A Liar Paradox Without Self-Reference
The Liar Paradox ("This sentence is false") seems to require self-reference. But in 1985, philosopher Stephen Yablo constructed a paradox that appears to have no self-reference - only an infinite chain of sentences, each referring forward.
→ continuing infinitely → Sn: "All Sk where k > n are false"
No sentence refers to itself. Each only talks about sentences after it. Yet this innocent-looking chain leads to paradox!
Let's try to assign truth values to these sentences. Pick an assumption:
No consistent truth assignment exists. The sentences can be neither all false, nor can any be true!
Each statement Sn depends on the truth values of all statements after it. Click on any node to see what it references.
Watch how references cascade infinitely forward. Select a starting statement and see which statements it claims are false.
Step through the formal proofs interactively. Choose a proof strategy and navigate through each logical step.
Suppose there exists some k such that Sk is true.
By definition, Sk states: "All statements Sm where m > k are false."
Since Sk is true, its claim must be correct. In particular, Sk+1 must be false.
Sk+1 claims: "All statements Sm where m > k+1 are false." Since Sk+1 is false, this claim is incorrect.
If not all statements after Sk+1 are false, then at least one Sj (where j > k+1) must be true.
But Sk claimed ALL statements after it (including Sj) are false. Yet Sj is true. This contradicts our assumption that Sk is true!
Suppose ALL statements in the sequence are false.
Consider S1, which claims: "All statements Sm where m > 1 are false."
By our assumption, S2, S3, S4, ... are all indeed false. So S1's claim is actually correct!
Since S1's claim is correct (all statements after it ARE false), S1 itself must be true.
We assumed S1 was false (as part of "all statements are false"), but we just proved S1 must be true!
Define the Yablo sequence formally.
We will show there is no consistent truth assignment for {S1, S2, S3, ...}.
Proof by contradiction: If T(Sn), then forall k > n: F(Sk). But F(Sn+1) implies exists j > n+1: T(Sj), contradicting forall k > n: F(Sk).
If forall n: F(Sn), then S1's claim (forall k > 1: F(Sk)) is satisfied, making S1 true, contradiction.
By Lemma 1, no statement can be true. By Lemma 2, not all can be false. Therefore, no consistent truth assignment exists.
No consistent truth assignment exists for the Yablo sequence. The statements can be neither all false, nor can any individual statement be true.
The revolutionary claim: no sentence in Yablo's chain refers to itself. Each Sn only talks about sentences with larger indices (k > n).
L: "This sentence is false"
Self-referential cycle
No cycles, only forward reference
"I conclude that self-reference is neither necessary nor sufficient to arrive at a paradox like that of the liar."
— Stephen Yablo, 1993
Not everyone agrees! Philosopher Graham Priest and others argue that Yablo's paradox is still self-referential, just in a hidden way.
| Yablo's View | Priest's Challenge |
|---|---|
| No sentence refers to itself | The definition of the sequence is self-referential |
| References only go forward (k > n) | The whole system has a "fixed point" structure |
| Acyclic reference pattern | Non-well-founded, which is the real culprit |
| Proves paradoxes don't need loops | Proves our notion of "self-reference" is too narrow |
Some philosophers describe Yablo's paradox as ω-circular (omega-circular). While no finite chain of references loops back, the entire infinite structure creates a kind of "circularity at infinity."
It's like a spiral staircase that never literally repeats, but the pattern itself is defined in terms of itself.
Stephen Yablo first constructs the paradox in unpublished work, showing that semantic paradoxes don't require explicit self-reference.
Yablo publishes "Paradox without Self-Reference" in Analysis, sparking widespread debate in philosophical logic.
Graham Priest responds with "Yablo's Paradox," arguing the paradox does involve implicit self-reference through the use of fixed points.
Roy Sorensen explores the broader "Yabloization" technique: transforming finitary paradoxes into non-circular infinite versions.
Debate continues. Yablo's paradox has become a key case study in the philosophy of logic, truth, and self-reference.
Shows that problems with the concept of truth run deeper than simple self-reference. Any theory blocking self-reference alone won't solve all semantic paradoxes.
The real issue may be non-well-founded structures - infinitely descending chains of dependence - rather than cycles per se.
Demonstrates that infinite structures can create paradoxes impossible in finite logic. Raises questions about using infinity in formal systems.
Related to Cantor's diagonalization and Gödel's incompleteness. All exploit a kind of "looking at the whole from within."
Yablo's construction can be applied to transform many paradoxes. The technique (dubbed "Yabloization" by Sorensen) works as follows:
Curry's Paradox: "If this sentence is true, then anything."
Yabloized version:
If Yablo's paradox truly lacks self-reference, then many proposed solutions to the Liar Paradox are insufficient:
Whatever causes semantic paradoxes, it's something deeper than simple self-reference. The culprit seems to be semantic instability - truth values that can't consistently settle.
"Every sentence after this one is false."
— S∞, eternally waiting its turn to speak