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The Liar's Paradox

A Most Merry and Illustrated Celebration

 

Paul and Epimenides

Suppose you are taking a midday respite from your onerous labors of sitting on your תחת at your desk and surfing the net while your boss is out of town. So you and a bunch of your jolly comrades begin the lunchtime conversation with weighty matters of the day like who will win the pennant, why the BCS system stinks, and how long will it be until the Super Bowl finally gets pushed to being played in April. Then, apropos of nothing, the office egghead tosses out a puzzle.

Is he lying, he asks, when he says "I am lying"?

Of course, the reasoning is supposed to go as follows. If you are lying when you say "I am lying", then you are lying about your lying. But if you lie about your lying, then you must be telling the truth. But if you are telling the truth when you say you are lying, then you must be lying.

So if you're lying, you're telling the truth, and if you're telling the truth, you're lying.

At this point, the poser of the problem (that's poser of the problem, not poseur) will expect you to fall down in astonished genuflection. What wit! What ingenuity! What intelligence!

What's really astonishing is that anyone can believe it is the height of sophisticated wit to pose a problem and not give an answer, particularly an answer that can be deduced with skills that are taught in your average middle school. Of course as recent events have shown, at least in the United States our political system requires its leaders to show lack of reasoning skills that are taught in your average middle school.

An average middle school? Hell, it requires lack of the reasoning skills learned by your average Entamoeba histolytica - although this comparison might be a bit of an insult to the Entamoeba. So it is with a sense of patriotic obligation that CooperToons - in its declared purpose to fight ignorance and superstition - offers what is perhaps the simplest explanation of the famous Liar's Paradox.

So if you are a true patriot who honors freedom, democracy, and the 'Merkin Way (and even if you're not a 'Merkin) - read on!

The Problem(s)

The Liar's Paradox has a venerable history which is strong in tradition but lean on facts. So for now we won't worry about rehearsing the various "true" original tellings which are complete balderdash.

On the other hand, we do know that both Aristotle and Cicero knew of the paradox. They wrote as if it was well known to the reader. So it goes back well in to the previous two millenia. But the most famous version is derived from the New Testament. This is from the letter of Paul to Titus where in Chapter 1, Verse 12 Paul states:

"One of themselves, a prophet of their own, said, 'Cretans are always liars, evil beasts, lazy gluttons.' This testimony is true."

In other words, a Cretan tells us that Cretans are always liars. And given our source, we cannot doubt this testimony is true.

So what can you conclude when a Cretan says, "All Cretans are liars"? Or equivalently, what do we conclude when anyone says "I am lying"? Is he lying? As the reasoning above showed, if he is, he isn't, and if he isn't, he is.

At this point, you might go into a spittle flinging diatribe and ask just what's the bloody point? Why worry about puerile tricks with language that have absolutely no import in our lives? Surely we have better things to do.

Weeeelllllll, maybe not. Developments in mathematics in the twentieth century have shown how crucial it is to find what lies behind the way paradoxes keep cropping up. For one thing, the study of paradoxes has led to the notion of undecideability and incompleteness which in turn have applications in finding which problems can and cannot be solved by computable systems. So there really is a point to such endeavors.

All right. What is a paradox? Well, paradoxes are statements that make it appear you can derive false conclusions from true statements. Why we don't want paradoxes is that if you can derive only one falsehood from one true statement, then any statement is both true and false (the proof of this statement is, like the mathematics books say, left for the reader). So unless we can demonstrate we have erred in deriving the Liar's Paradox, then you can not only believe anything you want (which is easy), but you can prove anything you want (which is supposed to be impossible). You can prove Abraham Lincoln was a vampire hunter, that George Washington was a British spy in cahoots with Eamon de Valera during the Irish Rebellion of 1916, and you can even, should you wish, prove God is a Penguin.

So we better find a resolution, right away!

Resolution(s)

Of course, if people have been talking about the Liar's Paradox for thousands of years, you know many explanations have been given. Actually, most of them are correct - or at least not entirely wrong. One of the simplest explanations is just to point out that any statement implicitly asserts that it is true. After all, saying "The Cardinals will win the pennant" is exactly the same as saying "It is true that the Cardinals will win the pennant." So then any statement that asserts its own falsehood is by definition a contradiction. So it's not anything you have to worry about.

Case closed, ¿verdad?

Well not quite. Although not strictly speaking incorrect, this "dissmissive" explanation doesn't satisfy many people - certainly not philosophy professors who like take up pages if not volumes of "learnéd" academic journals with articles so replete with obscure symbols that you can't tell if it's an explanation of the Liar's Paradox or whether you accidentally opened a JPEG file with a text editor. So we better not declare victory just yet.

Not My Type

Probably one of the most popular methods to resolve the Liar's Paradox is to point out that there are sentences that appear to make sense, but are really meaningless. The Liar's paradox is, then, one such sentence, so we don't need to worry, do we?

Well, eins mehr (as Schopenhauer might have said) this explanation requires you to find some way to clearly identify a meaningless sentence even though, as we said, it looks like it does mean something. This approach was first tried in the early 20th century by Bertrand Russell, in his Theory of Types.

What Bertie did was rank statements into classes or hierarchies. For instance, you had statements about ordinary things. Then you had statements about statements about ordinary things. Then you have statements about statements about statements about ordinary things. And so on. It was then by incorrectly mixing classes of sentences you end up with paradoxes

Bertie's approach wasn't universally accepted. In fact, he wasn't too sure about it himself. It appears to be just too much of an ad-hoc adjustment. You would think fixing a simple paradox shouldn't take adding never ending and infinite layers of complexity.

The Theory of Types, though, was one of the first ways to avoid confusing what a statement asserts within the system which created it with talking about the statement outside of the system. In other words, saying "It is true that S" where S is a statement is saying something about the statement, S. It is not the same as making the statement, S, which asserts its truthfulness within its system. So the Theory of Types says the paradoxes arise by improperly equating a statement that denies the truth of a statement with the negation of the statement itself. But as we said for many people this is not a very satisfying explanation. It certainly doesn't show exactly how we can create the paradox.

Watching Your Language

So what we'd like to do is attack the paradox using the simplest possible methods. That is, we will transform our ordinary speech into an unambiguous expression that will reveal the error - if indeed error there is (Who knows? Maybe George did join Eamonn at Boland's Mill.) Admittedly our approach does require some math background, but it should not tax anyone who has successfully completed our middle school curriculum. But we will still proceed carefully for those for whom middle school is a few years back.

First, we must ask just what do we really mean when we say "I am lying." Or to take a different tack, consider what would happen if you go up to Joe Blow on the street and say, "I am lying".

Rather than express astonishment about you posing an amazing philosophical problem that has taxed the greatest minds in history, Mr. Blow will simply respond, "Lying about what?".

That's because our man (or woman) on the street will instinctively recognize what the statement "I am lying" is. In ordinary language "I am lying" is an abbreviation of the statement "I am lying about something or other". So if you simply say "I am lying" without reference to what you are lying about - either stating it explicitly or having understood by context - you are actually misusing natural language. Making the distinction between proper use and misuse of natural language was something the 20th century philosopher Ludwig Wittgenstein recognized as being important in resolving logical contradictions and paradoxes

A Little Formality, Please!

But we can go even further. Note that the sentence "I am lying" is actually an abbreviated specific instance of a more general statement. That is, "I am lying" is derived from the general expression "Someone is lying about something."

This is where we resort to proper reasoning methods where statements can be proven to be true or false with the certainty of mathematics. To do so we turn to what is called formal logic. Formal logic does not mean you discuss philosophy dressed up in tuxedos or evening dresses. No, it means we must represent "Someone is lying about something" as a formula. Lest the mathematically challenged panic, this is actually quite simple to do.

First we substitute the words "someone" and "something" with algebraic variables, x and y, like we did in middle school math. So our sentence

"Someone is lying about something"

then becomes

"x is lying about y."

This is turn can be even more succinctly represented by the formula

L(x, y)

All right. We now see that the sentence we wrote as "I am lying" is from a two-variable relation. This relation requires a variable which represents the liar, x, and the other variable which is a statement which is lied about, y. For the formula, L(x, y), to have any meaning, you need both variables to be assigned values. For example, let's have x = "I" and y = "I stayed late at the office." Then the formula becomes:

L("I", "I stayed late at the office")

This is translated as "I lied about 'I stayed late at the office.'" Or in ordinary English, "I lied about staying late at the office". Which we don't recommend.

It is important at this stage to precisely define the domains of our variables, x and y. From how we defined L(x, y), we can see that x must be an element taken from the set of all beings capable of lying. The variable, y, on the other hand, must be drawn from the set of all statements which can be lied about. In other words, the variable y must be taken from statements that can unambiguously be assigned values of TRUE or FALSE.

It's also important to realize that not all sentences belong in the latter domain. That is, not all sentences are statements with a value of TRUE or FALSE. Questions, for instance, clearly have no TRUE or FALSE value. "How are you?" "What time is it?" "Are you lying?" None of these sentences are true or false. As long as we keep to our proper domains, our sentence L(x, y) can also be unambiguously assigned a value of TRUE or FALSE.

An Open Policy

But there is one type of sentence in logic - that is, a type of formula - that does not have a value of TRUE or FALSE. These are in fact the formulas themselves with unassigned variables. These are called open formulas as contrasted to closed formulas where the variables have been given actual assignments.

In other words, our basic formula, L(x, y), is an open formula. But L("I", "I stayed late at the office") is closed. It is only the latter formula that represents an actual statement which has a value of TRUE or FALSE. L(x, y), though, has no TRUE or FALSE value. To say an open formula is true or false is meaningless or nonsense.

[For the philosophical hotshots, we will admit we are ignoring certain types of formulas which are true or false without explicitly assigning variables. For instance x = x is true, regardless of what x is. x > x is always false. But we don't need to worry about these formulas here.]

Now a formula is open as long as at least one variable is unassigned. For instance, let's write

L("I", y)

This is still an open formula even though the variable x has been assigned a value. But since y is still unassigned L("I", y) is still an open formula and you cannot assign it a value of TRUE or FALSE. And it is this last formula that leads us to the actual resolution of the Liar's Paradox

The Formal Liar's Paradox

All right. How do you formally write "I am lying" as the Liar's Paradox? Well, we have to take the formula, L(x, y), and assign the values of x and y. The first is easy, x = "I". But remember, in the Liar's Paradox, y must refer to the statement itself. In other words we have to figure a way to make y = "I am lying".

But remember "I am lying" is an abbreviation of "I am lying about [something]" which is written L("I", y). So in the precise, proper, and formal language, the best we can do to is to write the Liar's Paradox as

L("I", L("I", y))

Hah! But look what we have. For one thing we've drawn y from the wrong domain since L("I", y) can't be assigned a value of TRUE or FALSE. Secondly, we see that when "I am lying" is written as the Liar's paradox - that is, if it refers to itself - is actually a somewhat obfuscated natural language version of an open formula. Therefore you cannot legitimately assign the entire expression a value of TRUE or FALSE!

But that - and pardon us if we shout - IS EXACTLY WHAT THE LIAR'S PARADOX DOES! Remember that Mr. Smartypants at the beginning of this essay asked whether "I am lying" is TRUE or FALSE. He didn't ask if it is TRUE or FALSE or MEANINGLESS. What we can show now is that by forcing us to treat a statement that is neither TRUE or FALSE as one or the other, we arrive at the Liar's Paradox.

That is not logical, Captain

At this point we need to delve into a bit of the algebra of logic. Although - lamentably - this is a topic that is not taught in middle school as often as it should be. Happily the algebra of logic is far simpler than the algebra you do get in middle school, and it's easy enough to learn. So rather than go into an in-depth introduction, we'll just give the necessary symbols in a table and amplify their meaning as we go along.

SymbolMeaningOperation
¬NOT
"It is false that"
¬T = F
¬F = T
¬¬T = T
¬¬F = F
ANDT∧T = T
F∧T = F
T∧F = F
F∧F = F
ORT∨T = T
F∨T = T
T∨F = T
F∨F = F
IF-THENP→Q = ¬P∨Q

(For those who have to have it spelled out, T means TRUE and F means FALSE.)

We will now simplify things a bit. Instead of writing

L("I", L("I", y))

we will write

L

The point to remember is that L - since it is the same as L("I", L("I", y)) - is an open formula.

Now, as the $500 an hour psychiatrist said after two months of twice weekly 3 hours sessions, we can begin, yes?

OK. Suppose we want to say that L cannot be assigned a value of TRUE. After all, if you can't assign it a value of TRUE or FALSE, you certainly can't assign it a value of TRUE. So it's quite legitimate to state:

It is false that L is TRUE.

But note that since L is an open formula you can't assign a value of FALSE to it either. So we can also say:

It is false that L is FALSE.

Now note it is not incorrect to use open formulas with the logical operators. It's just incorrect to say the resulting formula is true or false. So we can put our two statements together as an AND statement.

It is false that L is TRUE AND it is false that L is FALSE.

Well, then, how would you write our last statement using our symbolic logic? Well, you'll be tempted to simply write:

It is TRUE that L.

is the same thing as simply saying

L.

Similarly, if we say

It is FALSE that L.

... we will say this is

¬L.

That means we can write:

It is false that L is TRUE AND it is false that L is FALSE.

simply as

¬L ∧ ¬ ¬L

At this point we need to point out that the statements in English "It is true that L" or "It is false that L" are statements about the statement, L. And yet the statements L, ¬L, and ¬L ∧ ¬ ¬L are statements within the symbolic language. We need to keep in mind we are equating statements as being equivalent statements about the statements.

Making the Big One

At this point our derivation of this last formula is only valid if L is meaningless and cannot be assigned a value of either TRUE nor FALSE. Only with meaningless statements can we legitimately say a sentence is not true and also that it is not false. But remember, just because a meaningless statement is not true doesn't mean you can say it is false. Similarly even though we can say a meaningless statement is not false, that doesn't mean it is true.

But note what happens when you pose the Liar's Paradox. We are asked if L is TRUE or if it is FALSE. We don't have the option for giving a NEITHER category which is actually the correct answer. Instead we forge ahead in our reasoning and use the formal transformations in our table - which we must emphasize are really only valid for statements that are only TRUE or FALSE.

And what happens if we do follow the rules in the table? Well, we start off with:

¬L ∧ ¬ ¬L

But since in the table, the two NOT's (¬¬) cancel each other, this formula becomes:

¬LL

ERROR! ERROR!

Now in the formal logic of statements - where sentences are only TRUE or FALSE - the expression ¬LL is a contradiction. A contradiction is always FALSE. After all, if you saw "I am Socrates and I am not Socrates", this can never be true.

So what we've done is show that if you assume the rules for TRUE or FALSE statements can also be used with meaningless statments - such as open formulas - then you can (incorrectly) "prove" a statement that is always false in normal logic is TRUE! This is the reason for the Liar's Paradox. But our contradiction, ¬LL, is not quite the Liar's Paradox. At least not just yet.

The Irritating If-Then Logic

To proceed we must now resort to using the last line of the table, which is one of the most irritating logical transformations ever invented. That's the material implication statement which uses the symbol, →. As the table shows this is just the IF-THEN statement used to conclude something from another statement. (A full discussion of why material implication is irritating is something we need not go into here. Fortunately, the transformation, though irritating, is correct.)

The most famous (and boring) example of the IF-THEN reasoning is "IF Socrates if a man AND all men are mortal, THEN Socrates is mortal." A trivial conclusion, yes, but that's the way it goes. It is, after all, the IF-THEN statements that allow us to reason properly and draw correct conclusions.

A trivial conclusion

But note that the table says that an IF-THEN statement is the same as a NOT-OR statement. That seems a bit strange, but a little reflection shows the two operations are equivalent.

Suppose I say "IF I go to town THEN I will pick up a loaf of bread". Then consider what I mean if I say instead, "I will NOT go to town OR I will pick up a loaf of bread". What can we conclude from the two sentences?

Now in either statement, the IF-THEN or the NOT-OR, I have the option of not going to town. I also have the option of picking up a loaf of bread somewhere else (say at a store in a non-incorporated commercial center like Cookietown, Oklahoma). But with either case IF I do go to town, THEN I better pick up a loaf of bread, that is if I want to tell the truth. If I DO go to town and DON'T pick up a loaf of bread, then I lied. Since both the IF-THEN and the NOT-OR statements are true or false under the exact same circumstances, they are really the same statement.

Liar! Liar!

Now we can proceed step-by-step to derive the Liar's paradox. This requires us to use the logical transformations as defined in the table in a way that may seem a bit arbitrary, but is nevertheless correct algebra.

We start off with our derived formula:

¬LL

Now we use a trick where you can take a formula - whether closed or open - and connect it to itself using the OR operator. Doing so does not change whether it is true or false. In other words:

¬L is the same as ¬L∨¬L

And we can write:

L is the same as LL

So we can transform the formula ¬LL to the equivalent formula and write:

L ∨ ¬L) ∧ (LL)

Now remember that an IF-THEN statement is the same as a NOT-OR statement. So making this switch, we end up with:

L ∨ ¬L) ∧ (LL) is the same as (L → ¬L) ∧ (¬LL)

and we can finally write:

(L → ¬L) ∧ (¬LL)

But as Flakey Foont said to Mr. Natural, "What does it all mean?" Well, remember that L is the same as L("I", L("I", y)). So substitute the former formula for the latter we get:

(L("I", L("I", y)) → ¬L("I", L("I", y))) ∧ (¬L("I", L("I", y)) → L("I", L("I", y)))

Finally we substitute L("I", L("I", y) with it's original abbreviated natural language equivalent, "I am lying". We now end up with:

"If I am lying then I am not lying, and if I am not lying then I am lying."

Hey, presto! - we have the Liar's Paradox.

In Conclusion

So where do we get the Liar's paradox?

In brief, the Liar's Paradox arises because we take a sentence, "I am lying" and say it refers to itself. But when we do that we are left with what is actually an open formula and so cannot be assigned a value of true or false. We then (correctly) say that "I am lying" is not true and "I am lying" is not false.

¬L ∧ ¬¬L

But then we err. The statement

¬L ∧ ¬¬L

can be considered true as long as we are talking about L as an open formula. But we erroneously converted it to

¬LL

using logical transformations which are valid for statements which are TRUE or FALSE. We make the error because we are mixing statements in English about the formal statements with the actual statements withing the formal system. So we can derive a contradiction that "I am lying" is simultaneously true and false. Then with more transformations, we end up with the Liar's Paradox.

 

References

"Liar's Paradox", Standford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/liar-paradox/

Tractus Philosophicus, Ludwig Wittgenstein. Ludwig dealt with the idea of nonsense statements in philosophical discourse when he pointed out that even sentences as apparently meaningful as "There are things in existence that are beyond our human ability to imagine or conceive" really are merely assemblages of words no part of which refers to anything with a real meaning. Hence, the sentence is meaningless and is neither true or false. Interestingly, in the book Letters to Malcolm, Chiefly on Prayer, C. S. Lewis - Jack, to his friends - also addresses nonsense in philosophy when he wrote "Can a mortal ask questions which God finds unanswerable? Quite easily, I should think. All nonsense questions are unanswerable. How many hours are there in a mile? Is yellow square or round? Probably half the questions we ask - half our great theological and metaphysical problems - are like that." A strangely Wittgensteineque philosophy for Jack although Ludwig would have said all theological and metaphysical problems are like that.