The use of definitions and their logical representation in paradox derivation

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We start by noting that the set-theoretic and semantic paradoxes are framed in terms of a definition or series of definitions. In the process of deriving paradoxes, these definitions are logically represented by a logical equivalence. We will firstly examine the role and usage of definitions in the derivation of paradoxes, both set-theoretic and semantic. We will see that this examination is important in determining how the paradoxes were created in the first place and indeed how they are to be solved in a uniform way. There are three features that are special about these definitions used in the derivation of most of the above paradoxes. The first is the use of self-reference between the definiens and the definiendum, the second is the generality of the definiendum, and the third is the under-determination and over-determination of concepts that usually occur as a result of these definitions. We will examine the impact of these three features on the logical representation of definitions and show how this representation then leads to a uniform paradox solution using an appropriate logic that is both paraconsistent and paracomplete. However, it is the paracompleteness, exhibited through the rejection of the Law of Excluded Middle, together with the rejection of contraction principles, that enables the solution of the paradoxes to go through. We characterize definitions as involving syntactic identity and/or meaning identity. We point out that some paradoxes do not have explicit self-reference or circularity and some may not utilize the generality of the definiendum, but the general characterizations of definitions that we give will still apply. We also look beyond all this to paradoxes that rely on illicit definitions between objects that are essentially different to start with.