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
- Future contingents (Aristotle's sea battle)
- 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?