Do you agree with my solution of the Liar Paradox?

Question

The paradox is as follows. Consider the following sentence:

'THIS SENTENCE IS NOT TRUE'.

Is the above sentence true or false? It seems to be true if it is false, and vice versa. This is a long-standing philosophical problem.

My solution:

To begin with, there is nothing by which the truth-value of 'This sentence is not true' (let's call this statement P) can be ascertained. It is neither true nor false, hence it is not true. So it has an initial element of non-truth (N). Since P asserts that it is not true, its truth-value can now be judged with reference to this first element, and in this context it is true. So it has a second element, i.e. 'true' (T). Thereafter it has an infinite sequence of alternating elements N and T.

There is no contradiction here, since the elements are logically sequential, not simultaneous. They refer to different questions: element 1 answers the question 'Is P true at the initial stage of analysis?' whereas element 2 gives the answer to 'Is P true, given element 1?', and so on. The earlier elements remain correct even after the later ones have been added; indeed, the later ones depend on them.

P differs from most sentences in that it has a complex pattern of truth-values; but mere complexity does not constitute a paradox.

I would like an expert in logic to give me a detailed assessment of the above argument. Can anyone help here, or direct me to a website where I can pursue this matter?

Answer 1

I have a problem with 'logically sequential, not simultaneous'. Your argument assumes sequentiality; in the case of simultaneity there can be no 'initial' stage of analysis.

You are arguing linguistically, not logically.

It's just struck me that "This is a resolution to a paradox" is in itself paradoxical.

Answer 2

The problem with the Liar paradox is that it threatens the Logic law of 'bivalence' (sometimes called the Law of Excluded Middle) that says a statement can only ever be true or false.

Your solution breaks up the original statement into two separate elements: "this sentence" and "is not true". You then attach truth-values to both. The problem is that, in these two separate elements are not grammatically or logically coherent sentences and so cannot be assigned truth values at all.

It sounds sometimes as if you are offering one solution, which some philosophers adopt, that we have to have a three-tiered truth value: true, false and "paradoxical". The problem with this is that, as Russell tells us, we can just reconstruct a 'reinforced liar paradox:' as such:

"This sentence is either not true or paradoxical"

In this case, if it is 'paradoxical' it becomes true...

One solution is that we simply cannot assign truth values to self-referential sentences. This is too strict, however, since we often do, happily, self-refer e.g. "this sentence is typed". The most widely accepted solution to the paradox, which you sound very close to, is Tarski's. Like you he divides statements into elements. Tarski held that truth propositions (your second 'element') could ONLY be adequately defined for a language which is not 'semantically closed', that is, for one which does not contain either its own truth predicate or, crucially in your P example, the means to refer to its own expressions.

Instead we need to create a "meta-language" with which to do so which would follow this (known as his "T-Schema":

'P is true-in-L (the object language e.g. English) if and only if Q"

Your solution elegantly highlights how we need to consider how we predicate truth-values, and this is exactly what Tarski tells us.

Answer 3

I think that alot of paradoxes are paradoxes because of the language one uses. I think that language affects how someone thinks logically.

I aggree with your solution to this 'paradox', but i think its not much of a complicated paradox compared to ones such as "there is no absolute truth"

for this one i would add the word 'correct' in the equation.

you are only using "true" and "false"...1 or 0 etc

the statement "this sentence is not true" is correct, and if one assumes it is correct, it is therefore a lie.

would you say that "it is true that it is a lie" is hard to understand?  

Answer 4

What makes P a paradox is the fact that it does not have a consistent truth value.  And that is precisely what your explanation demonstrates, that the truth value for P is an infinite sequence of elements.

Answer 5

There are so many things that I have discovered simply by avoiding complexity. I seek ease and simplicity in everything, for I believe therein lies the health, wellbeing and joy of the mind, a thought, I believe should be as simple it is to be a solution worthy of living with.

No amount of sophistication in the logical argument can lead us to any final answer, and as in your own words an infinite sequence of alternating elements would keep presenting themselves are possible solution, each to be held true only for moment.

The sentence could have been: This sentence is not correct. Then we would be taking only about its grammatical form, to realise that what sentence means is not the same as what it actually is. The sentence is correct grammatically, but what it means to be is incorrect. The situation here is a little simply but there still is not one solution, i.e. we have to split the structure of the sense from its meaning, or form for its function. The question now is: does form follow function, or is it that form that suggests the function. In our common experience we realise that it is the forms are purposed by their functions, i.e. all things are build to perform some purposeful functions. The sentence in this sense is useless and its form does not hold true to its functions.

Then in case of your proposed sentence the matter is not of being it correct of incorrect, but of it being true or false, as in the cases of Boolean logic. The sense I would say is true by all means of definitions when beheld as a form functioned to perform a function of its own. The sentence is true in say what it means to say within the instance of its being.

This is obvious from the repetitiveness of the logical loop of the argument that once its has started it never ends but stay suspended in a persistently perpetual state of  yes and no. This proves that the sense has created a instance of its own, and within its own world, it holds true, beyond which it serves no comprehensive purpose. I seek simplicity as then I find it possible for me to operate within wider ranges of meanings and understanding of to the people, where this has been a little taxing on my mind.

Socrates: Whatever Plato is going to say next would be a lie.

Plato: Whatever Socrates has just said is true.

Answer 6

Resolving the "THIS SENTENCE IS FALSE" paradox

by saying "it's a dog chasing it's tail" is clever, but inaccurate

Because the TRUTH is there's no dog.

To EVALUATE we must have a VALUE.  

The sentence "THIS SENTENCE IS TRUE"  isn't true or false because it isn't about anything.  

Truth is an accurate description of some 'thing'.  But there is no 'thing' here.   The sentence's meaninglessness is apparent if you replace 'truth' with it's definition.

Saying:

"THIS SENTENCE IS AN ACCURATE DESCRIPTION OF THIS SENTENCE."  

Is neither accurate (true) or inaccurate (false) because nothing has been described.

Rephrase "This sentence is false" you get

"THIS SENTENCE IS AN INACCURATE DESCRIPTION OF THIS SENTENCE."  

OMG you say.  If it IS inaccurate that means it is accurate, but if it is accurate that means it is  inaccurate.   So which is it?

Neither.   Because there is no DESCRIPTION.

It's not a dog chasing it's tail, because there is no dog.

Answer 7

Another similar paradox is the sentence "I am a liar".

It is neither true nor false, as a person is a liar from a perspective (or an opinion).  I cannot tell you what your opinion of me is, and so the sentence is neither true nor false it is an opinion.

NOT TRUE = FALSE logically.  However, a sentence has two meanings in the way you describe it.  There is the word sentence within the sentence, and the actual sentence itself.  The word "sentence" is defined as NOT TRUE or FALSE within the sentence. The sentence itself is a definition, which holds no logical value.  

In other words, you are defining the word sentence with a logical value. There is no logical value attached to that definition, as it is whatever you say it is!

Answer 8

u need to get laid.