• False existential presupposition (The king of france is bald.)

• 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."

Category.LogicAndMaths

Explanation: truth-values require proper grounding

• Conjunction: T&N,F&N; Disjunction: T∨N,F∨N

• What about the semantics?

Semantics

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

• Three questions
  • When do TV gaps arise?
  • How is 3-valued logic different from classical logic?
  • Does it really solve the liar paradox?

Where gaps come from

Similar cases?

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

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

Explanation #1 : truth-values require proper grounding

• Three-valued non-classical logic: T, F, N

• Future contingents (Aristotle's sea battle)

• False existential presupposition (The kind of france is bald.)

• 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.

• Giving up bivalence. Claim: L1 is not true and L1 is not false. (Lacks a truth value)
• Similar cases?
  • Future contingents
  • Vagueness
  • False existential presupposition
• But L1 is different from all these cases. What explains the tv gap?

• Truth-maker principle: The truth (or falsity) of a sentence depends on something distinct from itself (truth-makers).

• This also deals with L2:

@(L2) L2 is true.@

Claim: L1 is not true and L1 is not false. (Lacks a truth value)

@(L1) L1 is false.@

Liar paradox - truth-value gap

Solutions?

• One might argue (though not everyone would agree) that a sentence fails to be true or false if (a) it has a false presupposition, or (b) the subject term fails to refer.
• But L1 does not seem to fall into either case. So we still need to explain why L1 does not have a truth value.

Explanation #1 : truth-values require proper grounding

• See Sainsbury for details.
• Assumption #3 : 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. Consider "Penguins do not have 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 (Assumption #4).
  • L1 fails this test.
  • Also sentence L2 : "L2 is true."

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.

Exercises

• 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?