Steven Ericsson-Zenith (Institute for Advanced Science & Engineering):
Conceptions Of Locality In Logic And Computation,
A History

Is the logical assembly of the parts sufficient to determine the whole? Or is it, rather, the inverse, that the parts are determined as logical differentiations of the whole? And if so does anything remain of the whole that is not characterized by the logical relations between the parts? Are these two views equivalent, i.e., does the fact of the matter make any difference to logical results?

These questions relate to the mechanics of the structure that is the natural basis of logical operations. A way to view the question is to ask if this base mechanics is either differentiation in dynamic manifolds of sense, as suggested by our biology and experience, and in which the mechanisms of sensory characterization may play a role; or the construction of discrete logical parts and the relations between them. Before the current era, resplendent with computing machinery constructed upon the latter view, versions of the manifold argument were often preferred, notably in the algebraic logic of Peirce-Schröder.

The necessity to provide a computable explanation of sensory characterization and bound response potential in biophysics justify a review of the matter. A review is further justified by the search for a computable solution to general recognition, a ready operation in biology that is imperfect and costly, and may be intractable in its general form, in current computing architectures.

Central to the question is the nature of the inference mechanism determined by the basis structure of logical systems. Can we say, for example, that presented with A and B that the logical combination of the two loses nothing of their natural implication? Can we extend such a claim when presented with A and B and C etc..?

We may also question whether it is always reasonable to assume that the logical product of A and B combined is immutable in the presence of all other terms? In other words, are logical compositions side-effect free as the atomic view insists, or are there hidden factors that are not accessible to it?

In the atomic, analytic, view all derived logical values have equal standing to their premises and a logical expression describes the composition of atoms. The interpretation of combining these atoms in a logical expression is either independent of any a priori binding or it is considered to capture or impose such a binding.

The manifold view can be given clear geometric correspondents, "shapes" ("symbols" or "signs") upon the manifold characterize the parts. The manifold provides a natural, continuous, and unifying dynamics binding them, inference in this view is a transformation of the manifold. We must consider therefore whether the atomic view is sufficient to characterize the behavior of such manifolds and to do so we must identify the difference that this makes to results.

Certainly, Charles Sanders Peirce (1839-1914) did not consider the atomic view sufficient, unlike Frege he was not concerned with mere statements of truth but rather the differences the apprehension of such statements identify in the world, and he was critical of directions being taken by his contemporaries (notably Bertrand Russell) that ignored these considerations. Of Russell's algebra, as of 1904, Peirce wrote to Victoria Welby:

"The criticism which I make on that algebra of dyadic relations ... is that the very triadic relations which it does not recognize it does itself employ. For every combination of relatives to make a new relative is a triadic relation irreducible to dyadic relations."
Charles Sanders Peirce. Letter to Victoria Welby. (October, 1904)

Peirce is essentially arguing that the interpretant manifold of the logician (called "thirdness" in Pierce's vocabulary, hence "triadic") is a necessary addendum for the correct interpretation of such logical text, i.e., the effective results, to be viewed as behavioral outcomes according to Peirce's Pragmaticism, between a purely mechanical interpretation of the dyadic relations according to the principles of modern computable inference differ from the natural manifold interpretation because there are factors inaccessible to the interpretation of purely atomic (dyadic) relations.

Christine Ladd-Franklin (1847-1930), Peirce's doctoral student at John Hopkins University, continued this criticism against the "Russellisation" of logic long after Peirce passed. We will explore Peirce's view in some detail and briefly discuss the contributions of his students.

In this presentation we will consider the question and the diverse historical views related to it during the period from Charles Sanders Peirce (1839-1914) to Alan Turing (1912-1954), especially including a review of the contributions of Ernst Schröder (1841-1902) and Rudolf Carnap (1891-1970) compared to Gottlob Frege (1848-1925), Bertrand Russell (1872-1970), and Alfred North Whitehead (1861-1947), and the influence of Clarence Irving Lewis (1883-1964) and Cooper Harold Langford (1895-1964).