Bootstrapping system defined by inconsistent relation between boolean and non-boolean algebra

We claim the using the formal biological system as a model evoking life must be much more unpredictable than Langton's guiding principle for artificial life, because any interaction is assumed to be computable in Langton's idea. In order to estimate nonlogical aspects of interactions which are not computable, we concentrate here on the relationship between a part and a whole. In other words, nonlogical aspects of interactions are articulated into the "prescription" described by a local observer and the " postscription" by a nonlocal observer. In this framework, we cannot deduce a prescribed local rule from a wholeness, and a local rule cannot be reduced from postscribed wholeness. We first show that such articulations with hierarchical structures result from descriptions involving measurements and/or observations. We propose a Bootstrapping system in which a local rule has two modes of descriptive manner: prescription, described in a logic a priori and a postscription, described in a logic a posteriori. We define cellular-automata fashioned model with a Bootstrapping system in adopting restricted Pseudo-Boolean algebra and Boolean algebra as a logic a priori and a posteriori, respectively, and show unpredictability underlying in the relation between a part and a whole, or deduction and reduction.