How to create language without self-contradiction?

[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 let's 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, let's 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 can't derive the Liar Paradox!

 

[ThomasT]

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 let's 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?)

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]

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.

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

 

[Hurkyl]

What's the Liar Paradox?

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 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.

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]

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. (a)"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: (b)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'?)

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 can't 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 can't study the anatomy of paradoxes in a system that doesn't 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]

There's an obvious problem (at least, superficially) with this approach: you can't prove anything true at all! With these semantics (at least, the form stated), we can consistently assign the truth value "false" to every proposition.

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)

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.

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]

(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.

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.

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]

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

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]

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

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]

No, I wasn't. The only context I used "if this statement is true" in is my description of Curry's paradox.

My mistake on that part then, sorry!

 

[lugita15]

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.

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!

 

[Hurkyl]

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

Fortunately, we've already done that, because we've assigned Q a truth value.

A truth valuation is a function that assigns truth values to propositions, respecting the rules of logic. There is no requirement that the function be expressible as a recursive computation with atomic propositions as the base case -- in fact, as you point out, such functions can't even exist if we allow propositions that aren't well-founded.

As an analogy, your objection is of the same form as this complaint about algebra:

If we know x = 4-x, you can't assign the value 2 to x, because to assign a value to x, you first have to assign a value to x, and doing that requires that we assign a value to x, ...​

Any function that fails to assign a truth value to each proposition can't be called a truth valuation anyways, at least in a typical formulation of logic.



Of course, there are other variations on logic than classical logic. I mentioned computability theory: that the alternatives are {true, false, infinite loop} is rather important to the theory. e.g. it gives a way out to the specific construction of the Liar's paradox I mentioned earlier: to refresh:

Q(R) := not R(R)
P := Q(Q)​

The naive implementation of the predicate Q and the sentence P clearly results in P evaluating to "infinite loop". The liar's paradox is a proof there isn't a more clever way to go about things that would allow P to be assigned the value "true" or "false".

 

[lugita15]

By the way, the "neither true nor false" resolution to the Liar paradox can be easily defeated with the sentence "This statement is not true." Because if it's neither true nor false, then it's not true, which is what it says, so it is true, etc.

And the "both true and false" resolution can be defeated by "This sentence is only false", because if it's both true and false then it's true, but it says that it's only false, so it's only false, etc.

 

[sigurdW]

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?

Never thought of Currys Paradox before... This is my first formulation of it:1 if sentence 1 is true then 1+1 = 3 (supposition 1)

2 sentence 1 is true (supposition 2)

3 1+1 = 3 (from 1 and 2 by modus ponens) (false sentence!)

4 sentence 2 is not true. (from 3) (denying supposition 2)

5 sentence 1 is not true (from 4 and 2) (denying supposition 1)

6 sentence 1 is true (from 5 and 1)

7 sentence 1 is true and sentence 1 is not true. (contradiction from 5 and 6)Is the derivation above a derivation of Currys paradox?

And if so...can you see the missing supposition the contradiction in sentence 7 should act upon?

 

[lugita15]

Never thought of Currys Paradox before... This is my first formulation of it:


1 if sentence 1 is true then 1+1 = 3 (supposition 1)

2 sentence 1 is true (supposition 2)

3 1+1 = 3 (from 1 and 2 by modus ponens) (false sentence!)

4 sentence 2 is not true. (from 3) (denying supposition 2)

5 sentence 1 is not true (from 4 and 2) (denying supposition 1)

6 sentence 1 is true (from 5 and 1)

7 sentence 1 is true and sentence 1 is not true. (contradiction from 5 and 6)


Is the derivation above a derivation of Currys paradox?

And if so...can you see the missing supposition the contradiction in sentence 7 should act upon?

I don't really follow your steps, so let me do my own version:
1. If statement 1 is true, 1+1=3.
2. Suppose statement 1 is true.
3. From 1 and 2, we get 1+1=3.
4. From 3, we can get 1+1=3 from the assumption that statement 1 is true.
5. From 4, if statement 1 is true then 1+1=3
6. From 5, statement 1 is true.
7. From 6 and 1, 1+1=3.

More informally, within brackets I am operating under the assumption that statement 1 is true: [It is true that statement 1 implies that 1+1=3. But statement 1 is true. So 1+1=3.]. Now outside the brackets, we can say that since the assumption that statement 1 is true allows us to conclude that 1+1=3, it follows that it is true that "If statement 1 is true the. 1+1=3", which is statement 1. So statement 1 is true, which means that if statement 1 is true then 1+1=3, but statement 1 is true, so 1+1=3.

 

[sigurdW]

1. If statement 1 is true, then 1+1=3. (assumption 1)
2. Statement 1 is true. (assumption 2)
3. 1+1=3. (From 1 and 2) (False statement!)

Our first three lines are identical, but I took the liberty of editing yours a little.

In your sentence 4 (see below) you use the false statement 3 together with assumption 2,
and I can't accept that:
Since statement 3 is false, statement 2 must be denied!

Which gives:
4 Statement 2 is not true (from 3, denial of assumption 2)
Which gives:
5 Statement 1 is not true (from 4 and 2, denial of assumption 1)

Since the right side of the implication in statement 1 is not true,
then for statement 1 to be not true the left side must be true!
Which gives:
6 Statement 1 is true.

Now; 5 and 6 contradict each other...To escape paradox something must be denied, but what??

4. From 3, we can get 1+1=3 from the assumption that statement 1 is true.

 

[sigurdW]

A word on the topic while waiting for the structure of Currys Paradox to become clear...

The topic should have been wider: How to solve paradoxes related to the Liar Paradox...

Or something even wider...Perhaps: The Philosophy and Anatomy of Paradoxes.

From watching reactions of my writings I think I should make some effort to present the fundaments of my thinking on the subject...

To begin with the question of what logic to use while studying the anatomy of paradoxes:

I see no alternative to use my own interpretation of Classical Logic...

I have tried searching for a Standard formulation of Classical Logic but failed.

There are three laws:

1 x = x (Law of Identity)
2 It is not true that x and not x (Law of Contradiction)
3 Either x is true, or not x is true (Law of excluded middle)

To this I add the modern definition of truth: x is true if and only if x.

Everything else I left outside the system... Since we have it in our understanding of our Natural Language!

In most logics its customary to restrict the variable x in order to avoid complications like paradoxes...

The logic I use is wide open to paradoxes... my only defence is to solve them!

I have a strategy derived from my solution of the Liar Paradox, simply put:

I search the derivation for its Liar Sentence and Liar Identity and then...

 

[lugita15]

Our first three lines are identical, but I took the liberty of editing yours a little.

In your sentence 4 (see below) you use the false statement 3 together with assumption 2,
and I can't accept that:
Since statement 3 is false, statement 2 must be denied!

Which gives:
4 Statement 2 is not true (from 3, denial of assumption 2)
Which gives:
5 Statement 1 is not true (from 4 and 2, denial of assumption 1)

Since the right side of the implication in statement 1 is not true,
then for statement 1 to be not true the left side must be true!
Which gives:
6 Statement 1 is true.

Now; 5 and 6 contradict each other...To escape paradox something must be denied, but what??

OK, let me try to be more precise. Let me make Sentence 0 say "If Sentence 0 is true, then 1+1=3." Note that sentence 0 is not a sentence of the proof, just a sentence referred to by the proof.

1. If sentence 0 is true, then if sentence 0 is true, then 1+1=3.
2. If sentence 0 is true, then by sentence 1 and modus ponens 1+1=3.
3. By sentence 2, sentence 0 is true.
4. By sentence 3, sentence 0, and modus ponens, 1+1=3.

Is that clear enough?

 

[sigurdW]

OK, let me try to be more precise. Let me make Sentence 0 say "If Sentence 0 is true, then 1+1=3." Note that sentence 0 is not a sentence of the proof, just a sentence referred to by the proof.

1. If sentence 0 is true, then if sentence 0 is true, then 1+1=3.
2. If sentence 0 is true, then by sentence 1 and modus ponens 1+1=3.
3. By sentence 2, sentence 0 is true.
4. By sentence 3, sentence 0, and modus ponens, 1+1=3.

Is that clear enough?

I got stuck on sentence 2. A sentence seems to be missing...
Modus ponens involves two sentences to get a third,say:
1 if a then b
2 a
3 b (conclusion, from 1 and 2 by modus ponens)

Is this how u mean?
1. if sentence 0 is true, then 1+1=3. (assumption 1)
2. sentence 0 is true (assumption 2)
3 1+1=3 (conclusion, from 1 and 2 by modus ponens)

Maybe I can shortcut this search for the proper derivation by checking out if the sentence 0 really exists?

1 sentence 0 = " if sentence 0 is true then 1+1=3 " (assumption)

let x = sentence 0:

2 x = " if x is true then 1+1=3 "

By definition of truth " x is true " reduces to x :

3 x = " if x then 1+1=3"

"if x then 1+1=3" means the same as " x is not true " !

4 x = " x is not true "

Sentence 4 is a liar identity and it is easy to prove that no liar identity is true so:

5 sentence 0 is not "if sentence 0 is true then 1+1=3"

 

[sigurdW]

Note: My system differs from most in having only one primary truth value: "true"!

The second value is defined as the negation of the first: "not true".

In logics in ordinary use the terms "false" and "not true" means the same...

Not so here in a system with a universal domain...Proof:

1 "false" and "not true" means the same (assumption)
2 "one,two,three!" is not true. (obviously so...but can be strictly proven)
3 "one,two,three!" is false. (False statement! Only statements can be false)

Note:Perhaps I should prove that Liar identities are false?

Definition: x is a Liar identity if and only if x = "x is not true".
1 x = x (Law of identity)
2 not( x = not x) (from 1 by double negation)
3 not x = "x is not true" ( can be checked by a truth table)
4 Liar identities are negations of the Law of Identity!

Note: Self referential sentences are a good model of the Correspondence Theory of Truth:

From Wikipedia:
The correspondence theory of truth states that the truth or falsity of a statement is determined only by how it relates to the world, and whether it accurately describes (i.e., corresponds with) that world. The theory is opposed to the coherence theory of truth which holds that the truth or falsity of a statement is determined by its relations to other statements rather than its relation to the world.
Correspondence theories claim that true beliefs and true statements correspond to the actual state of affairs. This type of theory attempts to posit a relationship between thoughts or statements on the one hand, and things or facts on the other. It is a traditional model which goes back at least to some of the classical Greek philosophers such as Socrates, Plato, and Aristotle. This class of theories holds that the truth or the falsity of a representation is determined solely by how it relates to a reality; that is, by whether it accurately describes that reality. As Aristotle claims in his Metaphysics: "To say that that which is, is not or that which is not is, is a falsehood; and to say that that which is, is and that which is not is not, is true".

Definition: x is a selfreferential sentence if and only if there is a predicate Z such that x= xZ.

To every sentence, xZ , there is a Reference Identity,making correspondence possible!

Lets see how the sentence "This sentence contain five words." corresponds with reality:

1 This sentence contain five words. (assumption)
2 This sentence = #This sentence contain five words.# (Referemce Identity)
3 #This sentence contains five words.#contains five words. (Correspondence! from 1 and 2)

For ordinary sentences the procedure is the same but in most cases neither the reference identity nor the correspondence are sentences:

1 The sun is hot.
2 The sun = #the sun# (here the right side of the identity is supposed to be the sun itself!)
3 #The sun# is hot. (left side is the sun itself)

 

[lugita15]

I got stuck on sentence 2. A sentence seems to be missing...

OK, let me modify my sentence 2 slightly.

1. If sentence 0 is true, then if sentence 0 is true, then 1+1=3.
2. If sentence 0 is true, then by sentence 1, sentence 0, and modus ponens 1+1=3.
3. By sentence 2, sentence 0 is true.
4. By sentence 3, sentence 0, and modus ponens, 1+1=3.

 

[lugita15]

Here's another paradox related to the Liar Paradox. There is a two-player game called Hypergame, which starts out by Player 1 saying the name of some two-player game. Then the two players proceed to play that game, with Player 2 taking the place of Player 1 and Player 2 taking the place of Player 1 within that subgame. (For instance, if Player 1 says Chess then player 2 will play white and player 1 will play black.) Then whoever wins the subgame wins the whole game. But there is an important restriction: player 1 cannot say any two-player game, it must be a finite game, meaning regardless of what moves the players make it must always terminate within a finite amount of time.

So now the question is, is Hypergame a finite game? It might seem obvious that it is, because player 1 is required to name a finite game, and then that finite game will be over in some finite time, so the whole game will be over in a finite time, so Hypergame must be finite. But then, if it is a finite game, then it is a valid game to be called during Hypergame. So player 1 says "Hypergame", and then the players start playing Hypergame with player 2 taking the role of player 1, so player 2 says "Hypergame", then within that game player 1 says "Hypergame", then player 2 says "Hypergame", etc. So the game can go on infinitely long, and thus Hypergame is not finite! But if it's not finite, then it's not a valid game to be called during Hypergame, so player 1 cannot call out "Hypergame" and thus the game cannot go on for an infinite amount of time, and thus Hypergame is finite!

Can you spot the similarity to the Liar paradox? Can you spot the similarity to Turing's proof that the Halting Problem is computationally undecidable?

 

[sigurdW]

OK, let me modify my sentence 2 slightly.

1. If sentence 0 is true, then if sentence 0 is true, then 1+1=3.
2. If sentence 0 is true, then by sentence 1, sentence 0, and modus ponens 1+1=3.
3. By sentence 2, sentence 0 is true.
4. By sentence 3, sentence 0, and modus ponens, 1+1=3.

I have a complaint... now its not an error to do as you do...and perhaps your technique is a new standard
but the old style is to keep reasoning outside the numbered sentences if possible!

Before reading I will rewrite:

1 If sentence 0 is true, then 1+1=3 (assumption 1)(sentence 1 IS by definition sentence 0)

Note that "(assumption)" functions as the left side of the outer implication in your sentence 1!
Reducing complexity.

In your sentence 2, two lines are reduced to one line, the old custom is to brake down all complex statements into its basic componentents! Which means more lines and that your later sentences will need to change their numbers...


2 sentence 0 is true (assumption 2)(making modus ponens together with sentence 1 possible)

3 1+1=3 (false statement from 1 and 2 by MP)

Perhaps you will resist this rewriting because your next argument does not work!

You argue that sentence 2 is true therefore sentence 0 is true (your sentence 3)

Your statement 2 ends with the false statement 3 above...And the deduction can't go on until the reason for the deduction of the falsity is removed!

Both you and I want to find (for different reasons) what is considered to be a correct deduction of Currys Paradox!

Every time you deduce the statement "1+1=3" you deduce a falsehood and you must immediately deny something causing the falsehood!

Accept that and see what happens: If statement 1 is not true then statement 1 is true!

 

[lugita15]

I have a complaint... now its not an error to do as you do...and perhaps your technique is a new standard
but the old style is to keep reasoning outside the numbered sentences if possible!

OK, then let me rewrite it in the old style.
1. If sentence 0 is true, then if sentence 0 is true, then 1+1=3.
2. If sentence 0 is true, then 1+1=3 (by sentence 1, sentence 0, and modus ponens).
3. Sentence 0 is true (by sentence 2).
4. 1+1=3 (By sentence 0, sentence 3, and modus ponens).

 

[sigurdW]

Here's another paradox related to the Liar Paradox. There is a two-player game called Hypergame, which starts out by Player 1 saying the name of some two-player game. Then the two players proceed to play that game, with Player 2 taking the place of Player 1 and Player 2 taking the place of Player 1 within that subgame. (For instance, if Player 1 says Chess then player 2 will play white and player 1 will play black.) Then whoever wins the subgame wins the whole game. But there is an important restriction: player 1 cannot say any two-player game, it must be a finite game, meaning regardless of what moves the players make it must always terminate within a finite amount of time.

So now the question is, is Hypergame a finite game? It might seem obvious that it is, because player 1 is required to name a finite game, and then that finite game will be over in some finite time, so the whole game will be over in a finite time, so Hypergame must be finite. But then, if it is a finite game, then it is a valid game to be called during Hypergame. So player 1 says "Hypergame", and then the players start playing Hypergame with player 2 taking the role of player 1, so player 2 says "Hypergame", then within that game player 1 says "Hypergame", then player 2 says "Hypergame", etc. So the game can go on infinitely long, and thus Hypergame is not finite! But if it's not finite, then it's not a valid game to be called during Hypergame, so player 1 cannot call out "Hypergame" and thus the game cannot go on for an infinite amount of time, and thus Hypergame is finite!

1 Can you spot the similarity to the Liar paradox?
2Can you spot the similarity to Turing's proof that the Halting Problem is computationally undecidable?

1 The few paradoxes i have already checked are not similar to the Liar since their deduction only contained a Liar Identity... like Russells Paradox: The set that contains all sets not containing themselves can not be demonstrated because its infinite if it exists...and that set is what amounts to the Liar Sentence...What is there is its Referential Identity which can be shown to be a Liar Identity which solves the paradox.

2 I am not a Mathematician so unless I get interested or gets cash i won't yet check the details of the Halting Problem... Id prefer to tell the experts (are you one?) what to look for, so they can do the job themselves. My problem is how to express my thoughts so they get understood: I never had to defend my theory since nobody understood enough to spot any weak points...Sigh!

 

[lugita15]

1 The few paradoxes i have already checked are not similar to the Liar since their deduction only contained a Liar Identity... like Russells Paradox: The set that contains all sets not containing themselves can not be demonstrated because its infinite if it exists...and that set is what amounts to the Liar Sentence...What is there is its Referential Identity which can be shown to be a Liar Identity which solves the paradox.

2 I am not a Mathematician so unless I get interested or gets cash i won't yet check the details of the Halting Problem... Id prefer to tell the experts (are you one?) what to look for, so they can do the job themselves. My problem is how to express my thoughts so they get understood: I never had to defend my theory since nobody understood enough to spot any weak points...Sigh!

OK, but regardless do you understand the Hypergame paradox?

 

[sigurdW]

OK, then let me rewrite it in the old style.
1. If sentence 0 is true, then if sentence 0 is true, then 1+1=3.
2. If sentence 0 is true, then 1+1=3 (by sentence 1, sentence 0, and modus ponens).
3. Sentence 0 is true (by sentence 2).
4. 1+1=3 (By sentence 0, sentence 3, and modus ponens).

You are improving ,but its late and i have to stop... but I felt like telling you so instead of

just letting you wait for my eventual next complaints or the solution... Coming tomorrow!

As it l o o k s at the moment your definition of sentence 0 is the liar identity,

and sentence 1 the liar sentence but i must look closer to be sure.

 

[sigurdW]

OK, but regardless do you understand the Hypergame paradox?

I think i do...But I am not infallible.

So far my reaction is superficial like:
H is not really a game since it takes an infinity to get it started...
I mean its not well defined as a subgame...
Its like looking at chessplayers rotating the board and never make a first move.
But its not a real objection since what to do at the end of eternity could be added to the definition of the game.

You present new things and I thank you for that!

 

[lugita15]

I think i do...But I am not infallible.

So far my reaction is superficial like:
H is not really a game since it takes an infinity to get it started...
I mean its not well defined as a subgame...
Its like looking at chessplayers rotating the board and never make a first move.

OK, but if it's not a finite game, then it's not a valid choice for Player 1 to call out. Since Player 1 cannot call out Hypergame, Hypergame always terminates in a finite amount of time, and thus Hypergame is a valid choice for player 1 to call out!

 

[sigurdW]

OK, but if it's not a finite game, then it's not a valid choice for Player 1 to call out. Since Player 1 cannot call out Hypergame, Hypergame always terminates in a finite amount of time, and thus Hypergame is a valid choice for player 1 to call out!

If its infinite then its finite and vice versa...It is a paradox and i will probably solve it since it seems familiar...
The only paradox yet where I don't find my tecniques appliable is the Sorites Paradox.

Tomorrow it is! Good Night.

Edit: Now it is tomorrow... The plan is to finish off your subjects one at a time at a leisurely pace.

But I think wed better spend some time checking my solution of the Liar Paradox,
its good also for new comers who tend to read only the last few entries thereby missing important information.

1 Sentence 1 is not true. (assumption)
2 Sentence 1 = "Sentence 1 is not true." (Empirical truth from 1 by inspection)
3 Sentence 1 is true. (The negation of 1, by substitution from 2 to 1 and simplifying)

Here the core of the Liar Paradox is exposed!

Informally:If sentence 1 is true then it is not true, and if so then again its true and so on.
And since everything is either true or not true then sentence 1 is both true and not true!

This state of affairs contradicts the Law of contradiction and makes Classical Logic inconsistent! The Logicians abandoned Classical Logic and formulated Logics that excluded self referential sentences from the domain of their logic. Thereby excluding sentences like: I think this thought therefore I am !

Formally there is yet no contradiction arrived at, so let's add it:

4 Sentence 1 is not true and sentence 1 is true. (contradiction from 1 and 3)

Here the road to the paradox consist in denying the assumption expressed in sentence 1,

and that results in an affirmation instead of a denial...let us leave the road to defeat and

check the remaining alternative: Denying sentence 2!

 

[sigurdW]

On denying sentence 2! (Part One)

Let us listen to the opposition: But sigurdW, you yourself affirm that sentense 2 is true so IF you deny it you are contradicting yourself!

sigurdW: I claim that sentence 2 is an empirically true contradiction!

Thats to say: Sentence 2 is empirically true and logically not true and that is not to contradict myself!

Proof:

10 Sentence 1 = " Sentence 1 is not true." (ASSUMPTION!)
11 Sentence 1 is true if and only if "Sentence 1 is not true." is true (from 10)
12 Sentence 1 is true if and only if Sentence 1 is not true.(from 11)

Sentence 12 is a contradiction and the assumption in sentence 10 must be denied!

13 It is not true that Sentence 1 = " Sentence 1 is not true." (Logical Truth)
14 Sentence 1 = " Sentence 1 is not true." (Empirical Truth = sentence 2)

The extraordinary fact that an empirical truth and a logical truth contradicts each other must be explained...

 

[sigurdW]

Edit:

Today I make it simpler:

Definition:

y is a Liar Identity if and only if y is of the form: x = "x is not true",
and if y is true then x is a Liar Sentence defined by y.

THESIS:No liar identity is Logically true.

Proof (Based on: (a=b) implies (Ta<-->Tb) )

1. Suppose x="x is not true" (assumption)

2. Then x is true if and only if "x is not true" is true (from 1)

3. And we get: x is true if and only if x is not true (from 2)

4. Sentence 3 contradicts the assumption. (QED)

The logical form of the foundation of the Paradox:

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

Some values for x makes the liar Identity Empirically true:

1. Sentence 1 is not true. (Liar Sentence)
2. Sentence 1 = " Sentence 1 is not true." (Liar Identity)

To get to the paradox one must produce " 3. Sentence 1 is true." from sentences 1 and 2.
But since sentence 2 is BOTH Empirically true and Logically false it can not be a well formed sentence!
Therefore no paradox can be derived from sentence 1,or any other liar sentence.


And here is how the thread originally started:

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 let's 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?)

 

[John Creighto]

How to prove him wrong?

English is a semantically closed language so let's 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?)

I find this thread hard to follow but I am returning to your first post in this thread to say that what you are trying to solve in the last post of this thread is not the liar's paradox. In sentence 2 you are using the symbol "=" in the normal scene but at the same time you are using it to mean "If and only If". I think the inconsistent use of the symbol "=" is confusing. Also I don't think what you start with in your final post in this thread:

x if and only if (x is not true)

is the liars paradox as self reference is completely removed.

 

[sigurdW]

I find this thread hard to follow but I am returning to your first post in this thread to say that what you are trying to solve in the last post of this thread is not the liar's paradox. In sentence 2 you are using the symbol "=" in the normal scene but at the same time you are using it to mean "If and only If". I think the inconsistent use of the symbol "=" is confusing.

I know the subject is difficult so I haven't been surprised that comments are few.

Suppose we have the identity "a=b" then from it we can get the equivalence "a is true if and only if b is true". The identity IMPLIES the equivalence but they are not identical. So you see I am not using the identity sign to mean anything but what it normally means!

Besides:Note that I am analysing what is supposed to be the beginning of a legitimate derivation of the liar paradox:

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

You must take care so you yourself doesn't solve the paradox by making the derivation of sentence 3 impossible.
(Thats my job: showing sentence 3 to be not derivable from sentences 1 and 2)

I thank you for your interest in this unbelievably (yes I am NOT joking.) difficult matter,
you are mistaken but you are an adventurous person honestly trying to check my argument.

Dont let my objection to your first attempt stop you from digging deeper into the matter :)

Also I don't think what you start with in your final post in this thread:

x if and only if (x is not true)

is the liars paradox as self reference is completely removed.

Im looking for this sentence in my post but I don't find it: "x if and only if (x is not true)"

Perhaps you can quote the post and make the objectional sentence (if its there) bold or something? The sentence "x if and only if (x is not true)" is indeed not wellformed and if I wrote it there's some correctioning (and self flagellation) that needs to be done.
Perhaps you read it while I still was editing the post? That would explain it. Cya ;)

 

[sigurdW]

I find this thread hard to follow

I take your comment very seriously! I am adressing a problem that is over two thousand years old and I should make every effort to present my solution in a clear manner aimed, not at the expert, but the general reader. So I will now start again from the beginning: What is the Liar Paradox?

I guess the dictionary will say something like: The Liar Paradox arises when you try to find out whether a sentence that says of itself that it is not true (Liar Sentence) is true or not.

Since, in my view there IS no Liar Paradox I define the objects believed to cause the paradox

My Definition:

y is a Liar Identity if and only if y is of the form: x = "x is not true",
and if y is true then x is a Liar Sentence defined by y.

The most common way to introduce the LP is to start with a "Liar" definition:

A Liar Definition: Let the words "The liar" be a name of the sentence "The liar is not true."

Then it is assumed that:

1 The liar is not true. (Liar Sentence)

And from the definition is gotten:

2 The liar = "The liar is not true." (Liar Identity)

And from the above we deduce:

3 The liar is true.

A formal proof takes a few lines more but its not necessary,
the reader should be able now to foresee the disastrous result that follows!
Sentence 1 shows itself to be true if it is false, and false if it is true .
Thereby showing English together with Classic Logic to be inconsistent.

 

[sigurdW]

So what is my solution?

Whenever you look at a definition you should ask yourself if it is valid!

In this case you should ask: Isnt the definiens in the definiendum?

The General Liar Definition: x = "x is not true."

The proposer of the paradox will then say:
The paradox can be produced by other means... the definition is not necessary.
And there are other circular definitions accepted by scientists...
For example the definition of simultaneity in special relativity.

You should not,as a student of Logic, accept that answer... your cool reaction should be:

Oh well, let us assume the liar definition is valid then we get:

1 x = "x is not true"

Since (a=b) implies that (Ta <->Tb) then from sentence 1 we get:

2 x is true if and only if "x is not true" is true

The right side can be simplified and we get a contradiction:

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

Therefore the liar definition is NOT valid after all...

By what other means can the paradox be demonstrated did you say?

(But first let us rest, so the eventual reader might catch up, and raise objections!)

 

[John Creighto]

I know the subject is difficult so I haven't been surprised that comments are few.

Suppose we have the identity "a=b" then from it we can get the equivalence "a is true if and only if b is true". The identity IMPLIES the equivalence but they are not identical. So you see I am not using the identity sign to mean anything but what it normally means!

I know what you are doing. You are applying a truth function to each side of the equation. However, I don't think you get a=b. Instead you get A=A which isn't particularly useful.

I thank you for your interest in this unbelievably (yes I am NOT joking.) difficult matter,
you are mistaken but you are an adventurous person honestly trying to check my argument.

I actually wanted to start my own tread on this and their are things I want to say on this topic but before doing so we need to make the proceeding discussion much clearer.

Dont let my objection to your first attempt stop you from digging deeper into the matter :)

It will be easier if the target doesn't keep shifting. I see a lot of very similar posts and it is hard to know which to critique.

Im looking for this sentence in my post but I don't find it: "x if and only if (x is not true)"

My appologies. This was from sigurdW post (Post #38).

 

[sigurdW]

I know what you are doing. You are applying a truth function to each side of the equation. However, I don't think you get a=b. Instead you get A=A which isn't particularly useful.

Instead of defending proof 1 right now...
You should give your contra argument in better detail I am not at all sure why you think i get A=A.
Ill just prove the same thing differently, you can't use the same argument so what is your next contra?

Proof 2
From the Law of Identity we get:
1 x = x
By Double Negation we get:
2 It is not the case that x = "x is not true"

And here's another one for your third contra.

Proof 3
Suppose:
1 x = "x is not true"
Straight from the definition of truth we get:
2 "x is not true"is true if and only if x is not true
And now a contradiction is derivable:
3 x is true if and only if x is not true
Therefore:
4 Its not true that x = "x is not true"

And to make the fact finally obvious:

Proof 4
Suppose:
1 x = "x is not true"
Let x be "water is wet" then we get:
2 "water is wet" = ""water is wet" is not true = "water is not wet"
Now let x be "Water is not wet" then we get:
3 "water is not wet" = ""water is not wet"is not true" = "water is wet"
Neither a true sentence nor a false sentence makes sentence 1 true. Therefore sentence 1 must be a contradiction. QED

PS Make a truth table!

Please state your contra arguments so anyone (including me) can understand them.

 

[sigurdW]

Heres the Logical form of The Liar:

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

You should by now understand what the function of sentence 2 is?

It makes a statement identical to its own negation! Only a statement being both true and false can satisfy it.

And if sentence 1 gets selfreferential it CAN satisfy the Liar Identity.

So Logic forbids it.

Proof:

Suppose:
1 x = Zx
Then:
2 Zx = ZZx
And the logical conclusion (tautology) is:
3 (x = Zx) implies (Zx = ZZx)
Let:
4 Z = is not true
Then we get:
5 (x = x is not true) implies (x is not true = x is true)
and its a Logical Demand that:
6 Its not true that x = x is not true
So any time we construct a sentence that says of itself that it is not true then we defy the laws of logic.
And this EXPLAINS and SOLVES the Paradox of the Liar.

 

[John Creighto]

I want to return to posts #14 to # 16 with regards to prior’s solution. I’m not sure that prior solution is the correct solution. However, I think that by focusing on his solution we may better clarify the rules of logic which we are using. I believe this will help to clarify the posts which have been presented hitherto. I know there are different types of logic and when approaching such difficult paradoxes if we aren’t explicit about what rules of logic we are using then any complex derivations will be difficult to follow. Prior’s solution defines a rule of logic which isn’t universally agreed on. This rule is that any statement implicitly affirms its own truth.

I’m going to quote lugita15 (post #15) as a possible way to apply Prior's rule:

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.

As this seemed to produce some agreement. Now the criticism given by Wikipedia of Prior’s solution is as follows:

“ But the claim that every statement is really a conjunction in which the first conjunct says "this statement is true" seems to run afoul of standard rules of propositional logic, especially the rule, sometimes called Conjunction Elimination, that from a conjunction any of the conjuncts can be derived. Thus, from, "This statement is true and this statement is false", it follows that "this statement is false" and so we have, once again, a paradoxical (and non-conjunctive) statement. It seems then that Prior's attempt at resolution requires either a whole new propositional logic or else the postulation that the "and" in, "This statement is true and this statement is false", is a special type of conjunctive for which Conjunction Elimination does not apply. But then we need, at least, an expansion of standard propositional logic to account for this new kind of "and".[6]
...
6- Kirkham, Theories of Truth, chap. 9

http://en.wikipedia.org/wiki/Liar_paradox#Arthur_Prior
My response to the criticism which is cited from Kirkam, is that the word, "this", references the entire construct which is:
, "This statement is true and this statement is false"

And hence direct conjunction elimination is not possible. Now applying prior’s implicit assumption to sigurdW’s post #38:

we see that step two is superfluous since it is implied in step 1.

However, sigurdW is using propositional logic which deals more with reducing logical propositions then the assertion of truth. In contrast the laws of thought attempt to get more at the heart of what is true and false in the world.

We can certainly try to use propositional logic to prove a truth value of the liar paradox but I suspect that if possible, that it will be challenging to do so in a self consistent way which maintains the self reference.

and hence direct conjunction elimination is not possible. For purposes of propositional logic we could distinguish between: an "independent And" and a "dependent And" which is analogous how in statistics; we distinguish between independent and dependent random variables. Now applying prior’s implicit assumption to sigurdW’s post #38:

Step two is superfluous as it is implied in step 1.

Yet sigurdW is using propositional logic which deals more with reducing logical propositions then the assertion of truth. In contrast the laws of thought attempt to get more at the heart of what is true and false in the world.

We can certainly try to use propositional logic to prove a truth value of the liars paradox but I suspect that if possible, that it will be: challenging to do this in a self consistent way and at the same time maintain self reference -- since propositional logic distinguishes the atoms (things which we can assert as true or false) from the propositions.

 

[Hurkyl]

You still really don't get it.


First, you seem to have any concept of idea of "consistency". Or, more precisely, you seem to be unable to grasp what it would mean for something to be inconsistent.

Your entire point seems to be based entirely on an inability to comprehend that, in the presence of inconsistency, one can produce two valid arguments with contradictory conclusions, which has led you to the nonsensical rebuttal "there is no contradiction -- you can't do that because it would lead to a contradiction".

To resolve a pseudo-paradox, one must demonstrate that one (or both) of the arguments are flawed. To resolve a true paradox, one must actually abandon the inconsistent theory and create a new one in which is free from that contradiction.



Secondly, the problem the liar's paradox brings to light is that various rules for forming formulas conflict with logical semantics. If one is not to alter logic, one must instead alter the grammar by which formulas are constructed.

One could take the approach of rejecting formulas based on whether or not they lead to contradictions, but there are two serious flaws with this approach:

• An argument involves many formulas . One needs a rule to decide which of the many formulas are disallowed
• We can be faced with situations such as the possibility that "P and Q" might be a disallowed formula, even when "P" and "Q" are both allowed. Without rules to guarantee that one is allowed to combine formulas in various ways, it would be nearly impossible to reason at all

Third, I think it would be interesting to point out that in the logic of computation, there is no problem with there being a sentence P satisfying "P = P is not true". Here's an implementation in python:

def P():
    return not P()

however, your computer probably cannot do this computation: an equivalent implementation that will not overflow your stack is:

def P():
    while True:
        pass

 

[John Creighto]

sigurdW, I re-labeled your statements in the following:

Proof 3
Suppose:
s1: x = "x is not true"
Straight from the definition of truth we get:
s2: "x is not true"is true if and only if x is not true
And now a contradiction is derivable:
s3: x is true if and only if x is not true
Therefore:
s4: Its not true that x = "x is not true"

Now let's try to connect them:

SS1: s1&s2->s3

SS2: Not S3 -> Not S1 or Not S2We know that not S2 is false because S2 is true (by the law of identity). S3 appears to violate the law of identity so should be false. Consequently not S3 should be true. Thus Not S1 must be true and hence S1 must be false.

However, the conclusion we arrive at should be obvious so why did we assume the opposite in the first place? Also what does it have to do with the liar's paradox?

Now with regards to Hurkyl above post. I am not sure who he is responding to but I certainly admit I don't have a good grasp on this stuff but I'm not trying to resolve the liar's paradox. Rather I am only trying to see if any of the attempts to do so in this tread have made any sense. I think Hurkly's following comment sheds the most light on this:

"Secondly, the problem the liar's paradox brings to light is that various rules for forming formulas conflict with logical semantics. If one is not to alter logic, one must instead alter the grammar by which formulas are constructed."

Perhaps this is what the previous posters were trying to do but in doing so then there is no paradox -- and hence there is nothing to prove. However, if we are left with nothing to prove then all attempts to do so are superfluous.

 

[sigurdW]

I want to return to posts #14 to # 16 with regards to prior’s solution. I’m not sure that prior solution is the correct solution. However, I think that by focusing on his solution we may better clarify the rules of logic which we are using. I believe this will help to clarify the posts which have been presented hitherto. I know there are different types of logic and when approaching such difficult paradoxes if we aren’t explicit about what rules of logic we are using then any complex derivations will be difficult to follow. Prior’s solution defines a rule of logic which isn’t universally agreed on. This rule is that any statement implicitly affirms its own truth.

I accept that x implies "x is true", but I am not sure of what that may commit me to :)
Ive looked at the discussion between hurky and logita, and it seems that priors solution may resemble mine...Since we both seem to claim a sentence having two truth values is not well formed

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.

But I don't get the conclusion that the Liar Sentence is false! It becomes a sentencefunction since its liar identity is not well formed!

Now applying prior’s implicit assumption to sigurdW’s post #38:

we see that step two is superfluous since it is implied in step 1.

You mean that the sentence (2. Then x is true if and only if "x is not true" is true ) is implied by priors assumption from (1. Suppose x="x is not true") ?
Or do you mean that (2. Sentence 1 = " Sentence 1 is not true.") is implied by(1. Sentence 1 is not true.)
My first impression is that I disagree in both cases.

#38:
y is a Liar Identity if and only if y is of the form: x = "x is not true",
and if y is true then x is a Liar Sentence defined by y.

THESIS:No liar identity is Logically true.

Proof (Based on: (a=b) implies (Ta<-->Tb) )

1. Suppose x="x is not true" (assumption)

2. Then x is true if and only if "x is not true" is true (from 1)

Here Priors rule on 1 will give "x="x is not true" is true", but 2 is an equivalence!

3. And we get: x is true if and only if x is not true (from 2)

4. Sentence 3 contradicts the assumption. (QED)

The logical form of the foundation of the Paradox:

1. x is not true.
2. x = "x is not true".
And here Priors rule on 1 will give "x is not true" is true, but 2 is an identity!

Some values for x makes the liar Identity Empirically true:

1. Sentence 1 is not true. (Liar Sentence)
2. Sentence 1 = " Sentence 1 is not true." (Liar Identity)

To get to the paradox one must produce " 3. Sentence 1 is true." from sentences 1 and 2.
But since sentence 2 is BOTH Empirically true and Logically false it can not be a well formed sentence!
Therefore no paradox can be derived from sentence 1,or any other liar sentence.y is a Liar Identity if and only if y is of the form: x = "x is not true",
and if y is true then x is a Liar Sentence defined by y.

THESIS:No liar identity is Logically true.

Proof (Based on: (a=b) implies (Ta<-->Tb) )

1. Suppose x="x is not true" (assumption)

2. Then x is true if and only if "x is not true" is true (from 1)

3. And we get: x is true if and only if x is not true (from 2)

4. Sentence 3 contradicts the assumption. (QED)

The logical form of the foundation of the Paradox:

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

Some values for x makes the liar Identity Empirically true:

1. Sentence 1 is not true. (Liar Sentence)
2. Sentence 1 = " Sentence 1 is not true." (Liar Identity)

To get to the paradox one must produce " 3. Sentence 1 is true." from sentences 1 and 2.
But since sentence 2 is BOTH Empirically true and Logically false it can not be a well formed sentence!
Therefore no paradox can be derived from sentence 1,or any other liar sentence.

Lets look into the details of my version of the Correspondence Theory of Truth:
Liar Identities are a special case of Referential Identities.

Definition:
y is a Referential Identity if and only if y is of the form: x is the object the "x" in the sentence "Zx" refers to.

Most referential identies are not sentences, say we have the sentence: The Sun is shining.Then the referential identity contains the words "The Sun" and the object that IS the Sun and there's a virtual equality sign joining them together.This makes the definition of truth work: The sentence "the Sun is shining." is true if and only if the Sun is shining.This is easier to understand if we only consider the set of self referential sentences...lets pick one for inspection: 1. Sentence 1 contains five wordsIts referential identity is a sentence!2. Sentence 1 = "Sentence 1 contains five words"And all we have to do is to count the words in the quote at the right side of the identity.

So I think any similarities to Priors theory are superficial. and youll have to convince me that Priors assumption is equivalent to my referential identities.

I like talking to you,I need to practise defence,so let's not be in any hurry,lets face the facts together :)

 

[sigurdW]

sigurdW, I re-labeled your statements in the following:

Originally Posted by sigurdW
Proof 3
Suppose:
s1: x = "x is not true"
Straight from the definition of truth we get:
s2: "x is not true"is true if and only if x is not true
And now a contradiction is derivable:
s3: x is true if and only if x is not true
Therefore:
s4: Its not true that x = "x is not true"
Now let's try to connect them:

SS1: s1&s2->s3

SS2: Not S3 -> Not S1 or Not S2


We know that not S2 is false because S2 is true (by the law of identity). S3 appears to violate the law of identity so should be false. Consequently not S3 should be true. Thus Not S1 must be true and hence S1 must be false.

However, the conclusion we arrive at should be obvious so why did we assume the opposite in the first place? Also what does it have to do with the liar's paradox?

Let me check if I understand you:
Do you accept that its not true that x = "x is not true"?
Its the cornerstone of my thinking:
y is a Liar Identity if and only if y is of the form: x = "x is not true",
and if y is true then,and only then, x is a Liar Sentence defined by y.
So if y is not true then x is not a Liar sentence claiming itself to be not true!
And how then can there be a paradox?

1 Sentence 1 is not true (assumed Liar Sentence)
2 Sentence 1 = "Sentence 1 is not true" (logically false and empirically true Liar Identity)

An extraordinary fact is now coming up to the surface!
How CAN a sentence be logically false and empirically true??
Arent logical truths and falsehoods supposed not to ever collide with empirical reality? Logic was thought to be barren but it has brought forth a contradiction, Poincare said... Is this even worse?...Or is there a satisfactory explanation?

Now with regards to Hurkyl above post. I am not sure who he is responding to but I certainly admit I don't have a good grasp on this stuff but I'm not trying to resolve the liar's paradox. Rather I am only trying to see if any of the attempts to do so in this tread have made any sense. I think Hurkly's following comment sheds the most light on this:

"Secondly, the problem the liar's paradox brings to light is that various rules for forming formulas conflict with logical semantics. If one is not to alter logic, one must instead alter the grammar by which formulas are constructed."

Perhaps this is what the previous posters were trying to do but in doing so then there is no paradox -- and hence there is nothing to prove. However, if we are left with nothing to prove then all attempts to do so are superfluous.

Yes... I have a problem with hurkyl too, he doesn't seem to back up his cl...Whatever they are.
I think your statement in blue shows unbiased thinking.