Four years before Schrödinger wrote his wave equation, Emil Post produced an account of m-valued truth systems. In standard binary logic, a proposition is defined as any statement that can be ascertained to be either true or false: the statement must have one of two possible truth-values. Post demonstrated that logics could be constructed, without contradiction, such that a proposition could have one of an infinite number, m, of possible truth values. This was not applied to the multivalued wave function of quantum physics because, even though Post demonstrated there is an infinite number of possible truth-values, any given proposition still can be ascertained to be single-valued. Since that time, however, fuzzy logic has shown there can be a “fuzziness” between true and false, and this logical property can be used to make machines such as refrigerators more self-regulative. Moreover, G. Spencer Brown, while elaborating proofs of Sheffer’s postulates for Boolean algebras, demonstrated that the notion of “distinction” is more fundamental than that of “truth-value”. If we consider that the notion of “identity” is more fundamental, yet, than that of “distinction”, we may arrive at an expanded understanding of Post’s orders of logical-value and postulate full-blown m-valued logics, wherein a proposition may simultaneously and validly embrace many logical values.

A reinterpretation of the significance and meaning of orders of logical-value is encouraged by recent developments in the field of mesoscopic physics -- widely reported in the newspapers as laying a foundation for quantum information units
(q-bits) and quantum computing -- where it has clearly been demonstrated that some atomic entities (not only elementary particles) are multivalued and can exist in more than one place at a time. The identity of such entities cannot be “simple”, cannot be absolutely self-identical. Such entities possess “complex” identity: that is, distinction between the given entity and other entities cannot be absolute, an opaque wall, but must be a matter of degree, must exhibit one or another order of identity transparency. This means that an order of logical-value (the multiple values of which a proposition can simultaneously and validly express) represents an order of identity transparency: one of the transfinite set of states between “no A is not-A” and “A is absolutely
not-A”. The complete logical domain, implied by multiple orders of logical-value, contextualizes the multivaluedness of the quantum wave function such that, for instance, transposing the temporal operators from Maxwellian to Schrödinger form necessarily involves the hypothesis that operator-time is the logical operator which determines the order of logical-value configuring the reference frame (or informational ground) of the given system dynamic. Active operator-time, as authentic topological operator (not merely a measurable), would decompose and recompose (as a mathematical involute) a multivalued reference space.

A point in Hilbert space (the multidimensional function space used in quantum theory), even though it has many degrees of freedom representing the intersection of many spatial dimensions, is still logically single-valued. Hilbert space was constructed with a calculus of propositions wherein each proposition must be either true or false. The same function space, however, can repeatedly be reconstructed with each order of logical-value permitted by an m-valued calculus of propositions interpreted in relation to the notion of identity transparency. A given point, then, would not only have many degrees of freedom, but, being logically m-valued, would also have many shadow selves holographically peppered throughout the space. Simple-locality and simple-identity would not be inherent properties of entities mapped into this densely stacked function space, which may be called a multivalued reference space (and is the hyparxic domain to which Julian Barbour’s term “Platonia” refers).

Gödel’s discoveries, concerning the limitations of mathematical systems constructed with a single-valued logic, can be viewed as opening a window upon m-valued logics, identity transparency, operator-time as logical operator, and so on. Moreover, when atomic entities are demonstrated to be in more than one place at a time, as has recently been done, the single-valued geometrical relations underlying molecular stereochemistry cannot have absolute definability. From the perspective of single-valued logic, these relations are definable only in an indeterminate fashion, as implied by the theory of molecular indeterminacy described by Isaacs and Lamb. A relation seen as parallel from the perspective of single-valued logic would also simultaneously manifest infinite degrees of skewness when viewed from the perspectives of the various orders of m-valued logics. Indeed, J. G. Bennett and R. L. Brown in the early 1950s constructed skew-parallel figures and diversely identical skew-cubes with null-vectors in developing a geometrical representation of the identity-in-diversity explicitly characterized by m-valued logics.

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