Laws of Form - Related Work

Related Work

Gottfried Leibniz, in memoranda not published before the late 19th and early 20th centuries, invented Boolean logic. His notation was isomorphic to that of LoF: concatenation read as conjunction, and "non-(X)" read as the complement of X. Leibniz's pioneering role in algebraic logic was foreshadowed by Lewis (1918) and Rescher (1954). But a full appreciation of Leibniz's accomplishments had to await the work of Wolfgang Lenzen, published in the 1980s and reviewed in Lenzen (2004).

Charles Sanders Peirce (1839–1914) anticipated the pa in three veins of work:

1. Two papers he wrote in 1886 proposed a logical algebra employing but one symbol, the streamer, nearly identical to the Cross of LoF. The semantics of the streamer are identical to those of the Cross, except that Peirce never wrote a streamer with nothing under it. An excerpt from one of these papers was published in 1976, but they were not published in full until 1993.
2. In a 1902 encyclopedia article, Peirce notated Boolean algebra and sentential logic in the manner of this entry, except that he employed two styles of brackets, toggling between '(', ')' and '' with each increment in formula depth.
3. The syntax of his alpha existential graphs is merely concatenation, read as conjunction, and enclosure by ovals, read as negation. If pa concatenation is read as conjunction, then these graphs are isomorphic to the pa (Kauffman 2001).

Ironically, LoF cites vol. 4 of Peirce's Collected Papers, the source for the formalisms in (2) and (3) above. (1)-(3) were virtually unknown at the time when (1960s) and in the place where (UK) LoF was written. Peirce's semiotics, about which LoF is silent, may yet shed light on the philosophical aspects of LoF.

Kauffman (2001) discusses another notation similar to that of LoF, that of a 1917 article by Jean Nicod, who was a disciple of Bertrand Russell's.

The above formalisms are, like the pa, all instances of boundary mathematics, i.e., mathematics whose syntax is limited to letters and brackets (enclosing devices). A minimalist syntax of this nature is a "boundary notation." Boundary notation is free of infix, prefix, or postfix operator symbols. The very well known curly braces ('{', '}') of set theory can be seen as a boundary notation.

The work of Leibniz, Peirce, and Nicod is innocent of metatheory, as they wrote before Emil Post's landmark 1920 paper (which LoF cites), proving that sentential logic is complete, and before Hilbert and Lukasiewicz showed how to prove axiom independence using models.

Craig (1979) argued that the world, and how humans perceive and interact with that world, has a rich Boolean structure. Craig was an orthodox logician and an authority on algebraic logic.

Second-generation cognitive science emerged in the 1970s, after LoF was written. On cognitive science and its relevance to Boolean algebra, logic, and set theory, see Lakoff (1987) (see index entries under "Image schema examples: container") and Lakoff and Núñez (2001). Neither book cites LoF.

The biologists and cognitive scientists Humberto Maturana and his student Francisco Varela both discuss LoF in their writings, which identify "distinction" as the fundamental cognitive act. The Berkeley psychologist and cognitive scientist Eleanor Rosch has written extensively on the closely related notion of categorization.

The Multiple Form Logic, by G.A. Stathis, "generalises into Multiple Truth Values" so as to be "more consistent with Experience." Multiple Form Logic, which is not a boundary formalism, employs two primitive binary operations: concatenation, read as Boolean OR, and infix "#", read as XOR. The primitive values are 0 and 1, and the corresponding arithmetic is 11=1 and 1#1=0. The axioms are 1A=1, A#X#X = A, and A(X#(AB)) = A(X#B).

Other formal systems with possible affinities to the primary algebra include:

• Mereology which typically has a lattice structure very similar to that of Boolean algebra. For a few authors, mereology is simply a model of Boolean algebra and hence of the primary algebra as well.
• Mereotopology, which is inherently richer than Boolean algebra;
• The system of Whitehead (1934), whose fundamental primitive is "indication."

The primary arithmetic and algebra are a minimalist formalism for sentential logic and Boolean algebra. Other minimalist formalisms having the power of set theory include:

• The lambda calculus;
• Combinatory logic with two (S and K) or even one (X) primitive combinators;
• Mathematical logic done with merely three primitive notions: one connective, NAND (whose pa translation is (AB) or—dually -- (A)(B) ), universal quantification, and one binary atomic formula, denoting set membership. This is the system of Quine (1951).
• The beta existential graphs, with a single binary predicate denoting set membership. This has yet to be explored. The alpha graphs mentioned above are a special case of the beta graphs.