In Support of Trivialism

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.

The word is doesn’t necessarily mean equal to. It can be used to provide an incomplete description of a thing, as the following example exemplifies.

Example. Tom is handsome. However, Tom does not equal handsome. The word handsome is an adjective that modifies, but does not completely describe, the noun Tom. This concludes the example.

It seems I have long considered that in statements such as “a rectangle is a square,” “a square” is not necessarily a complete description of “a rectangle,” but is rather used as an adjective phrase that modifies the noun phrase “a rectangle.”

I understand that: at least one rectangle is a square does not imply all rectangles are squares.

It should be clear throughout this thread that when I say “a rectangle is a square,” I mean “one particular rectangle is a square.” You don’t seem to be paying enough attention to context and what I’ve already said in this thread.

I agree that: some rectangles are squares implies at least one specific rectangle is a square. However, with respect to all I know, “at least one specific rectangle is a square” may not be all I can conclude from “some rectangles are squares.”

I am aware the argument you gave there is invalid.

I don’t agree. What I said there is clearly true. You don’t seem to have morally analyzed what I have said.

I know that some of my conclusions are paradoxical. But just because a conclusion is paradoxical, doesn’t mean it’s not true.

My reasoning at least appears to be correct. Thus, trivialism at least appears to be true.

It is not just my conclusions that should be analyzed, but it is also the reasoning I employ to arrive at my conclusions that should be analyzed. One sound proof alone is enough to successfully make my case. In order to show that I’m wrong, a fallacy within the reasoning I have already provided must be presented.

That is incorrect grammar; convenient slang, for lack of better lingo. Tom’s appearance is handsome. Tom is not handsome even if his appearance is handsome.

You’re struggling to obfuscate by juggling words in order to shove a rectangle peg in a square hole. Words are merely labels and just because you’ve worked some magic with them doesn’t mean you’ve done anything with the underlying objects they represent. You cannot claim a rectangle is a square and no amount of semantic hocus pocus can change that. I don’t mean to seem obtuse, but surely you understand this acute angle is right :wink:

A square is a complete descriptor of a specific type/subset of rectangle.

Polygon/quadrilateral/parallelogram/rectangle/square

A square has all the properties of a rectangle, plus some, so a square is a complete descriptor.

Then why are you asserting that it does?

I’ve only given the thread a cursory skimming in light of my perception of the silliness of the premise that a rectangle is a square. Surely you understand my reticence to burn the midnight glucose on the matter.

Some X are Y. All you can conclude from that is at least one member of X is a member of Y. You certainly couldn’t conclude X is Y. You’re conflating categories with members.

I don’t see how my morality is in question here; I’ve given you every benefit of objectivity and I’m not biased in any way such that I’d have incentive to be immoral concerning this.

Well, your reasoning may in fact be ok, but you’re conflating sets with members and struggling to justify with semantic hocus pocus rather than saying “Oh I see now! Well, back to the drawing board! Thanks!”

And the trivial thing is that I’ve no clue what that even is.

Well, I posted a picture of a rectangle and asked you if it is a square; that’s about as empirical as it gets.

No, it is not incorrect grammar. It is basic English. It is common knowledge.

On page 1 of Discrete Mathematics and Its Applications, Sixth Edition by Kenneth H. Rosen, copyright 2007, Rosen implies all propositions are sentences. So, an essential part of every proposition is syntactic. Words thus provide a required component of propositions. If there are no sentences, then there are no propositions.

There is a difference between a proposition and a sentence, but there is no difference, at least in how I use them, between a proposition and a statement.

Rosen suggests on page 3 that sentences with variable subjects are technically not propositions unless a particular subject is ascertained. This is not a problem for the proposition “a rectangle is a square” because the particular subject is a particular rectangle.

What I have asserted is that some rectangles are squares implies a rectangle is a square. What I mean by that is at least one rectangle is a square implies a particular rectangle is a square.

This doesn’t seem true. As you yourself have said, in the same post,

There’s an encyclopedia article about trivialism at en.wikipedia.org/wiki/Trivialism.

See, this is where you’re being immoral by not conceding “Tom is handsome” means “Tom’s appearance is handsome” and that the former is shorthand relying on people to be able to interpolate meaning without strict grammatical adherence and you’re resorting to dogmatic denial (immoral) to avoid conceding that “is” is “=” because it would undermine your premise (motive).

Is this an appeal to authority argument? So what if Rosen asserts all propositions are sentences? If a sentence did not accuracy describe the proposition, then they are not equivalent. Further, a sentence is a concept to describe a proposition rather than a sentence being a proposition.

A sentence is a conceptual tool to convey other concepts which can take the form of a proposition. The tool of conveyance is not the proposition just like a raft for crossing a river is not the same as the passengers seeking to cross.

So that illustrates the point I made above in that you have to explain what your sentence means in order to capture the proposition and that should be your first clue that your sentence is inaccurate.

If you say: “a rectangle is a square” and then proceed to have to explain what you mean by that, then your statement does not embody nor represent your proposition.

What you should have said is: “some rectangles are squares” because that requires no further explanation.

Because I honestly admitted that I perceive the premise as being silly makes me immoral? Where have I dogmatically refused to acknowledge a point?

How can a ball be red and not red without splitting perspectives? This reminds me of the tetralemma I’ve been discussing here viewtopic.php?f=5&t=193673&start=300#p2697559

If a ball is red, then it’s a subjective interpretation. If the same red ball is also not red, then it is an objective interpretation by realizing that color does not exist objectively. The perspectives are split. Within the same frame of reference, both cannot be true.

I agree that “Tom is handsome” means “Tom’s appearance is handsome,” but that does not defeat my point. Now, instead of having handsome be an incomplete descriptor for Tom, you have handsome as an incomplete descriptor for Tom’s appearance. Tom’s appearance does not equal handsome. Handsome is a general, abstract quality that other peoples’ appearances sometimes have and that some peoples’ appearances have in theory or fiction. There may be other men in addition to Tom, such as Caleb and Fabian, whose appearances are also handsome. Handsome, alone, does not completely, exclusively, and uniquely describe Tom’s appearance.

As I had explained earlier in this thread in my posts at viewtopic.php?p=2695954#p2695954, there are multiple senses of equality. Two rectangles can be equal in one sense, but unequal in another sense. The word is and the equality symbol = are not as straightforward in their meanings as you may think they are.

Yes, it is. It’s an argument that has cited and is backed by a well respected source.

Rosen’s assertion is good evidence that all propositions are sentences.

Rosen and I both agree that not all sentences are propositions.

Rosen has claimed that a sentence can be a proposition. They can be the same thing. While I am surprised by that idea, I do not object to it. If you do, perhaps you could give some counterexamples of propositions that are not sentences.

I did say that, in my original post, at viewtopic.php?p=2695519#p2695519, first sentence of the third paragraph.

The statement “a rectangle is a square” is a statement of basic, common English. Statements like that, including “a quadrilateral is a parallelogram,” “a cucumber is green,” “one angle is congruent to a second angle,” and “two lines are parallel,” are statements of basic, common English. They are used in high school geometry textbooks here in the United States, including the aforementioned Geometry (2004) and Larson Geometry (2012). In some of my previous discussions involving my argument for trivialism, I have cited some grade school geometry textbooks, other than the aforementioned two, that use simple statements of the described type. Those discussions are available through a link I have provided in the original post for this thread.

I quote you from ilovephilosophy.com/viewtopic.ph … 0#p2697559, which you previously cited.

If trivialism is true, then a light can be on and not on simultaneously.

I used to have a postulate in my Action-Reaction Theory called the Postulate of Temporal Extensionality. There’s a video about the postulate that involves a light being simultaneously on and off; it’s at facebook.com/Paul.E.Mokrzec … 008356517/.