How to solve the Liar Paradox

sigurdW

Alfred Tarski diagnosed the Liar Paradox as arising only in languages that are "semantically closed", by which he meant a language in which it is possible for one sentence to predicate truth (or falsehood) of another sentence in the same language (or even of itself).

To avoid self-contradiction, Tarski says it is necessary when discussing truth values to envision levels of languages, each of which can predicate truth (or falsehood) only of languages at a lower level. So, when one sentence refers to the truth-value of another, it is semantically higher. The sentence referred to is part of the "object language", while the referring sentence is considered to be a part of a "meta-language" with respect to the object language. It is legitimate for sentences in "languages" higher on the semantic hierarchy to refer to sentences lower in the "language" hierarchy, but not the other way around. This prevents a system from becoming self-referential.

How to prove him wrong?

English is a semantically closed language so lets begin by stating the conditions for the Liar to arise:

1 Sentence 1 is not true.
2 Sentence 1 ="Sentence 1 is not true.

Being careful I will not accept sentence 2 on its face value, perhaps its not true?

If it IS true then no harm is done if we declare it to be true...so this is what you should work with:

1 Sentence 1 is not true.
2 Sentence 1 = "Sentence 1 is not true"
3 Sentence 2 is true.

Now try to derive the Liar Paradox! (I predict you will fail to do so! Will you prove me wrong?)

sigurdW

To see what is involved, lets replace "Sentence 1 " with a variable:

1 x is not true.
2 x = " x is not true."

Sentence 1 is then no longer a sentence; its a sentence-function it has no truth value unless x is replaced with a name of a sentence, or a sentence inside quote signs.

But sentence 2 is an identity, and we can get an equivalence:

3 x is true if and only if "x is not true" is true.

Simplifying the right side we get a contradiction:

4 x is true if and only if x is not true.

And we must deny sentence 2:

5 It is not true that x = "x is not true"

Sentence 5 is a logical truth... its the law of identity:

6 x=x (law of identity)

7 -(x = -x) (from 6 by double negation)

Sentences 5 and 7 has the same logical form since (-x) = "x is not true"

Now let us again look at the foundation of the Liar Paradox:

1 Sentence 1 is not true.

2 Sentence 1 = "Sentence 1 is not true"

Sentence 2 is a denial of the law of identity so it is logically false...and empirically true!

This is because we were violating the law of identity when we created sentence 1!

Sentence 1 is identical with its negation thereby making the logically false sentence 2 empirically true!

So we can neither deny nor assert sentence 1 since its very existence is forbidden by Logic!

The Laws of logic are prescriptions that CAN be broken... They are NOT Natural Laws!

SO: Unless you violate the Laws of Logic you cant derive the Liar Paradox!

ThomasT

What's the Liar Paradox?

sigurdW

He he! Here I am showing that there IS no Liar Paradox... and you want me to tell you what it is that isnt:

The Liar Paradox arises when you try to find out if sentence 1 below is true or not:

1 Sentence 1 is not true.

Informally speaking: Sentence 1 is not true if it is true and true if it is not true...

It must be either true or not true so it is both!

And that is the paradox...

My point is that it is wrong to search for its truth value since logic forbids the existence of sentence 1.

ThomasT

Thanks. I'm going to have to think about this.

Hurkyl

The liar's paradox is the supposition that the following equation can be used to implicitly define a proposition P.

P = not P

Its importance to logic is this sneaky (and rigorous!) construction of a solution to the equation, which is naive formal logic's analog of Russel's paradox from naive set theory: (Q and R are predicates in one variable)

Q(R) := not R(R)
P := Q(Q)

And the fix to formal logic is similar in spirit to how Zermelo fixed set theory: in higher order logic, predicates aren't allowed to take arbitrary predicates as arguments. Instead, each predicate has an order, and it is only allowed to operate on variables of lower order.

So, in the construction above, R(S) is only allowed when the order of S is less than that of R. So R(R) is forbidden, because the order of R is equal to the order of R.


Edit: I should add that the above isn't the only significance of the paradox. Having the solution for formal logic already, it's easy to forget the more general issue. It clearly demonstrates issues in the semantics of language. "This sentence is false," is perfectly good English, and by the rules of English, the sentence itself really is the referent of the phrase "this sentence", and you run into difficulty when you suppose that we can assign truth values to English propositions.

Second edit: the paradox shows up in the theory of computation too, but with different consequences. With Turing machines (i.e. computer programs), it's fairly straightforward that programs can reference themselves, and we can enact the construction of the liar's paradox: the ensuing argument, however, doesn't yield a paradox: instead, it results in proof by contradiction that there are no algorithms for solving a certain class of problem (e.g. "Does this function return 'true' when given input 'x'?", or its more famous relative, the halting problem).

(certain cases of the problem can be solved, of course, but there cannot be an algorithm capable of solving every case, even when allowed unlimited time and memory)

sigurdW

Thanx for taking an interest in the matter of paradoxes!

I should have added some lines

"The Liar Paradox arises when you try to find out if sentence 1 below is true or not:
1 Sentence 1 is not true.
Informally speaking: Sentence 1 is not true if it is true and true if it is not true...
It must be either true or not true so it is both!
And that is the paradox..."

The consequence of sentence 1 is/was taken to show that the Classic Laws of Logic are inconsistent!

One way to overcome the difficulty was to reformulate the laws ...here Brouwer is a good example.

The other way is to prevent selfreference a la Tarski or Russell.

If I am correct this was/is unnecessary!
Logic admits self referential sentences in most cases, and forbids in some cases... Heres a Test Method:

1 x = xZ (assumption)
2 xZ = xZZ (from above)

3 if (x = xZ) then (xZ = xZZ) (conclusion) (Logical Truth!)

For some predicates the right side of the implication gets false witch means that the left side is false as well.
Example: Let Z = "is not true"

3 if (x = "x is not true") then ( "x is not true" = " "x is not true" is not true")
4 ... then ("x is not true" = "x is true")

5 The predicate "is not true" may not be the predicate Z in the selfreferential sentence xZ.

Well then...two things should be done:

1 Check the solution for errors.
2 Check the solution for consequenses!

sigurdW

The above is a way of EXCLUDING paradoxes... I use English as both object and meta language to show that the liar paradox cannot be logically correctly derived; thereby SOLVING it!
Related paradoxes can then be solved in the same manner.

sigurdW

You are interesting to read.

(a)
1 This sentence is false.
2 This sentence = "This sentence is false"

Since sentence 2 contradicts the law of identity sentence 2 is false.
Therefore sentence 1 either has no defined subject, or breaks the law of identity.
SO: The paradox cant be derived.
Note. A computer should use the test to exclude the predicate "false" from self referencential use.

(b)
Its probably too tecnical for me to really understand... But I suspect my results (if correct) will affect this class of problem.

PS This insight of yours is unusual:
"Having the solution for formal logic already, it's easy to forget the more general issue."

One cant study the anatomy of paradoxes in a system that doesnt allow self referemce

lugita15

My preferred resolution to the Liar Paradox is Prior's, summarized here. The idea is that the liar sentence, like all sentences, asserts its own truth. So a sentence that asserts both its truth and its falsity must be false.

One of the nice things about Prior's resolution is that it can be easily applied to many similar paradoxes, for instance Curry's paradox. Consider the sentence "If this statement is true, then 1+1=3." Or if you prefer, statement 1: "If statement 1 is true, then 1+1=3." Suppose that statement 1 is true. Then what statement 1 says is that if it is true, then 1+1=3. So supposing it is true, 1+1=3. In other words, it is correct to say that if statement 1 is true, then 1+1=3 would be true. In other words, "If statement 1 is true, then 1+1=3" is a true statement. But that is precisely statement 1. So statement 1 is true. But statement 1 says that if it's true 1+1=3. We have just shown that statement 1 is true. So we can conclude that 1+1=3. Can you apply Prior's resolution to solve this?

lugita15

I don't know what you mean. All this approach saying is that "snow is white" is the same as saying "'snow is white' is true." It doesn't affect the semantic structure or model validity at all.

Hurkyl

(note: a previous version of this post was deleted by me. The reply above was a response to that post)

There's an obvious problem: if we assign semantics in a way that every statement implicit asserts its own truth, then we cannot prove anything true! To wit, we can consistently assign the truth value "false" to every proposition.

For clarity: if we assign the truth value "false", then the statement's implicit assertion of its own truth is false, and therefore the statement is false.


Your version still runs afoul of the liar's paradox. If we interpret "this sentence is false" as a proposition satisfying

P = (P --> not P)

Then

• Case 1: P is true. In this case, we have True --> False, which is false, and therefore P is false.
• Case 2: P is false. This can only happen if the hypothesis is true and the conclusion is false: that is, we can conclude P is true and not P is false. Therefore, P is true.

lugita15

But rather we should analyze it as follows. P = (P --> not P) = P & (P --> not P) = P & not P = false.

The reason your case 2 doesn't work, in Prior's approach, is that P asserts not only that not P, but also that P. Therefore the mere fact that it's correct about not P is not good enough, because in order to be true it must be correct about both P and not P, which is impossible.

In general, the rule is that the statement "Q" is true if and only if Q is true and "Q and 'Q'" is satisfiable.

Hurkyl

I was referring to the semantics you had described in your post: that you interpret the phrase "this statement is false" as being a statement P with the property

P = (P --> not P)

(because you kept describing it as implicitly meaning "if this statement is true, then ...")

That part of my post doesn't apply to Prior's version, where he interprets it as a P satisfying

P = P and not P

This version suffers from the criticism I made in my first half of my post: no matter what proposition Q is, because we have interpreted it as satisfying

Q = Q and <something else>

we can assign the truth value "false" to Q.

lugita15

No, I wasn't. The only context I used "if this statement is true" in is my description of Curry's paradox. Rather, I'm saying that "This statement is false" is the same as saying "This statement is false and "This statement is false" is true."" Or to put in terms of P, P says "P is false", so it's implicitly saying "P is false and "P is false" is true", which is equivalent to saying "P is false and P is true", which is a contradiction.

Hurkyl

My mistake on that part then, sorry!

lugita15

No you can't, because Q is not an atomic proposition. In order to assign Q a truth value, we need to first assign Q a truth value, so we need to first assign Q a truth value, so we need to first assign Q a truth value, ad infinitum. The way this infinite regress is resolved is, as I said before, to say that "Q" is true if and only if Q is true and "Q and "Q is true"" is satisfiable. This criterion is of practical use, because it entails that as long as Q is not self-referential, then "Q and "Q is true"" is automatically satisfiable, so the truth of "Q" is equivalent simply to the (by assumption) non-referential content of Q. Confusing, but it works!