The Liar’s Paradox

Paradoxes involve accepting something that in no way appears problematic, and even appears entirely acceptable or obviously true, yet leads to an absurdity.   

The version of the Liar’s Paradox I’ll focus on goes:

“This sentence is false”

Why is this a paradox?

As indicated by the “This sentence” bit, the sentence refers to itself, and makes a claim about itself; the claim is that the sentence is false.

Now all claims are, I would agree, either true or false (this is called the Principle of Bivalence).

And so, what about the claim that the sentence makes: that the sentence is false. Is this claim true, or is this claim false?

Well, if the claim is true, then what the claim says about the sentence is that it is false; thus, this claim, that it is false, would be what is true. And so, the claim must be false.

However, if the claim is false, then what the claim says about the sentence is that it is false. Thus, the claim that it is false is correct after all; that is, the claim says what it actually is. And so, the claim must be true.

Since all claims are either true or false, via the Principle of Bivalence, we can reason with nothing more than some obviously valid rules of deduction (from first-order logic) and derive the contradiction that the claim is both true and false. Want to see how? Let’s derive the contradiction.

Our true sentences about the claim “This sentence is false” include the following:

If the claim is true, then it’s false (If T, then F).
If the claim is false, then it’s true (If F, then T).
The claim is either true or false (T or F). The Principle of Bivalence

Proof of a contradiction:

Because either T or F, assume for a sec that it’s T. Well, if T, then F (From our true sentences). Thus, we derive F from T. Now, let’s assume the other side of T or F, F. From F, we can obviously derive F. Thus, from either T or F, we get F.

Next, because either T or F, assume for a sec that it’s F. Well, if F, then T (From our true sentences). Thus, we derive T from F. Now, let’s assume the other side of either T or F, T. From T, we can obviously derive T. Thus, from either T or F, we get T.

Combining the two derivations, from either T or F, we get T and F, a contradiction. That is, the claim “This sentence is false” is both true and false.

Is it absurd to think that a claim is both true and false? Yes. To show this most effectively, we should get clear what being true and what being false amount to in the context of true and false claims. Let’s give an account of truth and falsity of claims.

First note that claims say something about how something or other is. The old chestnut goes, “The chestnut is on the mat” (or something like that). The sentence in quotes is a claim because it says that something (a chestnut) is a certain way (that it is on the mat). Conversely, if the claim fails to say something about something (e.g. “The chestnut is on the”), then it is not actually a claim. This is because nothing has been said about how the chestnut is. More specifically, no property (let’s call it 𝟇) has been specified about the chestnut that can either be or fail to be.

On the other hand, if the claim says or seems to say something, but it’s not about anything at all (e.g. “Is on the mat”), then it is not actually a claim. This is because nothing has been specified that is on the mat. More specifically, no thing (let’s call it 𝒖) and no property (𝟇) (or scope, as in “There is . . .”) has been specified, since properties (and scopes) are properties of things, and nothing at all has been mentioned that might have a property. It might be thought that being on the mat is a property. It is, but only if it is tacitly thought that something or other is on the mat (e.g. is equivalent with “Something or other is on the mat”). Without this qualification, however, no property has even been specified in “Is on the mat”. But if you don’t agree on this point, so be it, since it affects nothing that follows.

Thus, claims say something about something. That is, to be perfectly clear, claims specify either some property or some scope (𝟇) about some thing (𝒖), where “some” is meant to be “at least one”, and “thing” is a very loose term for any specifiable thing at all (cats, numbers, sets, words, forces, events, etc.). This will suffice to account for claims.

How are claims true? The most straightforward notion of truth given things and their scope or properties is truth as correspondence.

Truth as correspondence: Saying something about how something or other is is true when the something it is about is actually the way it is said to be.

More specifically, a specification of some scope or property (𝟇) about something (𝒖) is true when and only when (𝟇) about (𝒖) is actual or instantiated.

This is called truth as correspondence because the specification corresponds with reality, what is actually there.

Falsity as correspondence: Saying something about how something or other is is false when the something it is about is not actually the way it is said to be.

More specifically, a specification of some scope or property (𝟇) about something (𝒖) is false when and only when (𝟇) about (𝒖) is not actual or instantiated.

Truth and falsity as correspondence is, I think obviously, the right theory of truth for all claims. It captures our intuitive sense of what it is for a claim to be true or not–let how things are decide for what the claim says. We would need good reasons to reject a correspondence theory in favor of some other theory of truth for claims.

If correspondence is right, “The chestnut is on the mat” specifies that a chestnut (𝒖) is on the mat (𝟇), and this is true if and only if a chestnut is on the mat. (This is also how a disquotational theory of truth works since notice how I drop the quotes of the original claim at the end of the “if and only if”; however, I will save going into disquotation and the Liar’s Paradox for another time, since more would need to be said).

We can now make a general notion about claims clear that we have already covered. Claims specify truth conditions; that is, claims specify something about something, and this specification is either true or false given how what is said corresponds with how things are.

Thus, the contradictory result of the claim “This sentence is false” is that its specified truth conditions simultaneously correspond and fail to correspond with how things actually are, or, which is the same thing, that the property (𝟇) about something (𝒖) it specifies is both actual and not actual. This is an absurd result if anything is. It’s form in first order logic is P & ~P (the truth of a claim P and the claim’s negation ~P), and from this the truth of any claim at all is provable.

And so, the Liar’s Paradox is indeed a paradox. We start with the innocuous-looking claim, “This sentence is false”, and we arrive at the absurd conclusion that it is both true and false.

Solving the Liar’s Paradox

I’m not going into how others have tried to deal with the Liar’s Paradox, but here are one or two of my opinions on solving it.

My opinion is that throwing out the Principle of Bivalence is almost as absurd as the result, and so shouldn’t be considered without really good reasons that are independent from the Liar’s Paradox (go figure that some philosophers have gone this route–my advice is leave them to that hopelessness!).

A good starting place I take from John Searle. I remember from lecture that his response is, “Which sentence?” That is, the claim goes “This sentence is false”, but which sentence does it refer to? It may even appear that no sentence–that is, nothing capable of being either true or false, has been provided at all. If nothing at all has been provided that could be either true or false, then “This sentence is false” is neither true nor false.

This is a good way to start. However, there’s a problem. Since the sentence is about itself, the claim it makes, if it makes one at all, is best rendered “‘This sentence is false’ is false”. Given this, there is the nagging consideration that, if the claim is actually neither true nor false, then it would therefore be false that the sentence “This sentence is false” is false, and, if so, the paradox would be off and running all over again. To see this, it would be false the claim that “This sentence is false”. Thus, it would be true what the sentence claimed about itself all along; but in that case it would have to be false, and so on.

John Searle’s response seemed to me to be obviously correct when I first heard it years ago, but only recently have I made it sufficiently clear how it is to be exposed that no claim at all has been made, not even a claim that says that it’s false. That is, in order to go the route that “This sentence is false” is neither true nor false, you need need to dispel the appearance that “This sentence is false” claims to be false. This is because on the face of it a claim is made, one that amounts to the claim that “This sentence is false” is false. And this at least appears to be a legitimate claim.

For the benefit our sanity, I will now provide a solution to the Liar’s Paradox by showing that it is not a claim, even about itself. I provide two proofs that each suffice to show this. The first proof shows that there is no way to determine whether or not the sentence is true or false. This is enough to show that are no truth conditions given by such a sentence. Thus, nothing is given that could be either true or false; and so, no claim is made. The second proof shows that the sentence fails to specify any property (𝟇) at all, about itself or otherwise. Since claims must specify some property (𝟇) in order to be either true or false, no claim is made.

Proof I

Is “This sentence is false” a claim? If so, then it must say something about how something or other is. That is, there must be some truth conditions given by the claim.

Since, if it’s a claim, “This sentence is false” is a claim about itself, the claim is more explicitly rendered, “‘This sentence is false’ is false”. However, if this is to have truth conditions, we must have some way to find out whether or not “This sentence is false” is actually false. The only way to figure this out would be by some suitable account or definition for falsity of claims. Fortunately, I have already provided an adequate (and I think obviously right) account for falsity of claims. Thus, let’s return to our account of falsity of claims:

Saying something about how something or other is is false when the something it is about is not actually the way it is said to be.

That is,

Saying something about how something or other is (“This sentence is false” is false) is false when the something it is about (“This sentence is false”) is not the way it is said to be (false).

Thus, the question we need to answer as our only means of determining this is: is “This question is false” false? Let’s turn to our account of falsity of claims for help:

Saying something about how something or other is is false when the something it is about is not the way it is said to be.

We have obviously already done this and would continue to repeat ourselves indefinitely, with no determination for whether or not “This sentence is false” is false. Note: the same result occurs with “This sentence is true”, using our account of truth–try it out for fun!

And so, our account of truth or falsity as correspondence grants us no way to determine whether or not “This sentence is false” is true or false. However, there would not be other way by correspondence for a claim to be either true or false. That is, the definition of truth and falsity is clearly not lacking in anyway, it is complete. Thus, there is no way to determine the truth or falsity of “This claim is false” or, what is equivalent, “‘This sentence is false’ is false”, given by what such sentences say (i.e. there are no truth conditions given by it). That is, there is nothing given by “This claim is false” that could be either true or false; thus “This claim is false” is not a claim, not even about itself.

Proof II

Why does “This sentence is false” appear to be a legit claim about itself (perhaps even despite the proof above)? It appears legit because it appears to indicate a property (𝟇) of the sentence (𝒖), that of its being false (just like the property “green” of “this cup” in “This cup is green”). Let me dispel this appearance once and for all! I will show that no property (𝟇) at all is specified by the sentence “This sentence is false”. Once more, this will prove that “This sentence is false” fails to be a claim.

Let’s once more take a look at what “This sentence is false” says given our correspondence notion of falsity that saying something about how something or other is is false when the something it is about (𝒖) is not the way it is said to be (𝟇):

“This sentence is false.”

Which is to say:

“This sentence says something (𝟇) about something or other (𝒖) that is not the way it is.”

Since the “something or other (𝒖)” referred to is the sentence itself, let’s make this explicit:

“This sentence says something (𝟇) about this sentence that is not the way it is.”

However, what does the sentence say about itself that’s not the way it is? It’s now perfectly clear that nothing at all (𝟇) has been said about itself. Since claims must specify some property (or some scope) (𝟇) of the thing it is about (𝒖), no claim at all has been made. That is, “This sentence is false” is logically equivalent with “The chestnut is on the”. In both, a thing (𝒖) is specified, but no property (𝟇) of that thing is specified. Each of them, for the same reason, fails to be a claim.

We might be (once more) tempted to suppose that some property (𝟇) is said–namely, the property that it’s false. However, we just showed that this alone specifies no property. Falsity is always a broader notion that what is said fails to be the way it is said to be. If no way is ever said to be, it fails to be something that could ever fail to be that way.

If it be beneficial to belabor this point, I will make it clear once more. We might suppose that what is said is that the sentence is false. That is, resuming where we left off:

“This sentence says that this sentence is false and that is not the way it is.”

From here, we would inquire all over again what being false amounts to (or simply reflect that being false amounts to the something it is about not being the way it says it is, and note once more that nothing has been said about how it is). I’ll put the explication of the notion of falsity within brackets [] for ease of presentation; that is, [] makes explicit the meaning of the predicate “is false” in the previous bold sentence:

“This sentence says that this sentence [says something (𝟇) about something or other (𝒖) that is not the way it is] and that is not the way it is.”

From here, we would merely repeat the steps I previously made, except within the brackets, and within potentially more brackets, ad infinum. This suffices to show that no property (𝟇) is ever specified in the sentence “This sentence is false” that could ever correspond or fail to correspond with how things are. Thus, “This sentence is false” is not a claim.

Fittingly, we’d similarly be able to show that “This sentence is true” is not a claim, for failing to have truth conditions (via Proof I), and also for failing to specify any property (𝟇) (via Proof II).

If you’re wondering why it is that a claim like “‘The chestnut is on the mat’ is false” is a legit claim, it’s because there is already a claim made that is either true or false for specifying the property of being on the mat for the chestnut. If all that a claim specifies is that it is true or false, however, it necessarily fails to specify something that could be true or false, since truth and falsity are broader notions that relate a specification with how things are. A claim that a claim is false requires the instantiation of the property of falsity, which involves a claim that specifies a property or scope of a thing that fails to be instantiated. Falsity cannot be the only property specified by a claim, since falsity itself relies on a prior specified property failing to be instantiated.

With all of this, we can readily see why “I am now lying” also fails to be a claim that can be true or false. “I am now lying”, let’s say in order to avoid explaining what exactly constitutes lying, is equivalent with “I am now speaking a falsehood”. Under my analysis, “I am now speaking a falsehood”, if it says anything, says the equivalent of “The property that I am specifying fails to correspond with reality”. However, just as clearly, no property at all has been specified. It thus fails to be something that could be either true or false, a claim. 

This is how to solve the Liar’s Paradox–you prove that no claim at all has been made. This way, you don’t have to suspect the principle of bivalence, or think there must be something wrong with truth as correspondence. Don’t be a fool.

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