The Final Solution Of The Liar

[Fluxus]

There actually has been quite a bit of investigation into the Liar Paradox since the early 90s. Some of it is due to recent developments in formal logic (notably paraconsistent logics), and some instigated by Graham Priest's work on "dialethias." Perhaps I overstated things a bit, but It's about as well-understood as any other paradox.

 

As I mentioned in #13, there is a "solution" to the Liar Sentence, but it requires something like fuzzy logic. In classical logic, only two truth-values are allowed -- T and F, or 1 and 0. Fuzzy logic allows for gradations, which means you can assign numeric values as your truth-values. For example, let's say we have a 100ml beaker that contains 50ml of liquid.

 

G

= "the beaker is full"

In bivalent logic, we can only answer 1 or 0. The answer will be 0, but that doesn't really capture the situation. You'd need a new statement, e.g. "the beaker is 50% full." In fuzzy logic, we can say that G is 0.5 true.

 

So, when we review

 

P

=

P

is false

in classical logic, it is inconsistent because if it's 1 then it must be 0, and if it's 0 then we've shown it is 1, i.e. it oscillates and can't hold a truth-value.

 

But in fuzzy logic, we can say P is 0.5. A natural language rendition might be that P is neither true nor false, both true and false, partially true and/or partially false. It's a bit messy but it is consistent. (It also happens to resemble what Buddhist logicians calls the "Fourfold Negation.")

 

So in this sense, the specific formulation of the Liar Sentence can be shown to be consistent, as long as you understand how that 0.5 truth-value works, and you don't treat it to mean "exclusively true" or "exclusively false." (Another way to view it is that fuzzy logic is much more adept at handling multiple set-memberships than classical logic.)

 

But, as I said earlier, you can just figure out a way to say

 

P'

=

P'

is inconsistent

 

or

P''

=

P''

is neither consistent nor inconsistent

and per Gödel, you've still got an inconsistent proposition based on negative recursion that you need to exclude from your axiomatic formal system. And that is why ZFC opted to rule out certain recursive sets, as it favors an incomplete mathematics over an inconsistent one.

 

 

On a side note, I'm also very unclear on the grounds with which you disagree with Gödel's Incompleteness Theorem.

[sigurdV]

This entry contains my General Method of solving Paradoxes...

Of course it gets a minus sign with no motivation for it!

On the sidenote then...

 

On First reading skip this: A summary of The Incompleteness Theorem.

 

By translating (parts of) a Natural Language into a Formalized Arithmetics, Goedel produced a formula,G, stating that it has no proof in the system used to express it. Reasoning from outside the system (the story goes) we can understand that G must be true because if it were not so, then there would be a proof in the system that G has no proof in the system.

Either the system is inconsistent, or, there are undecidable formulas (like G) making the system incomplete.

 

(Here we go:)Second: Assuming "x" is a name or description, and "Z" a quality ... It is easily verified that

( x = "xZ" ) implies ( xZ = "xZ"Z ).

Now we can test qualities like non-truth: (x = "x is not true") implies (x is not true = "x is not true" is not true... = x is true)

 

Meaning there is no x such that x = "x is not true"... Logic allows no sentences stating themselves as not true.

 

Checking "can not be proven" will similarly lead to the conclusion that there are no self referential sentences stating that they cannot be proven.

It is in general assumed that any subject , x , and any predicate , Z , can form the sentence "xZ" ... but here practise must change: Logic demands that some predicates may not form self referential sentences.

 

Which should mean that any system containing a translation of logically unrestricted natural language is risking inconsistency.

 

The conclusion is that (contrary to what Alfred Tarski claimed) Natural Language using Classic Logic is consistent!

 

There are no paradoxes

[LaurieAG]

I saw 'the Enigmatic Giant', an anime episode, on tv the other day that describes the mechanics of an interresting dilemma.

 

The giant in the title guards a bridge and only allows people to cross if they answer his question correctly.

 

His question is "If you lie, I will run you through with my sword, but if you tell the truth, I will strangle you with my bare hands, what do you say?"

 

The correct response is "You will run me through with your sword".

 

If the giant runs the answerer through with his sword then, by his own stated rules, he implies that the answerer lied but if that was the case then he was telling the truth in the original answer and then should be strangled by the giants bare hands.

[Ken]

I have a philosophical background, which means you are free to make fun of my butcherings of scientific theories. ;)

 

First, let's get a little clearer understanding of the Liar's Paradox.

 

In bivalent symbolic logic, you only have two options with a semantic variable: T and F. If a logical proposition can only return T, it's a tautology (e.g. A = A). If it can only return F, it's a contradiction (e.g. A = ~A). Both tautologies and contradictions are well-formed statements as long as they can hold stable truth-values. If we can't settle on a set of truth-values for the operations, then the statement is "inconsistent." In classical logic, we use a "truth table" to track the results of a logical formula, like so:

 

 

So, here's the compact version:

 

P

=

P

is false

 

The problem here is not that P is a contradiction; it's that P cannot hold a stable truth value, and is thus inconsistent.

 

If you assert that "P is true," then that means P is actually false, so the truth-value flips to false. However, if we say that "P is false," then we have proven it to be true, and the truth value is forced to flip again. This process would repeat indefinitely as the truth-values "oscillate," and this means that P is inconsistent.

 

 

SigurdV's initial solution, if I'm reading it right, is apparently to create a property of "self-reference." Unfortunately, it is trivial to see why this doesn't work, as follows.

 

x = Z(x)

 

This is a statement of identity, which means that every time we have x, we can substitute Z(x). This triggers an infinite regress:

 

x = Z(x) = Z(Z(x) = Z(Z(Z(x)) = Z(Z(Z(Z(x))) = .....

 

The idea that "the odds are true" doesn't work, because logically the "odds" and "evens" are supposed to hold the same truth-value. Thus, this ends up being inconsistent, because you're equating T and F, or we can say that as above, the truth-value constantly oscillates between T and F (which is a mark of inconsistency).

 

This is not a solution to the Liar's Paradox. In fact, it's the very problem Gödel identified at the start of his proof of the Incompleteness Theorem.

 

 

His second solution appears to be some form of trivalence: e.g. truth-values of T, F, and "neither." This doesn't work either, because it's just an assignment that P is inconsistent, which means it cannot be allowed within a consistent axiomatic formal system. It's like saying "I solved the Liar's Paradox by saying it can't be solved." See the problem? :D

 

 

The closest thing I've seen to a "solution" was the use of fuzzy logic, and I'm not sure this genuinely resolves the issue either. In bivalent logic, we have T and F. In fuzzy logic, we assign gradations of truth-values ranging from 0 to 1 (or 0 to 100), which captures the common-sensical notion of a "partial truth."

 

For example, if Z = "the glass is full," and we see that the glass is 25% full, then Z will be false but this doesn't capture the fact that it's partially full. We'd have to test another proposition (Z' = "the glass is 25% full") instead. In fuzzy logic, the truth-value of Z is .25.

 

So, let's look at:

 

P

=

P

is false

 

The consistent fuzzy logic answer to this is .5. This is not especially easy to translate into natural language, since we might say that .5 means "neither true nor false" and/or "both true and false."

 

The reason I don't think this works is because we'd just turn to another variant:

 

Q

=

Q

is inconsistent

 

I'm not entirely sure how you'd express this in symbolic logic, but the natural-language version should suffice. If we claim that Q is true (1), or false (0), or is neither/both true/false (.5), then as long as that truth-value is stable, we've proven that Q is in fact consistent. However, it cannot be consistent, because Q is inconsistent. Thus, no matter what truth-value we assign to Q, it is inconsistent. Fuzzy logic may allow a handful of self-referential statements.

 

In short, I'm fairly confident that Gödel's Incompleteness Theorem holds, and there is no "solution" to the Liar Paradox in the context of any axiomatic formal system. If you state that it is inconsistent and/or not a well-formed statement, then you're stating that the Liar's Paradox cannot be solved. If you allow it in your system, then per the Principle of Explosion your entire formal system turns into gobbledygook, because you can simultaneously prove that any statement or formula as true, false, neither true nor false, both true and false, and any combination thereof.

 

So, there is no way to solve the Liar's Paradox. Since formal systems like ZFC seem to be just fine without them, the only "solution" is to recognize that certain types of statements can't be expressed in axiomatic formal systems.

 

Does that clarify things, without repeating too much of other people's posts? :D

 

I admit I have no experience with symbolic logic, and this discussion is over my head.

 

Having confessed I think the following paradox is an example of what you call bivalent symbolic logic. I used this as an example in part of a lecture in my Research Methods course.

 

Consider the quote box as a "real" box.

 

Every statement in this box is false

 

Based on your comments I would assume you would see this as an oscilatory truth value. I'd have to dig out my faded lecture notes to see what I called it back before I retired. <grin>

 

Comments?

[sigurdV]

Hi all!

And especially Newcomers LaurieAG and Ken!!!

 

Im happy that some ppl wants to figure out the foundations of thought (=logic)... It is not easy to find a more resistant subject so please enter your thoughts without fear of ridicule.

 

The "Giant Paradox" was known to me and is relevant to our subject matter,also the "box variant"... but I will however yet try to refrain from immediately diving deep in every matter brought to attention.

 

At least until its clear to participators what my eventual contribution to our understanding of logic is...

(As applied to the Liar Paradox)

 

Im still waiting for Fruxus to return but I feel that something should be said about fuzzy logic:

 

I see it as essentially the theory of probability... With an infinitude of "truth values", value 0 and 1 being equivalent to falsehood and truth.

 

I am somewhat suspicious here since infinity seems to be at heart of most paradoxes, one should form a theory containing an infinitude of truth values carefully...

 

Also: My basic idea is that Classic Logic is not in need of revision! All other solutions of paradox consists in restrictions of logic or language , and (however fruitful and interesting they may be) are not necessary to get rid of paradox.

 

PS As in Mathematics there should be nothing in Formal Language that cannot be (however cumbersome) said and understood in normal language ,so let the formal logicians speak out as they wish... Hopefully they will (if necessary) later aid in expressing their results in ordinary language.

[sigurdV]

There are many various formulations of the paradox , as the missionary caught by cannibals who is asked to produce a statement... if it is true then he will be fried, and if it is not true he will be cooked ... so he tells them they will cook him.

 

Now lets see where the examples given lead us astray B)

 

The fault is in the definitions!! I will show how it works in the case of The Liar.

 

(The Liar Definition) Let "The Liar" be a name of the sentence "The Liar is not true".

 

(Derivation of the Liar Paradox)

1 The Liar is not true. (assumption) (liar sentence)

2 The Liar = "The Liar is not true." (true according to definition) (LIAR IDENTITY)

 

And from here the negation of sentence 1 is easily derived, and the paradox follows!

 

But IS the Liar Definition correct??

 

Here is a derivation proving it leads to a contradiction:

 

1 The Liar = "The Liar is not true." (assumption)(Liar Identity)

2 The Liar is true if and only if "The Liar is not true." is true (from 1)

3 The Liar is true if and only if the Liar is not true (contradiction from 2)

 

(For easier reading I left out the (obvious) motivations of each step above,if asked for I will provide them. Here I should perhaps warn opponents that a contradiction from 1 can be reached in several different ways.)

 

You have now seen a special solution of the liar paradox... let us generalize:

 

No Liar Identity is logically true!

 

In the stories of the giant and the missionary a future action is defined...I will leave it to the interested reader to verify that those definitions indeed are logically incorrect :D

[sigurdV]

No actual description remains of the perhaps oldest paradox.

 

But the earliest formulation of its solution was made by the greek philosopher Parmenides, who told us not to belive that nothing is a "thing" :

 

Say you are a christian in the purgatory, and that you are given the job to check all things there are

and present a list for inspection by God... -Is nothing left? He asks...

 

And if you say "yes" he will throw you back saying -Dont come back unless the job is done!

And if you say "no" he will throw you back saying I cant find it in here, you must have lost it somewhere,

dont come back unless you find it...

 

According to my theory of truth there is some liar identity hidden in a paradox somehow,somewhere!

There remains only the matter of finding it:

 

What God doesnt understand here is that there is no x such that x = nothing.

And: "x=nothing" argumentably was the first liar identity in history.

[sigurdV]

And here I am at the end of my story. I see no good reason to go through the remaining cases of paradoxes even if ill be interested to check their solutions provided I come across them.

 

But some summary should be made for the new visitor who certainly will jump here when he finds the thread starting with the demand to check a complicated derivation...

 

All we need are the three classical laws of logic:

 

1 Every a is identical with itself.

 

2 It is not the case that a and not a.

 

3 Either a or not a.

 

Any teacher of logic will inform you that it is not the case that a = not a... -It follows from the laws of logic he says!

 

I agree and add that "a = not a" is an example of what I call "Liar Identity"

 

And the CASE is that he will not agree that what the Liar Sentence " This sentence is not true." is dependent on is

 

A LIAR IDENTITY: The word "This" in the sentence "This sentence is not true" refers to the sentence

 

"This sentence is not true.".

 

Which is a very strange behaviour thinks sigurdV

[sigurdV]

And here I am at the end of my story. I see no good reason to go through the remaining cases of paradoxes even if ill be interested to check their solutions provided I come across them.

 

But some summary should be made for the new visitor who certainly will jump here when he finds the thread starting with the demand to check a complicated derivation...

 

All we need are the three classical laws of logic:

 

1 Every a is identical with itself.

 

2 It is not the case that a and not a.

 

3 Either a or not a.

 

Any teacher of logic will inform you that it is not the case that a = not a... -It follows from the laws of logic he says!

 

I agree and add that "a = not a" is an example of what I call "Liar Identity"

 

And the CASE is that he will not agree that what the Liar Sentence " This sentence is not true." is dependent on is

 

A LIAR IDENTITY: The word "This" in the sentence "This sentence is not true" refers to the sentence

 

"This sentence is not true.".

 

Which is a very strange behaviour thinks sigurdV

I think the following link is a good treatment of the Liar... The only thing missing is my solution:

 

http://www.iep.utm.edu/par-liar/

 

Ill quote the important parts:

 

Aristotle offered what many philosophers consider to be a partially correct answer to our question about truth. Stripped of its overtones suggesting a correspondence theory of truth, Aristotle proposed what is essentially sentence (T):

 

(T) A declarative sentence is true if and only if what it says is so.

 

This article began with a mere sketch of the Liar Argument using sentence (L). To appreciate the various proposed solutions to the paradox, and the central role of (T), we need to examine more than just a sketch of the argument. The argument actually requires the following assumptions in addition to (T):

 

(2) Any declarative sentence “S” says that S.

(3) The Liar Sentence L is a grammatical and meaningful declarative sentence.

(4) The Liar Sentence L is true or false.

 

(5) The usual naming convention holds so that

the phrase “This sentence” in L refers to L(=The Liar Identity), and

(L) = “This sentence is false”.(= The Liar Sentence)

 

What I do is proving no Liar Identity is logically true!

Proof: Suppose that x = "x is not true" , then

x is true if and only if "x is not true" is true, which reduces to

x is true if and only if x is not true, and therefore

there is no x such that x = "x is not true"

 

Creating a Liar Sentence makes its Liar Identity empirically true,

therefore we broke the laws of logic when we created The Liar Sentence :)

[sigurdV]

In classical logic, only two truth-values are allowed -- T and F, or 1 and 0.

So, when we review

 

P

=

P

is false

in classical logic, it is inconsistent because if it's 1 then it must be 0, and if it's 0 then we've shown it is 1, i.e. it oscillates and can't hold a truth-value.

 

 

Note that Fluxus understands that the Liar Identity:

P

=

P

is false

is not true!

 

His problem is that he can find empirical verification that the same Liar Identity is (empirically) true provided he assumes P. Which gives:

1 P is false

2 P = Pis false

3 Contradiction

 

But he cant see that therefore logic forbids the EXISTENCE of P...

 

(A result that also can be gotten by my testing method)

 

He is in good company.