In Support of Trivialism

I agree.

There is still a contradiction because

is still true.

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.

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

Well then I claim trivialism is false :wink:

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.

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.

No.

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.

No contradiction.

Irrelevant.

They are separate senses.

No

Nope

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.

I have a proposal that might help you browser32… attempt to present your deductive argument using symbolic logic.

You’ll find that it’s very hard to mask the sophistry without the ability to equivocate using ambiguous language…

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.

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

The following is an excerpt from my reply to phyllo at viewtopic.php?p=2695577#p2695577.

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 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.
[i]R/i is the statement “x is a rectangle.”
[i]S/i is the statement “x is a square.”
P is a statement.
Statements (Reasons)

  1. x([i]R/i ˄ [i]S/i) (Premise)
  2. [i]R/i ˄ [i]S/i (Existential instantiation from (1))
  3. x([i]R/i ˄ ¬[i]S/i) (Premise)
  4. [i]R/i ˄ ¬[i]S/i (Existential instantiation from (3))
  5. d = g (Premise)
  6. [i]R/i ˄ ¬[i]S/i (Substitution property of equality from (4) and (5))
  7. [i]S/i (Conjunction elimination from (2))
  8. ¬[i]S/i (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.
[i]R/i is the statement “x is a rectangle.”
[i]S/i is the statement “x is a square.”
P is a statement.

Given: ꓱx(([i]R/i ˄ [i]S/i) ˄ ([i]R/i ˄ ¬[i]S/i))
Prove: P

Statements (Reasons)

  1. x(([i]R/i ˄ [i]S/i) ˄ ([i]R/i ˄ ¬[i]S/i)) (Given)
  2. ([i]R/i ˄ [i]S/i) ˄ ([i]R/i ˄ ¬[i]S/i) (Existential instantiation from (1))
  3. [i]R/i ˄ [i]S/i ˄ [i]R/i ˄ ¬[i]S/i (Simplification from (2))
  4. [i]S/i (Conjunction elimination from (3))
  5. ¬[i]S/i (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.

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.

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.

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.

I’m a human. The Pope is a human. Therefore I am the Pope.

Now go in peace and sin no more. You can cos as much as you like.

Sin and virtue are attributes of action and are therefore equal :smiley:

That’s right. If there exists a rectangle that’s a square, we symbolize this as

$$\exists x ( R(x) \land S(x))$$

We say that x is now a bound variable. As Wiki puts it:

That is, free variables become bound, and then in a sense retire from being available as stand-in values for other values in the creation of formulae.

en.wikipedia.org/wiki/Free_vari … _variables

So now if there’s some other rectangle that’s not a square, and we want to express this fact in conjunction with the earlier fact, we write

$$\exists x ( R(x) \land S(x)) \land \exists y (R(y) \land \neg S(y))$$

It’s not a flaw, it’s a feature. It’s how logic works. The purpose of symbolic logic is to be crystal clear in our meaning; to avoid the ambiguity of natural language. That’s the entire point.

Certainly we can use the same name for different objects in different contexts. But in formal logic, we first choose the context, or domain, or universe – different words for the same idea – and then within that context, names must refer uniquely to objects.

Well sure, a cat is a furry four-legged handwarmer; and a cat is a Caterpillar tractor; and a cat is a person with a hip demeanor, as in a cool cat.

What of it? Again, the entire point of symbolic logic is to remove the ambiguity of natural language, so that we may be sure that we are reasoning correctly. Of course natural language is ambiguous, that’s so poets will have something to do. The fog comes on little cat feet.

In poetry, we exploit the ambiguity of natural language; in formal logic, we avoid it. Poetry and logic. Two different human activities.

Math is good for doing math, and everyday natural language is good for doing everyday natural things. I don’t understand why you think one is “flawed.” We use hammers to hammer nails, and flashlights to illuminate the dark. We don’t say hammers are flawed because they’re not flashlights. We use different tools for different tasks. Surely you agree.

No, it’s an inevitable or necessary feature of natural language. You are confusing the names of things with the things themselves. You are confusing the words we use to talk about nature and to navigate the world; with the world itself.

This is a philosophical error. There are things, and there are names. The names of things are not the things.

I commend you for recognizing that this step is problematic.

But surely you see that just because I am a human and the Pope is a human, that I am not necessarily the Pope!

Here “is” does not mean “equals,” which would allow you to use transitivity of equality: if A = B and B = C then A = C.

Rather in this instance, “is” means, “is a member of some class.” So you have some horse that “is” an animal; and you have a cat that “is” an animal. But a horse is not a cat.

In this case “is” is being used as set membership, if you like; but not as equality.

Regardless of whether the rectangle referred to in “a rectangle is not a square” is the same rectangle referred to in “a rectangle is a square,” the former statement is the negation of the latter statement.

The following Argument 3 is a third attempt to present my argument in symbolic logic.

Argument 3.

Given: p = “A rectangle is a square,” p, ¬p, q is a statement.
Prove: q

Statements (Reasons)

  1. p (Given)
  2. ¬p (Given)
  3. ꓕ (ꓕ introduction from (1) and (2))
  4. q (ꓕ elimination from (3))
    This concludes the argument.

Premise (1) is justified by the fact that some rectangles are squares. Premise (2) is justified by the fact that some rectangles are not squares.

I know it sounds odd, but in a sense you are the Pope. You are the Pope in the sense that you and the Pope each is a human. There may be contexts in which you would use such discourse.

Example. You are talking to aliens that are not humans and are not from earth. You indicate to them that, among the numerous species of life on earth, you and the Pope are of the same species. You told the aliens you are not a cat, you are not a spider, you are not a giraffe, and you are not a penguin. You told the aliens, however, that you are the Pope. The aliens understand that you meant that there are some differences between you and the Pope, but that you and the Pope are of the same species. This concludes the example.

Again, there are multiple senses of equality. Since you are a human and the Pope is a human, there is a sense of equality in which “you = a human” and “the Pope = a human.” This sense of equality may be syntactic only, or it may be syntactic and semantic. You and the Pope have properties in common that can be used to pair you two up and regard you two in the same way.

Natural language is a part of nature.

It doesn’t matter what language a contradiction is asserted in; if a contradiction exists, then through the principle of explosion, all statements of all languages are true. Thus, if a contradiction exists in natural language, then by the principle of explosion, all statements of first-order logic, all statements of all natural languages, and all statements of all unnatural languages are true.

The names of things are not always the things. The names of things can be considered properties of the things. It actually seems the name of a thing is often considered a property of the thing. For example, in computer software, the name of an object is often considered one of the most important properties of the object.

Yes, and in biology a cat is a four legged furry handwarmer, while in popular culture it’s an especially hip hipster. What of it?

Regarding your equivocation of “is”, consider a mathematical example.

2 is a number and 3 is a number, but 2 is not equal to 3.

If we let N be the set of natural numbers, then we say 2 ∈ N and 3 ∈ N. That means “2 is a member of the set of natural numbers, and 3 is a member of the set of natural numbers.”

In this case the natural language “is” refers to set membership.

When we say that 2 = 1 + 1 and 1 + 1 = 5 - 3, that’s equality. It’s a transitive relation, so that we may conclude that 2 = 5 - 3.

You are simply equivocating “is” as set or class membership, and “is” as the equality relationship.

Some words have multiple senses. Equality is one of those words.

2 is not equal to 3 in a well known mathematical sense. But both are, as you have suggested in your post, natural numbers. So, in the sense that 2 and 3 each is a natural number, 2 is equal to 3.

The expressions 2 and 1 + 1 are not equal; there is an obvious syntactic inequality. In another sense of equality, however, the expressions 2 and 1 + 1 are equal; the value of the expressions are equal. This example is similar to the example I previously gave involving two dimes. Each pair of expressions (1 + 1, 5 - 3) and (2, 5 - 3) can be used in another similar example.

I am not equating naively. My equations have been justified.

I am aware of the two different uses of is that you have mentioned. I described the set membership use in my earlier post at viewtopic.php?p=2697589#p2697589, in my first paragraph after the first quotation from Serendipper that I provided.

2 is equal to 3 in that sense only. You cannot make further conclusions.

  1. things
  2. shapes
  3. more-specific shapes
  4. even more-specific shapes.

You cannot say because everything is a thing, therefore everything is equal in all senses.

If you can’t see this by now, maybe you can’t see it. Or perhaps you’re not giving it moral consideration because you need your proposition to pan out for some reason. Either way, this is going to go on forever because you can’t or won’t see reason. Isn’t there some other topic that you’d rather dive into? You seem like a nice guy. Why not comment on something else of interest and abandon this?

I think we might have found the problem…
I agree in reality we can have many different things that are identified by the same name… A kid could name a rock “bob”, he could name an imaginary friend “bob”, there is no end to the number of things we could call “bob” on a whim.

Language is a human invention, the naming conventions we employ are more for practical functionality than accurate representation… You’d be perfectly happy to simply call Bob your work friend “Bob” and Bob your childhood friend “Bob”… but if you were speaking to a friend who knew both Bobs then you’d feel the need to distinguish them by name for the sake of clarity… you’d say “work Bob” or “Original Bob” or use their last names… or simply bob1 and bob2

We rename things and add extra identifiers to how we name things all the time for the sake of clarity and communication…

You have correctly identified a failure of you own chosen naming convention… the term “A rectangle” is insufficient to distinguish distinct objects and therefor misrepresents them as the same thing.
Like with the two Bobs, the solution is to rename them for clarity… given how the “square/not square” aspect is generating the contradiction and not the rectangle part, perhaps including that detail in the naming convention would be useful.

Perhaps you could name them as you did above and stick to that naming convention: “The Rectangle that is a square” and “The Rectangle that is not a square”

At which point you’d be spouting a tautology “The Rectangle that is a square, is a square” and “The rectangle that is not a square, is not a square”… There is no contradiction when employing a more clear use of language.

Mad Man P:

I don’t have to rename the rectangles. I don’t have to give each rectangle a unique name. It is permitted that both rectangles simultaneously have the same name. That is how it is in real life.

I understand…
But what I’m suggesting is that your refusal to be precise or even practical with your use of language does not logically constitute sufficient cause to abandon the law of non-contradiction.

You are not demonstrating a contradiction… merely poor language skills.

Mad Man P:

I have not abandoned the law of non-contradiction. If there is a contradiction, then through the principle of explosion, the law of non-contradiction is true. If there is not a contradiction, the law of non-contradiction is true. Either way, the law of non-contradiction has not been abandoned.

My language use in my argument may be poor, but it does seem to be permitted. One concern I have is that my use of the indefinite article a whenever referring to one of the two rectangles may be grammatically prohibited. However, if the use is prohibited, my earlier given Argument 3, and even Arguments 1 and 2, still succeed since they do not violate that linguistic prohibition.

The word “permitted” implies a standard, system or authority that is “permitting”… From who or what are you seeking this permission?

I contend that it cannot be “logic”, as logic strictly prohibits that kind of ambiguity in language, equivocation is a fallacy, after all…
If you are operating by some alternative in which equivocation is not a fallacy, then you have successfully shown that this alternative generates contradictions.

I am seeking this permission from the rules of the English language.

As my opponent had suggested to me in the comments section for a past debate I participated in, a debate that was cited indirectly in the second paragraph of the original post for this thread and is located at viewtopic.php?p=2695639#p2695639,

The statement “a rectangle is a square and a rectangle is not a square” is, in some sense, a syntactic contradiction. Since the syntactic contradiction exists, a contradiction exists. So, the principle of explosion brings about trivialism. Also, the syntactic contradiction suggests some semantic contradiction. A semantic contradiction, through the principle of explosion, brings about trivialism.