but in that case referring to any other rectangle and saying that it is not a square does not produce a contradiction.

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."

You see all you're saying is "there is at least one rectangle that is a square and there is at least one rectangle that is NOT a square" there is no contradiction in that.

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.

[fuse]

The statement "a rectangle is a square" is not logically equivalent to the statement "any rectangle is a square."

What you must do to make your case is explain exactly the meaning of a rectangle is a square. Logical equivalence is effectively determined only after statements have been properly clarified/formalized.

There is no implied any in my premises.

Your argument will be quite confusing to anyone who speaks/writes/comprehends proper English. See below.

Some rectangles are squares. So, a rectangle is a square.

The indefinite article (a, an) is used before a noun that is general or when its identity is not known.

If by "a rectangle" you mean rectangles in general, then the first two premises are already in contradiction. If by "a rectangle" you mean some particular rectangle whose identity is not known, then your argument is poor since your terms are not well defined, and so remains inconclusive and proves no contradiction.

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.

No, it doesn't. You've just explained that the statements reference two different, particular rectangles ("the rectangle in the quoted statement..."). Thus, you would not faithfully capture the original meaning of either statement were you to substitute in the general definition: "one quadrilateral that has four right angles."

[browser32]

What you must do to make your case is explain exactly the meaning of a rectangle is a square. Logical equivalence is effectively determined only after statements have been properly clarified/formalized.

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.

You've just explained that the statements reference two different, particular rectangles ("the rectangle in the quoted statement...").

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.

Thus, you would not faithfully capture the original meaning of either statement were you to substitute in the general definition: "one quadrilateral that has four right angles."

Every rectangle is one parallelogram that has four right angles. Each rectangle is one parallelogram that has four right angles.

[Mad Man P]

You see all you're saying is "there is at least one rectangle that is a square and there is at least one rectangle that is NOT a square" there is no contradiction in that.

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."

Argument. It is given that some rectangles are squares. The given statement is conventionally taken to mean that at least one rectangle is a square. Otherwise, since the word rectangles is plural, it is taken to mean that at least two rectangles are squares. In either case, at least one rectangle is a square. Since at least one rectangle is a square, it is also true that one rectangle is a square. The implication of the previous statement is similar to the implication that if at least five peaches fell off of a peach tree today, then it is also true that five peaches fell off of the tree today. Since one rectangle is a square, one rectangle that is a square is a rectangle. So, a rectangle is a square. This concludes the argument.

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

[browser32]

Mad Man P:

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

[Mad Man P]

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.

[browser32]

You cannot perform a deduction without a second premise

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.
S(x) is the statement "x is a square."
Statements (Reasons)
1. ꓱx S(x) (Premise)
2. S(d) (Existential instantiation from (1))
This concludes the example.

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.

What you have said is unclear.

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

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.

at least one rectangle is a square. Since at least one rectangle is a square, it is also true that one rectangle is a square. The implication of the previous statement is similar to the implication that if at least five peaches fell off of a peach tree today, then it is also true that five peaches fell off of the tree today. Since one rectangle is a square, one rectangle that is a square is a rectangle. So, a rectangle is a square.

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.

[Mad Man P]

That is not true. A deduction does not have to involve at least two premises.

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.

[browser32]

You would have to show the SAME rectangle that is a square is also not a square for there to be a contradiction.

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,

"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.

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.

"A rectangle is a square" does not contradict "a rectangle is not a square" unless both statements were referring to the SAME rectangle

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.

[Mad Man P]

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.

[browser32]

you are not allowed to do this by the laws of logic

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.

I don't know who or what you think is "allowing" you to make that move.

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.

[browser32]

Regarding my post at viewtopic.php?p=2695954#p2695954, I think I may have been wrong; the referents of the subjects of the hypotheses of (1) and (2) are not any rectangle. If they were, then it seems I should be able to substitute the statement "any rectangle is a square" for "a rectangle is a square" and substitute the statement "any rectangle is not a square" for "a rectangle is not a square." However, those substitutions cannot validly be done. As I've said in my post at viewtopic.php?p=2695639#p2695639,

"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.

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.

You would have to show the SAME rectangle that is a square is also not a square for there to be a contradiction.

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 https://twitter.com/paulemok/status/975234801409118208.

[wtf]

Regarding my post at ...

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 http://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. https://en.wikipedia.org/wiki/Existenti ... tification

[browser32]

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

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.

(1) If a rectangle is a square, then it is regular.

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.

[wtf]

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?

[browser32]

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,

I am not speaking generally.

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.