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How to solve the Liar Paradox 
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#1
Mar1112, 06:57 PM

P: 27

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 selfcontradiction, 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 truthvalue 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 "metalanguage" 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 selfreferential. How to prove him wrong? English is a semantically closed language so lets 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?) 


#2
Mar1912, 10:26 AM

P: 27

To see what is involved, lets 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 sentencefunction 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 cant derive the Liar Paradox! 


#3
Mar1912, 10:48 AM

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#4
Mar1912, 11:00 AM

P: 27

How to solve the Liar Paradox
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. 


#5
Mar1912, 11:40 AM

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#6
Mar1912, 12:30 PM

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P = not PIts 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)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) 


#7
Mar1912, 12:34 PM

P: 27

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! 


#8
Mar1912, 12:59 PM

P: 27

Related paradoxes can then be solved in the same manner. 


#9
Mar1912, 01:19 PM

P: 27

(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 cant 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 cant study the anatomy of paradoxes in a system that doesnt allow self referemce 


#10
Mar1912, 02:56 PM

P: 1,583

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? 


#11
Mar1912, 03:14 PM

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#12
Mar1912, 03:16 PM

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(note: a previous version of this post was deleted by me. The reply above was a response to that post)
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



#13
Mar1912, 03:41 PM

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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. 


#14
Mar1912, 03:53 PM

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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 PThis 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. 


#15
Mar1912, 04:00 PM

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#17
Mar1912, 04:09 PM

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#18
Mar1912, 04:30 PM

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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 wellfounded. As an analogy, your objection is of the same form as this complaint about algebra: If we know x = 4x, 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)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". 


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