Liar's paradox

[Antone]

Talk (thought) is cheap.

http://en.wikipedia.org/wiki/Liar_paradox"

The illiterates of the 21st century will not be those who cannot read and write, but those who cannot learn, unlearn, and relearn." —Alvin Toffler

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“Silence [Stillness] is the true friend that never betrays.” —Confucius
arc-us

The last attmep wasn't very encouraging, but Just for laughs... here's another attempt to present a logical argument--and see what kind of comments I get.

The simplest version of the liars paradox is probably just the sentence:

(L1) The statement I am making is false.

or more succinctly, simply:

(L2) I am lying.

These sentences are paradoxical because, on the one hand, if the person who utters them is telling the truth then they must be truthfully saying that they are lying. But if they are indeed lying then how can they be telling the truth? On the other hand, if the person who utters them is telling a lie, then what they are saying must be the opposite of the truth. Since they are sying that they are lying, this means that they must be telling the truth--which is incompatibel with the original premise that they are lying.

My Resolution
This version of the paradox can be resolved rather simply by understanding that sentences which are intended to convey [truth and mean] are always defined by the two reciprocal aspects--[truth] and [meaning]. What is important to understand about this relationship is that

the meaning aspect does not convey a sense of truth; and
the truth aspect does not convey a sense of meaing.

For an example of a statement that has [meaning] but no [inherent truth value], consider the following statement:

(H) He is in love.

On its own, (H) is neither [true] nor [false]. If the pronoun [he] is understood to refer to a [particular Tom who is in love], then (R) is understood to be [true]. But if [he] refers to a [particular John who is not in love], then (R) is understood to be [false]. The important thing to understand is that (H) only takes on a truth value when it is being used to refer to a specific person.

There are also examples of statements that have an [inherent sense of truth] but no [inherent meaning]. For example,

(X) x = x ...............(Read, x is absolutely equal to x.)

Obviously, the symbols themselves have meaning. We know that [x] is a vague variable; and we know what [is absolutely equal to] means. But the sentence doesn't tell us anything that we don't already know before we read it. It is something that can be known a priori.

The reason this is important (with respect to the Liar's Paradox) is because it means that (L) the liar's statement (in whatever version) is neither [true nor false] until we apply it to a specific situation. This is evidenced by the fact that if (L1) or (L2) are not being spoken by someone they they cease to be paradoxical. When not spoken (L2) simply conveys a meaning that might be expressed as:

The first person speaker is lying

It makes no sense to even consider assigning a [truth value] to the statement until someone actually utters the phrase.

Now, we have already seen that [truth] and [meaning] are independent--or, in other words, we can have a sentence with one aspect and not the other. Most of the time, these two aspects are united by making the statement refer to a specific instance. The cleverness of the Liar's Paradox is that it is structured in such a way that it becomes impossible for these two aspects to become united--as they normally are. This is done by making (L1) a negative statement about itself.

Now, in order to determine if the meaning of (L1) is [true], we must refer back to (L1) as the confirming or disconfirming instance. The reason this is so terribly problematic is because it violates a fundamental principle of sentence structure. As we stated earlier, some sentences have [inherent meaning] and others have [inherent truth], but in both cases, the sentence is given it's [sense of the reciprocal aspect] by comparing it to something that is [not the sentence]. For example,

(H) He is in love.

has inherent meaning--and we give it a [sense of truth], by making (H) about a [specific someone]. That someone is outside of (H). They are other than (H). There is nothing about (H) that implies who that someone should be. Similarly,

(X) x = x

has inherent truth--and we give it a [sense of meaning], by making (X) about something that isn't in (X). For instance, we might replace [x] with [apple1]. Thus (X) becomes:

(A) apple1 = apple1

Now, (X), which was without a specific meaning has taken on the specific meaning that [an apple is an apple].

The Liar's Paradox is created by violating this fundamental rule of reciprocal structure. Now, instead of [one aspect being derived from within the sentence] and [one aspect being derived from outside the sentence]--both aspects are derived from within. The necessary (and expected) result is that (L) can no longer function as a bridge that unites the two reciprocal aspects of a sentence into one [meaningful truth]--the way most practical sentences do. Instead, it creates a sentence that is incapable of carrying a [sense of truth].

This is not a paradoxical situation, because it is exactly what we should expect. (At least if we assume that all things are defined by reciprocal aspects, as I do.)

When we look to (L) as the [confirming instance that provides a sense of truth] to [L's meaning], there is nothing to provide the desired [truth] because the instance is simply [L's meaning--which is devoid of truth]. If (L) simply stated that the sentence was true, then there wouldn't be an apparent paradox. It would still only be a statement that had meaning but no truth. We understand this at an intuitive level, but it is easy to overlook when we're using our rational minds to scrutinizing it carefully. Thus it seems easy to suppose that [This statement is true] might actually be a true statement. And even if not there isn't any contradiction. But becasue the meaning of (L) is that it is false, it seems reasonable to assume that there is a contradiction--even though the statement still only meaning and no truth.

Finding this [absence of truth] unsatisfactory, we can choose to arbitrarily assume that (L) will have one truth value or the other. But as the paradoxical result confirms, this is an invalid assumption. Because we have broken the rules of forming a sentence--we have made it impossible for (L) to carry a [value of truth].

[altonhare]

Just because I can say something is not what it is, doesn't make it true

Premise: A is A

Question: Is A not A?

Answer: No, A is A.

Done.

[altonhare]

I actually think you did a fairly good job on the analysis in this one. Saying it another way, a specific statement has no meaning if it has no context (doesn't refer to something outside itself). It may be true, as you said, that x=x; but this has no meaning unless x refers to something in reality.

On the other hand, a statement can refer to something in reality but still have no meaning if it contradicts itself. "This apple is not this apple" means "This apple is not itself nor is it anything else" begs the question "Then what is it?"

So, if I say "I am lying" it begs the question "about what?".

If I say "I am lying in saying that I'm lying" that's just an explicit contradiction i.e. "Statement A is neither true nor untrue" begs the question "Then what is it?"

[bboyer]

I actually think you did a fairly good job on the analysis in this one. Saying it another way, a specific statement has no meaning if it has no context (doesn't refer to something outside itself). It may be true, as you said, that x=x; but this has no meaning unless x refers to something in reality.

On the other hand, a statement can refer to something in reality but still have no meaning if it contradicts itself. "This apple is not this apple" means "This apple is not itself nor is it anything else" begs the question "Then what is it?"

So, if I say "I am lying" it begs the question "about what?".

If I say "I am lying in saying that I'm lying" that's just an explicit contradiction i.e. "Statement A is neither true nor untrue" begs the question "Then what is it?"

Well done. The highlighted portion above is what has nagged at me in several posts where you present the choice of (paraphrasing) "Does, or do you agree that, A=A; yes or no?" Which always struck me as a bit of the False Dilemma (http://en.wikipedia.org/wiki/False_dichotomy).

So, in a specific "snapshot" context of both, or many of us, observing a specific apple at a specific and precise point in time, yeah that apple is itself (Identities are identical). However, roll the cameras, and over a period of time that object we have agreed upon as embodying our concept of "apple" will dynamically change it's shape and appearance (morphological transformation) and will eventually ... lose ... its transitional, temporal identity as apple, dis-integrating back to its many amorphous, composite parts (amorphous, compared to its previous morphology as integral apple; or "apple entity" as I think you might term it).

Physical, objective identity - identity as a specific object - is transitory, transitional, temporal and completely conditional. It is only the illusory, mental concept of a symbolic identity where conceptual, symbolic A=A is unconditionally and "forever" true.

Anyways, for whatever that's worth.

[altonhare]

Well done. The highlighted portion above is what has nagged at me in several posts where you present the choice of (paraphrasing) "Does, or do you agree that, A=A; yes or no?" Which always struck me as a bit of the False Dilemma (http://en.wikipedia.org/wiki/False_dichotomy).

Did what I say affirm or refute your misgivings that it was a false dichotomy/dilemma?

In response to what you said about the apple. Yes we may point to an apple and say "apple". Then as we watch it, it changes (decays etc.). We see it as "not an apple". But we know that A=A so our goal is not to accept a contradiction but to explain the *observation* that an apple changes into a blob.

The explanation is something like, the apple is not really a single continuous thing but rather is composed of atoms. The atoms interact with other atoms in the air (or whatever). The entity "apple" is transforming because of external influences.

When we extend this logic down the chain (har har) we watch a single atom. We point at it and say "carbon". But as we watch it, it changes too! Maybe it started as C14 and now it's C12. Or maybe it was sp3 hybrid and now it's sp2. Whatever. Why does the carbon atom we originally identified, become something else?

The answer is the same as for the apple. The atom is made of constituents too. Now we watch a constituent. If it changes we watch its constituents... and so on and so forth until we have discovered the "fundamental constituent". This constituent is unique in that, unlike the apple or the atom, it is truly immutable. We watch it, and it doesn't change. It doesn't compress or break (if it breaks or compresses its made of constituents and we keep looking ). Therefore, all entities are ultimately identified in a single way (A is A) by their fundamental constituents. Therefore I distinguish between entities (aggregates of fundamental constituents) and objects (the fundamental constituent itself, continuous, unbreakable, and undeformable). Each entity now has a specific identity (A is A) by virtue of a specific arrangement of its fundamental constituents. The apple is defined as some specific sets of arrangements. When these constituents are no longer at one of these arrangements, the apple is no more! But it didn't turn into "nothing", the constituents just changed location. Instead of apple we have "apple2" or "glob" or "apple juice" etc. This new entity has a new, specific identity.

Make sense arc-us?

[bboyer]

Well done. The highlighted portion above is what has nagged at me in several posts where you present the choice of (paraphrasing) "Does, or do you agree that, A=A; yes or no?" Which always struck me as a bit of the False Dilemma (http://en.wikipedia.org/wiki/False_dichotomy).

Did what I say affirm or refute your misgivings that it was a false dichotomy/dilemma?

In response to what you said about the apple.... <snip>.

Make sense arc-us?

In the context here, I have no problem with what you've stated.

[altonhare]

Well done. The highlighted portion above is what has nagged at me in several posts where you present the choice of (paraphrasing) "Does, or do you agree that, A=A; yes or no?" Which always struck me as a bit of the False Dilemma (http://en.wikipedia.org/wiki/False_dichotomy).

Did what I say affirm or refute your misgivings that it was a false dichotomy/dilemma?

In response to what you said about the apple.... <snip>.

Make sense arc-us?

In the context here, I have no problem with what you've stated.

You have a problem in a different context?

[bboyer]

You have a problem in a different context?

Guess that would depend on the specificity of the different context. Just saying that the way you explained yourself, the explanatory context as stated, sounded fine by me. Full stop. I'm done, in other words.

[Antone]

It may be true, as you said, that x=x; but this has no meaning unless x refers to something in reality.

Wrong.

[x=x] means that [any x] is [that x]. For example, if we let [x=apple] then [x=x] becomes [an apple is an apple]. If we let [x= dog] then it becomes [a dog is a dog]. No matter what we substitute for x, the statement remains true and that is where the truth and meaning of the sentence is contained... in the structure of the sentence itself.

This is a reciprocal kind of meaning and truth to a statement such as [x = y + z]. In this case, the truth and meaning of the sentence depends entirely on what terms are substituted for [x], [y] and [z]. We can only know that the statement is true after the substitutions have been made. This is a definition. In a sense, all definitions make the claim that [some x] is [something other than x]... or in other words that [x] equals [not x]. This is inherent in the fact that it is a definition. [y + z] is not [x]. The only way we can understand it as being [x] is in the sense that [y + z] conceptually define what it means to be [x]. And this is only true to the extent that the terms we substitute are conceptually appropriate--and there are a number of reasons that the substitutes can never be absolutely appropriate: (1) real things have vague borders. (2) each person has a slightly different notion of what a term means, and so forth.

Thus, there are two reciprocal kinds of meaning and truth that a statement can have. The structural truth: [x=x] has absolutely no definitional meaning. It's truth does not depend on what terms are substituted. And there is no vagueness whatsoever about the fact that it is a truth. It cannot help being a truth, no matter what substitution is made.

By contrast, the definitional truth [x=y+z] has absolutely no structural truth. It's truth depends entirely on on the specific terms that are substituted. And there is typically a good deal of vagueness about whether or not the definition is true. For instance, we might say an [apple] is a [crunchy fruit]. This is true, as far as it goes, but there are other fruits that can be called crunchy. So the definition is inadequate to fully define [what a fruit is not]. Even when a definition appears to be fully defined, it will always be inadequate--precisely because only [x=x]. Anything other than [x] is (at least in some sense) [not x]. So any definition will always be (at least partially) inadequate to [fully explaining what x is].

Once again, we see the reciprocal nature of reality.

[StevenO]

Once again, we see the reciprocal nature of reality.

I like that observation. Now if one realizes that time is the reciprocal of space physics becomes a lot more logical.

[altonhare]

It may be true, as you said, that x=x; but this has no meaning unless x refers to something in reality.

Wrong.

[x=x] means that [any x] is [that x]. For example, if we let [x=apple] then [x=x] becomes [an apple is an apple]. If we let [x= dog] then it becomes [a dog is a dog]. No matter what we substitute for x, the statement remains true and that is where the truth and meaning of the sentence is contained... in the structure of the sentence itself.

This doesn't make any sense. As soon as you say "[x=x] means that any particular x is itself, you are referring to something outside the statement (a particular...). The statement itself refers to nothing particular. As soon as you insert apple or dog etc. the statement now refers to something outside itself. By itself x=x is just an internally consistent statement without meaning (physical/existent referent). Otoh when I point at *this* keyboard and say "this keyboard is itself" the statement is both internally consistent (true) and has meaning (existent referent).

[Drethon]

x=x can also be false if time is not taken into account.

x=x is true if x=x means x@time 1=x@time 1. Another possibility is x=x means x@time 1=x@time 2 which is false.

Even worse, without appropriate precision time1!=time1...

[altonhare]

x=x can also be false if time is not taken into account.

x=x is true if x=x means x@time 1=x@time 1. Another possibility is x=x means x@time 1=x@time 2 which is false.

Even worse, without appropriate precision time1!=time1...

If you "bring time into account" we are looking at a dynamic problem. X=X is a static scenario.

[Antone]

It may be true, as you said, that x=x; but this has no meaning unless x refers to something in reality.

[x=x] means that [any x] is [that x]. For example, if we let [x=apple] then [x=x] becomes [an apple is an apple]. If we let [x= dog] then it becomes [a dog is a dog]. No matter what we substitute for x, the statement remains true and that is where the truth and meaning of the sentence is contained... in the structure of the sentence itself.

This doesn't make any sense. As soon as you say "[x=x] means that any particular x is itself, you are referring to something outside the statement (a particular...). The statement itself refers to nothing particular.

This is not entirely true. The statement refers to something quite specific. It is the idea that anything is absolutely identical to itself.

You are correct that there isn't any physical instance of this statement that is exactly the statement--but that is only because ideas are not physical. They are reciprocal opposites. When you turn the statement into it's opposite: i.e. something physical, then you replace the [x] with whatever the physical thing is. At this point, the statement is the opposite of what it was. For example, it is specific where once it was general; it refers to something physical where once it referred to something that was necessarily conceptual.

As soon as you insert apple or dog etc. the statement now refers to something outside itself.

Yes, being [physical]--an apple refers to something other than the [idea of anything]. That has always been my point--and that is why it is so crucial to acknowledge the reciprocal aspects of the [physical] and the [conceptual]. Without them both you get impossible paradoxes such as the one you just presented.

By itself x=x is just an internally consistent statement without meaning (physical/existent referent).

Wrong. [x=x] has meaning but no physical reference. The meaning comes from our essential understanding of what terms like [anything] or [everything] are. These terms have no physical reference, because if you take [all the physical things that exist], you haven't referenced what is meant by [everything] or [anything]. Suppose, for example that a caveman created an abstract sculpture out of wood. Over time the wood rotted and the sculpture disappeared. It no longer exists, and so it is not a part of what has a physical reference. Even if we found a scroll that made reference to this piece of wood, there is no way that we can know what it looked like. We have no way to reference it except as a vague and abstract concept. Similarly, ideas do not have physical references. Yet both ideas and the abstract sculpture are clearly defining aspects of the term [everything]. If we want to ignore them we have to refer to some other concept, such as [every physical thing that currently exists]. But this is not [everything], it is a [limited subset of everything].

Otoh when I point at *this* keyboard and say "this keyboard is itself" the statement is both internally consistent (true) and has meaning (existent referent).

Ironically, in the most absolute possible sense, this slightly statement is false. The [keyboard] is made of atoms, those atoms are in constant motion. Thus, at any given instant, the [make-up of the keyboard] is ever so slightly different than it is at any other given moment. Thus, in the most absolute sense, the [keyboard at time1] is not the same [keyboard at time-2]. Thus, this [keyboard is this keyboard] is a necessarily vague statement that has no absolutely specific meaning or truth. It is considered to be true only because we interpret the statement using concepts that artificially define the [keyboard at time-1] to be identical to the [keyboard at time-2] even though, in reality, we know that they are not the same at all.

You may argue that the [keyboard at time-1] is a physical reference that we can use to make a specific incident of [x=x] true, but this is not so, either. [Keyboard at time-1] is necessarily a concept, because time is infinitely divisible. Thus, any specific time that we use for [time-1] is only a vague approximation of an absolute time. Thus, there isn't any physical thing we can refer to that makes [x=x] absolutely true in the specific physical sense.

By contrast, the concept [x=x] is absolutely true in even the most absolute sense possible. This is possible precisely because we are not referring to something physical. We are referring to a concept, which is exactly what it is.

Concepts, however, are vague in a reciprocal sense--because no two people understand the same concept in exactly the same way. Thus, while [x=x] is necessarily true, the meaning that it has is also necessarily vague to a certain degree.

[Antone]

X=X is a static scenario.

The reason [x=x] is 'static] is because it is a concept. But as I mentioned last time, each person has a slightly different understanding for any given concept. And similarly, the same person's concept will change gradually over time as they experience more of physical reality and engage in thought--which molds their understanding of what things mean.