1. The Liar Paradox

2. The Liar
2 + 2 = 17

3. 2 + 2 = 17 The first sentence on this slide is false.
The Liar
2 + 2 = 17 The first sentence on this slide is false.

4. The first sentence on this slide is false.
The Liar
The first sentence on this slide is false.

5. Let’s abbreviate the sentence on the last slide as ‘L’ for “liar”
Let’s abbreviate the sentence on the last slide as ‘L’ for “liar”. Let’s ask whether ‘L’ is true or not.

6. Possibility #1: ‘L’ is true
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.

7. Possibility #1: ‘L’ is true
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.

8. 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.

9. Possibility #1: ‘L’ is true. If ‘L’ is true, then L. L = ‘L’ is false
Possibility #1: ‘L’ is true. If ‘L’ is true, then L. L = ‘L’ is false. So ‘L’ is false.

10. Bivalence Principle
Every (declarative) sentence (that makes sense) has exactly one truth-value among these two: true, false.

11. Possibility #1: ‘L’ is true. If ‘L’ is true, then L. L = ‘L’ is false
Possibility #1: ‘L’ is true. If ‘L’ is true, then L. L = ‘L’ is false. So ‘L’ is false. Add in bivalence  Contradiction!

12. Possibility #2: L is false
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.

13. 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.

14. Possibility #2: ‘L’ is false. If L, then ‘L’ is true. ‘L’ is false = L
Possibility #2: ‘L’ is false. If L, then ‘L’ is true. ‘L’ is false = L. So ‘L’ is true. Add in bivalence  Contradiction!

15. The Strengthened Liar

16. Potential Solution: Deny Bivalence
Some things are neither true nor false: Rocks Trees Questions Meaningless declarative sentences Perhaps the liar is in this category?

17. Potential Solution: Deny Bivalence
“Snow is green.”
“Grass is green.”
“What time is it?”
“Dogs moo.”
“Dogs bark.”
“This sentence is false.”
True
Neither
False

18. 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.

19. Possibility #1: L is true
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  Contradiction

20. 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.

21. The Law of Excluded Middle
“Snow is green.”
“Grass is green.”
“What time is it?”
“Dogs moo.”
“Dogs bark.”
“This sentence is false.”
True
Not True

22. Solutions Give up excluded middle Give up disjunction elimination
Give up disquotation
Disallow self-reference
Accept that some contradictions are true

23. 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.”

24. 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-C

25. 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.

26. 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 true

27. Circular Reference
‘B’ is true.
‘A’ is false. A B

28. Assume ‘A’ Is True
‘B’ is true.
‘A’ is false. A B

29. Then ‘B’ Is Also True
‘B’ is true.
‘A’ is false. A B

30. But Then ‘A’ is False!
‘B’ is true.
‘A’ is false. A B

31. Assume ‘A’ Is False
‘B’ is true.
‘A’ is false. A B

32. Then ‘B’ Is Also False
‘B’ is true.
‘A’ is false. A B

33. But Then ‘A’ Is Also True
‘B’ is true.
‘A’ is false. A B