So all you're saying is that in Euclidean geometry the angles of a triangle add up to 180 degrees. And they do so because of the axioms of Euclidean geometry.

Imperfect triangles are imperfect by definition. I'm focused on absolutes. — Philosopher19

What do you mean by an "imperfect" triangle?

or the angles in a triangle add up to 180 degrees — Philosopher19

You should see non-Euclidean geometry where the angles in a triangle can be more or less than 180 degrees.

If definitions aren't subject to truth apt, then can I say, "Let 'X' mean a married bachelor," and that this sentence is not truth apt? — Brendan Golledge

The sentence isn't truth apt, but "married bachelor" is a contradiction.

But neither "if this sentence is true then I am 30 years old" nor "if this sentence is true then I am not 30 years old" is a contradiction, or at least not obviously so.

Is this statement false? If I've done the truth table right, then it means that the first line of the proof is wrong. — Brendan Golledge

The first line is a definition, not a premise, and so not truth apt. It is simply saying this:

Let "A" mean "if A is true then B is true".

What has not been shown to me is how this logically obliges us to view the set of all sets as contradictory. — Philosopher19

It doesn't. This is your misunderstanding. Russell's paradox only shows that the axiom schema of unrestricted comprehension leads to a contradiction, and so that naive set theory is inconsistent.

But the paper went on further to prove that if 6 is false, then 1 must also be false. So, it is a bad definition. — Brendan Golledge

It’s not that the definition is bad, it’s that when we apply the normal rules of logic to some self-referential sentences then we lead to a contradiction. It’s the paradox of all liar like sentences and there’s no agreed upon resolution.

If A is false, then B is not false. Given the definition of the sentence you are using, A is false (or meaningless) and B is true. — Brendan Golledge

Consider these sentences:

1. if this sentence is true then Germany borders China
2. if (2) is true then Germany borders China

Do you accept that (1) and (2) are materially equivalent?

If so then consider these sentences:

2. if (2) is true then Germany borders China
3. if (2) is true then Germany borders China

Do you accept that (2) and (3) are materially equivalent?

If so then (1) and (3) are materially equivalent.

As for your formal logic, I think I am confused about whether you are asserting logic or truth. For instance, I cant tell whether you mean, "if X is true, then Y is true" (I agree with this logic) or "X IS true, and therefore Y is true" (I disagree with this because I think X is either false or meaningless). — Brendan Golledge

Hopefully this is clearer:

1. X means if X is true then Y is true (definition)
2. If X is true then X is true (law of identity)
3. If X is true then if X is true then Y is true is true (switch in the definition of X given in (1))
4. If X is true then Y is true (from 3 by contraction)
5. X is true (switch out the definition of X given in (1))
6. Y (from 4 and 5)

Although one thing to consider is that A → B is equivalent to ¬B → ¬A, and so these are equivalent:

1. if this sentence is true then Germany borders China
2. if Germany does not border China then this sentence is not true

(2) appears to be a more complex version of the standard liar sentence.

When you say the axioms of naive set theory, are you referring to those notations that I asked you to put in clear language. — Philosopher19

Yes.

If so, it seems to me you left half way through trying to clarity on it. — Philosopher19

I'm not interested in teaching you mathematics. I am simply explaining to you that Russell proved that the axioms of naive set theory are inconsistent. That's it. It's not a debatable issue.

See Dunning–Kruger effect.

and that you cannot say x is bigger than y without some measurement/count involved to compare the sizes of the two. — Philosopher19

If you were a mathematician then you would know that this is false.

You're just in no position to argue against Cantor.

Can we establish set x as being bigger than set y without counting the number of items in x and y? If yes, how? — Philosopher19

Yes, we can establish set X as being "bigger" than set Y without counting the number of items in X and Y. We can establish this by using Cantor's diagonal argument. It is a well-accepted mathematical proof. If you were a mathematician you would understand it.

Again, 1 is contradictory. Put it in clear language as to why the contradictoriness of 1 obliges us to reject 2 or to view the set of all sets as contradictory. — Philosopher19

The axioms of naive set theory entail (1). Therefore, the axioms of naive set theory are inconsistent.

This is all Russell's paradox shows. Again, you're showing that you don't even understand the problem.

That is not an answer. — Philosopher19

It is an answer. You just don't understand it because you're not a mathematician.

disagreed that a and 2 are equivalent — Brendan Golledge

Two statements are materially equivalent if either both are true or both are false:

1. A if and only if B

If (1) is true then "A" and "B" are materially equivalent.

So, in the above case:

A) if this sentence is true then Germany borders China
B) if (A) is true then Germany borders China

If (B) is true then (A) is true. If (B) is false then (A) is false. Therefore, (A) and (B) are materially equivalent.

When you used formal logic, you didnt prove that x is true — Brendan Golledge

Are you referring to step 5? As it explains, it simply takes step 4 and replaces X → Y with X, which is allowed given the definition in step 1.

or that x->y is true — Brendan Golledge

Are you referring to step 4? As it explains, it follows from step 3 given the rule of contraction.
X → (X → Y) entails X → Y.

Can we establish set x as being bigger than set y without counting the number of items in x and y? If yes, how? — Philosopher19

Cantor's diagonal argument.

I've seen cantor's diagonal argument and the following objection applies to it: — Philosopher19

It doesn't. If you were a mathematician then you would understand it. Your question simply shows your ignorance of mathematics. You're really in no position to argue against Cantor.

This is where you show that you don't understand the problem.

Naive set theory accepts both the axiom of extensionality and the axiom schema of unrestricted comprehension.

The axiom schema of unrestricted comprehension entails that there is a set that only contains all sets that are not members of themselves (the Russell set). In conjunction with the axiom of extensionality this is a contradiction, and so naive set theory is shown to be inconsistent.

This is all Russell's paradox does.

In response to this, naive set theory had to be changed. Specifically, it had to reject the axiom schema of unrestricted comprehension.

ZFC replaces this axiom with two others; the axiom of regularity and the axiom of pairing. These axioms entail that no set is a member of itself, and so that there is no universal set.

NF restricts the axiom schema of comprehension to permit only stratifiable formulas. This does allow for a universal set.

It's not entirely clear what you're trying to argue. Is it that Russell's paradox doesn't prove naive set theory is inconsistent? Is it that ZFC allows for a universal set? You'd be wrong on both counts. Is it that NF is "better" than ZFC? I'm not entirely sure such a claim would make sense.

Or are you trying to argue that, independently of any set theory, a set that contains all sets that are not members of themselves is possible, therefore only set theories that allow for this are "correct"? That's putting the cart before the horse.

How would a difference in size be established between them when there is no counting involved? — Philosopher19

See Cantor's diagonal argument, which proved that there are higher-order cardinal numbers.

You've already admitted that you're not a mathematician, so it's strange that you think you know mathematics better than Cantor (and Russell).

Let A be the set of all integers.
Let B be the set of all positive integers.

Every member of B is also a member of A, but some members of A are not members of B (i.e. the negative integers).

B is a subset of A.
A is a superset of B.

1 is contradictory if you say set B only contains all sets that are not members of themselves. — Philosopher19

Yes, that's the premise. Let R be the set of all sets that are not members of themselves. This is the Russell set.

If R is not a member of itself then it is a member of itself. This is a contradiction. Therefore we must abandon the axiom schema of unrestricted comprehension.

ZFC replaces the axiom schema of unrestricted comprehension with the axiom of regularity and the axiom of pairing. These entail that no set is an element of itself, and so doesn't allow for a universal set.

Whereas NF restricts the axiom schema of comprehension by allowing only stratifiable formulas. This doesn't allow for a Russell set but does allow for a universal set.

There's really nothing to argue here. Russell's paradox is an undeniable proof that naive set theory is inconsistent. The "resolution" is to replace the problematic axiom with others, which is what has been done. ZFC does it one way, NF another way, and others in other ways.

Yes. Given the axiom schema of unrestricted comprehension there exists a set B whose members are sets that are not members of themselves.

This leads to a contradiction. If B is not a member of itself then it is a member of itself.

Therefore, we must reject the axiom schema of unrestricted comprehension (or the axiom of extensionality, but that would be far too problematic).

is predicate φ "A and B are equal if every member of A is a member of B and every member of B is a member of A"? If not, what is it? — Philosopher19

In Russell's paradox, $φ$ is "sets that are not members of themselves".

It says that A and B are equal if every member of A is a member of B and every member of B is a member of A.

The axiom of extensionality
Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B:
${\displaystyle \forall x\,\forall y\,(\forall z\,(z\in x\iff z\in y)\implies x=y)}$

The axiom schema of unrestricted comprehension
There exists a set B whose members are precisely those objects that satisfy the predicate $φ$:
${\displaystyle \exists y\forall x(x\in y\iff \varphi (x))}$

Russell's paradox
Let $φ(x)$ be ${\displaystyle x\notin x}$ ($x$ is not a member of $x$):
${\displaystyle \exists y\forall x(x\in y\iff x\notin x)}$

Therefore:
${\displaystyle y\in y\iff y\notin y}$

This is a contradiction. Therefore the axiom of extensionality and the axiom schema of unrestricted comprehension are inconsistent.

ZFC replaces the axiom schema of unrestricted comprehension with the axiom of regularity and the axiom of pairing and as such is consistent. This doesn't allow for a universal set.

New Foundations restricts the axiom schema of comprehension by allowing only stratifiable formula for $φ(x)$. This allows for a universal set.

I believe I understand Russell's paradox very well — Philosopher19

So you understand the below?

Axiom of extensionality:
${\displaystyle \forall x\,\forall y\,(\forall z\,(z\in x\iff z\in y)\implies x=y)}$

Axiom schema of unrestricted comprehension:
${\displaystyle \exists y\forall x(x\in y\iff \varphi (x))}$

Substitute ${\displaystyle x\notin x}$ (the Russell set) for ${\displaystyle \varphi (x)}$:
${\displaystyle \exists y\forall x(x\in y\iff x\notin x)}$

Therefore:
${\displaystyle y\in y\iff y\notin y}$

The conclusion is a contradiction. Therefore the premises are inconsistent. In this case, the problematic premise is the second premise, which is the axiom schema of unrestricted comprehension.

ZFC is, I believe, set up specifically so that "a list can't list itself". That's how it avoids the various paradoxes. — Banno

Yes, the axiom of regularity and the axiom of pairing entail that no set is an element of itself.

Russell's paradox is a mathematical proof that the axiom schema of unrestricted comprehension leads to a contradiction. As such, early naive set theories had to be abandoned.

Zermelo–Fraenkel set theory is the most used replacement, and doesn't allow for a universal set.

New Foundations is an alternative replacement that does allow for a universal set.

This isn't really anything to do with philosophy. It's just about the internal consistency of some set of mathematical axioms. Some lead to a contradiction, as Russell's paradox shows, and so their axioms must change.

If you're trying to argue that a "correct" set theory must allow for a universal set then I don't think you really understand mathematics.

I think we can show this by considering the complement of a liar sentence:

1. This sentence is true

If (1) is true then there is no paradox. If (1) is not true then there is no paradox. But is (1) true or not true? — Michael

Curry's paradox is an interesting extension of this.

1. Let (a) be the sentence "if this sentence is true then Germany borders China"
2. If (a) is true then Germany borders China
3. Given that (2) is true, and given that (a) and (2) are materially equivalent, then (a) is true
4. Therefore, Germany borders China

In formal logic:

1. X := (X → Y)
2. X → X
3. X → (X → Y)
4. X → Y (from 3 by contraction)
5. X (substitute 4 by 1)
6. Y (from 4 and 5)

Except you can’t break it down that way because “This sentence contains 36 characters” is true but “The sentence in point A contains 36 characters” is false.

Its just a bad contraction. If we break out the sentence into its full meaning, its fine.

A. This is a sentence. True
B. The sentence in point A is a false sentence. False.

There ya go. — Philosophim

This sentence contains 36 characters

Should we break the above sentence into the below?

A. This is a sentence
B. The sentence in point A contains 36 characters

Or else, some people are using the words "moral" or "أخلاقي" wrongly. — baker

What determines the right way? Is it how most speakers of the language use the word? If the vast majority of Arabic speakers use the word "أخلاقي" to describe acts which are condoned by the Quran, and if the meaning of a word is determined by the things most speakers of the language use it to describe, then it would seem to follow that being condoned by the Quran is part of the meaning of the word "أخلاقي".

Google translates أخلاقي as "moral", "ethical". What is the basis of this translation? — baker

The argument the other person made was that the meaning of a word is determined by the things it is used to describe.

The things Arabic speakers describe using the word “ أخلاقي” often aren’t the things English speakers describe using the word “moral”.

Therefore if we accept the other person’s reasoning then the words “أخلاقي” and “moral” don’t mean the same thing.

If the words “ أخلاقي” and “moral” do mean the same thing then the other person’s reasoning is wrong, and the meaning of a word is not determined by the things it is used to describe.

Let's assume the correspondence theory of truth: that a sentence is true is that it corresponds to a fact. We can use this to rephrase the liar sentence:

1. This sentence does not correspond to a fact.

Does (1) correspond to a fact?

I think I understand @flannel jesus's point, and that it can be explained like this:

A box contains 100 balls. Either all 100 are blue or 1 is blue and 99 are red, determined by a fair coin toss. A ball is picked at random from the box and it is blue.

Do you believe that the box now contains 99 blue balls or 99 red balls?

The probability of the first ball being blue is 100% if the box contains 100 blue balls and 1% if the box contains 1 blue ball and 99 red balls. As such, given that the first ball was blue, it is much more likely that the box now contains 99 blue balls.

The first ball being blue is strong evidence that the box now contains 99 blue balls. However, the first ball being blue is compatible with the box now containing 99 red balls, i.e. the first ball being blue does not prove that the box doesn't now contain 99 red balls.

Your approach seems to be the same as that of Kripke. See here.

In general, if a sentence such as (1) asserts that (all, some, most, etc.) of the sentences of a certain class C are true, its truth value can be ascertained if the truth values of the sentences in the class C are ascertained. If some of these sentences themselves involve the notion of truth, their truth value in turn must be ascertained by looking at other sentences, and so on. If ultimately this process terminates in sentences not mentioning the concept of truth, so that the truth value of the original statement can be ascertained, we call the original sentence grounded; otherwise, ungrounded.

Liar sentences are "ungrounded". Them being true or false isn't meaningful.

I think we can show this by considering the complement of a liar sentence:

1. This sentence is true

If (1) is true then there is no paradox. If (1) is not true then there is no paradox. But is (1) true or not true?

My mistake.

If they couldn't both be true at the same time, then you would be certain John was approaching. — flannel jesus

Either "John is approaching" is true or "John is not approaching" is true.

This doesn't entail that I am certain that "John is approaching" is true.

Perhaps a distinction needs to be made between firsthand knowledge and secondhand knowledge. I do not have firsthand knowledge of history, for example. The skeptic seems to be arguing that sensory experience isn't firsthand knowledge of an external world. I think there's some truth in that. But I think it a fallacy to argue that all knowledge must be firsthand knowledge.

The question, then, is whether or not our secondhand knowledge of an external world is reliable. The answer to that likely depends on whether or not the naive realist view of perception is correct.

Why do so many make moral propositional statements if they are not truth-apt? — Chet Hawkins

Moral non-cognitivists will say that a sentence such as "this is wrong" actually means something like "don't do this", and that the sentence "don't do this" isn't truth-apt.