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I'm Tackling the Liar's Paradox

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A skeptic made the following observation:

Some famous dialetheists have argued that "This sentence is false" is both true and false. To get around this example, you need to go the lengths of providing a semantics which avoids this result.

I'm considering a response as follows:

When we are born, we do not have language. We need to create words to think and communicate, so over the course of our lives we make many words to represent the things around us, either by naming individual objects or by creating abstractions that subsume large numbers of instances. We combine words for the same reason- to represent the world for the purpose either of thought or communication. A subset of these combinations of words are sentences. Therefore we see that the concept of a sentence is formed in the context of man's need to convey information about the world around him. In saying a sentence, you affirm that the content of the sentence represents the world. To create a sentence that explicitly does NOT represent the world is to drop the context in which the concept of "sentence" is formed. "This sentence is false" is not false nor true, simply because it is not a sentence.

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"This sentence is false" certainly is a sentence, it just represents an invalid concept. It's no different than if I were to ask you to "please draw me a square circle." The concepts these sentences represent both violate the Law of Excluded Middle, which states that a thing either posses a given attribute or it does not posses it. A sentence cannot posses the attribute of truthfulness and untruthfulness at the same time, just as a 2D shape cannot posses the attribute of circularity and non-circularity at the same time.

According to Definr, a sentence is:

a string of words satisfying the grammatical rules of a language;

I think the answer is not that "this sentence is false" is true or false, or that it is not a sentence. It is that "This sentence is false" is just a sentence, without any truth value. The concept it represents is invalid, so its affirmation in reality cannot be validated.

Edit: Clarified some language.

Edited by cilphex
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"This sentence is false" certainly is a sentence, it just represents an invalid concept.

I disagree. ^Y is correct, it is not a sentence (despite what dictionaries etc may say). It is instead a mere word salad, as someone else once put it. Grammatical rules exist as means to expressing a thought about the world. They are not ends in themselves. Unless there is a bona fide attempt to describe the world - and the word-salad in question most certainly is no such thing - then grammatical structure is irrelevant as it is not being put to its proper purpose.

It's no different than if I were to ask you to "please draw me a square circle."

Again, incorrect, in that they are wrong partly for different reasons - but also correct, in that they also exhibit instances of the same error (yet not the one you claim). In the liar paradox, no grammatical unit is an invalid concept. In your example, the grammatical unit "square circle" is not a concept at all but is instead itself just another word salad tossed up in perverted glee, just as the liar paradox is.

The concepts these sentences represent both violate the Law of Excluded Middle, which states that a thing either posses a given attribute or it does not posses it.

Which again presupposes a genuine attempt to describe the world, as is the case for all principles of logic. The liar paradox is not a sentence, and hence the LEM does not even apply.

JJM

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Again, incorrect, in that they are wrong partly for different reasons - but also correct, in that they also exhibit instances of the same error (yet not the one you claim). In the liar paradox, no grammatical unit is an invalid concept. In your example, the grammatical unit "square circle" is not a concept at all but is instead itself just another word salad tossed up in perverted glee, just as the liar paradox is.

It is neither gleeful nor perverted. It is a caution concerning self-reference.

Here is a non-gleeful non-perverted version of the same paradox (the Russell-Frege Paradox).

There are sets which contain themselves (as elements) and there are sets which do not contain themselves as elements. For example the set of all abstractions is an abstraction. Therefore the set of all abstractions contains itself as an element. On the other hand the set of all little boys is not a little boy so this set does not contain itself as an element.

The notion of a set containing or not containing itself as an element is therefore meaningful.

Now (pay attention!) consider the set of all sets that are not elements of themselves. Is this set an element of itself or not?

Either way leads to a contradiction. It is a genuine paradox. It has been dealt with by modifying set theory in such a way that collections of the above sort are NOT sets, so the paradox is prevented. See any text on formal set theory for the details.

A variant of this paradox showed up 2300 years ago in the writing of Plato and Aristotle. It is called the Third Man problem. It too can be dealt with.

The moral of the story is that self reference can be tricky (although not necessarily paradoxical). The Go"del Incompleteness Theorem is a variant of the "liar paradox" that is not a paradox at all.

Bob Kolker

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What you need here is the concept of broken units. "This sentence is false" is a broken unit of the concept "sentence" : it is something that "tries" to be a sentence, and does have the formal appearance of a sentence, but lacks something that a sentence needs in order to function as a sentence: it does not actually say anything about reality; in other words, it has no meaning. In this respect, it is not essentially different from a random, incoherent string of words that starts with a capital letter and ends with a period: it's a failed attempt (or pretense) at communicating a fact about existence.

BTW, ctrl y, I hope you don't mind if I ask what your name stands for. Does it mean "Yank" ? :thumbsup:

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A skeptic made the following observation:

I'm considering a response as follows:

When we are born, we do not have language. We need to create words to think and communicate, so over the course of our lives we make many words to represent the things around us, either by naming individual objects or by creating abstractions that subsume large numbers of instances. We combine words for the same reason- to represent the world for the purpose either of thought or communication. A subset of these combinations of words are sentences. Therefore we see that the concept of a sentence is formed in the context of man's need to convey information about the world around him. In saying a sentence, you affirm that the content of the sentence represents the world. To create a sentence that explicitly does NOT represent the world is to drop the context in which the concept of "sentence" is formed. "This sentence is false" is not false nor true, simply because it is not a sentence.

See also

http://en.wikipedia.org/wiki/Liar_Paradox

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

The Quine Variant avoids blatant self-reference but still produces a paradox.

The problem is not with the world, but with languages that march "a bridge too far" and permit self reference. Self reference either blatant or subtle (as with Quine's Paradox) presents a problem and clever methods of avoidance must be formulated. If you think all this is a lot of trickiness, then consider Graham Priest's approach; Dialethism a kind of paraconsistent logic which tolerates blatant contradictions. Fortunately, such systems inhibit inference strongly to prevent ex falsi quodlibet. Caution! Studying such systems can cause one's head to explode. Other than being a formal curiosity I cannot see any use for them.

Bob Kolker

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I disagree. ^Y is correct, it is not a sentence (despite what dictionaries etc may say). It is instead a mere word salad, as someone else once put it. Grammatical rules exist as means to expressing a thought about the world.
Okay, I have to disagree, professionally. The sentence is a sentence. It is generated by the grammar of English, it follows the syntactic rules of English sentence form, and that is what is required for a string of words to be "a sentence". Similarly, "Colorless green ideas sleep furiously" is a setence, and "Furiously sleep ideas green colorless" is not. Being a well-formed sentence is not the same as expressing a true proposition. Grammatical rules exists as a means of distinguishing word-strings generated by the grammar from those not generated by the grammar (thus not part of English), and also play a role in regulating certain meaning relations.

The cognitive mechanisms regulating the purpose "express a thought about the world" vastly exceed the bounds of grammar, indeed, they cover all of cognition. It includes not just grammar (sentence structure), but semantics (concepts and their heirarchical relationship within a sentence -- why "Tom sees Bill" is not the same as "Bill sees Tom"), conventionalized implicature (the reason why "There are biscuits on the table if you want some" is not a ridiculous thing to say) and all aspects of your real-world knowledge. The problem is not that this isn't a sentence, the problem with this sentence is, simply, that it does not assert a proposition with a truth value. "Being a sentence" does not mean the same thing as "being an attempt to communicate a true proposition".

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cliphex,

The concepts these sentences represent both violate the Law of Excluded Middle, which states that a thing either posses a given attribute or it does not posses it. A sentence cannot posses the attribute of truthfulness and untruthfulness at the same time, just as a 2D shape cannot posses the attribute of circularity and non-circularity at the same time.

Appeal to the law of the excluded middle is begging the question against the dialetheist, whose thesis is precisely that a proposition can be both true and false - that contradictions exist. The liar paradox is used by dialetheists as a consideration to motivate their position, as a purported example of a true contradiction.

ctrl y,

In saying a sentence, you affirm that the content of the sentence represents the world. To create a sentence that explicitly does NOT represent the world is to drop the context in which the concept of "sentence" is formed. "This sentence is false" is not false nor true, simply because it is not a sentence.

That's not an adequate answer to the dialethist. First, DavidOdden is right - it really is a sentence of English. Second, the dialethist will argue that the sentence does, in fact, represent the world. Under their view, the proposition that "this sentence is false" really is false - because it is both true and false. Third, saying that utterance of a sentence is affirmation of a representation of the world isn't as uncomplicated as you make it sound. You need to make sense of sentences about fictional beings, like "Howard Roark laughed", as well as lies, sentences which the speaker does not intend to represent the world. Not to say that those things can't be done - folks make a living off these kinds of problems - but you can't just go around making expansive claims about the nature of language without negotiating past these sorts of issues.

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Loath as I am to quote Chomsky in my first post to this forum, his example of a perfectly grammatical but meaningless sentence, "Colorless green ideas sleep furiously" is worth mentioning. A sentence only requires correct syntax. There is also a difference is the form of meaninglessness in his example and "This sentence is false", his meaningless, the other perhaps invalid, but I'm not quite sure of how to term it.

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I'm going to present another perspective, from that of a programmer.

First, I will establish the necessary functions to process this (in very simply terms)

function grammatically_correct(sentence)
{
return true if "sentence" is grammatically correct.
return false if "sentence" is not ungrammatically correct.
}
function evaluate_truthiness(proposition)
{
extract the subject and predicate from "proposition".
call the predicate's associated function passing the subject as the parameter
}
[/codebox]

I will show that sentences presented "This sentence is false" and "This sentence is true" are both infinitely recursive (self-referencing).

But before that, let's consider the sentence "This sentence is a properly formed sentence," which is also recursive (self-referencing), but not infinitely so, as I will show.

So, we'll make the call

evaluate_truthiness("This sentence is a properly formed sentence");

When we do this, inside evaluate_truthiness will perform the following tasks:

[codebox]extract the subject and predicate as "This sentence" and "is a properly formed sentence."
The subject "This sentence" is a reference to the whole sentence "This sentence is a properly formed sentence".
"is a properly formed sentence" is long-hand for the function "grammatically_correct", so we make the call:
grammatically_correct("This sentence is grammatically correct"), which returns true, because the sentence is grammatically correct.
evaluate_truthiness returns true.

When we try to process the original sentence: "This sentence is false" this is what happens:


evaluate_truthiness("This sentence is false");

which invokes the following inside evalute_truthiness:

extract subject and predicate as "This sentence" and "is false."
The subject "This sentence" is a reference to the whole sentence "This sentence is false".
"is false" is the negation of the reference to the function evaluate_truthiness, so we make the call:
evaluate_truthiness("This sentence is false"), which brings us exactly back to the first step, where we loop forever[/codebox]

Ultimately, this proposition can be neither true nor false. "This sentence is false" will forever try to evaluate itself. It's unanswerable, except in saying that by that standard, it is not a proper proposition, as it has neither a true nor false evaluation.

(I'm no linguist, I'm a programmer, and these are the terms that make the most sense to me)

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