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.

I said we should not trust men to be around kids as much as we trust women. — RogueAI

You didn't just say that. You also said:

Also, the fact that he's 50 makes me suspect he's perving on them. — RogueAI

I'm a teacher and I'm always a little suspicious of male teachers in elementary school settings (and the priesthood). I think the reasons are obvious. I would not let a man or teenage boy babysit my daughter. I am equally suspicious of gay and straight men. — RogueAI

No, but we're about nine times more likely to sexually abuse children. — RogueAI

That biological men are nine times more likely than biological women to sexually abuse children isn't that biological men are likely to sexually abuse children.

Until that changes, men should not be trusted to be around kids as much as women. — RogueAI

Even if they shouldn't be trusted as much it doesn't then follow that they shouldn't be trusted.

I'm much more likely to die in a car crash than being eaten by shark, but I'm not going to assume that I'm going to die in a car crash. In fact, I'm very unlikely to die in a car crash.

It seems a bit of a strange and abstracted formulation to me but I am only looking at the paragraph. — Tom Storm

With my post above in mind, consider a slightly more realistic scenario: I knowingly have COVID, but nonetheless attend a party. Someone else catches COVID from me. I fully recover but they are soon to die. They can be saved with a partial lung transplant but none will be available in time. Am I morally obligated to donate a part of my lung to save their life?

My question is is this a valid analogy for the moral permissibility of abortion? — Captain Homicide

I'm not sure it's specifically meant as an analogy, but as a reductio ad absurdum against the premise that one person is obligated to save the life of another.

Of course, @Tzeentch raises the point that (at least when sex is consensual), the mother bares some responsibility for the situation in which abortion is considered, and so perhaps a better example would be to ask if Henry Fonda would be obligated to touch her brow if her illness was caused by Henry Fonda himself (even if unintentionally). But then we have to ask; what if he must do more than just touch her brow? What if he must donate a kidney to save her life? Is that too much? Does his obligation to help her only extend so far?

Although as for abortion, the very premise that the foetus has a right to life can also be questioned, and so even if Henry Fonda is obligated to touch her brow to save her life, it wouldn't then follow that abortion is morally impermissible, especially as bearing a child for 9 months and giving birth to it is a much greater burden than just touching someone's brow.

Biological men, however, cannot be trusted around kids to the same degree. — RogueAI

So just sexism. Biological men must be assumed to be child-molesting paedophiles. :roll:

Also, the fact that he's 50 makes me suspect he's perving on them. — RogueAI

So you're a transphobe, got it.

Would you assume that a 50 year old gay man using the men's changing room is perving on any 13 year old boys who also happen to be changing? Or a 50 year old lesbian in the women's changing room?

You think a 50 year old biological male should be sharing a changing room with teenage girls? You don't see any problem with that? What if the girls are 13? 8? — RogueAI

Is your concern their age or that they're transgender? Would you mind if it were a 50 year old cisgender woman, or a 16 year old transgender woman?

Amadeus has a good point about motive. Biological men are much more likely to have nefarious motives than biological women. — RogueAI

That biological men are more likely to have nefarious motives than biological women isn't that there is a good reason to believe that a transfemme swimmer has nefarious motives for competing in women's swimming competitions.

AmadeusD doesn't have a good point, it's simply transphobia. Much like it would be homophobia to suggest that a gay swimmer has nefarious reasons for competing in a men's swimming competition for anyone over the age of 16.

I'm unsure this makes it any better... The idea that 50 year old male is given unfettered (in context) access to vulnerable young females undressing and competing is — AmadeusD

Is your concern her age, that she's trans, that she's competing, or that she uses the women's changing room?

Regarding the competition, it is an open age category. Anyone over the age of 16 could compete. There is never a maximum age for these things. In fact, in terms of competitiveness, it is better for the other participants that she is 50 rather than say 18, as a 50 year old is much less fit.

Regarding the changing room, would you care if it was a 50 year old cisgender woman, or an 18 year old transgender woman?

you've got to question motive. — AmadeusD

She wants to compete in swimming competitions, like everyone else competing.

Your apparent suggestion that transgender people have some nefarious motives for being transgender is straight up transphobia.

I am only in this thread like ... for ... moral objectivity. But if there is something called moral realism that is different for glossy technical reasons, I am trying to understand so that I can either agree or disagree there. — Chet Hawkins

It's useful to separate a moral theory out into its constituent parts. There are, roughly speaking, three considerations when discussing meta ethics:

1. Are moral propositions truth-apt?
2. If moral propositions are truth-apt then are any true?
3. If there are true moral propositions then are they objectively true?

We can set out these three considerations as affirmative claims that are then either accepted or rejected:

a) Moral propositions are truth-apt
b) Some moral propositions are true
c) Some moral propositions are objectively true.

(c) entails (b) entails (a).

If you reject (a) then you are a moral non-cognitivist. If you reject (b) then you are an error theorist. If you reject (c) then you are a non-objectivist.

Some say that you must accept (c) to be a realist, others say that you need only accept (b) to be a realist, and that to accept (c) is to be a "robust" realist.

Although I wouldn't get too caught up in labels, they're just pragmatic tools with no real philosophical significance. What matters is whether or not (a), (b), and (c) are true.

I mean, I agree. — Chet Hawkins

If some moral propositions are objectively true then:

a) moral propositions are truth-apt and b) some moral propositions are true.

Suggestion: let's vote against him — Relativist

Would that I could.

To my understanding, the subset issue was because you could have a set of all sets that are members of themselves. Since you could have this you should also have been able to have a set of all sets that are not members of themselves — Philosopher19

I don't see how that follows.

Wanting to have a set of all sets that are not members of themselves that is itself not a member of itself is a contradictory thing to want. — Philosopher19

That's exactly what Russell was proving.

• that the J6 conspiracy trial will have concluded before the election, but even if it is - pending appeals will keep him out of prison. If he's elected, he'll pardon himself and put an end to that. — Relativist

He can't do that for the Georgia case.

The Russell set is the set of all sets that are not members of themselves.

Yeh, it's good that we're invitation-only. — Jamal

Thanks, Marco.

Self-Sacrifice and the Trolley Problem, Ezio Di Nucci

Does the 1st amendment extend to this philosophy forum? — alan1000

This is the First Amendment:

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.

Given that this philosophy forum isn't the United States Congress, the answer is no.

Also, as per the Terms of Service:

Except to the extent applicable law, if any, provides otherwise, this Agreement, any access to or use of the Website will be governed by the laws of the state of England and Wales.

in Canada a 50-year-old man really is being allowed to compete in swimming competitions alongside 13- and 14-year-old girls. — RogueAI

It's hard for me to believe this is true, but it sounds like it might be. — RogueAI

What is true is that a 50 year old transgender woman was allowed to compete in the women’s events for athletes 16 and older.

Morality is objective. — Chet Hawkins

What does this mean if not "some moral propositions are objectively true"?

The golden rule? Do unto others as you would have them do unto you.

I am absolutist in that regard. — NOS4A2

Well, Trump isn't, so they're just applying the golden rule.

Given that I haven't said what you've accused me of saying, your questions are misplaced.

Well in that case you are claiming that 'good' involves flourishing, but that flourishing does not exhaust goodness. — Leontiskos

Nowhere in saying "some people use the word 'good' to describe chastity" am I saying anything about flourishing.