The Liar 2 2 17 The Liar 2 2 17 The first sentence on this slide is false The Liar The first sentence on this slide is false Lets abbreviate the sentence on the last slide as L for liar ID: 579014
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Slide1
The Liar ParadoxSlide2
The Liar
2 + 2 = 17Slide3
The Liar
2 + 2 = 17
The first sentence on this slide is false.Slide4
The Liar
The first sentence on this slide is false.Slide5
Let’s abbreviate the sentence on the last slide as ‘L’ for “liar”.
Let’s ask whether ‘L’ is true or not.Slide6
Possibility #1: ‘L’ is true.
A declarative sentence describes the way the world is.
If a sentence is true, then the world is the way it describes it.
‘L’ says that the world is this way: ‘L’ is false.
So ‘L’ is false.Slide7
Possibility #1: ‘L’ is true.
A declarative sentence describes the way the world is.
If a sentence is true, then the world is the way it describes it.
‘L’ says that the world is this way: ‘L’ is false.
So ‘L’ is false.Slide8
Disquotation Principle (1)
A declarative sentence describes the way the world is.
If a sentence is true, then the world is the way it describes it.
If ‘P’ is true, then P:
If “Today is Friday” is true, then today is Friday.
If “Michael is hungry” is true, then Michael is hungry.Slide9
Possibility #1: ‘L’ is true.
If ‘L’ is true, then L.
L = ‘L’ is false.
So ‘L’ is false.Slide10
Bivalence Principle
Every (declarative) sentence (that makes sense) has exactly one truth-value among these two: true, false. Slide11
Possibility #1: ‘L’ is true.
If ‘L’ is true, then L.
L = ‘L’ is false.
So ‘L’ is false.
Add in bivalence
Contradiction!Slide12
Possibility #2: L is false.
A declarative sentence describes the way the world is.
If the world is the way a sentence describes it, then the sentence is true.
L says that the world is this way: L is false.
So L is true.Slide13
Disquotation Principle (2)
A declarative sentence describes the way the world is.
If the world is the way a sentence describes it, then the sentence is true.
If P, then ‘P’ is true.
If today is Friday, then ‘Today is Friday’ is true.
If Michael is hungry, then ‘Michael is hungry’ is true. Slide14
Possibility #2: ‘L’ is false.
If L, then ‘L’ is true.
‘L’ is false = L.
So ‘L’ is true.
Add in bivalence
Contradiction!Slide15
The Strengthened LiarSlide16
Potential Solution: Deny Bivalence
Some things are neither true nor false:
Rocks
Trees
Questions
Meaningless declarative sentences
Perhaps the liar is in this category?Slide17
Potential Solution: Deny Bivalence
“What time is it?”
“This sentence is false.”
“Grass is green.”
“Dogs bark.”
“Snow is green.”
“Dogs moo.”
Neither
True
FalseSlide18
Problem: The Strengthened Liar
Liar sentence (L): The first sentence on this slide is false.
Strengthened Liar (L*): The second sentence on this slide is
not true
.Slide19
Possibility #1: L is true.
A declarative sentence describes the way the world is.
L says that the world is this way: L is not true.
If a sentence is true, then the world is the way it describes it.
So L is not true.
L is true and not true
ContradictionSlide20
The Law of Excluded Middle
LEM: A or not-A
Everything is either blue or not blue.
Everything is either a dog or not a dog.
Everything is either true or not true.Slide21
The Law of Excluded Middle
“What time is it?”
“This sentence is false.”
“Grass is green.”
“Dogs bark.”
“Snow is green.”
“Dogs moo.”
True
Not TrueSlide22
Solutions
Give up excluded middle
Give up disjunction elimination
Give up
disquotation
Disallow self-reference
Accept that some contradictions are trueSlide23
1. Giving up Excluded Middle
The problem with giving up the Law of Excluded Middle is that it seems to collapse into endorsing contradictions:
“According to LEM, every sentence is either true or not true. I disagree: I think that some sentences are not true and not
not
true at the same time.”Slide24
2. Give up Disjunction Elimination
Basic logical principles are difficult to deny. What would a counterexample to disjunction elimination look like?
A or B
A implies C
B implies C
However,
not-CSlide25
3. Give up Disquotation
Principle
Giving up the
disquotation
principle
P = ‘P’ is true
Involves accepting that sometimes
P but ‘P’ is not true
or accepting that
not-P but ‘P’ is true
.Slide26
4. Disallow Self-Reference
The problem with disallowing self-reference is that self-reference isn’t essential to the paradox.
A: ‘B’ is true
B: ‘A’ is not trueSlide27
Circular Reference
A
B
‘B’ is true.
‘A’ is false.Slide28
Assume ‘A’ I
s True
A
B
‘B’ is true.
‘A’ is false.Slide29
Then ‘B’ Is Also True
A
B
‘B’ is true.
‘A’ is false.Slide30
But Then ‘A’ is False!
A
B
‘B’ is true.
‘A’ is false.Slide31
Assume ‘A’ Is False
A
B
‘B’ is true.
‘A’ is false.Slide32
Then ‘B’ Is Also False
A
B
‘B’ is true.
‘A’ is false.Slide33
But Then ‘A’ Is Also True
A
B
‘B’ is true.
‘A’ is false.