One way of assigning logical meaning to the initial equations is known as the entitative interpretation (En). Under En, the axioms read as follows.
Another way of assigning logical meaning to the initial equations is known as the existential interpretation (Ex). Under Ex, the axioms read as follows.
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The formal system of logical graphs is defined by a foursome of formal equations, called initials when regarded purely formally, in abstraction from potential interpretations, and called axioms when interpreted as logical equivalences. There are two arithmetic initials and two algebraic initials, as shown below.
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Logical graphs are next presented as a formal system by going back to the initial elements and developing their consequences in a systematic manner.
Logical Graphs • First Impressions gives an informal introduction to the initial elements of logical graphs and hopefully supplies the reader with an intuitive sense of their motivation and rationale.
The next order of business is to give the precise axioms used to develop the formal system of logical graphs. The axioms derive from C.S. Peirce’s various systems of graphical syntax via the calculus of indications described in Spencer Brown’s Laws of Form. The formal proofs to follow will use a variation of Spencer Brown’s annotation scheme to mark each step of the proof according to which axiom is called to license the corresponding step of syntactic transformation, whether it applies to graphs or to strings.
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Logical graphs are next presented as a formal system by going back to the initial elements and developing their consequences in a systematic manner.
Logical Graphs • First Impressions gives an informal introduction to the initial elements of logical graphs and hopefully supplies the reader with an intuitive sense of their motivation and rationale.
The next order of business is to give the precise axioms used to develop the formal system of logical graphs. The axioms derive from C.S. Peirce’s various systems of graphical syntax via the calculus of indications described in Spencer Brown’s Laws of Form. The formal proofs to follow will use a variation of Spencer Brown’s annotation scheme to mark each step of the proof according to which axiom is called to license the corresponding step of syntactic transformation, whether it applies to graphs or to strings.
The formal system of logical graphs is defined by a foursome of formal equations, called initials when regarded purely formally, in abstraction from potential interpretations, and called axioms when interpreted as logical equivalences. There are two arithmetic initials and two algebraic initials, as shown below.
One way of assigning logical meaning to the initial equations is known as the entitative interpretation (En). Under En, the axioms read as follows.
Another way of assigning logical meaning to the initial equations is known as the existential interpretation (Ex). Under Ex, the axioms read as follows.
All the initials 
The actual business of proof is a far more strategic affair than the routine cranking of inference rules might suggest. Part of the reason for this lies in the circumstance that the customary types of inference rules combine the moving forward of a state of inquiry with the losing of information along the way that doesn’t appear immediately relevant, at least, not as viewed in the local focus and short run of the proof in question. Over the long haul, this has the pernicious side‑effect that one is forever strategically required to reconstruct much of the information one had strategically thought to forget at earlier stages of proof, where “before the proof started” can be counted as an earlier stage of the proof in view.
This is just one of the reasons it can be very instructive to study equational inference rules of the sort our axioms have just provided. Although equational forms of reasoning are paramount in mathematics, they are less familiar to the student of the usual logic textbooks, who may find a few surprises here.
To familiarize ourselves with equational proofs in logical graphs let’s run though the proofs of a few basic theorems in the primary algebra.
The first theorem goes under the names of Consequence 1 (C1), the double negation theorem (DNT), or Reflection.
The proof that follows is adapted from the one George Spencer Brown gave in his book Laws of Form and credited to two of his students, John Dawes and D.A. Utting.
One theorem of frequent use goes under the nickname of the weed and seed theorem (WAST). The proof is just an exercise in mathematical induction, once a suitable basis is laid down, and it will be left as an exercise for the reader. What the WAST says is that a label can be freely distributed or freely erased anywhere in a subtree whose root is labeled with that label. The second in our list of frequently used theorems is in fact the base case of the weed and seed theorem. In Laws of Form it goes by the names of Consequence 2 (C2) or Generation.
Here is a proof of the Generation Theorem.
The third of the frequently used theorems of service to this survey is one Spencer Brown annotates as Consequence 3 (C3) or Integration. A better mnemonic might be dominance and recession theorem (DART), but perhaps the brevity of dominant form theorem (DFT) is sufficient reminder of its double‑edged role in proofs.
Here is a proof of the Dominant Form Theorem.
Using no more than the axioms and theorems recorded so far, it is possible to prove a multitude of much more complex theorems. A couple of all‑time favorites are listed below.
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Discussion of the topic continues at Logical Graphs • Formal Development.
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Speaking of algebra, and having just encountered one example of an algebraic law, we might as well introduce the axioms of the primary algebra, once again deriving their substance and their name from the works of Charles Sanders Peirce and George Spencer Brown, respectively.
The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems. As it happens, the example of an algebraic law we noticed first, “a ( ) = ( )”, as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms.
We might also notice at this point a subtle difference between the primary arithmetic and the primary algebra with respect to the grounds of justification we have naturally if tacitly adopted for their respective sets of axioms.
The arithmetic axioms were introduced by fiat, in a quasi‑apriori fashion, though it is of course only long prior experience with the practical uses of comparably developed generations of formal systems that would actually induce us to such a quasi‑primal move. The algebraic axioms, in contrast, can be seen to derive both their motive and their justification from the observation and summarization of patterns which are visible in the arithmetic spectrum.
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Experience teaches that complex objects are best approached in a gradual, laminar, modular fashion, one step, one layer, one piece at a time, especially when that complexity is irreducible, when all our articulations and all our representations will be cloven at joints disjoint from the structure of the object itself, with some assembly required in the synthetic integrity of the intuition.
That’s one good reason for spending so much time on the first half of zeroth order logic, instanced here by the primary arithmetic, a level of formal structure Peirce verged on intuiting at numerous points and times in his work on logical graphs but Spencer Brown named and brought more completely to life.
Another reason for lingering a while longer in these primitive forests is that an acquaintance with “bare trees”, those unadorned with literal or numerical labels, will provide a basis for understanding what’s really at issue in oft‑debated questions about the “ontological status of variables”.
It is probably best to illustrate the theme in the setting of a concrete case. To do that let’s look again at the previous example of reductive evaluation taking place in the primary arithmetic.
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(16) |
After we’ve seen a few sign-transformations of roughly that shape we’ll most likely notice it doesn’t really matter what other branches are rooted next to the lone edge off to the side — the end result will always be the same. Eventually it will occur to us to summarize the results of many such observations by introducing a label or variable to signify any shape of branch whatever, writing something like the following.
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(17) |
Observations like that, made about an arithmetic of any variety and germinated by their summarizations, are the root of all algebra.
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Let 
For example, consider the reduction which proceeds as follows.
|
(16) |
Regarded as a semiotic process, this amounts to a sequence of signs, every one after the first serving as an interpretant of its predecessor, ending in a final sign which may be taken as the canonical sign for their common object, in the upshot being the result of the computation process.
Simple as it is, the sequence exhibits the main features of any computation, namely, a semiotic process proceeding from an obscure sign to a clear sign of the same object, being in its aim and effect an action on behalf of clarification.
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The axioms of the primary arithmetic are shown below, as they appear in both graph and string forms, along with pairs of names which come in handy for referring to the two opposing directions of applying the axioms.
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This much preparation allows us to take up the founding axioms or initial equations which determine the entire system of logical graphs.
Though it may not seem too exciting, logically speaking, there are many reasons to make oneself at home with the system of forms represented indifferently, topologically speaking, by rooted trees, balanced strings of parentheses, and finite sets of non‑intersecting simple closed curves in the plane.
This space of forms, along with the pair of axioms which divide it into two formal equivalence classes, is what Spencer Brown called the primary arithmetic.
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![\begin{array}{ccccc} \mathrm{I_1} & : & \text{true} ~ \text{or} ~ \text{true} & = & \text{true} \\[4pt] \mathrm{I_2} & : & \text{not} ~ \text{true} & = & \text{false} \\[4pt] \mathrm{J_1} & : & a ~ \text{or} ~ \text{not} ~ a & = & \text{true} \\[4pt] \mathrm{J_2} & : & (a ~ \text{or} ~ b) ~ \text{and} ~ (a ~ \text{or} ~ c) & = & a ~ \text{or} ~ (b ~ \text{and} ~ c) \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bccccc%7D++%5Cmathrm%7BI_1%7D++%26+%3A+%26+%5Ctext%7Btrue%7D+%7E+%5Ctext%7Bor%7D+%7E+%5Ctext%7Btrue%7D++%26+%3D+%26+%5Ctext%7Btrue%7D++%5C%5C%5B4pt%5D++%5Cmathrm%7BI_2%7D++%26+%3A+%26+%5Ctext%7Bnot%7D+%7E+%5Ctext%7Btrue%7D++%26+%3D+%26+%5Ctext%7Bfalse%7D++%5C%5C%5B4pt%5D++%5Cmathrm%7BJ_1%7D++%26+%3A+%26+a+%7E+%5Ctext%7Bor%7D+%7E+%5Ctext%7Bnot%7D+%7E+a++%26+%3D+%26+%5Ctext%7Btrue%7D++%5C%5C%5B4pt%5D++%5Cmathrm%7BJ_2%7D++%26+%3A+%26+%28a+%7E+%5Ctext%7Bor%7D+%7E+b%29+%7E+%5Ctext%7Band%7D+%7E+%28a+%7E+%5Ctext%7Bor%7D+%7E+c%29++%26+%3D+%26+a+%7E+%5Ctext%7Bor%7D+%7E+%28b+%7E+%5Ctext%7Band%7D+%7E+c%29++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
![\begin{array}{ccccc} \mathrm{I_1} & : & \text{false} ~ \text{and} ~ \text{false} & = & \text{false} \\[4pt] \mathrm{I_2} & : & \text{not} ~ \text{false} & = & \text{true} \\[4pt] \mathrm{J_1} & : & a ~ \text{and} ~ \text{not} ~ a & = & \text{false} \\[4pt] \mathrm{J_2} & : & (a ~ \text{and} ~ b) ~ \text{or} ~ (a\ \text{and}\ c) & = & a ~ \text{and} ~ (b ~ \text{or} ~ c) \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bccccc%7D++%5Cmathrm%7BI_1%7D++%26+%3A+%26+%5Ctext%7Bfalse%7D+%7E+%5Ctext%7Band%7D+%7E+%5Ctext%7Bfalse%7D++%26+%3D+%26+%5Ctext%7Bfalse%7D++%5C%5C%5B4pt%5D++%5Cmathrm%7BI_2%7D++%26+%3A+%26+%5Ctext%7Bnot%7D+%7E+%5Ctext%7Bfalse%7D++%26+%3D+%26+%5Ctext%7Btrue%7D++%5C%5C%5B4pt%5D++%5Cmathrm%7BJ_1%7D++%26+%3A+%26+a+%7E+%5Ctext%7Band%7D+%7E+%5Ctext%7Bnot%7D+%7E+a+++%26+%3D+%26+%5Ctext%7Bfalse%7D++%5C%5C%5B4pt%5D++%5Cmathrm%7BJ_2%7D++%26+%3A+%26+%28a+%7E+%5Ctext%7Band%7D+%7E+b%29+%7E+%5Ctext%7Bor%7D+%7E+%28a%5C+%5Ctext%7Band%7D%5C+c%29++%26+%3D+%26+a+%7E+%5Ctext%7Band%7D+%7E+%28b+%7E+%5Ctext%7Bor%7D+%7E+c%29++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
Inquiry Into Inquiry 