Main.LiarGap History

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October 09, 2008, at 11:18 AM by 147.8.231.123 -
Changed lines 19-20 from:
* False existential presupposition (The kind of france is bald.)
to:
* False existential presupposition (The king of france is bald.)
August 03, 2006, at 04:13 PM by 219.78.21.164 -
Changed lines 34-36 from:
** The proposal is not plausible if it is taken as a general definition. Consider "Penguins do not have the concept of truth."
to:
** 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."
July 30, 2006, at 05:10 PM by 219.78.20.110 -
Added lines 71-72:

[[Category.LogicAndMaths]]
July 20, 2006, at 10:11 AM by 219.77.134.183 -
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Added line 20:
Changed lines 23-24 from:
Explanation #1 : truth-values require proper grounding
to:
!!Explanation: truth-values require proper grounding
July 18, 2006, at 04:10 PM by 219.77.144.16 -
Added lines 3-4:
media:mindthegap.jpg
July 18, 2006, at 04:03 PM by 219.77.144.16 -
Changed line 39 from:
** Conjunction: T&N,F&N; Disjunction: TvN,FvN
to:
** Conjunction: T&N,F&N; Disjunction: T∨N,F∨N
July 18, 2006, at 04:03 PM by 219.77.144.16 -
Deleted line 5:
* Three-valued non-classical logic: T, F, N
Changed line 8 from:
** How is 3-valued logic different from classical logic?
to:
** What about the semantics?
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!!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.

July 18, 2006, at 04:00 PM by 219.77.144.16 -
Changed lines 7-14 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
to:
* 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
July 18, 2006, at 03:56 PM by 219.77.144.16 -
Added line 6:
* Three-valued non-classical logic: T, F, N
July 18, 2006, at 02:21 PM by 219.77.144.16 -
Changed line 7 from:
** Future contingents
to:
** Future contingents (Aristotle's sea battle)
Changed line 9 from:
** False existential presupposition
to:
** False existential presupposition (The kind of france is bald.)
July 17, 2006, at 07:00 PM by 219.78.20.121 -
Changed line 24 from:
** 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).
to:
** 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.
July 17, 2006, at 06:59 PM by 219.78.20.121 -
Deleted lines 2-3:
Claim: L1 is not true and L1 is not false. (Lacks a truth value)
Changed lines 5-7 from:
* 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.
to:
* 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?
Changed line 15 from:
* Assumption #3 : The truth (or falsity) of a sentence depends on something distinct from itself (truth-makers).
to:
* '''Truth-maker principle''': The truth (or falsity) of a sentence depends on something distinct from itself (truth-makers).
Changed lines 26-27 from:
** Also sentence L2 : "L2 is true."
to:
** This also deals with L2:
@@@(L2) L2
is true.@@@
July 17, 2006, at 06:55 PM by 219.78.20.121 -
Added lines 3-6:
Claim: L1 is not true and L1 is not false. (Lacks a truth value)

@@@(L1) L1 is false.@@@

July 17, 2006, at 06:48 PM by 219.78.20.121 -
Changed lines 1-2 from:
!!Solutions?
to:
!Liar paradox - truth-value gap
July 17, 2006, at 06:48 PM by 219.78.20.121 -
Added lines 1-46:
!!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?