Editing Laws of Form (section)

==Related work==
{{original research|section|date=December 2024}}
[[Gottfried Leibniz]], in memoranda not published before the late 19th and early 20th centuries, invented [[Boolean algebra (logic)|Boolean logic]]. His notation was isomorphic to that of ''LoF'': concatenation read as [[Logical conjunction|conjunction]], and "non-(''X'')" read as the [[Logical complement|complement]] of ''X''. Recognition of Leibniz's pioneering role in [[algebraic logic]] was foreshadowed by {{harvp|Lewis|1918}} and {{harvp|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 {{harvp|Lenzen|2004}}.

[[Charles Sanders Peirce]] (1839–1914) anticipated the ''primary algebra'' in three veins of work:
#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,{{sfnp|Peirce|1976|loc=101-15.1}} but they were not published in full until 1993.<ref>"Qualitative Logic", MS 582 (1886) in Kloesel, Christian et al., eds., 1993. ''[[Charles Sanders Peirce bibliography#W|Writings of Charles S. Peirce: A Chronological Edition]], Vol. 5, 1884–1886''. [[Indiana University Press]]: 323-71. "The Logic of Relatives: Qualitative and Quantitative", MS 584 (1886) in Kloesel, Christian et al., eds., 1993. ''Writings of Charles S. Peirce: A Chronological Edition, Vol. 5, 1884–1886''. Indiana University Press: 372-78.</ref>
#In a 1902 encyclopedia article,<ref>Reprinted in Peirce, C. S. (1933) ''[[Charles Sanders Peirce bibliography#CP|Collected Papers of Charles Sanders Peirce]], Vol. 4'', [[Charles Hartshorne]] and [[Paul Weiss (philosopher)|Paul Weiss]], eds. [[Harvard University Press]]. Paragraphs 378–383</ref> 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.
#The [[syntax]] of his alpha [[existential graph]]s is merely [[concatenation]], read as [[Logical conjunction|conjunction]], and enclosure by ovals, read as [[negation]].<ref>The existential graphs are described at length in Peirce, C. S. (1933) ''Collected Papers, Vol. 4'', [[Charles Hartshorne]] and [[Paul Weiss (philosopher)|Paul Weiss]], eds. Harvard University Press. Paragraphs 347–529.</ref> If ''primary algebra'' concatenation is read as [[Logical conjunction|conjunction]], then these graphs are [[isomorphic]] to the ''primary algebra''.{{sfnp|Kauffman|2001}}

''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''.

{{harvp|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 ''primary algebra'', 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 operators, [[Polish notation|prefix]], or [[Reverse Polish notation|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 [[David Hilbert|Hilbert]] and [[Jan Łukasiewicz|Łukasiewicz]] showed how to prove [[axiom independence]] using [[model theory|model]]s.

{{harvp|Craig|1979}} argued that the world, and how humans perceive and interact with that world, has a rich Boolean structure. [[William Craig (logician)|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 {{harvp|Lakoff|1987}} (see index entries under "Image schema examples: container") and {{harvp|Lakoff|Núñez|2000}}. 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.

Other formal systems with possible affinities to the primary algebra include:
*[[Mereology]] which typically has a [[lattice (order)|lattice]] structure very similar to that of Boolean algebra. For a few authors, mereology is simply a [[model theory|model]] of [[Boolean algebra (structure)|Boolean algebra]] and hence of the primary algebra as well.
*[[Mereotopology]], which is inherently richer than Boolean algebra;
*The system of {{harvp|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, [[Sheffer stroke|NAND]] (whose ''primary algebra'' translation is <math>\overline{A \ \ B \ |}</math> or, dually, <math>\overline{A |} \ \ \overline{B |}</math>), universal [[Quantification (logic)|quantification]], and one [[binary relation|binary]] [[atomic formula]], denoting [[Set (mathematics)|set]] membership. This is the system of {{harvp|Quine|1951}}.
* The ''beta'' [[existential graph]]s, 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.