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Hi, ET - Sorry for the very delayed response !

Here are some suggestions of solving the "Liar paradox":
(These are the explanations of the solutions you suggested, ET; Of Russelle's and Tarskian's)

1. By Alfred Tarskian:
"It's raining" - the "object's language" is English.
"this sentence is of two words" - "Mata-language": A language which is above the ordinary language.

Tarskian says, that the "liar paradox" - a semantic paradox, is within the natural/ordinary language, which allows us to submit words like: "true" or "false" to it's own sentences.
This possibility, allows the natural language to contain contradictions.

The words "true", "false" - the talking about the meaning - all these are semantic elements, which their existence within the language resolves in paradoxes.

The "liar paradox" is using the language to say something about the truth or false of it's own sentences. By that, two levels of language - the "object's language" level and the "mata-language" level, are mixed.

When the liar says that he's saying a lie, he's using the "mata-language". At this point, or on this level, he can point out a sentence in the "object's language", and to say that it is false.
He's simultaneously, speaking both levels of language.

The writer* claims, that Tarskian's suggestion is not a solution. Also Tarskian himself, did not think he had provided a solution to the "liar paradox", within the natural language.

2. Many solutions refer to the phrase "Self-Reference".
Which means, that the "liar paradox" would have been prevented if it hadn't contained self-reference.
So, one must avoid self-reference.

The writer* claims again, that it is not a solution, since the particular paradox was phrased in different versions, which avoided self-reference, but still contained paradoxes.

3. Kripke's solution:
Why not accept that a statement can be either true, or false, or paradoxical? - But naturally, this solution contradicts pure-classic logic, which claims that there are only two possibilities:
"true" or "false".


Paradox: "The set of all sets that aren't members of themselves. Is it a member of itself?
Russell's solution (An explanation)

Russell claimed, that paradoxes are created when one is trapped in a "Vicious Circle". In order to avoid that, one must pay attention that when phrasing qualities, what characterizes a whole collection, isn't one of the collection itself.

Russell developed the "Theory Of Types", which divides all existences in the world to existences of type A, B etc.
Instead of "existences", we'll use the word "objects".
Objects of type A, B, C - and the world is divided to these types; The lowest type are the individuals in the world. The kind which is above it, are the qualities of these individuals and the kind which is above that, are the qualities of the qualities of these individuals and so on.

Each language has a type, and we must recognize the signs that represent objects of the first, second and third types.
We mustn't mix the types. Therefore, first we must see if the sentences are meaningful and only afterwards, weather they are true or not.

Each sign of the language has, has it's own field of meaning, that it has to fit into, so we can understand what we're saying; If someone said: "Aristotle is a benefit", than he/she didn't claim a false statement, but said something which was meaningless, because he/she didn't apply his/her sentence on to levels, types and fields of meaning.

It is possible to apply sets or a quality, only on what's one level beneath them.
This is the solution to "Is it a member of itself?" paradox.

* The writer is Anat Biletzki and the book is "Paradoxes".

Here are some books or essays regarding the subject, from the book's ("Paradoxes") bibliography:

Chihara Charles S., "The Semantic Paradoxes: A Diagnosis", Philosophical Review, 88, 1979.

Goodman N., Fact, Fiction and Forecast, Cambridge, Mass: Harvard University Press, 1955.

Grelling, K., "The logical paradoxes", Mind 45, 1936.

Hempel, C., Aspects of Scientific Explanations and Other Essays in the Philosophy of Science, New York: The Free press, 1945.

Martin Robert L., ed., Recent Essays on Truth and the liar paradox, Oxford: Oxford University Press, 1948.

Popper, K., "Self-reference and meaning in ordinary language", Mind 63, 1945.

Russell, B., "Mathematical logic as based on the theory of types", American Journal of Mathematics 30, 1908.

I hope this list helps you ET, or at least interests you.
The book I read was in Hebrew and I don't think it was translated to English, sorry ! - I also couldn't find any suggestions/solutions to the "Barber paradox".

I also enjoy logic very much. It's sometimes hard to understand, especially at first, but little by little it gets clearer and clearer... That's the beauty of it !
And needless to say, that I would have never thought of these solutions or even suggestions, myself.
But still, I believe that: Poetry and Philosophy build minds !

Floothe, I'll think of a Natalie-Pizza paradox. More likely, I'll look for one, which may be suitable.

"How to get to your pizza"

Natalie was in her room reading a book. for a moment, she
raised her eyes, took a look at her watch and realized she had been reading for hours. Suddenly, she felt hungry. So, she layed down her book, went towards the kitchen, and then noticed a delicious slice of pizza on the kitchen's table.

She felt a sudden urge to grasp that slice, but how could she get to the table? - Although there were only four steps which distanced her from her current fantasy, but according to Xenon's paradox of dichotomy, one could never get from one point to the other, since infinity was between them.

Natalie feared, she would never be able to indulge herself in this way; She would never get to her pizza, since she would have to pass half of the way but before that half of the half..."God!" ,she thought to herself, "Xenon! - that stupid one who didn't have anything better to do, rather than to search for paradoxes" - What would Natalie do?

But, clever Natalie wasn't about to give up ! - She went back to her room, searched for a few books, and finally came across her salvation - A solution to Xenon's Paradox of Dichotomy !
She read that paragraph:

Solution - The term 'Limit'
'Limit' is a mathematical term or place, which can describe the end of an infinitive series. For example:
We say that even though the series 1+1.25+1.5+...+2, is infinitive it's limit is 2. With defying this term, we can clarify how a person gets from one point to the other.

Joy filled Natalie's heart, when she realized she could finally get to her pizza - It was her "limit" !
She ran to the kitchen, grasped her pizza, but before she took her first bite, she had said to herself:
"What a great contribution, Philosophy and Mathematics are, to our lives !"

Bon Appetite, Natalie !

"The thinker without a paradox is like the lover without
feeling: a paltry mediocrity" - Kierkegaard.

That's SO true....

Hi, ET:
Or maybe, the barber is naturally bald !
This Wittgenstein guy, sounds interesting.

Hi, floothe:
How did Natalie get to her room? - She ran! - HaHaHaHa...

Glad you enjoyed the story, jrash567!