All other sciences without exception depend upon the principles of mathematics; and mathematics borrows nothing from them but hints.
C.S. Peirce • “Logic of Number”
A principal intention of this essay is to separate what are known as algebras of logic from the subject of logic, and to re‑align them with mathematics.
G. Spencer Brown • Laws of Form
The duality between entitative and existential interpretations of logical graphs tells us something important about the relation between logic and mathematics. It tells us the mathematical forms giving structure to reasoning are deeper and more abstract at once than their logical interpretations.
A formal duality points to a more encompassing unity, founding a calculus of forms whose expressions can be read in alternate ways by switching the meanings assigned to a pair of primitive terms. Spencer Brown’s mathematical approach to Laws of Form and the whole of Peirce’s work on the mathematics of logic shows both thinkers were deeply aware of this principle.
Peirce explored a variety of dualities in logic which he treated on analogy with the dualities in projective geometry. This gave rise to formal systems where the initial constants, and thus their geometric and graph‑theoretic representations, had no uniquely fixed meanings but could be given dual interpretations in logic.
It was in this context that Peirce’s systems of logical graphs developed, issuing in dual interpretations of the same formal axioms which Peirce referred to as entitative graphs and existential graphs, respectively. He developed only the existential interpretation to any great extent, since the extension from propositional to relational calculus appeared more natural in that case, but whether there is any logical or mathematical reason for the symmetry to break at that point is a good question for further research.
Resources
References
- Peirce, C.S., [Logic of Number — Le Fevre] (MS 229), in Carolyn Eisele (ed., 1976),
The New Elements of Mathematics by Charles S. Peirce, vol. 2, 592–595. Excerpt. - Spencer Brown, G. (1969), Laws of Form, George Allen and Unwin, London, UK.
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and saw how it might be expressed in two different ways, under the entitative and existential interpretations, respectively. The next thing to do is see how our choice of interpretation bears on the patterns of proof we might find. A sense of the possibilities may be gotten by displaying the two styles of proof in parallel columns, as shown below.
Inquiry Into Inquiry 