Laws of Form (hereinafter LoF) is a book by G. Spencer-Brown, published in 1969, that straddles the boundary between mathematics and philosophy.
The preface states that the work was first explored in 1959, and Spencer Brown cites Bertrand Russell as being supportive of his endeavour.
In 1963 Spencer Brown was invited by Harry Frost, staff lecturer in the physical sciences at the department of Extra-Mural Studies of the University of London, to deliver a course on the mathematics of logic.
Key ideas of the LOF were first outlined in his 1961 manuscript Design with the Nor, which remained unpublished until 2021,[1] and further refined during subsequent lectures on mathematical logic he gave under the auspices of the University of London's extension program.
LoF also echoes a number of themes from the writings of Charles Sanders Peirce, Bertrand Russell, and Alfred North Whitehead.
What prompted such a claim, is obscure, either in terms of incentive, logical merit, or as a matter of fact, because the book routinely and naturally uses the verb to be throughout, and in all its grammatical forms, as may be seen both in the original and in quotes shown below.
[4] Ostensibly a work of formal mathematics and philosophy, LoF became something of a cult classic: it was praised by Heinz von Foerster when he reviewed it for the Whole Earth Catalog.
[5] Those who agree point to LoF as embodying an enigmatic "mathematics of consciousness", its algebraic symbolism capturing an (perhaps even "the") implicit root of cognition: the ability to "distinguish".
Stafford Beer wrote in a review for Nature, "When one thinks of all that Russell went through sixty years ago, to write the Principia, and all we his readers underwent in wrestling with those three vast volumes, it is almost sad".
LoF has influenced, among others, Heinz von Foerster, Louis Kauffman, Niklas Luhmann, Humberto Maturana, Francisco Varela and William Bricken.
LoF (excluding back matter) closes with the words: ...the first distinction, the Mark and the observer are not only interchangeable, but, in the form, identical.C.
There are just two atomic expressions: There are two inductive rules: The semantics of the primary arithmetic are perhaps nothing more than the sole explicit definition in LoF: "Distinction is perfect continence".
A1 and A2 have loose analogs in the properties of series and parallel electrical circuits, and in other ways of diagramming processes, including flowcharting.
The second quote seems to assert that statements in the object language are essentially commands addressed to the reader by the author.
Equational logic was common before Principia Mathematica (e.g. Johnson (1892)), and has present-day advocates (Gries & Schneider (1993)).
LoF asserts that concatenation can be read as commuting and associating by default and hence need not be explicitly assumed or demonstrated.
The primary algebra contains three kinds of proved assertions: The distinction between consequence and theorem holds for all formal systems, including mathematics and logic, but is usually not made explicit.
If the Marked and Unmarked states are read as the Boolean values 1 and 0 (or True and False), the primary algebra interprets 2 (or sentential logic).
Extending the primary algebra so that it could interpret standard first-order logic has yet to be done, but Peirce's beta existential graphs suggest that this extension is feasible.
Repeating this exercise for all possible assignments of the two primitive values to each variable, reveals whether the original expression is tautological or satisfiable.
Schwartz (1981) proved that the primary algebra is equivalent — syntactically, semantically, and proof theoretically — with the classical propositional calculus.
Likewise, it can be shown that the primary algebra is syntactically equivalent with expressions built up in the usual way from the classical truth values true and false, the logical connectives NOT, OR, and AND, and parentheses.
But an empty Cross is a well-formed primary algebra expression, denoting the Marked state, a primitive value.
Then every syllogism that does not require that one or more terms be assumed nonempty is one of 24 possible permutations of a generalization of Barbara whose primary algebra equivalent is
These 24 possible permutations include the 19 syllogistic forms deemed valid in Aristotelian and medieval logic.
The following calculation of Leibniz's nontrivial Praeclarum Theorema exemplifies the demonstrative power of the primary algebra.
Chapter 11 of LoF introduces equations of the second degree, composed of recursive formulae that can be seen as having "infinite" depth.
Charles Sanders Peirce (1839–1914) anticipated the primary algebra in three veins of work: LoF cites vol.
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 Berkeley psychologist and cognitive scientist Eleanor Rosch has written extensively on the closely related notion of categorization.