The Logical Death of the Liar's Paradox

I'm hoping for some feedback on the following arguments, largely inductive, about the Liar's Paradox from anyone well versed such arguments. If the argument fails, I would like to know exactly where.

The Liar's Paradox arises from statements like "This very statement is a false proposition." As the argument pointing out the paradox goes, if such a statement is taken as true, then it must be false as it claims; but, if it is false as it claims, then it has made a truthful statement about its falsity and must be taken as true, and thus the argument begins anew, ad infinitum.

Normally, as a matter of logic, a statement declared to be true or false is a statement that was compared to some referent fact or concept and either found to match or not to match.

Thus, when we say that a statement is true or false, we imply or show that such a comparative process can be and has been carried out.

With the correct process in mind for how to comparatively find a statement to be true or false, let us look closer at the exact meaning of a statement that gives rise to the Liar's Paradox:

"This very statement is a false proposition."

The subject-class of this proposition is the whole statement itself, and the proposition ascribes to its subject-class the predicate-class quality of being a statement that can be proven false by comparison to a referent fact or concept.

In judging the truth value of the statement, we need only compare the attributes of the subject class -- here the statement itself -- with the attributes of the predicate class -- here the comparative process of how we find statements to be true or false (inductively).

The subject class of the statement declares itself to be false, ipse dexit, but this is at variance with the predicate class which requires that statements be proven true or false by comparison to a referent fact or concept.

So, the statement has made a provably false proposition, and thus it correctly states that it makes a false proposition, and thus its final outside truth value is true.

Conversely, to clarify the point, the opposite statement that "This very statement is a true proposition," would have a final outside truth value of false because the statement asserts itself to be true rather than being found true. The subject class -- the statement itself -- falsely ascribes to itself the predicate-class quality of being found true by comparison to a referent fact or concept.

Someone will urge, of course, that we have not dispensed with the Liar's Paradox because we must change our final outside truth value because it is contradicted by the self-assigned truth value of the statement itself -- whether it be the positive version or the negative version of the statement (the negative version gets the most press).

We should counter, of course, that the self-assigned truth values of either the positive or negative versions of the statement were a part of the very matter that we judged with a correct, comparative process for determining the truth or falsity of propositions. Thus, we may reject as unsound the argument that we must change our final outside truth values based solely on the contradictory truth values alleged in the statements themselves.

Specifically, with the little-discussed positive version of the statement above (a.k.a. "The Truth Teller"), we have simply and correctly disagreed that it states a true proposition.

Likewise, with the negative version of the statement, we have come to correctly understand the statement and have agreed with its assertion that it makes a false proposition -- thus, there is either no contradiction in giving the statement a final outside truth value of true or, at the very least, we have correctly disagreed with the statement's self-asserted final outside truth value of being false.

In short, we prefer a sound process for determining the truth or falsity of propositions, not an unsound rule that changes our final judgments with a simple decree of disagreement.

This should be the logical death of the Liar's Paradox, and a very welcome death for those of us who value logic as a practical matter.

Many of us have been doubtlessly tortured by the endless sing-song chants of "if true, then false, and if false, then true," etc., printed endlessly onto the paper remains of ever more dead trees.

Let's work together to spread and even sharpen the logical necessity of the long-awaited demise of the Liar's Paradox.

Very Respectfully,
Ray Donald Pratt

 

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Great post will have to read this in more detail tomorrow.

Surely, this debate is based in the mind limited capacity to only ever see two options. Either something is correct or incorrect. However, there are many states in between and I think this fine example of that. Therefore, the only reason this is a debate at all is that humans by their very nature are black and white thinkers.

I think that this statement comes down to can it be falsified?

 

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Interesting thoughts Ray.

Unfortunately, there's a simple misunderstanding here that has caused some confusion.

What needs to be addressed is the difference between Truth-value and Validity: statements can be analyzed for truth-value, while arguments are analyzed for validity.
I mention this because you've mis-stated the 'Liar's Paradox' (more correctly known as the 'Cretan Liar Paradox').
The Paradox, properly stated, is in fact an argument, and not a statement:

All Cretans are Liars.
I am a Cretan.
Therefore....

What follows the "Therefore..." of course, is entirely contingent upon the truth-value of each of the premisses. And in either case, the conclusion is contradictory. In any case, the point is this: there is never at any point any paradox, per se, because we're dealing with an argument. An argument is either valid or invalid, and validity is a function of form, having nothing whatsoever to do with truth-value.

 

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the statement can be true, false, unknown, or unknowable.

a paradox is unknowable.

truth values between true and false are part of 'fuzzy logic' and it is better to think of the values as representing not the truth value of the statement itself but rather the degree of belief that you have in its truth.

 

[quote="glaucon]Interesting thoughts Ray.

Unfortunately, there's a simple misunderstanding here that has caused some confusion.

What needs to be addressed is the difference between Truth-value and Validity: statements can be analyzed for truth-value, while arguments are analyzed for validity.[/quote]

While this may be true, you are not speaking of anything besides truth and other semantics.

The point of the original post is to demonstrate how truth is false because it is self-assigned. All concepts are false because it does not exist in reality but only the self. On the other hand, a non-concept is real. This leads to the infinite regress no matter how much we redefine reality because we live in a world of self-assigned concepts. In order to reshape and remodel reality and still be able to create new concepts, reality must remain intact or coherent outside of our minds. If there is a "crack" in reality or an inconsistency, reality becomes imperceivable (dark matter?). Thus, there must be a binding for reality. That binding is pure cognition which has no form. The size and purity of this cognition is greater than the sum of its parts. But in order for the cognition to remain functioning at most every moment and instant of its existence, it necessarily must give rise to self-perception.

 

Truth is strictly a function of semantics.

It is you who have missed the point. All truth-value is assigned.

 

Thank you all for your thoughts -- they were actually thought out, and that's refreshing.

I was particularly intrigued by the post marking the difference between reality as it is as opposed to the concepts that we carve out of it. Indeed, the crux of my original argument is open to attack depending upon one's theory of truth (and the consistency of it as used in the argument), but I honestly think that one could substitute any theory of truth with minor changes and come to the same conclusions, but it had not occurred to me that false theories of truth might actually be the norm in all versions.

However, to function as a human being in a complex world, I choose to believe that we can come closer to truth by trying to get closer rather than by not trying.

I also enjoyed the post pointing out that the Liar's Paradox also has a syllogistic version known as the Cretan's Liar's Paradox. However, I would merely take the first premiss of "All Cretan's are Liar's" as not being truthfully utterable by a Cretan, and thus in that case false. As false, no truthful conclusion may be drawn from the argument even if the form of the argument is valid. Note that my conclusion as to the falsity of that premiss was inductively derived, which I think is the correct way to find the truth or falsity of any proposition or premiss, not deductively according to a rule that leads to paradox.

Very Respectfully,
Ray Donald Pratt

 

Its impossible to talk about yourself. Done.

 

Hey I put virtually the same solution forward a year or two ago here at sci somewhere, or philosophyforums. I think it was here though. I'm almost sure it was almost exactly the same point. Glaucon were you in on some of that? Harumph.

 

hai

its very nice

 

The Cretan Liar's paradox implies that the truth can be 'found' but only if it's 'lost'.

Which means you have to stop looking for the truth to see it, and you also have to look for it to see it, or to step back from it, I suppose. I thought once that, since we're essentially a conscience of two halves, maybe consciousness is the result, or the part 'between' the two, so the validity/falsity paradox is the same thing, and so is the search and giving the search away, to see something.

 

The logic of the paradox is the same as reversible computation vs universal computation. A logic gate, to be UC has to have 3 inputs and in reversible gates you keep at least 2 of them, or the input conditions. In the lie these are "I am a liar", and "I am not a liar". the "gate" is the operator "lie", it inverts either condition.

The liars paradox essentially is the contradiction: "I am therefore I am not". The logic is the NOT gate or inverter, which is trivially reversible but not UC.

 

Formal logic is not designed for self reflective statements. While I'm making no claim at being great shakes at formal logic the approaches I've seen for this tend to take the following approaches: the statement is ambiguous and therefore neither true nor false, or, the statement requires an iterative extension to formal logic to render it intelligible at iteration i. The iterative approach seems to either involve a temporal analysis or a separation of statement and referent, which you seem to be doing.

This seems to be introducing an unneeded empirical extension (correspondence to facts) to formal logic (which only requires tautology (i.e. a=a) as a base minimum for truth). As straight forward as correspondence to a fact might seem, there are those who balk at it as a logical basis for truth and I don't think you need this concept in order to proceed with a statement/referent dichotomy as your approach.

Adding iteration makes it no longer traditional formal logic.

Personally I prefer simply leaving it at first order logic is incomplete because it has no concept of ambiguity, self reference or iteration and the statement is one of many which are ambiguous in it because of this.

 

hi,

The logic of the paradox is the same as reversible computation vs universal computation. A logic gate, to be UC has to have 3 inputs and in reversible gates you keep at least 2 of them, or the input conditions. In the lie these are "I am a liar", and "I am not a liar". the "gate" is the operator "lie", it inverts either condition.

The liars paradox essentially is the contradiction: "I am therefore I am not". The logic is the NOT gate or inverter, which is trivially reversible but not UC.
its very nice

 

It is not just that contradiction. If it were, it would not be paradoxical.

Your example is not self referential, which is key to the paradox.

Here is way more than you probably want to know about the issues involved: http://www.iep.utm.edu/p/par-liar.htm

 

This very statement cannot be proven.

 

THIS very statement makes no sense.

 

The 'state' of the sentence is temporary at present (states change!) so it may become true at any time. Should it become true then it becomes "That permanence is true."

 

I never tell the truth.

 

.... well maybe just this once. honest.