The first thing to notice about this question is that it involves the process of attribution, and that the rules of attribution are set forth in stages by mathematical logic. The first stage is called sentential logic and contains the rules for ascribing the attributes true or false, respectively denoting inclusion or non-inclusion in arbitrary cognitive-perceptual systems, to hypothetical relationships in which predicates are linked by the logical functors not, and, or, implies, and if and only if. Sentential logic defines these functors as truth functions assigning truth values to such expressions irrespective of the contents (but not the truth values) of their predicates, thus effecting a circular definition of functors on truth values and truth values on functors. The next stage of attribution, predicate logic, ascribes specific properties to objects using quantifiers. And the final stage, model theory, comprises the rules for attributing complex relations of predicates to complex relations of objects, i.e. theories to universes. In addition, the form of attribution called definition is explicated in a theory-centric branch of logic called formalized theories, and the mechanics of functional attribution is treated in recursion theory.
In sentential logic, a tautology is an expression of functor-related sentential variables that is always true, regardless of the truth values assigned to its sentential variables themselves. A tautology has three key properties: it is universally (syntactically) true, it is thus self-referential (true even of itself and therefore closed under recursive self-composition), and its implications remain consistent under inferential operations preserving these properties. That is, every tautology is a self-consistent circularity of universal scope, possessing validity by virtue of closure under self-composition, comprehensiveness (non-exclusion of truth), and consistency (freedom from irresolvable paradox). But tautologies are not merely consistent unto themselves; they are mutually consistent under mutual composition, making sentential logic as much a “self-consistent circularity of universal scope” as any one of its tautologies. Thus, sentential logic embodies two levels of tautology, one applying to expressions and one applying to theoretical systems thereof. Predicate logic then extends the tautology concept to cover the specific acts of attribution represented by (formerly anonymous) sentential variables, and model theory goes on to encompass more complex acts of attribution involving more complex relationships.
Or in other, more easy to understand words: How is it that they are exclusive when they are necessarily encapsulated by the same rules of attribution and a common medium by which a difference relationship (or any relationship) can be affirmed? Maybe because these thoughts are merely manifestations of the natural and necessary process of their embedding entity and language: To process something, you need to have something to process, right? If the process is that of logical reasoning, then if there is no problem, how can a processor exist? No question ,no answer; no problem, no resolution. Humans process things thus, and herein lies the necessity of such things;paradox is a state of nature, otherwise it wouldn't need to actualize and move. It's not contradictory, merely at another level of attribution and embedment. Consider this : If both truth-values(true and false) didn't already exist, how could something be true or false? They are in unresolved truth-state before being processed, pretty much like words before being syntactically processed, just that the processing "language" is noetic, not lingvistic. Realistic acknowledgment of this fact and it's proper integration in models resolves many nasty paradoxes associated with short-circuiting their relationships.
posted 3 years ago. ( permalink )