Resolving the Liar's Paradox: "This Statement is False"

"I am a liar," is the original liar's paradoxical statement, but we're going to focus on one of its variations: "This statement is false." If true, then it is false. If false, it must be true. To resolve this paradox, we begin with the following premises:

1. If a statement is true or false, it is one or the other and not both, and, we can determine whether it is true or false.

2. If a statement is true and false, it is a challenge to give it a truth value.

Now let's define some statements: Statement A = "This statement has five words." Statements B and C = "This statement is false." Statement D = "This statement has four words."

Given the foregoing premises, we can determine the truth value of each statement using an AND-gate truth table and by following these steps:

1. Read the statement and determine if it is true or false. If this can't be done, assume it is true (or false if you prefer). Mark the left side of the truth table T for true or F for false.

2. Then ask, "If I assert the statement is true (false), does it become false (true)? If not, repeat your previous mark. If so, add the opposite mark.

If these steps are followed, the truth table for statements A, B, C, D should look like this:

For the true statement A, there is no contradiction or paradox so the two marks are TT. Statement B equals statement C. At B we start out assuming the statement is true, and, at C we begin assuming it is false. That leads to scores TF and FT, respectively. Statement D is determined to be false and remains false, so we have FF. At the far right column are the final truth values. The true statement is true; the false statement is false, and, the paradoxical statement is false. This last result is consistent with the law of non-contradiction. Statements that are contradictory or lead to a contradiction are not credible and should be considered false, notwithstanding any claim to the contrary.

References:

1. Mano, M. Morris and Charles R. Kime. Logic and Computer Design Fundamentals, Third Edition. Prentice-Hall, 2004. p. 73.

2. Epimenides paradox has "All Cretans are liars." Titus 1:12

3. Jan E.M. Houben (1995). "Bhartrhari's solution to the Liar and some other paradoxes". Journal of Indian Philosophy. 23 (4): 381–401.

4. Hájek, P.; Paris, J.; Shepherdson, J. (Mar 2000). "The Liar Paradox and Fuzzy Logic". The Journal of Symbolic Logic. 61 (1): 339–346.

5. Mills, Eugene (1998). "A simple solution to the Liar". Philosophical Studies. 89 (2/3): 197–212.