The Simplified Liar Paradox: "This Statement is False"

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An original proof by Dan Christensen

This proof was written with the aid of DC Proof 2.0.

Download it at http://www.dcproof.com

Informal Introduction

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(Note: Informal commentary is shown here in a blue font.)

Following is a resolution of the simplified Liar Paradox arising from the paradoxical "This statement is false." This is a statement famous for supposedly being true if and only if it is false.

The simplified Liar is NOT the logical equivalent of original Cretan Liar Paradox, as supposed by many writers. Unlike the original, the simplified Liar Paradox arises from a simple self-contradiction. (For a resolution of the original Cretan Liar Paradox, see http://www.dcproof.com/LiarParadox2.htm)

In a similar way to the Cretan Liar Paradox, we make use of the following predicates:

S(a) means 'a' is a (well-defined) statement

T(a) means 'a' is true

Here, we prove in three lines that there cannot exist a well-defined statement that is true if and only if it is false:

~EXIST(a):[S(a) & [T(a) <=> ~T(a)]]

Proof

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Suppose to the contrary that x is a well-defined statement that is true if and only it is false.

**
1**
S(x)
& [T(x)
<=> ~T(x)]

Premise

Splitting out the last part of this premise, we have the contradiction...

**
2**
T(x)
<=> ~T(x)

Split, 1

We conclude, generalizing on x, that the initial premise on line 1 is false...

As Required:

3 ~EXIST(a):[S(a) & [T(a) <=> ~T(a)]]

4 Conclusion, 1