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Logic

The Liar Lied

Neil Lefebvre and Melissa Schehlein give an intuitive solution to the famous Liar Paradox.

This Subtitle is False

This article is about a well-known paradox that dates back to ancient times, known as the Liar Paradox, or sometimes, Epimenides’ Paradox. It can take many forms, but one of the most common is the following sentence:

“This sentence is not true.”

The problem occurs when you try to determine if the sentence is true or not. If it is true, one reasons, then by its own claim, it must not be true, which is a contradiction. But if it is not true, then what it claims must not be true, so its claim that it is not true, is itself not true, making the sentence true. So, we reason, if it is not true, then it is true, but if it is true, then it is not true. This conclusion violates the most fundamental rules of logic, saying that if X is true, then X is not true! By applying rules of logic that seem reasonable to us, we arrive at an unacceptable conclusion, which is why it is called a paradox.

“My goodness”, you may think; “that is very profound. But is it important?”

Actually, this paradox has attracted the attention of great philosophers, and many writers have, even recently, written lengthy, complex, technical articles dealing with this problem, and with the associated problems of language and truth. You may wonder why experts would devote their time to such an exercise. The answer is that the Liar Paradox exposes what appears to be a flaw in our system of logic. If our system of logic is flawed, and allows us to draw absurd conclusions in this case, then it is possible that other conclusions we have drawn, may not be correct. Answers obtained using flawed logic may be incorrect. Our entire understanding of reason and truth may be at stake!

Past Attempts to Find a Solution

Philosophers have proposed solutions to the Liar Paradox, but in many cases the solutions have been shown to be susceptible to close variants of the same paradox. In other cases, the solution may in fact stand up to scrutiny, but requires that we make major changes to our understandings of language and truth. (An outline of the history of attempts to solve the Liar Paradox can be found in The Internet Encyclopedia of Philosophy.)

For example, the paradox used to be expressed as “This sentence is false”, and a solution was proposed that in fact this statement is neither true nor false, but that its truth-value lies in a gap between true and false, eliminating the paradox. Not only did this have the unfortunate consequence of having to deny that a statement that seems well-formed must be either true or false, but it is also not able to deal with the paradoxical sentence we are considering, “This sentence is not true”. If the sentence is not true or false, then it is not true. If we claim that it is not true, then since the sentence says that it is not true, we are claiming that it is not not true. Usually, a double negative cancels itself out, which would produce a paradox in this case, but some solutions have claimed that our word ‘not’ is inadequate. This type of solution has the problem that it claims that a word we have used consistently throughout history is not valid.

Another set of proposed solutions is based on Tarski’s theory of how truth should be expressed in languages. His theory claims that the predicate ‘true’ cannot be used to describe a statement that contains the predicate ‘true’. His system requires multiple levels of truth, so that, for example, one could rightly say that the sentence “The sky is blue” is true, but you could not say that the sentence “It is true that the sky is blue” is true. You would need a new truth-predicate, for example, super-true, and say something like “It is true that the sky is blue” is super-true. This would rule out the Liar Paradox, since it refers to its own truth-value. However, this does not agree with our intuitions about language. Surely it is fine to say that “It is true that the sky is blue” is true, or “The Pope’s claim that the Bible is true, is true”. We do not understand these uses of the word ‘true’ as being different predicates, so solutions based on Tarski’s theory are disappointing and unsatisfying.

Other methods claim that the Liar Paradox is somehow not a statement, because it violates some semantic rule, perhaps by using self-reference in an unacceptable way. However, clearly some sentences that refer to themselves are valid, such as:

“This sentence contains words.”

Also, anyone who denies that the Liar Paradox is a real statement will find himself pointing to the following sentence (based on an example in Paradoxes, by Mark Sainsbury):

“This sentence does not express a true statement.”

and saying “That sentence does not express a true statement.”

Look at the similarity between what the person is saying, and what he is saying it about. Perhaps some strict set of semantic rules could justify this, and there have been attempts at defining such a set of rules. But to be a good solution, the rules should have a justification. In addition, if possible, the rules would be consistent with our normal use of language. We long for a solution that explains and fixes the paradox, without forcing us to throw away our familiar concepts of language.

What Does It Really Mean?

The solution to this paradox requires that we first examine our understanding of what it means to make a statement. According to this theory, a statement consists of two parts:

1. An explicit meaning that all statements have, which is expressed as a sentence.
2. An implicit assertion that places the meaning in relation to what is, by declaring that it conforms with what is.

Surely, when any of us say something, we realize that we are implicitly asserting that what we said was true. It would be ridiculous for someone to make a unusual statement, then under cross-examination, declare that he was not asserting the truth of the statement but was asserting the falsity of the statement.

A sentence can have a meaning, which is an idea or ideas in the mind of anyone contemplating the sentence. This meaning has no comparison to reality, no truth-value, until someone actually places it in relation to reality, or what is. Until that happens, it has meaning, in the technical sense of the term used here, but it is not a statement about reality. The meaning becomes a statement when it is asserted, that is, when someone declares that the meaning of the sentence is a description of what is.

A statement can only be made as a combination of sentence and assertion. It is important to note that a sentence-assertion is not the same as adding “and this sentence is true” to the end of a sentence. Adding “and this sentence is true” to the end of a sentence is actually adding a second sentence to the sentence, rather than an assertion. If there is no assertion, one could still make the unexpected claim “I added ‘and this sentence is true’ because the sentence is false”. There has to be an original assertion to make the added phrase or the original phrase into a statement.

If this seems like a strange way to look at the sentence-statement relationship, we should realize that almost all sentences we encounter have been asserted. The sentences in this article are asserted by the authors, and spoken words are asserted by the speaker. It is hard to find an example of a sentence that is not asserted, but one example comes to mind. Imagine you are eating a bowl of alphabet soup, and by chance seven letters arrange themselves to spell “BAD SOUP”. This is a scary coincidence, but no one or no thing is actually making an assertion here, about the soup. The oddly-arranged letters merely form a sentence, without actually asserting that the soup is bad. So, it is possible for a sentence to have meaning, without being an assertion.

Intuitive Solution

The solution proposed here is to introduce the following rule for understanding sentences:

R: Any sentence whose meaning nullifies its own assertion cannot be asserted as a statement.

A meaning can nullify an assertion if it does either of the following:

1. Contradicts the assertion, that is, means that the sentence is false.
2. Means that the sentence can not be asserted.

This rule seems intuitively reasonable, and it doesn’t seem to lead to a further paradox. The rule says something about every individual sentence, and for it to be true, what it says about every sentence, including itself, must be true. About itself, one thing that R says is:

S: If this sentence nullifies its own assertion then this sentence cannot be asserted as a statement.

And therefore:

T: If this sentence means that this sentence is not true, then this sentence is not true.

This might seem problematic at first. After all, the second part of the statement threatens to say that T is not true, but only if T says that T is not true. Remember, our rule R holds if and only if T is a true statement. If we assume that S or T do in fact deny that they can make a claim, then they are making the claim that they cannot make a claim, which is a contradiction. However, if we assume that T does not mean that T is not true, then there is no paradox or contradiction. So, it seems consistent to say that R, S and T are true, and that none of them deny that they can form a statement.

So, rule R is based on the fact that when we make a statement, we are implicitly asserting its truth. If we have not asserted its truth, we have not made a statement. Similarly, if in attempting to assert its truth, we also deny its truth, then we have failed to assert its truth, which makes our sentence a non-statement.

Examples and Definitions

Consider the following example of a liar-type paradox, but with two sentences that refer to each other.

Mary: What Paul says is true.
Paul: What Mary says is not true.

If we try to determine the truth-values of these sentences, we are likely to become confused, particularly about what order they should be evaluated in. To resolve this confusion, we need to clarify the definitions of true, false, and the negation of true and false.

True: A property of any assertible meaning whose assertion conforms to what is.

False: A property of any assertible meaning whose assertion does not conform to what is.

A thing may be not true for many reasons. The thing may not be a meaning, or it may not be assertible, or, it may not conform to what is. It is not true if it does not possess the property true. Similarly, a thing may not be false because it is not a meaning, it is not assertible, or it conforms to what is. This use of ‘not’ agrees with how it is normally used in English. It also allows double negation, where ‘not not true’ is equivalent to ‘true’. A meaning that is false is not true, and a meaning that is true is not false, but a meaning can also be not true and not false, for example, if it is not assertible.

Using these definitions, we can resolve the problem of Paul and Mary’s sentences. Mary’s sentence means that Paul’s sentence can be asserted, and that its assertion conforms with what is. But if the assertion of Paul’s sentence conforms with what is, then it contradicts the assertion of Mary’s sentence. So Mary’s sentence cannot be asserted to form a statement.

The more cynical reader may point out that if Mary’s sentence means that Mary’s sentence is not true, then it in turn means that Paul’s sentence is not true, which would mean that Mary’s sentence is true. So, Mary’s sentence means that Mary’s sentence is not true, and it means that Mary’s sentence is true. It seems that we are faced with the same paradox, and that we have simply shifted the paradox from the realm of true and false to the realm of meaning. However, there is nothing paradoxical about a meaning that contradicts with itself. We have shown that a meaning can mean more than one thing, even its opposite, but that does not cast doubt on our rules of logic. It simply requires us to accept that meanings can contradict with themselves. If any of the possible meanings nullify its assertion, then the meaning is non-assertible, and cannot be compared to reality.

Paul’s sentence means that Mary’s sentence is not assertible or its assertion does not conform to what is. Another way of saying this is, if it is assertible, then its assertion does not conform with what is. If Paul’s sentence means the second term, that the assertion of Mary’s sentence does not conform with what is, then it is non-assertible. But, it isn’t clear which of the two phrases in Paul’s statement should apply here, to determine its truth-value. In order to evaluate this kind of case, we need an expanded system of logic that handles non-assertible meanings.

We can derive a system of logic that handles non-assertible meanings, if we take a closer look at the example of Mary and Paul. Basically, the problem is that Paul’s sentence means that either X or Y is true, where X stands for “Mary’s sentence is not assertible” and Y stands for “Mary’s sentence does not conform to what is”, which turns out to mean “Paul’s sentence is not true”, which is a non-assertible meaning. In true-or-false logic, a sentence that says ‘X or Y’ is considered true if either X or Y, or both, is true, and considered false only if both X and Y are false. In our logic, with true, false, and the third value, non-assertible, it seems reasonable to say that in cases of true and false, we will use the same rules. However, with non-assertible variables, we have a decision to make. If X is true and Y is non-assertible, is ‘X or Y’ true, or non-assertible? Consider the following example:

“Three is less than four, or this sentence is not true.”

It seems correct to say that the sentence is true, rather than non-assertible. Also, using that rule of logic in our three-valued system does not seem to lead to a paradox. We could come up with a whole logical system which took into account other operators, such as AND, IF-THEN, etc., in a similar way. So, knowing that Mary’s sentence from above is not assertible, if we apply our rule to Paul’s sentence, which means “Mary’s sentence is not assertible or this sentence is not true”, we can see that Paul’s sentence is true.

Finally, consider the following two sentences, which have identical meaning:

“If this sentence contradicts itself, then this sentence is not true.”

“This sentence does not contradict itself, or this sentence is not true.”

The first sentence expresses a standard rule of logic, that a contradiction is false. So, if our method is correct, the second sentence should be true. Since the sentence does not contradict itself, the first part is true, and clearly the second part is non-assertible. Using the rule we arrive at above, a true phrase OR a non-assertible phrase produces a true sentence. So in this case our rule produces the expected result.

Conclusion

The beautiful thing about this solution, is that the rule only rules out one type of sentence, that is, liar sentences. We do not have to worry that the Liar Paradox is symptomatic of some larger problem inherent in our language. As it should be, we have shown that the paradox of the liar sentence comes about only because it violates an intuitively obvious rule, a rule that can simply be ignored in all cases except for liar sentences.

Let’s take another look at the liar sentence that we’ve been considering in this article.

“This sentence is not true.”

What exactly does this theory say about the liar sentence? The sentence has a meaning, which is that the sentence is not true. However, it is incapable of being expressed as a statement, because when a person attempts to assert the sentence, it backfires on him, nullifying his attempted assertion. Since it is not assertible, the sentence can not properly be called either true or false. Admittedly, that means it is not true, but since it is a sentence that can not be given as a statement, stating that it is not true does not mean we are agreeing with what the sentence states, thereby making it true. The sentence does not state anything.

So, did the liar who originally spoke this sentence lie? It depends on a number of things. What is considered a lie? Is it a lie to say any sentence that is not true? We don’t think of someone who says a sentence that makes no sense as being a liar. Normally, we call someone a liar if they make a false statement. But the sentence under consideration can’t express a statement. But, what if the person who spoke that sentence also said:

“The sentence that I just spoke expressed a statement.”

This sentence would be a lie. Did the person who spoke the liar paradox also say the second sentence? That is unlikely. However, it is possible to make a statement without speaking words. Statements can be made with body language, for example, such as nodding one’s head. There are other ways to make a non-spoken statement, some of which are cultural. One example would be a waiter setting a plate of food before a person sitting at a table in a restaurant. The waiter might not say anything, but he is still very much making the statement “this food is for you, you may eat it”. Imagine if a waiter in a restaurant gave you a plate with poison on it. He couldn’t claim “I never said it was food, or that you should eat it”, because by handing it to you he made that very claim.

In the case of the person speaking the liar sentence, a strong argument can be made that when one speaks a sentence, at least in most contexts, one is making the non-spoken statement that the spoken sentence is in fact expressing a statement. Did the original speaker of the liar sentence make that non-verbal claim? His sentence has caused philosophers to spend thousands of years trying to figure out what he meant by it, so, I think we can make the assumption that it was indeed meant to express a statement. So, after all of this, we see that the liar did indeed lie.

Neil Lefebvre lives in Ottawa, Canada, where he studied electrical engineering and philosophy at Carleton University. He enjoys working with logic in the diverse fields of electronics, philosophy and software design.

Melissa Schehlein graduated from the University of Maryland-College Park and currently is an advertising executive.