Liar paradox - truth-value gap

@(L1) L1 is false.@

  • Giving up bivalence. Claim: L1 is not true and L1 is not false. (Lacks a truth value)
  • Three questions
    • When do TV gaps arise?
    • What about the semantics?
    • Does it really solve the liar paradox?

Where gaps come from

Similar cases?

  • Future contingents (Aristotle's sea battle)
  • Vagueness
  • False existential presupposition (The king of france is bald.)

But L1 is different from all these cases. What explains the tv gap?

Explanation: truth-values require proper grounding

  • See Sainsbury for details.
  • Truth-maker principle: The truth (or falsity) of a sentence depends on something distinct from itself (truth-makers).
    • Whether "Socrates had red hair" is true depends on the state of certain physical objects.
    • But whether L1 is true or not does not depend on anything else external to it.
  • Problems with this explanation :
    • What about "this is a sentence"?
    • What about logical truths?
  • Sainsbury : "an example" of proper grounding is a sentence whose truth depends on a fact that can be expressed without using the concept of truth.
    • If only an example, what is the general definition?
    • The proposal is not plausible if it is taken as a general definition.
    • e.g. "Penguins do not have the concept of truth."
      "The concept of proof is defined in terms of the concept of truth."
    • Perhaps what is meant is : the truth of a sentence has to depend on some fact that is independent of whether the sentence is true or false.
    • L1 fails this test.
    • This also deals with L2:

@(L2) L2 is true.@


  • Three-valued non-classical logic: T, F, N
  • How is 3-valued logic different from classical logic?
    • Conjunction: T&N,F&N; Disjunction: T∨N,F∨N
    • Conditionals: T→N, N→N, etc.

A further problem

  • If L1 does not have a truth-value, then we can draw two conclusions :
    • (a) L1 is not true.
    • (b) L1 is not false.
    • L1 says of itself that it is false. This is compatible with (a).
    • But since (b) is true, it is false that "L1 is false." So L1 is false after all.
  • So if L1 is neither true nor false, then L1 is both false and not false. So we have a contradiction, as Sainsbury points out.

The strengthened liar sentence

@(L3) L3 is not true.@

  • L3 is true → "L3 is not true" is true → L3 is not true.
  • L3 is false → L3 is not true → L3 is true
  • L3 is neither true nor false → L3 is not true → L3 is true.
  • Conclusion : L3 is both true and not true!

Two lessons to bear in mind ?

  • The proper grounding theory as it stands cannot solve the paradox.
  • It is not clear how solutions which postulate semantic defects in the liar sentences can deal with L3 since a sentence with semantic defects would not be true, which is what L3 says.


  • For you to think about : Footnote 3 of Sainsbury on page 113 says "This derivation shows that one could not regard L1 as a basis for a straightforward proof of G." What does he mean by this claim?