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?

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

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.

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.

@(L3) L3 is not true.@

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?