browser32 wrote:There seems to be some sort of contradiction regarding the statement "a rectangle is a square."

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.

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."

wtf wrote:In predicate logic we remove the ambiguity by use of either an existential quantifier, "There exists ..." or a universal quantifier, "For all ..."

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

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

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.

Serendipper wrote:This is a matter of accepted definitions:

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

Serendipper wrote:A square is a rectangle.

I agree. Your statement is always true.

Serendipper wrote:A rectangle is not always a square.

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=2695612#p2695612 proves, some rectangles are squares implies a rectangle is a square. Since the hypothesis "some rectangles are squares" is always true, it follows that the conclusion "a rectangle is a square" is also always true. Thus, a rectangle is always a square. So by conjunction introduction, a rectangle is always a square and a rectangle is not always a square. I previously pointed out this contradiction 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.

browser32 wrote:The statement "some rectangles are squares" is also always true. But as the argument I've given Mad Man P in my post at http://www.ilovephilosophy.com/viewtopi ... 2#p2695612 proves, some rectangles are squares implies a rectangle is a square.

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.

Since the hypothesis "some rectangles are squares" is always true, it follows that the conclusion "a rectangle is a square" is also always true.

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

Thus, a rectangle is always a square.

Square?

So by conjunction introduction, a rectangle is always a square and a rectangle is not always a square.

Serendipper wrote:No, "some" means "at least one". "Is" means "equal to".

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."

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

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

Serendipper wrote:You're taking liberties with language and conflating/equivocating sets of shapes with specific shapes.

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.

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

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."

Serendipper wrote:Some men are doctors. Some doctors are tall. Therefore, some men are tall.

That's false.

I am aware the argument you gave there is invalid.

Serendipper wrote:That's false so every conclusion based on it does not follow.

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

Serendipper wrote:Nonsense.

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.

Serendipper wrote:No, "some" means "at least one". "Is" means "equal to".

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.

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

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."

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

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

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

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

Then why are you asserting that it does?

Serendipper wrote:You're taking liberties with language and conflating/equivocating sets of shapes with specific shapes.

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'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.

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

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."

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.

Serendipper wrote:That's false so every conclusion based on it does not follow.

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

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.

Serendipper wrote:Nonsense.

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.

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!"

Thus, trivialism at least appears to be true.

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

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.

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

Serendipper wrote:That is incorrect grammar; convenient slang, for lack of better lingo.

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

Serendipper wrote: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.

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.

Serendipper wrote:Then why are you asserting that it does?

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.

Serendipper wrote: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.

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

Serendipper wrote: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.

Serendipper wrote:And the trivial thing is that I've no clue what that even is.

Serendipper wrote:That is incorrect grammar; convenient slang, for lack of better lingo.

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

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).

Serendipper wrote: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.

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.

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.

Serendipper wrote:Then why are you asserting that it does?

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.

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.

Serendipper wrote: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.

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

Serendipper wrote: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.

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

Serendipper wrote:And the trivial thing is that I've no clue what that even is.

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.

Serendipper wrote: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).

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=2695639#p2695639 and 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.

Serendipper wrote:Is this an appeal to authority argument?

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

Serendipper wrote:So what if Rosen asserts all propositions are sentences?

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

Serendipper wrote:If a sentence did not accuracy describe the proposition, then they are not equivalent.

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

Serendipper wrote:Further, a sentence is a concept to describe a proposition rather than a sentence being a proposition.

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.

Serendipper wrote:What you should have said is: "some rectangles are squares" because that requires no further explanation.

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.

Serendipper wrote:The only way a light can be on and not on is if there are two lights.

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 https://www.facebook.com/Paul.E.Mokrzec ... 008356517/.

browser32 wrote:I agree that "Tom is handsome" means "Tom's appearance is handsome,"

Thank you

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.

This is interesting but I'm inclined to believe there is more unspoken meaning embedded in the phrase because one attribute of Tom simply must be equal to handsome; we just haven't decided exactly what it is.

I agree that Tom may also be tall, or fat, or whatever in addition to handsome so it is therefore not a complete descriptor of his appearance, but on the other hand "handsome" is a complete descriptor of something. If that is true, then once again we find we haven't constructed the sentence properly to accurately represent the proposition.

"Tom's appearance is handsome." That doesn't work because the statement is objective while "handsome" is subjective.

"Sally finds Tom's appearance is handsome." Now we've aligned the subjectivity, but maybe Tom is not always handsome.

"Throughout the day, Sally finds Tom's appearance is handsome." Now we're aligned temporally, but can Tom be perfectly handsome?

"Throughout the day, Sally finds Tom's appearance is consistent with what she categorically regards as handsome." On and on until it seems like a police report as we zero-in on exactly what is being equivocated.

Two rectangles being equal in one sense and not another is just another example of what was stated above: there are embedded meanings that need to be unpackaged. What is equivalent between all rectangles is having 4 parallel sides and right angles, so it's not the rectangles that are equal, but aspects of them. If you want to state equivalence beyond that, then the rectangles in fact must be the same size.

Serendipper wrote:Is this an appeal to authority argument?

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

Are you aware of instances where respected sources have been wrong? If so, then how can authority ever be a definitive authority?

Serendipper wrote:So what if Rosen asserts all propositions are sentences?

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

An assertion is not evidence. Evidence backs the assertion.

Serendipper wrote:If a sentence did not accuracy describe the proposition, then they are not equivalent.

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

What sentences are not propositions? Questions? Statements state propositions and questions question them.

Serendipper wrote:Further, a sentence is a concept to describe a proposition rather than a sentence being a proposition.

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 gave you an example: a raft used for crossing a river is not the same as the passengers; it is just a means of getting across the river; a conveyance. Likewise, a sentence is not the proposition, but carries the proposition from one head to another head.

Serendipper wrote:What you should have said is: "some rectangles are squares" because that requires no further explanation.

You said "Some rectangles are squares. So, a rectangle is a square." You cannot make that leap. Alternatively, you could have said "Some rectangles are squares. So, a specific type of rectangle is a square."

browser32 wrote:Some rectangles are squares.

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 slang.

The color of the skin of specific varieties of ripened, but not over-ripened cucumbers are often green. Instead of saying all that, we simply say cucumbers are green, but it's not a precise statement. You cannot take advantage of linguistic conveniences to prove something about reality.

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.

It is either unfortunate that those textbooks use such language or the books rely on the student's ability to interpolate properly.

Serendipper wrote:The only way a light can be on and not on is if there are two lights.

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 https://www.facebook.com/Paul.E.Mokrzec ... 008356517/.

That assumes that all events that have ever happened or ever will happen actually exist, right now.

Start at about 4:00

Then toss in Many Worlds where all possibilities also exist, then we can say that everything and anything exists right now.

Now what?

Little Fluffy got hit by a car? Oh don't fret, there are infinite Fuffies still alive somewhere.

No, your honor, I didn't steal the money. The money is still there in the past and in multiple universes. There is no crime.

I suppose this means I haven't been born yet. Oh good! There are a few things I'd like to do differently

Serendipper wrote:It is either unfortunate that those textbooks use such language or the books rely on the student's ability to interpolate properly.

I believe the books are presenting the material correctly. So correctly, in fact, that I treat their presentation like a standard in my analysis.

browser32 wrote: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.

Both instances of the noun phrase "a rectangle" there do not have to refer to the same rectangle. They simply have to refer to a rectangle. What rectangle or rectangles they refer to does not matter. Whether they refer to the same rectangle or to different rectangles does not matter.

Serendipper wrote:It is either unfortunate that those textbooks use such language or the books rely on the student's ability to interpolate properly.

I believe the books are presenting the material correctly. So correctly, in fact, that I treat their presentation like a standard in my analysis.

Yes but that's just your opinion.

browser32 wrote: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.

Both instances of the noun phrase "a rectangle" there do not have to refer to the same rectangle. They simply have to refer to a rectangle. What rectangle or rectangles they refer to does not matter. Whether they refer to the same rectangle or to different rectangles does not matter.

The category of "rectangle" is different from a specific rectangle. You're conflating/equivocating categories with members of categories.

"A rectangle is a square" should be changed to "a (specific) rectangle is (in the category of) a square".

You can't say the category of rectangle is the same as the category of square nor that a specific rectangle is necessarily the same as a specific square.

Serendipper wrote:The category of "rectangle" is different from a specific rectangle.

I know that.

Serendipper wrote:"A rectangle is a square" should be changed to "a (specific) rectangle is (in the category of) a square".

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

browser32 wrote:Some rectangles are squares. So, a specific rectangle is in the category of square. Some rectangles are not squares. So, a specific rectangle is not in the category of square. It follows by conjunction introduction that "a specific rectangle is in the category of square and a specific rectangle is not in the category of square." But the quoted statement is a contradiction. Therefore, by the principle of explosion, trivialism is true.

Not quite.

"a specific rectangle (R1) is in the category of square and a specific rectangle (R2) is not in the category of square." No contradiction.

Whether each instance of “a specific rectangle” refers to the same specific rectangle or to different specific rectangles does not matter. Each instance simply refers to a specific rectangle.

Whether each instance of “a specific rectangle” refers to the same specific rectangle or to different specific rectangles does not matter. Each instance simply refers to a specific rectangle.

If each instance refers to a specific rectangle, then in that instance, the specific rectangle will either be a square or not. Still no contradiction.

It's like your example with the lamps: the lamp was on in the instance in the past and off in the instance in the present.

You may have some luck in the quantum realm where two particles can be in different states at the same time, but the particles could be waves rather than particles so there's some debate even then.

Why are you so determined to prove trivialism anyway?

Serendipper wrote:If each instance refers to a specific rectangle, then in that instance, the specific rectangle will either be a square or not.

I agree.

Serendipper wrote:Still no contradiction.

There is still a contradiction because

browser32 wrote:a specific rectangle is in the category of square and a specific rectangle is not in the category of square

is still true.

Serendipper wrote:Why are you so determined to prove trivialism anyway?

Proving trivialism gives us a proof for every claim that can possibly be made. No more proofs will ever be required. Instead of proving claims with possibly long, complicated, and tedious proofs, academics, scientists, and others can advance to other things.

browser32 wrote:There is still a contradiction because

browser32 wrote:a specific rectangle is in the category of square and a specific rectangle is not in the category of square

is still true.

It can only be a contradiction if it is the same rectangle that is and is not a square, which it cannot be by definition of the proposition.

Here is an image I found online

Members of the area labeled 5.1 do not mingle with members of the area labeled 3.4. The area 3.4 represents squares and 5.1+3.4 represents rectangles. You may as well be saying 3.4 = 5.1+3.4

Serendipper wrote:Why are you so determined to prove trivialism anyway?

Proving trivialism gives us a proof for every claim that can possibly be made. No more proofs will ever be required. Instead of proving claims with possibly long, complicated, and tedious proofs, academics, scientists, and others can advance to other things.

Serendipper wrote:It can only be a contradiction if it is the same rectangle that is and is not a square, which it cannot be by definition of the proposition.

The subject of the statement "a specific rectangle is in the category of square" and the subject of the statement "a specific rectangle is not in the category of square" are the same. That common subject is "a specific rectangle." Thus, "a specific rectangle is not in the category of square" is the negation of "a specific rectangle is in the category of square." Since, according to the modified argument, both are true, a contradiction exists.

The subjects are the same in at least one sense. This is like how there are two separate subjects in the sense that the subjects are of two separate statements, but there is exactly one subject in the sense that the subject of both statements is "a specific rectangle." Since the subjects are the same in at least one sense, "a specific rectangle is not in the category of square" is the negation of "a specific rectangle is in the category of square" in at least one sense. Since, according to the modified argument, both are true, a contradiction exists in at least one sense. So, a contradiction exists. Therefore, by the principle of explosion, trivialism is true.

Serendipper wrote:Well then I claim trivialism is false

I've already acknowledged, in a reply to fuse at viewtopic.php?p=2695577#p2695577, that if trivialism is true, then trivialism is also false. Since trivialism is true, I agree with you that trivialism is false. Nonetheless, trivialism is true.

Serendipper wrote:It can only be a contradiction if it is the same rectangle that is and is not a square, which it cannot be by definition of the proposition.

The subject of the statement "a specific rectangle is in the category of square" and the subject of the statement "a specific rectangle is not in the category of square" are the same.

No.

That common subject is "a specific rectangle."

R1 not equal to R2. They are defined to be different by your proposition that one is a square and the other is not. You cannot define a contradiction to exist and then claim that means something.

Thus, "a specific rectangle is not in the category of square" is the negation of "a specific rectangle is in the category of square." Since, according to the modified argument, both are true, a contradiction exists.

No contradiction.

The subjects are the same in at least one sense.

Irrelevant.

This is like how there are two separate subjects in the sense that the subjects are of two separate statements, but there is exactly one subject in the sense that the subject of both statements is "a specific rectangle."

They are separate senses.

Since the subjects are the same in at least one sense, "a specific rectangle is not in the category of square" is the negation of "a specific rectangle is in the category of square" in at least one sense.

No

Since, according to the modified argument, both are true, a contradiction exists in at least one sense. So, a contradiction exists. Therefore, by the principle of explosion, trivialism is true.

Nope

Serendipper wrote:Well then I claim trivialism is false

I've already acknowledged, in a reply to fuse at http://www.ilovephilosophy.com/viewtopi ... 7#p2695577, that if trivialism is true, then trivialism is also false. Since trivialism is true, I agree with you that trivialism is false. Nonetheless, trivialism is true.

If it's false it can't be true. This is the silliest way to use time.

All you're doing is claiming: X contradicts Y because you said so; therefore Z is true and because it's true, it's also false. It's a strawman. You set it up just to knock it down and then claim it means something.

I suppose we can keep going back n forth like school children: no huh uh, yes huh, no huh uh, yes huh, until one of us gets bored with it. It's slow around ILP now, so let's see what happens I guess.

Serendipper wrote:R1 not equal to R2. They are defined to be different by your proposition that one is a square and the other is not.

R1 may not be equal to R2 in some sense of equality. But in some sense of equality, R1 is equal to R2 because each refers to a specific rectangle. R1 = a specific rectangle. R2 = a specific rectangle. Therefore, by the substitution property of equality, R1 = R2.

R1 and R2 share a property in common with each other, and that common property alone can be used to equate the two objects in some sense of equality. R1 and R2 are equal in merely the sense that they share the same name. That name is "a specific rectangle."

Example. Let b be a United States dime and f be a United States dime. b does not equal f in some sense of equality because b was manufactured in 2003, but f was manufactured in 2007. However, b is equal to f in the sense that both share the property of being a United States dime. Furthermore, b is equal to f in the sense that both have the same monetary value, 10 cents. This concludes the example.

The existence of different senses of identity like in the example was discussed in a course I took my first semester of college called Minds and Machines. I did not graduate from college, but according to an unofficial transcript, I did get an A- in the course.

Serendipper wrote:They are separate senses.

I agree, and I am not conflating senses there. There, I am explaining how there are at least two senses.

browser32 wrote: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.

Mad Man P:

I don't believe my argument can be presented in first-order logic; at least not in a traditional way as I envision it. It is somewhat bold for one to think that all arguments can be presented in commonly accepted, formal, symbolic logic. The existence of second-order logic and other logical systems suggest that not all arguments can be presented in first-order logic. See the first two sentences of https://en.wikipedia.org/wiki/Second-or ... sive_power. My argument is presented in the language of symbolic logic known as English.

We previously talked about the logical rule of inference known as existential instantiation. With the rule of existential instantiation, I would be required to give the rectangle referred to by "a rectangle is not a square" a unique name that is different from the name I give the rectangle referred to by "a rectangle is a square." One potential flaw with existential instantiation is that it seems to be based off of the implicit assumption that two things that are unequal are not permitted to have the same name. In mathematics in general, that assumption seems to be made. However, in the real world, two things that are unequal can, and sometimes do, have the same name. So, it seems that traditional mathematics is flawed because it seems to have an unnecessary rule that reality does not abide by.

Conflation among different things with the same name may be an inevitable or necessary feature of nature.

Nonetheless, for the sake of advancement here, I make the following attempts to present my argument in symbolic logic.

Argument 1. Domain: All rectangles. R(x) is the statement "x is a rectangle." S(x) is the statement "x is a square." P is a statement. Statements (Reasons) 1. ꓱx(R(x) ˄ S(x)) (Premise) 2. R(d) ˄ S(d) (Existential instantiation from (1)) 3. ꓱx(R(x) ˄ ¬S(x)) (Premise) 4. R(g) ˄ ¬S(g) (Existential instantiation from (3)) 5. d = g (Premise) 6. R(d) ˄ ¬S(d) (Substitution property of equality from (4) and (5)) 7. S(d) (Conjunction elimination from (2)) 8. ¬S(d) (Conjunction elimination from (6)) 9. ꓕ (ꓕ introduction from (7) and (8)) 10. P (ꓕ elimination from (9)) This concludes the argument.

The most questionable step in the above argument is (5), which I provided as a premise. The premise is that the rectangle that is a square is equal to the rectangle that is not a square. The premise's truth is based off of the fact that d and g each have the same name, a rectangle. Notice that I did not indicate what was given and what was to be proved at the beginning of the argument. That was because premise (5) was not available at the beginning of the argument. Premise (5) involves and thus is dependent upon the context developed previously in the argument. The following argument integrates premise (5) and the other premises into a single premise.

Argument 2. Domain: All rectangles. R(x) is the statement "x is a rectangle." S(x) is the statement "x is a square." P is a statement.

Given: ꓱx((R(x) ˄ S(x)) ˄ (R(x) ˄ ¬S(x))) Prove: P

Statements (Reasons) 1. ꓱx((R(x) ˄ S(x)) ˄ (R(x) ˄ ¬S(x))) (Given) 2. (R(d) ˄ S(d)) ˄ (R(d) ˄ ¬S(d)) (Existential instantiation from (1)) 3. R(d) ˄ S(d) ˄ R(d) ˄ ¬S(d) (Simplification from (2)) 4. S(d) (Conjunction elimination from (3)) 5. ¬S(d) (Conjunction elimination from (3)) 6. ꓕ (ꓕ introduction from (4) and (5)) 7. P (ꓕ elimination from (6)) This concludes the argument.

In the second argument, it is postulated that there is a rectangle that both is and is not a square. The basis for that contradiction is that the rectangle that is a square and the rectangle that is not a square are the same because each is a rectangle. They each have the property of being a rectangle. They each have the name a rectangle. They each share the noun phrase a rectangle.

Serendipper wrote:R1 not equal to R2. They are defined to be different by your proposition that one is a square and the other is not.

R1 may not be equal to R2 in some sense of equality. But in some sense of equality, R1 is equal to R2 because each refers to a specific rectangle. R1 = a specific rectangle. R2 = a specific rectangle. Therefore, by the substitution property of equality, R1 = R2.

No, R1 = specific rectangle and R2 = a different specific rectangle. That's your definition in order to meet the requirement of one being a square and the other not. So now you can't turn around and make a conclusion based on your definition: "because I defined them to be different, but labeled them the same, they are therefore equal and contradictory, therefore trivialism is true."

You've taken different things by definition and labeled them the same then claimed a contradiction exists.

R1 and R2 share a property in common with each other, and that common property alone can be used to equate the two objects in some sense of equality. R1 and R2 are equal in merely the sense that they share the same name. That name is "a specific rectangle."

See? That's exactly what you have done. So I could take a cat and dog, label the dog a cat, then claim a cat is equal to a dog because they have the same name, and then claim a contradiction.

Example. Let b be a United States dime and f be a United States dime. b does not equal f in some sense of equality because b was manufactured in 2003, but f was manufactured in 2007. However, b is equal to f in the sense that both share the property of being a United States dime. Furthermore, b is equal to f in the sense that both have the same monetary value, 10 cents. This concludes the example.

The existence of different senses of identity like in the example was discussed in a course I took my first semester of college called Minds and Machines. I did not graduate from college, but according to an unofficial transcript, I did get an A- in the course.

I still maintain you are conflating senses. The value of a dime is equal to the value of another dime, but they are not the same dime.

A better example is a red block and blue block that are exactly the same other than color. We can say they are the same in one sense, but not in the other sense and therefore they are not equal except in their respective senses which you cannot conflate with arbitrary labeling.