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!