where Ξn is a sign system corresponding to a representation of a (design) problem at time t1, Ξn+1 is a sign system corresponding to a representation of the problem at time t2, t2 > t1, fn is a composition of semiotic morphisms that specifies the interaction of variation and selection under the condition of information closure, which requires no external elements be added to the current sign system; мn is a semiotic morphism, and Pn is the probability associated with мn, ΣPn = 1, n=1,…,M, where M is the number of the meaningful transformations of the resultant sign system after fn. There is a partial ranking – importance ordering – on the constraints of A in every Ξn, such that lower ranked constraints can be violated in order for higher ranked constraints to be satisfied. The morphisms of fn preserve the ranking.
The Semiotic Theory of Self-Organizing Systems postulates that in the scale hierarchy of dynamical organization, a new level emerges if and only if a new level in the hierarchy of semiotic interpretance emerges. As the development of a new product always and naturally causes the emergence of a new meaning, the above-cited Principle of Emergence directly leads us to the formulation of the first law of life-cycle semiosis as follows:
I. The semiosis of a product life cycle is represented by a sequence of basic semiotic components, such that at least one of the components is well defined in the sense that not all of its morphisms of м and f are isomorphisms, and at least one м in the sequence is not level-preserving in the sense that it does not preserve the original partial ordering on levels.
For the present (i.e. for an on-going process), there exists a probability distribution over the possible мn for every component in the sequence. For the past (i.e. retrospectively), each of the distributions collapses to a single mapping with Pn = 1, while the sequence of basic semiotic components is degenerated to a sequence of functions. For the future, the life-cycle meaning-making
process can be considered in a very general probabilistic sense only (e.g. in terms of probability distributions that are characteristic of a specific domain, social group, design approach, or the like).
It seems logical to assume that the successful (perhaps, in any sense) introduction of a product to the market effects the introduction and settlement of the corresponding meanings at the onto-, typo-, and phylogenic semiotic levels. Let us denote the number of relations between the product and its environment as ε. We can now formulate the second law of life-cycle semiosis as follows:
Although the above laws have been formulated with sufficient precision, it is recommended to apply them (alike Algebraic Semiotics in general) in an informal way, calling for details only in boundary and difficult situations. The main purpose of these as well as other not-yet-formulated laws of life-cycle semiosis is to guide the examination of the product development and usage processes, no matter which design theory or even paradigm is employed at the lower, applied level.