‘Mu’ is indeed a symbolic form or reference to śūnyatā. — Wayfarer

I didn't know that. Updated my database. :up:

Zen is not intellectual in the sense that Western philosophy is — Wayfarer

As far as I'm concerned you hit the nail on the head. What I meant to do was offer an explanation on how the Mu/sunyata state of mind is achieved.

Take into account the fact that Western philosophy, to my reckoning, has been and is by and large about thoughts (ideas, hypothesis, isms, and so on) and their relationships to each other and the world. This particular character of Western philosophy can be summarized, in a broader context, as the interaction between mind and thoughts - the mind taken as that which holds, deals, tinkers around, with thoughts.

As far as I can tell, the Mu (sunyata) state of mind as without or not bears a close resemblance to mysticism defined, by some, as "conscious without being conscious of anything." In fact, a case can be made that they're the same thing.

As you already know, the mind is constantly thinking, either logically, associatively or even randomly some times. There's not a moment that goes by when the awake person's mind is not having some thought or other. Thus, if we're to achieve Mu (sunyata), our first order of business is to empty the mind (Mushin) and only then can one be "conscious without being conscious of anything"

How can we "empty the mind"? you might ask. There maybe different ways of doing that of course but one that I suspect Nagarajuna developed was predicated on one particular property of thoughts that makes them, in a sense, mind-apt (capable of being held by the mind). This "particular property" is truth value. Consider the sentence, "I love hamburgers". It's true for me but may not be true for you for other people but what I want to emphasize is that the sentence "I love hamburgers" is mind-apt only if it has a truth value. It's worth noting that truth value maybe a surrogate for meaning i.e. semantics determines truth value. You know, from experience, that the meaningless i.e. the semantically empty sentences (utterances, writings) are not mind-apt - the mind experiences great difficulty translating the meaningless into thoughts, in fact what always/usually happens is the mind fails to generate a thought that corresponds to the meaningless. To make the long story short, the meaningless, those missing a truth value, can't be thought about i.e. they're not mind-apt i.e. they're unthinkable.

Nagarjuna's tetralemma, by denying every possible truth state for a sentence, any sentence, is attempting to strip sentences of their truth value which is one way, even though it may be going round Jack Robinson's barn, of saying that sentences, all of them, are meaningless, semantically empty and when that's done to all possible sentences that can be generated, the mind, since it's incapable of thinking about the meaningless, becomes empty - Mu/Sunyata/Mushin/"conscious without being conscious of anything"

In conclusion, yes it's true that "Zen is not intellectual" - it is after all the quest for the state of mind in which we're not thinking about anything (Mu). However, to get to Mu, our minds, habituated over generations and lifetimes of constant, unceasing thinking, must devise ingenious ways as a workaround, Nagarjuna's tetralemma being one of them.


It's like Useless Machines

The most well-known "useless machines" are those inspired by Marvin Minsky's design, in which the device's sole function is to switch itself off by operating its own "off" switch. — Wikipedia

I don't want to make this post longer than necessary but I think an analogy will help in understanding Nagarjuna's tetralemma. Imagine a world of objects and these objects can be only of two colors, black or white. Your eyes can only see these two colors. Imagine now that someone walks up to you and says, "there's an object in this world but it's not white, it's not black, it's not black and white, and it's not neither white nor black". Would you be able to see this object?

Substitutions for the analogy to work.
1. objects = sentences
2. white = true
3. black = not true/false
4. eyes = the mind
5. It's not white = it's not true
6. It's not black = it's not false
7. it's not black and white = it's not true and false
8. it's not neither white nor black = it's not neither true nor false
9. Would you be able to see this object? = would you be able to think about such sentences? (Mu)

The idea, it seems, is to take the mind beyond the possible (consistency) to the impossible (inconsistenct) and, Nagarjuna seems to be hinting, beyond that too (Mu).

I can't follow ... what does any of this has to do with "true" and "good" or the OP? (Here are my two bits on 'wisdom'.) — 180 Proof

I may have gone off on a tangent there. Do excuse the digression as I hadn't slept well.

Let me summarize my thoughts for your convenience.

a) The easiest, simplest way to never lie given a proposition p is to claim p v ~p, a tautology and ergo, always true but one must remember that truth in this case is a function of the logical form of p v ~p i.e. the specific content doesn't matter.

b) Thus one way of heeding the moral injunction to "always tell the truth" is to resort to p v ~p for a proposition p. You could, using this rather obvious technique, become an examplar of honesty; after all, you can't possibly utter a falsehood with p v ~p. It might be worth noting that the universal requirement for truthfulness is nowhere more emphasized than in religion and Kantian ethics.

c) From a certain perspective then, we can say that to lie, faleshood itself, deception to the nth degree, is the exact opposite of p v ~p i.e. falsity is ~(p v ~p) and what is ~(p v ~p) but p & ~p or a frank contradiction. In other words, the moral obligation to always tell the truth is a roundabout way of telling us, whosoever cares to listen, that we should avoid contradictions on pain of upsetting God himself in a religious context. There's probably more that can be said but my brain doesn't seem up to the task.

d) The other possibility is that all I've said above is balderdash, poppycock, utter nonsense. What "always tell the truth" is some kind of coded message that p v ~p though the most obvious way to follow the moral commandment is, instead of being the solution, actually the problem. Whatever else p v ~p is, it becomes important at this juncture to realize that it's the well-formed formula (?) version of the law of the excluded middle (LEM). LEM, true to its name, excludes a possibility ("middle") which Wikipedia informs me is the contradiction p & ~p.

@Wayfarer However, in the Mahayana tradition of Buddhism, Nagarjuna, the founder does something amazing with the LEM. He rejects it outright as part of Nagarjuna's tetralemma i.e. he states ~(p v ~p) which in plain English means neither p nor not p but then he makes it a point to also deny p & ~p (contradiction). With this he, to use a gaming analogy, unlocks the so-called middle of LEM but by rejecting p & ~p he wishes to warn us that p & ~p ain't the middle he's interested in. Please note this is my interpretation and may not reflect mainstream views which I discuss below.

Nagarjuna, as part of his famous tetralemma, rejects even ~(p v ~p) like so ~~(p v ~p) which means not that neither p nor not p, another way of saying to hell with the...er...middle as well. Thus for a proposition p he says no to p, no to not p, no to p and not p and, last but not the least, no to neither p nor not p (the middle). In essence, Nagarjuna takes us one step further from where we would've been if we'd only rejected the LEM. Please note that the logical notations employed herein may not match up perfectly with Nagarjuna's own ideas about what his tetralemma truly means. For example, I don't think ~~(p v ~p) evaluates to p v ~p as it does in classical logic.

Multi-valued logic, paraconsistent logic, fuzzly logic, etc. maybe relevant.

FYI, the greek skeptics like Pyrrho had remarkably similar thoughts, so uncanny is the resemblance is that it's been hypothesized the Greek skeptics borrowed heavily from Nagarjuna and his ilk. More on that below.

Nagarjuna's tetralemma in re skepticism.

For any proposition p,

1. p or ~p or (p & ~p) or ~(p v ~p)

2. ~p :down: says Nagarjuna

3. ~p or (p & ~p) or ~(p v ~p) [1, 2 DS]

4. ~~p :down: says Nagarjuna

5. (p & ~p) or ~(p v ~p) [3, 4 DS]

6. ~(p & ~p) :down: says Nagarjuna

7. ~(p v ~p) [5, 6 DS]

8. ~~(p v ~p) :down: says Nagarjuna

9. p v ~p [8 DN]

I wonder if the above is an accurate description of Pyrrho's logic for skepticism because Nagarjuna's negation rule seems to differ significantly from those of Pyrrho e.g. for Nagarjuna ~~(p v ~p) = p v ~p is false.

Anyway, if Pyrrho did arrive at p v ~p (line 9 above) via denial of every possibility for a proposition, it seems like a good stance for a skeptic to take; after all, a skeptic knows that either p or not p (p v ~p) but denies that it's possible to know that p or to know that not p. In other words, p v ~p sums up a skeptic's position regarding any proposition.

This immediately raises the question as to whether the ethical principle to always be truthful is just another name for skepticism? It's kinda like a person who can't bear the fact that fae doesn't have a penny to faer name but just to make faerself feel better fae tells faerself that fae "has" money, 0.00 dollars to be precise.

More can be said but I'm just too tired to think.

Thanks for replying and I just want to pick your brain on what I think is a link, tenuous though it may be between Nagarjuna's approach and the Zen notion of Mu (Not)

My understanding of Nagarjuna's tetralemma is sketchy at best but if it is what I think it is then, Nagarjuna (saint, scholar, philosopher) aims to create a state of mind very similar to the Zen Mu.

An excerpt from the Wikipedia page on what Mu is:

Some English translation equivalents of wú or mu 無 are:

"no", "not", "nothing", or "without"

nothing, not, nothingness, un-,

is not, has not, not any

Pure human awareness, prior to experience or knowledge. This meaning is used especially by the Chan school

A negative.

Caused to be nonexistent

Impossible; lacking reason or cause

Nonexistence; nonbeing; not having; a lack of, without

The 'original nonbeing' from which being is produced in the Tao Te Ching. — Wikipedia

That out of the way, take a look at what Nagarjuna says:

For any given proposition p, there are only four possible truth states that can be viz.

1. p (p is true)

2. Not p (p is not true)

3. p & not p (p is true and not p is true)

4. Neither p nor not p (not p is true and not not p is true)

Nagarjuna, if I read him correctly, wants us to refuse or deny all 4 possibilities. Let's take a proposition that you seem to be familiar with viz. the buddha exists after death. The conversation that takes place, as recorded in buddhiat texts I suppose, goes something like this:

Assuming E = the buddha exists after death.

1. E. No!
2. Not E. No!
3. E and not E. No!
4. Neither E nor not E. No!

In my humble opinion, this actually amounts to,

1. Refusing to assign a truth value to E

or

2. Proposing a "third alternative" to the usual habit of thinking about propositions in a binary way: affirm or deny a proposition, nothing else is possible.

Since our minds seem to deal only with propositions that have truth values, Nagarjuna's tetralemma by divorcing truth states (true/false) from propositions sentences maybe,

1. suggesting that we look at sentences (utterances, written) not just as true/false/affirmation/denial but as something more than that if that even makes sense.

2. proposing a third alternative to handling sentences i.e. there's one more, as of yet undiscovered (I'm not sure about this), way of comprehrnding sentences in addition to the two we're familiar with which are affirming and denying them.

3. attempting to deliberately, for reasons unknown to me, drive a wedge between thoughts and mind. From the little that I know, the mind seems capable only of tackling sentences that can be affirmed or denied or only those whose truth values can be ascertained or, more to the point, assigned in ways that are unproblematic.

Of course, there's fuzzy logic, paraconsistent logic, dialetheism, and other more exotic varieties of logic out there. Perhaps Nagarjuna's work anticipated these modern developments. I'm not certain.

Nevertheless, denying every possibility for a proposition as Nagarjuna does manage, in an in-your-face kinda fashion, to make sentences, in a sense, unthinkable. This seems to square with the Zen practice of emptying one's mind of, well, thoughts otherwise known as Mu.

Mind you, this is only a hunch, a tentative hypothesis about the link between Nagarjuna's tetralemma and Zen Mu and other ideas.

I think truth and knowledge is good for atheists, scientists and materialists, but it is detrimental to theists, Christians and the otherwise religious. — god must be atheist

Streetlight Effect/The Drunkard's Search Principle. Seeking truths where they're easy to find is not my idea of good faith. Different strokes for different folks.

Keep in mind the distinction between truth and validity, and the a priori and contingent. And do not ever confuse logic with communication, especially with a woman. — tim wood

I'm fairly onfident of my ability to distinguish those concepts you mentioned above but they don't seem to be relevant insofar as the tautology p v ~p is the issue. As for women I'm absolutely clueless.

My conception is, in order to optimize agency (eudaimonia), the latter (praxis) applies, or strives for, the former (arete). — 180 Proof

I don't know if this is relevant or not but I recall listening to a podcast many suns ago and the speaker defined wisdom as, and I quote, " the knowleddge of what is true and good". A perfect example, don't you think?, of the belief, perhaps intuition, that the good and the true are intertwined like snakes in a mating ball ( @tim wood :wink: ), quite possibly good & true are two sides of the same coin.

To All Of The Above Posters

The tautology p v ~p for any proposition p guarantees that one speaks the truth, that's for sure. However, what bothers me is that employing p v ~p to never lie also commits you to the law of the excluded middle which, to my understanding, is that there are only two possibilities for a given proposition p, either one must affirm it (p) or one must deny it (~p) which together is p v ~p. The law of the excluded middle is also considered to be just another way of saying contradictions p &~p are impossible because ~(p & ~p) = p v ~p. Understood differently, deny the law of the exluded middle is tantamount to claiming contradictions are true.

Noteworthy too is that some logics reject the law of the excluded middle i.e. ~(p v ~p) but that doesn't necessarily imply that one has to accept contradictions are true i.e. for some systems of logic, ~(p v ~p) =/= p & ~p

In summary,

1. The easiest way to speak the truth with respect to a proposition p always and everywhere to anybody who cares to listen is to utter/write p v ~p. This has been explained well enough in the OP.

Now for some detective work bordering on conspiracy theory. The long tradition of making truthfulness a big deal in ethics, especially in religion which seems to be the inspiration for Kantian ethics may be a clue put there by ethicists of days bygone (Buddha to Kant), either deliberately but elliptically or unwittingly but intuitively, to eventually guide us to the problem disguised as an easy solution for dishonesty/lying viz. the law of the excluded middle. In other words the (easy) solution for mendacity, the law of the excluded middle, is actually the source of all our problems, ethical or otherwise. Perhaps like any truther worth his salt, I'm reading too much into this.

2. What happens if make the obvious choice and reject the law of the excluded middle but within the bounds of classical logic? We must perforce accepts contradictions are true. What would that mean for us? Taoism which I believe is a treatise on paradoxes (contradictions); paraconsistent logic; dialetheism; Zeno of Elea; As you can see I'm out of my depths.

The last possibility listed in Nagarjuna's tetralemma is ~(p v ~p) i.e. it amounts to denying the law of the excluded middle. Unfortunately, I can't comment more on it because the Wikipedia page doesn't provide a good enogh explanation. Other resources don't seem to be to be up to mark either. Suffice it to say thqt Nagarjuna denies contradictions (p & ~p) and so he must've had something else in mind by, what I believe is a, rejection of the rejection of the law of the excluded middle.

It was never put in those dry scholastic terms in the early Buddhist texts. — Wayfarer

An old thread I know but I think there's something interesting going on.

Given any proposition p, there are 4 possible states it can be in, yes p, no p, yes p and no p, snd neither yes p nor no p.
1. p (yes p)
2. ~p (no p)
3. p &~p (yes p & no p)
4. ~(p v ~p) (neither yes p nor no p)

Nagarjuna calls them the 4 extremes. He negates them all

1. p: not p
2. not p: not not p
3. (p & ~p): not (p & not p)
4. ~(p v ~p): not ~(p v ~p)

shortest proof length for any given theorem ('length' means the sum of the lengths of the formulas that appear as lines in the proof)? — Shawn

I see a paradox. If I'm anywhere near the ballpark, finding a shorter proof (should) take(s) longer than finding a longer proof? A dilemma in the making? Should we save space (short proofs) or should we save time (long proofs)? :chin:

Money seems to have two dimensions to it with respect to its value. The only good analogy that comes to mind is philately [stamp collection]. A stamp, back in the days before email, had two kinds of value viz. 1) to send mail (a use that's dying out or is already nonexistent) and 2) as part of a philatelist's precious collection.

Money too has a more or less similar value pattern viz. 1) purchasing power (buy stuff) and 2) as a collector's item in a manner of speaking (hoarding greenbacks, gold, etc.)

As for the ethics of money, both what one spends money on and how much of it one collects can matter, right? The two seem related in quantitative terms as the amount one possesses will vary inversely with the amount one spends.

However, if you must know, to seek the link between ethics and money per se seems to be, in a sense, a misconception. One only has to turn the pages of a history book to realize that - ethical and unethical deeds motivated by economics predate money, by thousands of years in all likelihood.

That said, money is, all said and done, a unique economic invention and it could be that there ethical issues exclusive to it. Analyzing the ethics of money would then require us to get a handle on the modus operandus so to speak of money and what psychological aspects of humans are vulnerable to such.

You thought about it for at least half a minute? — TonesInDeepFreeze

Well, sorry. I have issues. I hope you'll cut me some slack and let my impudence slide. G'day. You've been very helpful.

Imagine a high security prison, like a set {...}. You're the prison warden and under your careful watch, there have been no untoward incidents - no prison breaks, fairly good behavior from the inmates, no riots, etc.

One day, you're at your desk and a prison guard comes up to you and announces the arrival of a new prisoner, his name is K. K is a very unique prisoner. What's unique about fae? Well, fae has the following relationship with prisons, K ={K}. In other words, K inside the prison {K} is equal to (is the same as) K outside the prison.

You don't think that's too much of a problem and imprison K like so, {K} and feel quite content with the arrangement. In what seemed almost instantaneous, a guard informs you that K is no longer in prison because K = {K}.

You're angry and surprised but you give the matter some thought and realize K = {K}. You decide that one "solution" is to build another prison outside your prison and if so {K} will become {{K}} and K will become {k}. K is now in prison or so you think.

{{K}} = {K} but {K} = K. The prison guards are mortified as K is again outside the prison (it couldn't be contained in a set).

You then decide to build a third prison like so {{{K}}} and feel confident that you've finally managed to solve the problem...once and for all.

Unfortunately for you, {{{K}}} = {{K}} = {K} = K. Again, K is no longer in prison.

You hold a meeting with your colleagues and after many, many hours of brainstorming you soon realize the gravity of the situation K = {...{...{...{K}...}...}...} i.e. even if you build an infinite number of prisons, K can't be imprisoned.

So you figured why this is not the case?: — TonesInDeepFreeze

No.

That is so blazingly incorrect that it scorches the core of this planet. — TonesInDeepFreeze

:ok: Thank you for your time.

{P} is a member of other sets. — TonesInDeepFreeze

Assume that it's impossible to make P a member of another set i.e. {{P}} is not possible or, more relevantly, always {P}.

Now if sets csn contain themselves, {P} = {{P}} = {{{P}}} =...

In other words, always P.

That means a set {P} that can't be contained in another set (always {P} never {{P}}) is the same as the set {P} that contains itself (even if {{P}} always {P}).

You see the point don't you? A set {P} that contains itself is the set that can't be a member of another set!

1. Take a set {P}. If it's impossible to make this set a member of another set, then {P} always. Any attempt {{P}} will result in {P}

2. P = {P} where P is the set that contains itself. Let's try to make {P} a member of another set like this, {{P}}. But {{P}} = {P}. In other words, {P} always

So,

A set that cannot be made a member of a another set (see 1 P always) is the same as a set that can be made a member of itself (see 2 P always).

Contradiction.

Hence, Impossible/false that P = {P}.

"something was done to it" is not a set theoretic predicate. — TonesInDeepFreeze

So, am I to think that putting, say, a list of items e.g. 1, w, # inside curly braces like so, {1, w, #} amounts to doing nothing?

1. There's a set N
2. {N} does something to N
3. N = {N} [a set can contain itself][assume for reductio ad absurdum]
4. If N = {N} then {N} does nothing to N
5. {N} does nothing to N [3, 4 modus ponens]
6. {N} does something to N and {N} does nothing to N [2, 5 conj]
7. False that N = {N} or no set can contain itself [3 - 6 reductio ad absurdum]

Thanks for starting this thread. I'd like to share with you my finding regarding Russell's paradox.

Russell's paradox boils down to whether the set of all sets that doesn't includes themselves in includes itself? or, for easier comprehension of my attempt at a resolution of the paradox, does the set of all sets that doesn't contain themselves contain itself?

As you can see, one of the central issues is whether a set can contain itself or not because Russell's paradox can be rephrased as,

1. If C doesn't contain itself then C contains itself
2. If C contains itself then C doesn't contain itself

where C = the set of all sets that don't contain themselves.

If sets can't contain themselves, the consequent in 1 above and the antecedent in 2 above become meaningless for sets can't contain themselves. At least that's what I think.

If sets can contain themselves, there should be a set N ={N}

1. N = {N}
2. N contains itself, {N} contains itself
3. If {N} contains itself, {N} = {{N}}

Did you see what happened there?

The outermost curly braces "{...}", in a sense, collapsed or behaves as if it didn't exist at all: {N} = { {N} }.

Likewise,

4. N = {N} = {{N}} = {{{N}}} = {{{{N}}}} =...

Same things's happening or rather not happenning as any attempt to make the set N an element of another set like {N} returns, to use a computer terminogy, the set N itself.

In short, the set N, though defined as {N} can't be contained in another set for the reasons provided above.

The entire series N = {N} = {{N}} =... is an illusion so to speak.

Thus, a set N such that N = {N} can't exist. In other words, no set can contain itself and so Russell's paradox is a none issue.

Suppose a set P

1. P = P [reflexivity. Nothing was done to set P]

2. {P}. P was made an element of the set {P}. Something was done to P]

3. P = {P} [P is the set that contains itself] [assume]

4. Nothing was done to P = Something was done to P [contradiction]

5. P =/= {P} [3 - 4 reductio ad absurdum]

But I don't get why someone would post arbitrary non sequiturs. — TonesInDeepFreeze

Work in progress.

Please stop using '=' to stand for the biconditional.

(1) N = {N} premise

(2) N e N <-> {N} e {N} from (1)

(3) ~ {N} e {N} non sequitur

Why do you waste our time? — TonesInDeepFreeze

Thanks for the clarification and I'm sorry if I've wasted my time but I suppose for people likey yourself who have to deal with those less knowledgeable than themselves, it's part of the territory.

he realized that the only thing worse than his depression was the psychiatric treatment he was receiving for it — baker

Sorry to hear that.

From inability to let well alone, from too much zeal for the new and contempt for what is old, from putting knowledge before wisdom, science before art and cleverness before common sense, from treating patients as cases and from making the cure of the disease more grievous than the endurance of the same, good Lord deliver us. — Sir Robert Hutchison (1871 - 1960)

First, do you understand my explanation that you just quoted? — TonesInDeepFreeze

Sorry, if it seemed as though I hadn't paid attention to it but my argument in the previous post seems to make your well-meaning explanation moot.

1. Take N = {N}, the set that contains itself.

2. N contains itself = {N} contains itself

3. {N} doesn't contain itself.

QED

It fallacious to argue that, because N is written as '{x N}' in '{x {x N}}' but as 'N' in '{x N}', we have that {x N} is not in {x N}. It is fallacious to argue from the mere happenstance of two different means of notating the set. — TonesInDeepFreeze

Ok. I've run out of options. Let's get straight to the brass tacks.

A set that doesn't contain itself = {1, &, :sad: }

Now, consider the set {...} = N1 where "..." stand for element(s). Suppose {...} = {N}

Let's try and make N1 contain itself, {...{...}} = {N, {N}}

But N2 =/= N1 as {...} =/= {...{...}} or {N} =/= {N, {N}}

In other words, N1 couldn't be made to contain itself.

Let's try to make N2 contain itself, {...{...{...}}} = N3

But N2 =/= N3 as {...{...}} =/= {...{...{...}}} or {N, {N}} =/= {N, {N, {N}}}

So and so forth.

In other words, no set can be made to contain itself as trying to do that results in the new set being different from the old set i.e. the (say) set A that contains the set A is different from the set A.

The only option then is to try out the empty set N1 = { }.

Let's try and put N1 inside N1, {{ }} = N2

Unfortunately, { } =/= {{ }}

So and so forth...

I've proved here that N = {N} isn't possible

@TonesInDeepFreeze

1. Sets can contain themselves. [assume for reductio ad absurdum]
2. Suppose N is a set that contains itself
3. Let N = {x, N} where x is either no elements or x is some elements
4. N = {x, {x, N}} [substituting N with {x, N}]
4. N - N = {x, N} - {x, N} = { } [N = {x, N} from 3]
5. N - N = {x, {x, N}} - {x, N} = {{x, N}} = {N} [N = {x, {x, N}} and N = {x, N} from 4 abd 3]
6. N - N = { } and N - N = {N} [from 4 and 5] [contradiction]
Ergo,
7. Sets can't contain themselves [1 - 6 reductio ad absurdum]

What do you think?

That went over my head. Thank you for trying though.

Keep it simple for me, ok.

1. Assume whatever axiom you want to/have to assume to prove the proposition, C = sets can contain themselves.

2. I'll assume, given your knowledge, C is proven or C

3. If C then there has to be a set N ="{...} such that it contains itself

4. There has to be a set N = {...} such that it contains itself [2, 3 modus ponens]

My request to you is express N as I express the set P (prime numbers less than 4), P = {2, 3}. In other words, I want to know if it's possible to have a set N = {x, N} where x = no, one, or more members of that set.

This seems impossible for the following reason.

1. There's a set N that contains itself [assume for reductio ad absurdum]
2. Suppose, N = {x, N}
3. N is a proper subset of {x, N} [proper subset of a set]
4. If N is a proper subset of {x, N} then, n(N) < n({x, N})
5. n(N) < n({x, N}) [3, 4 modus ponens]
6. n(N) = n(N) [true for all sets]
7. n(N) < n(N) [2, 5 substitution]
8. n(N) = n(N) and n(N) < n(N) [contradiction]
Ergo,
9. There's no set N that contains itself [1 - 8 reductio ad absurdum]

reflexivity — Wayfarer

Correct but it's not just self-reference, it's also negation of some kind. The self-referential sentence, "I exist" doesn't create problems like the self-referential negation, "I don't exist."

Let's look at the liar sentence L = this sentence (L itself) is false.

According to most books, the logic is as below,

Option 1
1. If L is true then L is false (the liar sentence)
2. L is true (assume)
3. L is false (1, 2 modus ponens)
4. L is true and L is false (2, 3 together, contradiction)
Ergo,
5. L is false (2 - 4 reductio ad absurdum)

Option 2
6. If L is false then L is true (the liar sentence)
7. L is false (assume)
8. L is true (6, 7 modus ponens)
9. L is true and L is false (7, 8 together, contradiction)
Ergo,
10. L is true (7 - 9 reductio ad absurdum)

Now, what's interesting is,

1. If L is true then L is false = L is false or L is false = L is false
That means the argument in option 1 becomes,
11. L is false = If L is true then L is false
12. L is true (assume)
13. L is true and L is false (11, 12 together, contradiction)
Ergo,
14. L is false (12 - 13 reductio ad absurdum)

However, notice line 11 (premise) = line 14 (conclusion). In other words, what was to be proved was assumed beforehand among the premises. The argument is circular which simply means 14. L is false is unwarranted.

Similarly, revisiting option 2, the statement 6. If L is false then L is true = L is true or L is true = L is true

The argument for option 2 then becomes,
15. L is true = If L is false then L is true
16. L is false (assume)
17. L is true and L is false (15, 16 together, contradiction)
Ergo,
18. L is true (16 - 17 reductio ad absurdum)

Notice here again that line 15 (premise) = line 18 (conclusion). Put simply, the conclusion has been assumed in the premises. Circular argument, which means we're not justified in concluding 18. L is true.


What does this all mean? We can arbitrarily assign a truth value to the liar sentence (true/false) but that's where it all stops - all logic beyond that is going to be circular and useless. Since the alleged contradiction of the liar paradox can only occur after an inference which begins with an assumption of a truth value for the liar sentence, and since, as explained above, all such arguments are circular, we're no longer justified to infer anything at all (that includes any further truth value for the liar sentence) from the initial assigned truth value for the liar paradox. Ergo, there being no inferrable truth value, there can be no contradiction. In short, the liar paradox doesn't entail a contradiction at all.

Kindly point out a flaws in my argument if there are any.

You mentioned that without the axiom of regularity, we can't prove ~Ex xex. As far as I can tell that means,

Can prove ~Ex xex -> Axiom of regularity.
No axiom of regularity -> Can't prove ~Ex xex

Since I've proved a set can't contain itself, it follows that I've assumed the axiom of regularity then.

Challenge for you: Can you prove that a set contain itself? Feel free to use any axiom of your choice.

Without the axiom of regularity, you can't prove

~Ex x = {x} = {{x}} = {{{x}}} ... for as finitely many iterations you want to make. — TonesInDeepFreeze

Then, there can't be a set that contains itself.

1. All sets are sets that don't contain themselves

2. No sets that contain themselves are sets [from 1]

In other words, sets that contain themselves aren't sets which simply means the sentence, "sets that contain themselves" is meaningless.

Let's visit Russell's paradox now.

Suppose C = set of all sets that doesn't contain itself

1. If set C doesn't contain itself then, set C contains itself.

On the face of it, statement 1 looks reasonable but, as shown above, the consequent of the conditional 1 (above) is meaningless ["set" and "contains itself" in the same sentence is a contradiction]. Ergo, Russell's conditional (1 above) is gibberish.

2. If set C contains itself then, set C doesn't contain itself.
Again for the same reason as above, the antecedent of this conditional is nonsense. Ergo, Russell's conditional 2 is balderdash.

Russell's paradox can't be a paradox because the two key conditionals have no meaning at all.

x = {x} — TonesInDeepFreeze

So, set X = {X} = X1
That means, I can substitute X with {X}.

Here goes, X = {{X}} = X2

X1 = X2 (should be) because both are X but {X} =/= {{X}}.

Here's a more direct reference to Godel's incompleteness theorem vis-à-vis the Liar paradox.

Gödel's incompleteness theorems are two fundamental theorems of mathematical logic which state inherent limitations of sufficiently powerful axiomatic systems for mathematics. The theorems were proven by Kurt Gödel in 1931, and are important in the philosophy of mathematics. Roughly speaking, in proving the first incompleteness theorem, Gödel used a modified version of the liar paradox, replacing "this sentence is false" with "this sentence is not provable", called the "Gödel sentence G" — Wikipedia

Please use the notation I prescribed viz. a set with its elements/members enumerated.

A set that doesn't contain itself: {1, y, $}

A set that contains itself: ???

Just give me one instance of a set that contains itself. I can't. Can you?

Here's a set that doesn't contain itself: A = {7, &, troll}

Here's a set that contains itself: Your turn

From the intro to Gödel's Theorem: An Incomplete Guide to Its Use and Abuse.

You might find the answers in there. — Wayfarer

Thank You Wayfarer

Relationship with the liar paradox [of Godel's Incompleteness Theorems]

Gödel specifically cites Richard's paradox and the liar paradox as semantical analogues to his syntactical incompleteness result in the introductory section of "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". The liar paradox is the sentence "This sentence is false." An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentence G for a system F makes a similar assertion to the liar sentence, but with truth replaced by provability: G says "G is not provable in the system F." The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar sentence. — Wikipedia

Please carry out your own investigations into the issue if you're interested of course. Good luck.

Set theory doesn't prove things in this kind of context by saying "the task cannot be completed". — TonesInDeepFreeze

Ok. There are a couple of different ways sets can be written down as:

1. Set notation: S = {x | x is a prime number less than 10}
2. In words: S = {all prime numbers less than 10}
3. As a list of members of the set: S = {2, 3, 5, 7}


1. Set notation, C = {x | x is a set that contains itself}. Nothing seems amiss
2. In words, C = The set that contains itself. Doesn't seem to be problematic
3. As a list of members of the set: C = ??? [Problem. No such set can be]

No, you know it.

(1) is set theory proving there is no set whose members are all and only those sets that are not members of themselves.

(2) is Tarski's theorem. — TonesInDeepFreeze

How about the following argument. Please go easy on me.

Suppose C = the set of all sets that don't contain itself

1. A set can contain itself like so A = {1, A} but there seems to be problem when you list down the elements. Suppose B is a set that contains itself:.{1, 2, {1, 2, {1, 2,..the task can't be completed. In other words, no set can contain itself. Ergo ALL sets are sets that don't contain themselves. Can we construct a set of all sets that don't contain themselves? Why not? Of course we can because ALL sets can't contain themselves. However, this set can't can't contain itself.

1. All sets don't contain themselves.
2. The set of all sets that don't contain themselves = The set of all sets.
3. The set of all sets is impossible because it can't be member of itself and so it can't be the set of all sets. (from 1)
4. The set of all sets that don't contain itself is also impossible (from 2 and 3).
5. For Russell's paradox, the set of all sets that don't contain itself must be a set.
6. The set of all sets that don't contain itself is impossible i.e. it isn't a set.
Ergo,
7. Russell's paradox is not a paradox.

Above my paygrade. I'm expecting a raise soon!

I thought Russell's paradox was meant to undermine set theory. As far as I can tell, it begins as C = the set of all sets that doesn't contain itself. Either C is itself in C or not. If C contains itself, C can't contain itself. If C doesn't contain itself, C contains itself. In short, C contains itself AND C doesn't contain itself. Contradiction.

Let's look at the issue as the conditionals they're said to be

1. If C contains itself then C doesn't contain itself
2. C doesn't contain itself or C doesn't contain itself (1 Imp)
3. C doesn't contain itself (2 Taut)

4. If C doesn't contain itself then C contains itself
5. C contains itself or C contains itself (4 Imp)
6. C contains itself (5 Taut)

Ergo,
7. C contains itself and C doesn't contain itself (3, 6 Conj) [Contradiction]

But then, Russell's argument boils down to,

3. C doesn't contain itself
6. C contain itself
8. C contains itself and C doesn't contain itself (3, 6 conj)
Ergo,
9. C contains itself (deny 3)
Or
10. C doesn't contain itself (deny 6)

Both 3 and 6 can't be denied...leads to a contradiction.

It's a tautology, C contains itself or C doesn't contain itsellf. There's no contradiction!

Thank you. Have a G'day.

@Amalac

I apologize for butting in but I want to run something by you.

There are three parts to time - past, present, future.

The future presents no problems as regards it being an infinity for the simple reason that it doesn't, in a sense, violate our mind's "integrity" which seems to, in this particular case, hinge on the Aristotelian distinction between actual and potential infinity - that the former kind is ontologically suspect while the latter kind is acceptable.

The past, however, is treated as something complete - done with so to speak - and thus, any talk of the past being an infinity immediately sets off alarm bells inside our heads.

It might be of some help hsre to look at the definition of infinity. I visited the Wikipedia page on infinity and among the various meanings of infinity, I find this:

but infinity continued to be associated with endless processes — Wikipedia

Concentrate all your fire on the nearest starship...er...I mean focus your attention on "endless". Infinity, it seems, is defined in terms of an end, to be precise, endless. There simply is no talk of the corresponding concept of an end viz. the beginning.

Could this mean we can't, shouldn't, apply the idea of infinity to beginnings? If yes, then infinity and a beginning to time or anything else for that matter can't be harmonized in a way that we can grasp.

Why, you may ask, is it that people (philosophers/scientists/theologists/etc.) continue to investigate this matter as if it makes sense?

For my money, this is the situation because we're reversing the arrow of time i.e. we're looking at the past as a time traveller. When we do that the past flips in a manner of speaking and morphs from an actual infinity to a potential infinity. How you look at the past (normally or as a time traveler) will cause it to flip-flop between an actual infinity and a potential infinity. This, in all probability, will play havoc with our minds. My puny brain at least is baffled beyond measure.

I defer to your better judgement. @Wayfarer :wink:

Sarvam mithyā bravīmi — Bhartrihari

I agree whole-heartedly.