Philosophical, mathematical and computational logic, linguistics, formal argument, game theory, fallacies, paradoxes, puzzles and other related issues.

The simplest version of the paradox is the sentence:
This statement is false. (A)

If (A) is true, then "This statement is false" is true. Therefore (A) must be false. The hypothesis that (A) is true leads to the conclusion that (A) is false, a contradiction.

If (A) is false, then "This statement is false" is false. Therefore (A) must be true. The hypothesis that (A) is false leads to the conclusion that (A) is true, another contradiction. Either way, (A) is both true and false, which is a paradox.

This statement is false, is not a proposition, ...a declarative sentence which is either true or false.
'This statement' is an incomplete description which has no reference.
'This statement' is not a statement at all. Therefore it cannot have the property of 'truth' or 'falsity'.

Thus: there is no paradox either.

Without adding referential context, 'This statement is false' has no sense or meaning.
The subject 'This statement' cannot have the predicate 'is false'.

Surely, ~p <-> (p, is false), but..
~(This statement) <-> (This statement, is false) is meaningless.

((p, is false) is false) <-> p, but, ((This proposition, is false) is false) <-> (This proposition) is gibberish.
Owen
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I think we can reach an explanation from a programming point of view
if we let 0='false' and 1='true'
the statement is as the following

A='A=0'; (Matlab language)
notice that

'A=0' is a string input (being put between quote marks') and is equivalent to 'anything!' for a computer
so the statement becomes A='anything!'
A can be right or false
since we can add next : A= 0 (true) or A=1 (false) and there won't be any paradox in either cases

the logical misconception that makes us think that this is a paradox is that we are trying to equate between A inside the statement which has no meaning for a computer because it's a part of a text, and A outside the statement which is a variable
So the two A's are two different kinds of inputs! which can't be equated logically

it looks like programming teaches us how to think right !
Last edited by Over-thinker on August 25th, 2011, 11:57 pm, edited 1 time in total.
Over-thinker
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Welcome to the forums.

Over-thinker wrote:I think the way you explained it is the most logical way we can reach from a programming point of view
if we let 0='false' and 1='true'
the statement is as the following

A='A=0';
notice that

...

I don't understand.

In the programming language C, if we take the code:
Code: Select all
`A = A = 0;`

This assigns the value 0 to A.

It's also discussed in chapter 1 of Kernighan and Ritchie's The C Programming Language

Over-thinker wrote:the logical misconception that makes us think that this is a paradox is that we are trying to equate between A inside the statement which has no meaning because it's a part of a sentence, and A outside the statement which is a variable
So the two A's are two different kinds of inputs! which can't be equated logically

it looks like programming teaches us how to think right !

I really don't understand exactly how you think programming resolves this paradox.

Can you provide a little program on codepad you think demonstrates your idea?

xcthulhu
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This is the MATLAB language , and in MATLAB language to add a text , quotation marks are used according to this:

http://www.mathworks.com/help/techdoc/ref/strings.html

in C language i think text is added as in the following link

i think saying A=A=0 is meaningless since we could have said A=0
on the other hand the best way to express the statement of the paradox in a program is by making the statement itself a text (string), which is done by by adding two quotation marks in MATLAB language:

'A=0'

and then calling this text (string) A :
A='A=0' (in MATLAB language)
and as you know, for a computer, a text is meaningless. we just add texts in programs for explanation
so the text 'A=0' is like any other text , for example 'anything'
so the letter A inside the text is meaningless unlike the letter A outside which is a variable identified by the computer

so A which is '----' (a text), can be 0 or 1 (false or true)
hence we can say A=0 or A=1 ---> either cases this has nohing to do with what inside the text which is not identified by the computer

so the whole statement is different from the word statement (mentioned in the statememnt)
Over-thinker
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Over-thinker wrote:This is the MATLAB language , and in MATLAB language to add a text , quotation marks are used according to this:

http://www.mathworks.com/help/techdoc/ref/strings.html

in C language i think text is added as in the following link

i think saying A=A=0 is meaningless since we could have said A=0
on the other hand the best way to express the statement of the paradox in a program is by making the statement itself a text (string), which is done by by adding two quotation marks in MATLAB language:

'A=0'

Okay, I honestly had no clue what programming language you were thinking of.

FYI, python uses quotes the same way (and it's free so you can use codepad to demonstrate your ideas).

Anyway, I'm afraid I don't think your analysis is very illuminating. For instance, it doesn't help me reason about the following program, which clearly has the liar paradox at the core:

Code: Select all
`#include <stdio.h> int eubulides() {        return ! eubulides();} int main(void) {         printf("0 && eubulides() is: %i\n", 0 && eubulides());         return 0;}`

Without running the thing, what do you think the output is?

------

Addendum: In case you are wondering where I am going with this, I'm building up to Kripke's theory of truth

xcthulhu
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Unfortunately, I work only on MATLAB language . but i think it's not a problem of the program as it is of the philosophy of it.
The program is so simple. It's just putting the statement in a text 'A=0' and then calling this text A.
A='A=0' don't you agree?

I haven't been into details of the kisper's theory yet but i have read tht it is an opposition to the Tarski's theory.
I think what iam saying is a little bit comparable to the tarski's theory though they are not the same
Tarski believes that language is an heirarchy system and only the higher in this heirarchy can evaluate the truth of the lower.
However, what iam saying is that when evaluating the truth, only one level of language has a meaning at a time, all other levels are meaningless. and as i mentioned b4 the A inside the statement and the A which is the whole statement belong to two different levels of language.

I think it's all a matter of ordering information and this what programming is all about: we first said that A=0 and then we called the whole statement A. so A NOW is the last A which is the whole statement and which can be true or false regardless of the statement
that's why the truth evaluation inside the text (between quotaion marks 'A=0' )is meaningless for the computer, unlike the real truth evaluation A=0 or A=1

Do you find that convincing?
Over-thinker
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Over-thinker wrote:Unfortunately, I work only on MATLAB language.

A pitty, because now you really can't share any code with me to illustrate your ideas.

Over-thinker wrote:but i think it's not a problem of the program as it is of the philosophy of it.
The program is so simple. It's just putting the statement in a text 'A=0' and then calling this text A.
A='A=0' don't you agree?

The program is simple, yeah; but I really think it's too simple to capture the puzzle in an interesting way.

Here, I'll try to do the program I did above in matlab for you (my computer is in the shop and I don't have matlab on any other computers, so bear with me a little). Suppose you declare a function like this:
Code: Select all
`function [o] = eubalides()~ eubalides()`

Now try at the prompt:
Code: Select all
`>> true || eubalides()`

What will it say?

Over-thinker wrote:I haven't been into details of the kisper's theory yet but i have read tht it is an opposition to the Tarski's theory.
I think what iam saying is a little bit comparable to the tarski's theory though they are not the same
Tarski believes that language is an heirarchy system and only the higher in this heirarchy can evaluate the truth of the lower.

However, what iam saying is that when evaluating the truth, only one level of language has a meaning at a time, all other levels are meaningless. and as i mentioned b4 the A inside the statement and the A which is the whole statement belong to two different levels of language.

Over-thinker wrote:I think it's all a matter of ordering information and this what programming is all about: we first said that A=0 and then we called the whole statement A. so A NOW is the last A which is the whole statement and which can be true or false regardless of the statement
that's why the truth evaluation inside the text (between quotaion marks 'A=0' )is meaningless for the computer, unlike the real truth evaluation A=0 or A=1

Do you find that convincing?

Not really.

I know a few theories of Truth (they made me study them in school); Tarski's treatment of the liar paradox is "You're not allowed to say it", which is boring and uninsightful in my opinion.

You can't convince me of one theory of Truth or another; there are multiple theories of Truth and each has its applications.

Since you only know one programming language and one theory of truth, you have to admit you have a narrow point of view. I am trying to show you some other perspectives.

Is that clear?

I'm not really arguing with you.

Kripke's theory is better for thinking about recursive functions and short circuit evaluation.

xcthulhu
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xcthulhu wrote:. Suppose you declare a function like this:
Code: Select all
`function [o] = eubalides()~ eubalides()`

Now try at the prompt:
Code: Select all
`>> true || eubalides()`

What will it say?

I didn't get what's the point of this program
since the answer will definitely be one becuz it contains (true || anything)
i can't really see what do you mean by this function file and this prompt
Over-thinker
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Eubalides() is the liar paradox, in program form.

In the programming language Haskell, which does a more elegant job of treating this sort of thing, I would write:

Code: Select all
`eubalides = not eubalidesmain = putStrLn \$ show \$ True || eubalides`

Except for the crazy syntax for printing things, it's obvious what this program is doing.

You can play with it here: http://codepad.org/armHdxgg

Since you think this all seems silly, let's try something more interesting, shall we?

Code: Select all
`a = f && eb = not (True && e) || (False && f) || dc = not (True || a) && (b || e)d = c || (not (c && not d) && (False && not b)) e = f || not d || not (b && not b) || bf = not (c || d) && Truemain = putStrLn \$ show \$ a`

What do you think the value of a is? What about the other variables?

What do you think it means for a variable in a system of equations like this to be behaving "like a liar sentence"?

Also, do you really think Tarski's theory of truth will help you understand this puzzle all that well?

xcthulhu
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Hello world, can anyone see the forest for the trees?

Example 1:
I love mi amigo cum gravitas c'est la vie.

Example 2:
Who cares? <== self-evident truth

Example 3:
From now on*, you are aware that every conspiracy that ever existed is true.

Please refer to categorical error, semantics, and conspiracy theory.

*cue spooky '50's sci-fi music

psionic11
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<Logos>
In plainspeak, lest I be labeled a troll, here's what it is:

The OP conflates semantic meaning and "truth" with mathematical and logical truths. Mathematics has the general tendency to describe the operating principles of physical objects. Logical truth is a self-contained and somewhat archaic abstraction reliant on certain arbitrary premises.

Semantics (or "Thought" -- as tribal/cultural behavior, whether acted, spoken, sung, performed, produced, popularized, regurgitated, traditionalized), as a set of meanings and/or truths, has changing premises that bear fleeting resemblance to physical realities. Word. In other words, Words are arbitrary and contradictory and powerful and specific and approximate all at the same time.

Math/logic need not apply.

</Logos>
Last edited by psionic11 on August 27th, 2011, 1:32 am, edited 7 times in total.

psionic11
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Simplest yet:

Dude, it's only a paradox if you get caught up in the trip...

Disbelieve.

Detach.

Use some "common" sense and realize you don't fix a car engine with a tape measure.

Don't use logic on words.

psionic11
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xcthulhu wrote:
What do you think it means for a variable in a system of equations like this to be behaving "like a liar sentence"?

I really liked this question, i think, as u suggested b4, there are many perspectives to look at this paradox

I think another way is as the fllowing .. let the whole statement be B so the paradox wud look :
B: "A is false" but A=B

then the 1-sentence program is as the following(MATLAB) (assuming again true=1, false=0)

B=1
A=(B&&0)

in words, this means that we have started by assumption that B is true and actuallyy this is whatt we are really doing when we think of the paradox .. reading the sentence the very first time means assuming that it's true

i believe that we don't start this program with B=0 becasuse if the whole statement was false we wudn't have read it at all

since the statement is read the very first time only assuming that it's true ..... we can rewrite the program as an if loop

if B=1, A=(B && 0);end

which means that only if the whole statement is right we can go to the next stage which is the statement itself

Now, typing B=1 would give an error (easy to see why!)
but typing next B=0 would give B=0 which means that the whole statement is false .. no need to read it!

so since B=1 gives an error it's not the best choice..on the other hand the best choice is to let B=0 from the beggining which makes the if loop end directly without passing by the statement and this is equivalent to NOT reading the statement

I guess iam back to Tarski's ideas .. I think Tarski is proving himself to me ..
Over-thinker
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@Psionic: I wish you wouldn't put posts like that in this forum; that stuff vaguely belongs in metaphysics.

And I'm sure you hate it, but the standards of this forum demand a level of technical sophistication you're not demonstrating.

@Over-tinker: First, you didn't answer my first question: What is the truth value of "a" in my puzzle? What are the truth values for the other variables?

And the exact issue with the liar paradox is that there is recursion going on (assuming you don't just drink the Tarski koolaid at time zero). Therein lies the problem with drawing intuition from MATLAB for this stuff: while you can write recursive programs, it's difficult. You haven't written anything recursive in what you say above...

I'll try to work in MATLAB for you. Like I said, I don't have MATLAB in front of me, but can you tell me if this is valid syntax?

Code: Select all
`B = ~ B;`

If so, can you tell me what MATLAB thinks the value of B is? Otherwise can you declare a recursive function for me that behaves like this?

As an aside, "ifs" aren't loops. They are control flow statements.

-----------

Addendum: I want you to know that the reason I am being pedagogical like this is because I really want you to figure out Kripke's theory of truth for yourself, since I think you are clever enough to do this.

xcthulhu
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xcthulhu wrote:@Psionic: I wish you wouldn't put posts like that in this forum; that stuff vaguely belongs in metaphysics.

Haha, quit forwarding the BS to me :P

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xcthulhu wrote:@Over-tinker: First, you didn't answer my first question: What is the truth value of "a" in my puzzle? What are the truth values for the other variables?

this is the way i though of it...i supposed that order is not important as could be if this is a loop (is that what you mean)

from 6 : f is true
from 1: a,e are also true
from 5: e=f (nuthing new)

Now the order matters :

1st way:
from 3: c false b false
from 2: d is false

i think the contradiction occurs in 4 since it contains (false & not b) though we know that b is false

2nd way:
from 4: d false b true c true (since d=not(c&&not d))
in this case the contradiction is in 2 since b must be false or d (false)

I'll try to work in MATLAB for you. Like I said, I don't have MATLAB in front of me, but can you tell me if this is valid syntax?

CODE: SELECT ALL
B = ~ B;

first we have to assign a value for the variable in order for the program to recognize it
let it be 1

B=1 B = ~ B running this program wud give a zero
because it's like asking matlab (is B different from B)
if yes it would turn 1, if No as is the case it wud turn 0

to make a recursive program .. i think the following is the best:

assuming that A is the whole statement and B is the word statement in the statement our ultimate goal for the paradox to be resolved is to have A=B since we mean the same statement by them

Code: Select all
` A=1; B=1; while(1)    if A==1;B=0,A=B,end     if A==0;B=1,end  if A==B,break;  else A=B,end  end`

running this program on an m-file will make matlab go crazy giving an infinite series of
B=0, A=0, B=1, A=1, (u can stop it by ctrl+c)

in words, this program first assumes that both A and B are true. Using while loop the program iterates until A and B (that are meant to be the same) are equal (this is the break command in the 5th row of the code) A=B is logically satisfying but it will never be reached.

i believe this program is exactly what goes on in our mind when thinking of the paradox. Do you agree?.. that's why i think that making a recursive program won't solve the misconception.

and for this reason, i preffered using the if statement as i did b4:
if B=1 ; A=(0&&B) ; end which gives an error if B=1 and ends the statement if B=0
Over-thinker
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Over-thinker wrote:
xcthulhu wrote:@Over-tinker: First, you didn't answer my first question: What is the truth value of "a" in my puzzle? What are the truth values for the other variables?

this is the way i though of it...i supposed that order is not important as could be if this is a loop (is that what you mean)

Well, it's a programming language... so you can run the program to find out the answer:

Order doesn't matter.

And yes, anything you can do with loops you can do with recursion, and anything you can do with recursion you can do with loops. Generally speaking, as long as a programming language has one or the other, it will be Turing complete. That's a fancy way of saying you can just about program anything you want.

Over-thinker wrote:
Code: Select all
` A=1; B=1; while(1)    if A==1;B=0,A=B,end     if A==0;B=1,end  if A==B,break;  else A=B,end  end`

running this program on an m-file will make matlab go crazy giving an infinite series of
B=0, A=0, B=1, A=1, (u can stop it by ctrl+c)

...

i believe this program is exactly what goes on in our mind when thinking of the paradox. Do you agree?.. that's why i think that making a recursive program won't solve the misconception.

There's no one way to understand the paradox.

Now you are reinventing another theory in the literature - the Gupta/Belnap Revision Theory of Truth. Since you are an engineer perhaps you will appreciate that this is how a ring oscillator in VLSI design functions.

But Kripke's (basic) theory is more sophisticated than Tarski's but less crazy than Gupta & Belnap's (his theory gets more complicated and eventually becomes just as complicated as crazy as the revision theory). I will tell you that in Kripke's logic, instead of two values, there are three: $0,1,\bot$. I think you already could tell me what $\bot$ means... in the following two equations both the variables take on the value $\bot$:

a = not a
b = b

I think now you could naturally give a name to $\bot$.

What do you think Kripke meant by $\bot$?

Over-thinker wrote:and for this reason, i preffered using the if statement as i did b4:
if B=1 ; A=(0&&B) ; end which gives an error if B=1 and ends the statement if B=0

I didn't realize it was looping.

Do you know JAVA? It really would be helpful if you put code I could run on codepad for explaining yourself (also that would help any of the other people who might be reading this and want to play with the programs to see understand what each of us is thinking).

xcthulhu
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xcthulhu wrote:Well, it's a programming language... so you can run the program to find out the answer:

.

I think i missed the paranthesis in this part:
d = c || (not (c && not d) && (False && not b))

i think the third value of truth is neither true nor false. So maybe it means that we can't even evaluate the truth of the sentence. Maybe the reason for that in :
a = not a
b = b

is the fact that in each of these two statements a variable is related to itself, so kind of we can't decide if it's true or false

-------------
I tried to put the code i did on codepad. I wrote it in perl language which is similar to MATLAB

but i got a (timeout) i think this is due to the infinite loop and the fact that codepad doesn't handle infinite loops.
I think the only way is to trust me on what i got on matlab:
Code: Select all
`B =     1A =     1B =     0A =     0B =     1A =     1B =     0A =     0  .  .  .??? Operation terminated by user during ==>paradox at7`

There's no one way to understand the paradox.

i totally agree !!
I don't think this recursive program makes us understand the paradox but it just makes us describe it. or express what goes on in our mind. but the paradox is still there!

i think the revision theory of truth in its simple form is similar to the code i did though the order might not be the same.
Over-thinker
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Over-thinker wrote:I tried to put the code i did on codepad. I wrote it in perl language which is similar to MATLAB

but i got a (timeout) i think this is due to the infinite loop and the fact that codepad doesn't handle infinite loops.

Thanks a lot man for indulging me like this!

I hacked your code a bit. I turned your while(1) loop into a for(;;) loop, and put in print statements so you can see that state of A and B (and I cleaned up your indentation for my own readability). As you can see your code doesn't seem to be getting the alternating truth values right:

Here's a simpler program that I believe does a better job of representing the paradox:

It has a crazy bug where when \$A is "false" it doesn't print. That's because perl represents false as "" (see perlmonks.org for a discussion of this). This kind of nonsense is why I quit perl years ago (also this XKCD comic).

I will tell you the skinny on Kripke's theory - rather than worrying about alternating truth values, if you can never determine the truth value of a proposition then it gets value $\bot$ (meaning: undefined). The logic of the operators AND, OR, and NOT is given by (strong) Kleene Three valued logic (although in this SEP entry they write $\bot$ as "1/2"... logicians love being inconsistent with their notation).

Kripke's theory gets more complicated if you allow for "for all" and "there exists" in your langauge, or if you declare that statements like "This statement is true" may be either true or false instead of just undefined.

xcthulhu
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xcthulhu wrote:I hacked your code a bit. I turned your while(1) loop into a for(;;) loop, and put in print statements so you can see that state of A and B (and I cleaned up your indentation for my own readability). As you can see your code doesn't seem to be getting the alternating truth values right:

I think the reason for that is that you put only one print command . to make it identical to the matlab code, i added another "print". since two if-decisions take place at each loop. Here's how i arranged it:

same results as matlab!
actually the code is exactly the same, but in matlab we don't need to put a print in order to display, we only have to avoid suppressing the statement with (;) in order to display it

Here's a simpler program that I believe does a better job of representing the paradox:

I think your program is smart ! but it looks at the paradox from outside.. it's like a summary to the paradox. on the other hand mine goes through the steps that human mind passes when reading the paradox, taking into account that i used 2 variables and then made them equal since i think this is how it happens in our mind. So i think each of the programs suggests a different type of solution; since each of them looks at the paradox from a different perspective.

and you drew my attention to the difference between A!=A and A=!A .. i think i mixed them up at an earlier stage

regarding kripke's theory. so do you believe it solves the paradox by the third value of truth or by suggesting that the statement is (either true or false) or are both equivalent?
Over-thinker
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Over-thinker wrote:I think your program is smart ! but it looks at the paradox from outside.. it's like a summary to the paradox. on the other hand mine goes through the steps that human mind passes when reading the paradox, taking into account that i used 2 variables and then made them equal since i think this is how it happens in our mind.

You know... we really can't know what the human mind is doing by introspection alone. Not everyone thinks the same way.

Here's a little anecdote by Nobel laureate Richard Feynman (physicist) about the different ways people think on a rather basic level:

Over-thinker wrote:So i think each of the programs suggests a different type of solution; since each of them looks at the paradox from a different perspective.

Not really; A and B both alternate between 0 and 1 at a frequency of one flip per time step. Up to a phase shift, they are the same sequence. Mine is the same too up to a change of phase.

If you want a much more interesting system that isn't the same as these models of the liar sentence under phase shift, you might check out linear feedback shift registers.

"The Revision Theory of Truth" has been reinvented at least twice outside of philosophical logic as far as I can tell. For instance, Kauffman's Boolean network approach to understanding genetic regulatory networks is basically the revision theory of truth you have reinvented here. Likewise, digital timing diagrams in electrical engineering are again just the revision sequences in the revision theory of truth by another name. I would clump all of these things under the same category, although I guess EEs don't know genetics, geneticists don't know EE, nobody knows philosophical logic and philosophical logicians don't know anything.

Over-thinker wrote:regarding kripke's theory. so do you believe it solves the paradox by the third value of truth or by suggesting that the statement is (either true or false) or are both equivalent?

It's not a matter of "believing". I didn't make this stuff up. Kripke's approach, and Kleene's before him, is to introduce a third value for "failed computation".

This approach relaxes bivalence, which is the principle in logic that truth values of variables are either 0 or 1. A lot of logics are not bivalent. You can read about them on the Stanford Encyclopedia of Philosophy's entry on Many-Valued Logics.

xcthulhu
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Owen wrote:'This statement' is an incomplete description which has no reference.
'This statement' is not a statement at all. Therefore it cannot have the property of 'truth' or 'falsity'.

Thus: there is no paradox either.

I actually disagree with you that "this statement" does not refer to anything.
"This" as opposed to "that" is posessive by nature of proximity, the proximity is referring to itself. "This statement" IMO is clearly referring to itself. I believe "this statement" is in fact a statement, it's a statement about itself as being a statement about itself.

False as a concept is loosely, "Other than what it is actually concluded to be".

If we take false at face value, it is not a statement about itself but a statement about nothing or something else or everything. But I think we can agree that it is in fact a statement about itself because of how we commonly use this and that.

It would then be false that "This statement is false" (other than what it's actually concluded to be), and that it is true (what it's concluded to be)... a statement about itself.

This is about evidence based common usage. We know from evidence based common usage that "this statement" refers to itself. We know by this common usage that it's false that it doesn't refer to itself. And we know that if we say it's false we're making the claim that it doesn't refer to itself. The claim fails on the evidence.
It is false that the statement is false. That I believe resolves the paradox.

With the statement, "I am representing other than what I know to be true linguistically" (I am lying), I think the calculations get harder as the statement isn't so clearly referring to itself based on what we understand as common usage. In fact, what does it even mean to parse this statement as if its referring to itself? Personally, I'd have to think on that statement more to really understand it.
0oqpo0
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Hello 0oqpo0,

0oqpo0 wrote:False as a concept is loosely, "Other than what it is actually concluded to be".

Would you accept Tarski's criterion of truth: that

"Snow is white" is true in English if and only if snow is white.

?

I think this is compatible with yours, it just seems a bit clearer to me.

0oqpo0 wrote:This is about evidence based common usage. We know from evidence based common usage that "this statement" refers to itself.

It might depend on quite how we define "statement"; many philosophers take a statement to be a sentence with a truth-value, or something like that. In such a case

1. "this statement"

refers only if

2. "this statement is false"

has a truth value, so I think we get ambivalence: if we take (2) to be a statement then (1) refers, and if not, not. So I actually don't mind whether to take your approach or Owen's.

Anyway, my preferred explaining-away of the paradox owes to my friend Martin Castro-Manzano, who believes that such paradoxes are just camouflaged contradictions:

Martin Castro-Manzano wrote:Take the following statement, S1:

<<P and ~P>>

Is S1 false? Of course it is, for its output, using a truth table, is the vector: [0 0]. Now, take the following, S2:

<<Q if and only if ~Q>>

Is S2 false? Of course it is, for its output, using a truth table, is the vector: [0 0]. Then, S1 and S2 are equivalent. And we know both statements are false. There is no paradox in them.

DEVELOPMENT
Now take S3:

<<This statement is false if and only if it is true>>

Which is an instance of the liar paradox; however, what is the big deal about S3? Consider the following argument, A1:

1. S2 is false (just as S1).
2. S3 is equivalent to S2.
C. S3 is false.

However, it is a common assumption that S3 is a paradox. But then, we have the next argument A2:

2. S3 is equivalent to S2.
3. S2 is equivalent to S1.

CONCLUSION
Thus, if S3 is a paradox, then S1 should be a paradox, but S1 is not paradoxical at all, it is simply a well defined contradiction.
So, what is the big deal about S3, if it is a plain contradiction just as S1? What I suspect is that "paradox" is a psychological term rather than a logical one.

This, to me, seems satisfactory, and saves us the pains of a theory of types, or stratification, or trying to figure out just what the hell Wittgenstein meant anyway.

Lomax

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Lomax wrote:Hello 0oqpo0,

0oqpo0 wrote:False as a concept is loosely, "Other than what it is actually concluded to be".

Would you accept Tarski's criterion of truth: that

"Snow is white" is true in English if and only if snow is white.

?

I think this is compatible with yours, it just seems a bit clearer to me.

Sure, except we know with different clustering and aspects of perceptual actuity that snow like polar bear hairs are actually clear.

Lomax wrote:
0oqpo0 wrote:This is about evidence based common usage. We know from evidence based common usage that "this statement" refers to itself.

It might depend on quite how we define "statement"; many philosophers take a statement to be a sentence with a truth-value, or something like that. In such a case

1. "this statement"

refers only if

2. "this statement is false"

has a truth value, so I think we get ambivalence: if we take (2) to be a statement then (1) refers, and if not, not. So I actually don't mind whether to take your approach or Owen's.

I think it's the reverse, this "statement is false" can only have a truth value is "this statement" has a truth value. I believe, "this statement" has a truth value.

Lomax wrote:Anyway, my preferred explaining-away of the paradox owes to my friend Martin Castro-Manzano, who believes that such paradoxes are just camouflaged contradictions:

Martin Castro-Manzano wrote:Take the following statement, S1:

<<P and ~P>>

Is S1 false? Of course it is, for its output, using a truth table, is the vector: [0 0]. Now, take the following, S2:

<<Q if and only if ~Q>>

Is S2 false? Of course it is, for its output, using a truth table, is the vector: [0 0]. Then, S1 and S2 are equivalent. And we know both statements are false. There is no paradox in them.

DEVELOPMENT
Now take S3:

<<This statement is false if and only if it is true>>

Which is an instance of the liar paradox; however, what is the big deal about S3? Consider the following argument, A1:

1. S2 is false (just as S1).
2. S3 is equivalent to S2.
C. S3 is false.

However, it is a common assumption that S3 is a paradox. But then, we have the next argument A2:

2. S3 is equivalent to S2.
3. S2 is equivalent to S1.

CONCLUSION
Thus, if S3 is a paradox, then S1 should be a paradox, but S1 is not paradoxical at all, it is simply a well defined contradiction.
So, what is the big deal about S3, if it is a plain contradiction just as S1? What I suspect is that "paradox" is a psychological term rather than a logical one.

This, to me, seems satisfactory, and saves us the pains of a theory of types, or stratification, or trying to figure out just what the hell Wittgenstein meant anyway.

Lomax
[/quote]

I don't understand how finding a vector of 0,0 makes a statement false. It seems much closer to Owens use of non reference, as if matter and anti matter are colliding in perfect proportions to cancel each other out so that there's nothing to determine truth or falsity of.
0oqpo0
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I think with the statement "I am lying" we are into territory where the liar statement would actually be "this is false" instead of "this statement is false". This gets tricky because we don't assume in common usage that "this person" refers to the person speaking, but the person closest in proximity besides the person speaking determined by how we're measuring proxomity.

So it's not equivilent by common usage to say "This person is lying" and "This statement is false", aside from the differences between person and statement.

I would probably side with Owen in determing that "this is false" doesn't have a truth value because we don't know by common usage what "this" refers to without a reference, whereas with "this statement" we have a common perception that it is referring to itself as a statement, and thus formulate it as a statement with a truth value, a reference.
0oqpo0
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I decided upon re-examination as happens that the last post was weird, upon more time to think this through though:

Using the loose approximation for false, and how I think people actually fill in the blanks for this statement when they consider the paradox, I came up with:

This statement is referring to itself but is actually other than what it is concluded to actually be.
For "This statement is false"

The only references we have to go on are "statement that is referring to itself" and negation of the only reference being considered.
What happens here are two simultaneous actualities for opposite statements.

This is a statement that refers to itself.
This is not a statement that refers to itself.

I still maintain unlike in the OP that "This statement" is in fact a statement as people conceive the paradox, namely, "this statement is referring to itself as a statement that refers to itself as a statement", that it does have a referrant, namely itself and that the non-referrant argument doesn't apply when considering how people have considered this paradox. It is self-referrant. Not non-referrant.

When considering those two opposing sentences that I believe derive from the expanded liar statement I suggested above, it is worth noting that "This is not a statement that refers to itself" is false, as every statement minimally refers to itself, even if it points also to another statement.

The first statement is always true, the second statement is always false.
0oqpo0
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Lomax – I don't understand this:

Lomax wrote:S1 is not paradoxical at all, it is simply a well defined contradiction.

Stanford Encyclopedia of Philosophy: "Most paradoxes – but not all – involve contradictions; for such cases, we often use the word "contradiction" as well."

For what it is worth, here is my take on the liar paradox:

1. The sentence "This statement is false" has a grammatical subject and predicate, so (in common parlance at least) it is a statement, regardless of whether it has a truth-value or not. It may be a nonsensical statement, but it's still a statement.

2. Hence it is legitimate to construe "This statement is false" as being equivalent to "The statement 'this statement is false' is false". This, I suggest, is how it is usually construed in discussions of the liar paradox.

3. At the first stage of the analysis, there are no existing facts to make "This statement is false" either true or false. Hence its initial truth-value is "neither"; or, alternatively, it can be said to have no truth-value at the initial stage.

4. Hence it falsely implies that it is false-at-the-first-stage. But it consequently acquires a truth-value, i.e. "false". So now there is a second stage, at which it is false. But the fact still remains that it was not false at the first stage; therefore there is no contradiction.

5. Because it was false at the second stage, it correctly states its second-stage truth-value, so it now acquires a third-stage value of "true". And so on, with infinite oscillation between "true" and "false". But the facts about any particular stage do not retrospectively alter those about previous stages. So, no contradictions.

To sum up:
(a) the statement is neither true nor false given the pre-existing facts;
(b) it is false in implying that it is false given the pre-existing facts;
(c) it is true in implying that it is false in implying that it is false given the pre-existing facts;
and so on. Statements (a), (b), (c) etc are all simultaneously true.

I don't know if I'm trying to reinvent the wheel here. Can you point me to any published 'solution' similar to the above?
Positor
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Hello Positor,

Stanford Encyclopedia of Philosophy: "Most paradoxes – but not all – involve contradictions; for such cases, we often use the word "contradiction" as well."

Yeah, well I think Martin is not using the terms as conceptually (only extensionally) interchangeable. I mean, a lot of philosophers treat paradoxes as something brain-baffling and mysterious; nobody would make the same fuss over (for example) "I am red and not red" as they do over the liar paradox. So Martin's claim is that the only difference between a paradox and a regular contradiction is that paradoxes somehow confuse us; this is why he says it's a psychological, rather than a logical issue.

Positor wrote:For what it is worth, here is my take on the liar paradox:

1. The sentence "This statement is false" has a grammatical subject and predicate, so (in common parlance at least) it is a statement, regardless of whether it has a truth-value or not. It may be a nonsensical statement, but it's still a statement.

2. Hence it is legitimate to construe "This statement is false" as being equivalent to "The statement 'this statement is false' is false". This, I suggest, is how it is usually construed in discussions of the liar paradox.

3. At the first stage of the analysis, there are no existing facts to make "This statement is false" either true or false. Hence its initial truth-value is "neither"; or, alternatively, it can be said to have no truth-value at the initial stage.

4. Hence it falsely implies that it is false-at-the-first-stage. But it consequently acquires a truth-value, i.e. "false". So now there is a second stage, at which it is false. But the fact still remains that it was not false at the first stage; therefore there is no contradiction.

5. Because it was false at the second stage, it correctly states its second-stage truth-value, so it now acquires a third-stage value of "true". And so on, with infinite oscillation between "true" and "false". But the facts about any particular stage do not retrospectively alter those about previous stages. So, no contradictions.

To sum up:
(a) the statement is neither true nor false given the pre-existing facts;
(b) it is false in implying that it is false given the pre-existing facts;
(c) it is true in implying that it is false in implying that it is false given the pre-existing facts;
and so on. Statements (a), (b), (c) etc are all simultaneously true.

I don't know if I'm trying to reinvent the wheel here. Can you point me to any published 'solution' similar to the above?

Xcthulhu pointed me to a work by Belnap and Gupta (which I haven't read, and forget the name of) that suggested the truth-value of the liar paradox oscillates in the way you suggest. Also, Douglas Hofstadter considers it a"strange loop" which is true when false and false when true. Those are the only similar positions I know of but Xct is obviously much better acquainted with the literature.

My only concern with your position is: if the first stage is meaningless and the second stage is false, then surely they are not equivalent statements, even if they seem to be? I'm assuming some kind of truth-conditional semantics though.

Lomax

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Lomax wrote:Xcthulhu pointed me to a work by Belnap and Gupta (which I haven't read, and forget the name of) that suggested the truth-value of the liar paradox oscillates in the way you suggest. Also, Douglas Hofstadter considers it a"strange loop" which is true when false and false when true. Those are the only similar positions I know of but Xct is obviously much better acquainted with the literature.

My only concern with your position is: if the first stage is meaningless and the second stage is false, then surely they are not equivalent statements, even if they seem to be? I'm assuming some kind of truth-conditional semantics though.

Belnap/Gupta's theory is the Revision Theory of Truth. If you scroll up the page a little you'll see that over-tinker basically reinvented it, too.

Admittedly, in my own research I just use Tarskian truth because these other crazy definitions are unweildy, even though I think they're fun.

xcthulhu
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Positor:

This is probably more how it's thought about:

Revised the Liar Statement (This statement is false):

-------------------------------
In referring to its own truth condition, this statement is other than what the actual answer is concluded to be.
-------------------------------

What this means is that if you conclude it's false, it must be something within the realm of a truth condition that is not false (true? undefined? etc..) If you conclude that it's true, it must be something within the realm of a truth condition that is not true (false? undefined? etc...)

I think that sentence above is exactly how the paradox is expanded when all the implications are added. The statement which refers to it's own truth condition must be other than what you actually conclude in your mind for it to be.

It continues to grow, as by leaving it undefined like this it effectively becomes defined, and if I define it like this is effectively becomes undefined. I actually think it's a true paradox, looking at it like this, I have to accept that there's probably not an answer besides that it's a paradox. I took me almost a year to reach that conclusion. Oh well. Maybe someone can enlighten me otherwise.
0oqpo0
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