Can a Sentence Flip Between True and False Forever?

The Sentence That Can’t Make Up Its Mind

Imagine a friend looks you in the eye and says, “I’m lying right now.” If she really is lying, then what she says is false — so she isn’t lying, which means she’s telling the truth. But if she’s telling the truth, then she is lying, so it’s false. Around and around it goes, like a dog chasing its tail. Philosophers call this the liar paradox.

The liar sentence is a sentence that says of itself, “This sentence is not true.” Write it down: (1) (1) is not true. Now ask: is (1) true? If it’s true, then it’s not true. If it’s not true, then it’s true. You’re stuck. For over two thousand years, this puzzle has bugged thinkers because it seems to break the ordinary idea that every statement is either true or false.

Some philosophers try to escape by saying the liar is neither true nor false — it’s a third thing. But in the 1990s, two philosophers, Anil Gupta (1949–) and Nuel Belnap (1930–), proposed a different move. Instead of blocking the paradox, they built a theory that lets the liar keep flipping, and they used that flipping to explain how truth works. They called it the Revision Theory of Truth, or RTT.

A New Idea: Revise Your Guess

To understand the RTT, start with a guess. Suppose you guess that (1) is not true. Then you use a basic rule about truth: a sentence ‘p’ is true exactly when p. This is called the T-biconditional. For (1), it says: ‘(1) is not true’ is true if and only if (1) is not true. From your guess that (1) is not true, and the T-biconditional, you can conclude that ‘(1) is not true’ is true — but since (1) just is the sentence ‘(1) is not true’, you now must guess that (1) is true! Your guess flipped.

Now start with the new guess that (1) is true. Using the same rule, you conclude that ‘(1) is not true’ is not true, so (1) is not true. You flip again. The process never stops: your guess about the liar toggles between true and false with every step. Gupta and Belnap’s key move was to say that truth isn’t a fixed label you stick on a sentence once and for all, but a rule for revising your guesses. They called this rule the revision rule: given a hypothesis about which sentences are true, you apply the T-biconditionals to generate a new, revised hypothesis. This rule is what the word “true” really gives us — not a list of true sentences, but a method for improving our guesses.

Running the Movie in Slow Motion

The RTT doesn’t just flip back and forth once. It imagines an endless sequence of hypotheses, one after another, like frames in a movie. This is called a revision sequence. Start with any guess at all, then apply the revision rule to get the next frame, then apply it again, and keep going forever. For the liar, the sequence alternates: not true, true, not true, true, and so on.

But not all sentences are so wobbly. Consider three sentences that talk about each other:

• (7) says: (8) is true or (9) is true.
• (8) says: (7) is true.
• (9) says: (7) is not true.

Now run the revision sequence. No matter what you guess at the start, after a few steps, (7) and (8) become stably true, and (9) becomes stably false. Try it: if you guess (7) is false, then (9) is true, so (7) is true — and then (8) is true and (9) false. From that point on, your guess stays fixed. Gupta and Belnap call this stability: a sentence is stably true in a revision sequence if, after some point, every hypothesis in the sequence says it’s true. If a sentence settles down to false, it’s stably false.

The RTT also handles infinite stretches. Sometimes you need to go beyond all the finite steps to a limit stage, like the “omega-th” frame. At that point, you keep the sentences that have stabilized and freely choose values for the ones that haven’t. This lets sentences that take forever to settle down eventually find a stable truth.

Stable Truths and Wobbly Ones

If a sentence is stably true in every possible revision sequence, no matter what crazy guess you start with, then Gupta and Belnap say it is categorically true. The liar is never categorically true or false — it’s unstable. That’s why it feels paradoxical. The RTT doesn’t say the liar is “neither true nor false” in the way a stoplight might have a third color. Instead, it says the liar’s truth value never settles; it’s always in motion.

Gupta and Belnap also defined a weaker idea called near stability. A sentence is nearly stably true if, after some point, it’s true almost all the time, except maybe for brief flickers right after limit stages. Near stability turns out to capture some patterns of reasoning that simple stability misses. For example, with near stability, you can say that the sentence “It is not true that A” is true exactly when A is not true — a principle that fails for simple stability in some tricky cases.

The Truth-Teller That Just Stares Back

Not every self-referential sentence is a liar. Consider the truth-teller: “This sentence is true.” If you guess it’s true, it stays true. If you guess it’s false, it stays false. The revision rule doesn’t change your guess; it just reflects it. So the truth-teller is not unstable, but it’s not categorically true or false either — its truth value depends on your starting hypothesis.

This matters because some theories of truth, like the three-valued approach associated with Saul Kripke, say the truth-teller is neither true nor false, a third status. But the RTT says no — it’s just that the word “true” itself doesn’t pick out a single final answer for the truth-teller. There are many possible revision sequences, some where it’s true and some where it’s false. The RTT respects the supervenience of semantics: the idea that which sentences are true should be fixed entirely by the non-semantic facts (like the meanings of words and the world), plus the definition of “true.” Since the truth-teller’s non-semantic facts don’t force one answer, the RTT doesn’t force one either — but it doesn’t invent a third truth value. It just leaves the question open.

Why It Still Matters

So why should you care about sentences that loop? The RTT isn’t just a toy for paradoxes. It offers a picture of how circular concepts work. Many ideas we use every day — “this rule applies to itself,” “this promise presupposes trust that the promise creates” — are circular in a similar way. Gupta and Belnap’s method of revising guesses can be applied to any concept introduced by circular definitions, from the notion of a “set” that contains itself to the rules of a game that refer back to the players’ expectations.

The RTT also forces us to think differently about truth in our own conversations. When you and a friend argue about whether a self-referential joke is funny, you might be running a tiny revision sequence in your head: you guess it’s funny, test the guess, revise, and maybe never settle. The revision theory says that’s not a failure — that’s just how some sentences behave. Truth isn’t always a still photograph; sometimes it’s a movie that keeps rolling.

Think about it

1. If a sentence can flip between true and false forever, does it still make sense to say it “means” something? Why or why not?
2. Suppose you made a video game where a character says “I am lying.” How could you program the game to handle that sentence without crashing?
3. Can you think of a real-life situation where a rule refers to itself and creates a loop? How would you revise your way out of it?