A Solution to the Liar’s Paradox and the Difference Between Being and Meaning, Part 1
By Dennis Loo (9/27/13)
The Liar’s Paradox has bedeviled philosophers, mathematicians, and logicians for more than two millennia. In possibly the earliest form of the paradox, in Ancient Greece, Cretan Epimenides declared, “All Cretans are liars.” The paradox, of course, is that his statement that all Cretans are liars cannot be true if he is always lying (since he’s a Cretan).
The paradox is best known in the form of “This statement is false” in which the same paradox arises. If the statement is true, then it’s false and vice-versa.
I’m going to offer a solution to the Liar’s Paradox here, but first some background.
When I was a senior at Harvard my roommate David was taking a course with Professor Wilard Quine and David introduced me to this paradox as a result of the course. I did not get a full explanation from David of what Quine’s solution to the Liar’s Paradox was, but from what I can find on the web, Quine’s solution appears to be in the form of recognizing that language has different levels and for meaning to be asserted about a statement, that would have to come from a meta-level sentence. In other words, it would have to come from a step above the sentence/statement in question, not from the same level. Bertrand Russell resolved the paradox slightly differently by disallowing self-referential statements. As discussed by the Internet Library of Philosophy under Possible Solutions to the Liar’s Paradox:
1. The Liar Sentence L is meaningless, so the Liar Argument can’t even get started because its main assumption (that the Liar Sentence exists or is meaningful) is faulty. Natural language is incoherent, and its underlying sensible structure is that of an infinite hierarchy of levels. Because the Liar Sentence would have to reside on more than one level simultaneously if it were a legitimate sentence, it’s not really a legitimate sentence. This way out of the paradox is taken by [Bertrand] Russell in his ramified theory of types and, following [Alfred] Tarski, by Quine in his hierarchy of meta-languages. For Russell, the referential phrase “This sentence” in L is the culprit because the phrase is not allowed to refer to the sentence in which the phrase itself occurs. For Quine, instead, the culprit is the phrase “is false” in L because the phrase must be satisfied by sentences in a language lower in the hierarchy and not by the very sentence in which the phrase occurs.
I want propose a different solution to the Liar’s Paradox. It is consistent with Quine and Tarski’s approach, but differs from theirs in its specificity and in what I argue flows from this solution. I included this solution in my undergraduate honor's thesis at Harvard (decades ago), but because my thesis was in the Government Department, none of the Government graduate students who read my thesis were qualified to comment upon it.
I don’t think that the Liar’s Paradox is meaningless exactly. Instead it is making an illegitimate statement. Its’ illegitimacy as stated is not apparent until you recognize something basic about what’s going on here.
The statement “This statement is false” is not actually referring to the meaning of itself when it says “This statement” but in fact referring - and can only sensibly be referring - to the lines and dots – markings - that make up the sentence that our eyes and brains interpret as a meaningful English sentence: “This statement is false.” In other words, it is not impermissible for a sentence to refer to itself, but if it is doing so it can only refer to its constituent elements and not what the sentence proports to mean.
Put another way, if we showed “This statement is false” to someone who does not know any English at all and has never seen the English alphabet or any permutation of that alphabet and further this person does not know that this sentence is supposed to be a meaningful language statement, they could reasonably look at it as a set of dark lines, some curvy and some straight, and dots. In other words, they could see it as a non-representational, abstract bit of artwork. What the sentence is saying when it appears to be saying something about itself is in fact about the lines, curvy, straight, and dots that make up the physical existence of the statement, not a meaningful sentence about its own truth or falsity. This is what sentences that are not seemingly self-referential are doing when we say, for instance, “That table is red (in color)” or “He is wearing a cowboy hat.” They are referring to something else other than themselves and are not paradoxes for that reason.
What else could “This statement is false” be referring to other than to a level below the level of a meaningful English sentence? It’s not in fact a statement about the statement’s meaning. It’s a statement about the dark lines etc. that make up the sentence. In that instance, then, “This statement is false” is not a paradox but a sentence that provokes trouble because of a misunderstanding of what it is actually saying. It shouldn’t actually be properly understood as referring to its own truth or falsity. It’s not grammatically or syntactically impermissible to say “This statement is false.” But the problem arises from the fact that people who read this statement misunderstand what “this statement” is referring to. They take it to mean that “this statement” is about itself on the level of its claim to being false. But those lines and dots aren’t true or false. They just are. They just exist. They don’t have any meaning. Thus, strictly speaking, “This statement is false” is wrong as a statement in terms of its content. You can’t say that a set of lines and dots on a page prior to and independent of interpreting what those lines and dots “mean” is true or false. It just is. Thus, a proper statement that can be read straightforwardly and that avoids the Liar’s Paradox would be something like “This statement is printed on this paper and is made up of lines and dots.” You cannot make a statement about the truth or falsity of a statement within the statement itself.
“Cretans always lie” is impermissible not because you can’t utter the words, but impermissible because it’s false on its face. No individual or group, Cretan or otherwise, could possibly always lie.
This illustrates another fundamental point: meaning isn’t something that exists within a system but only as something that is imposed upon that system from above that system. Meaning is a meta-level judgment. Systems – or levels of matter in motion – don’t have any meanings inherent in them. They only have meanings, or purposes, or goals, or what have you, if you are outside of that system or level and are imposing some meaning for the events inside of that system. What is, is. That which is can only be said to have “meaning” from a different level above that which “is.” Being, in other words, is different from Meaning.
Life is. It doesn’t have any purpose beyond itself. The Universe just is, and in fact, the Universe is EVERYTHING THERE IS. There is nothing outside of the Universe. Therefore the Universe has no meaning because for it to have meaning means that there has to be something outside of and above it which is impossible because the Universe is ALL THAT THERE IS.
Try to imagine something outside of an infinite universe. Imagine that space and stars extend outward continuously without end. The only way that you can imagine something outside of something infinite is to make that which is infinite finite with boundaries around it. Then you can imagine a deity or something else outside of the universe, but you can only do so by turning what is in fact infinite into something finite.
P.S. Since my field of training is not in Logic I would be happy to hear from anyone who can poke holes in what I have set forth here, or comment in any other ways. I plan to run this argument by people in Philosophy and Logic in academia and will report back in some form after those inquiries.
End of Part 1
1 In fact, my honors thesis was so unorthodox as a Government thesis (at Harvard Political Science is called Government) that the two readers assigned my thesis were at polar opposites in their grading of it and a third reader was brought in. This third reader confided to me that the low-ball grader told him that he purposely graded me low so as to compensate for what he expected to be a high grade from the other reader. His gambit, of course, was unethical, because he was trying to overcome the other grader’s evaluation and dictate by himself what my score should be. The third reader, fortunately, agreed with the high grader that my thesis was “brilliant” but unfortunately the Government Department decided to average all three scores instead of throwing out the outlier. If the reason you bring in another reader is because you’re trying to decide who is the outlier, then you don’t then equally weight all three scores, you throw out the outlier. When I decided to go to graduate school later on I was delighted to discover that there was a discipline - Sociology - that accomodated rather than frowned upon - as my Government Department did - the wide-ranging inquiries and approaches that fuel my intellectual curiosity.
2 Simiarly, no one who claims to be a relativist and who says that there is no such thing as objective truth can be taken seriously as a thinker because they are contradicting themselves both in their statements and in the way that they live their lives. Avowed relativists such as postmodernists who claim that there is no objective reality and that there are only differing perspectives, none of which are any better than any other, do not actually live their lives in that manner. They behave in the real world as if there is a real world because otherwise they’d immediately run into problems. They act as if the place that they parked their car in is still going to be there when they leave work and that their place of work isn’t “wherever they want it to be” but a stable location that has an objective existence independent of anyone’s subjective notions. So too the statement, for example, that relativists make that “there is no truth” is self-contradictory because if there is no truth then the statement that there is no truth is also not true.