The Liar Paradox

The Liar 2 + 2 = 17

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

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

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.

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.

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.

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.

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.

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

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!

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.

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.

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!

The Strengthened Liar

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

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

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.

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

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.

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

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

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.”

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

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.

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

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

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

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

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

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

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

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