Fuzzy Does Not Lie!

This paper is meant to be a brief review of the paper The Liar Paradox and Fuzzy Logic. In this review, it is intended to avoid technicality of proofs. However, the basic ideas of the proofs will be underlined. Original article follows afterwards. Background and Introduction Background In order to analyze the present paper The Liar Paradox and Fuzzy Logic ([LP] for short), Peano Arithmetic (PA for short) should be recalled first. Axioms of PA that were used in [LP] are equality (that is, S(x, y) → [S(x, z) → y = z]), functionality of successor (that is S(k, k + 1) where k is constant for all numeral k), addition and multiplication. As deduction rules, we will use modus ponens, generalization and the rule of induction which is: φ(0), (∀x)[φ(x)→ (S(x, y)→ φ(y))] ∴ (∀x)φ(x) Recall that, induction axiom scheme is not sound in many-valued case since the definition of implication might arise some problems1. We also know that nonstandard models of PA exists and we can obtain them by adding constants. Furthermore, it is possible to interprete the constants as the points of infinity.2 Moreover, as stated in [LP] following accomplishments are needed to be recalled: 1See [5]. 2Any text book on axiomatic set theory will help grasp the basic ideas of these facts. Since they are off-topic, we will not go into details.