Consider the statement (call it p):

This statement is false.

Is p true or false? If it is true, it contradicts itself. So it must be false. But that is what p states, so it must be true. Liar’s Paradox.

On the face of it, it seems that the problem with this paradox is either self-reference or bi-valued logic itself. If the problem is bi-valued logic, it seems that a lot of math is suspect, in particular proof by contradiction. Proof by contradiction is a staple of math as I know it, and giving it up would be difficult.

For a long time, I thought that the problem here was inappropriate self-reference. But not all forms of self-reference lead to contradictions. Self-referencing statements are even used in proofs of theorems, the Godel’s incompleteness theorem for example. The title of this post “This is my 200th post” is a self-referencing statement (incidentally, it *is* true). It does not lead to any contradiction. If the problem is inappropriate self-reference, what forms of self-reference are problematic? Change the liar’s paradox to “This statement is true.” and there is no contradiction. The form of self-reference is clearly the same. What then?

q: This statement is true.

If q is false, it contradicts itself. So q must be true. This does not lead to any contradiction. Assuming bi-valued logic is valid, q is true. But what does q *mean*? *What* is true? As I see it now, the problem with the liar’s paradox is that just like q, it is empty, devoid of any content. There is nothing in it that can be either true or false. In other words, it is not a statement at all because it does not state anything about the world. Mere adherence to rules of grammar does not produce statements.

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Filed under: Concepts | Tagged: Liar's Paradox, Logic, Meaning, Reference |

Krishnamurthy, on December 31, 2012 at 12:11 pm said:Hi K.M.,

Congrats on your 200th post. Just wanted to add in two bits:

1. Although proof by contradiction is widely used, more constructive schools seek to avoid it. An example is the intuitionistic school (Brouwer). This leads to what is known as “non-standard maths”, with hyperreals,infinitesimals, etc and where for example you can show that “all continuous functions are (infinitely) smooth”.

2. Do check out Yablo’s paradox, an extension(?) of the Liar’s paradox, designed to avoid self-reference.

http://en.wikipedia.org/wiki/Yablo%27s_paradox

Excited for the next 200.

Kris

K. M., on January 1, 2013 at 10:32 pm said:Hi Kris,

Thanks. Am happy to resume writing.

Had read about the Intuitionistic school on wikipedia but not in any depth. Will try to look into it again.

Checked out Yablo’s paradox.

One extension of the liar’s paradox is:

The following statement is true.

The preceding statement is false

This extension is circular instead of self-referencing.

Yablo’s paradox avoids the circularity too by having infinite statements.

I think all of these are essentially the same and the problem is neither self-reference nor circularity but that the statement(s) in all these cases are purely empty. None of them states anything about the world and therefore they cannot be either true or false.

I am a little surprised that this argument is nowhere to be found on the wikipedia page for the liar’s paradox although it does list some possible resolutions. Is there something badly wrong about my line of thought?

Krishnamurthy, on January 2, 2013 at 11:18 am said:Hi KM,

I think the issue with this argument is that you are appealing to something external to the formal system under consideration. To formalize your argument, one would have to introduce a notion of “statements with/without content”, and one of the axioms would be that only statements with content would have truth values. Now this is pure speculation, but I think this would create a superset system, with its own “liar’s paradox”; the intuition for this being something like the Goedel’s incompleteness theorem. Maybe something like “this statement is contentless or false” ??

Kris

K. M., on January 3, 2013 at 2:06 am said:Hi Kris,

Thanks for pointing that out. I believe that what I am appealing to is the correspondence theory of truth – a statement is true if it corresponds with the world. And that of course leads to questions like Are mathematical statements part of the “world”?

Which is exactly what one needs to answer to label your example as contentless/true/false.

I don’t have a fully thought out position on that yet, but I am inclined to believe that statements are indeed part of the world as long as they ultimately refer to something in the world that is not a statement. And that is what I (intuitively) mean by a statement having/not having content.

Try to resolve all references in a statement recursively. If one never encounters any reference to a non-statement, the statement is definitely contentless.

Will think more about this.

Just for fun: We have so much trouble with bivalued/trivalued logic. Imagine the Jains with their 7 valued “logic”: http://en.wikipedia.org/wiki/Eastern_epistemology

Krishnamurthy, on January 4, 2013 at 1:33 am said:Hi KM,

Thanks for the pointer to Jain epistemology; seems intimidating!

Although it seems intuitive to hold the correspondence theory, for this to make sense, one would require “the statements” in the “world” to have something special about them–different from other objects in the world. If not, one could add in the two statement version of the liar’s paradox to the world, and reach the paradox. But it is not clear what could be special about such statements; for correspondence theory, doesn’t “is true” mean “exists”?– and all objects in the world, by definition, exist.

Another issue with deferencing is as follows: consider the transfinite version of Yablo’s paradox:

S1 : for all k < 1, Sk is false

S(1/2): for all k< 0.5, Sk is false

…

S(1/n): for all k<1/n, Sk is false

…

S0: Some false statement.

Here all the deferencing eventually (after \omega-steps) leads to S0, an in-world false statement. Unless you object to transfinite steps, the deferencing does not solve the paradox. Let me know if I am mistaken.

Kris

K. M., on January 5, 2013 at 1:48 am said:Hi Kris,

Will try to think and write about the correspondence theory and statements later. For now, let me address the transfinite version:

Since there is a one-to-one correspondence between the sequences 1,2,3… and 1,1/2,1/3…, I find it difficult to accept that the second sequence terminates at zero. In other words, I fail to grasp the difference between the transfinite and the infinite. This is the first time I have read about transfinite numbers (I knew about the cardinal numbers and countability but had not encountered ordinal numbers before) so I might be missing something. As of now, I am inclined to think that the dereferencing never ends and so there is no paradox provided I am able to make the distinction between statements and other objects in the world.

Krishnamurthy, on January 6, 2013 at 8:22 pm said:Hi KM,

The difference between two sequence is that the order relation is not the same — the two sets with their order relations are not isomorphic. Another way to realize this would be as follows, if you do not want to talk about transfinite.

S0 == Some false statement.

Si == Statements Sk for k>i and S0 are false

Again, here, every finite sequence either terminates on some statement Si or terminates on S0. One can still object by saying that there is an infinite non-terminating deferencing sequence, although it seems to be a very technical objection.

Kris

Piracetam, on January 13, 2013 at 1:05 pm said:The third and fourth points relate to the problem of containment. To begin with, there is the problem of criteria. We finite human knowers are awash in a sea of mystery, and nowhere more so than in theology. Given this fact, what are the conditions that would warrant one in identifying a particular theological doctrine as truly contradictory as opposed to simply transcending our understanding? Without clear and unequivocal criteria, it is likely that the choice of true contradictions would be arbitrary, and raise the question of where to stop in identifying such contradictions. Without sufficient criteria, theology could soon become a truly trivial enterprise.