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More exactly, the use of <math>{}^{\backprime\backprime} \# {}^{\prime\prime}</math> refers to a 3-adic relation <math>L_X \subseteq X \times X \times X</math> that licenses the formula <math>a ~\#~ b = c</math> just when <math>(a, b, c)\!</math> is in <math>L_X\!</math> and the use of <math>{}^{\backprime\backprime} + {}^{\prime\prime}</math> refers to a 3-adic relation <math>L_Y \subseteq Y \times Y \times Y</math> that licenses the formula <math>p + q = r\!</math> just when <math>(p, q, r)\!</math> is in <math>L_Y.\!</math>

More exactly, the use of <math>{}^{\backprime\backprime} \# {}^{\prime\prime}</math> refers to a 3-adic relation <math>L_X \subseteq X \times X \times X</math> that licenses the formula <math>a ~\#~ b = c</math> just when <math>(a, b, c)\!</math> is in <math>L_X\!</math> and the use of <math>{}^{\backprime\backprime} + {}^{\prime\prime}</math> refers to a 3-adic relation <math>L_Y \subseteq Y \times Y \times Y</math> that licenses the formula <math>p + q = r\!</math> just when <math>(p, q, r)\!</math> is in <math>L_Y.\!</math>

In this setting the mapping <math>f : X \to Y</math> is said to be ''linear'', and to ''preserve'' the structure of <math>L_X\!</math> in the structure of <math>L_Y,\!</math> if and only if <math>f(a ~\#~ b) = f(a) + f(b),</math> for all pairs <math>a, b\!</math> in <math>X.\!</math>  In other words, <math>f\!</math> ''distributes'' over the additions <math>\#</math> to <math>+,\!</math> just as if it were a form of multiplication, analogous to <math>m(a + b) = ma + mb.\!</math>

In this setting the mapping <math>f : X \to Y</math> is said to be ''linear'', and to ''preserve'' the structure of <math>L_X\!</math> in the structure of <math>L_Y,\!</math> if and only if <math>f(a ~\#~ b) = f(a) + f(b),</math> for all pairs <math>a, b\!</math> in <math>X.\!</math>  In other words, the function <math>f\!</math> ''distributes'' over the two additions, from <math>\#</math> to <math>+,\!</math> just as if <math>f\!</math> were a form of multiplication, analogous to <math>m(a + b) = ma + mb.\!</math>

Writing this more directly in terms of the 3-adic relations <math>L_X\!</math> and <math>L_Y\!</math> instead of via their operation symbols, we would say that <math>f : X \to Y</math> is linear with regard to <math>L_X\!</math> and <math>L_Y\!</math> if and only if <math>(a, b, c)\!</math> being in the relation <math>L_X\!</math> determines that its map image <math>(f(a), f(b), f(c))\!</math> be in <math>L_Y.\!</math>  To see this, observe that <math>(a, b, c)\!</math> being in <math>L_X\!</math> implies that <math>c = a ~\#~ b,</math> and <math>(f(a), f(b), f(c))\!</math> being in <math>L_Y\!</math> implies that <math>f(c) = f(a) + f(b),\!</math> so we have that <math>f(a ~\#~ b) = f(c) = f(a) + f(b),</math> and the two notions are one.

Writing this more directly in terms of the 3-adic relations <math>L_X\!</math> and <math>L_Y\!</math> instead of via their operation symbols, we would say that <math>f : X \to Y</math> is linear with regard to <math>L_X\!</math> and <math>L_Y\!</math> if and only if <math>(a, b, c)\!</math> being in the relation <math>L_X\!</math> determines that its map image <math>(f(a), f(b), f(c))\!</math> be in <math>L_Y.\!</math>  To see this, observe that <math>(a, b, c)\!</math> being in <math>L_X\!</math> implies that <math>c = a ~\#~ b,</math> and <math>(f(a), f(b), f(c))\!</math> being in <math>L_Y\!</math> implies that <math>f(c) = f(a) + f(b),\!</math> so we have that <math>f(a ~\#~ b) = f(c) = f(a) + f(b),</math> and the two notions are one.

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Now, those who wish to say that these logical signs are iconic of their logical objects must not only find some reason that logic itself singles out one interpretation over the other, but, even if they succeed in that, they must then make us believe that every sign for Truth is iconic of Truth, while every sign for Falsity is iconic of Falsity.  Well, I confess that it strains my imagination, if not the over-abundant resources of theirs.

Those who wish to say that these logical signs are iconic of their logical objects must not only find some reason that logic itself singles out one interpretation over the other, but, even if they succeed in that, they must further make us believe that every sign for Truth is iconic of Truth, while every sign for Falsity is iconic of Falsity.

One of the questions that arises at this point, where we have a very small object domain <math>O = \{ \operatorname{falsity}, \operatorname{truth} \}</math> and a very large sign domain <math>S \cong \{ \text{rooted trees} \},</math> is the following:

One of the questions that arises at this point, where we have a very small object domain <math>O = \{ \operatorname{falsity}, \operatorname{truth} \}</math> and a very large sign domain <math>S \cong \{ \text{rooted trees} \},</math> is the following:

:* Why do we have so many ways of saying the same thing?

:* Why do we have so many ways of saying the same thing?

In other words, what possible utility is there in a language having so many signs to denote the same object?  Why not just restrict the language to a canonical collection of signs, each of which denotes one and only one object, exclusively and uniquely?

In other words, what possible utility is there in a language having so many signs to denote the same object?  Why not just restrict the language to a canonical collection of signs, each of which denotes one and only one object, exclusively and uniquely? Indeed, language reformers from time to time have proposed the design of languages that have just this property, but I think this is one of those places where natural evolution has luckily hit on a better plan than the sorts of intentional design that inexperienced designers typically craft.

Indeed, language reformers from time to time have proposed the design of languages that have just this property, but I think this is one of those places where natural evolution has luckily hit on a better plan than the sorts of intentional design that inexperienced designers typically craft.

The answer to the puzzle of semiotic multiplicity appears to have something to do with the use of language in interacting with a complex external world.  The objective world throws its multiplicity of problems at us, and the first duty of language is to provide some expression of their structure, on the fly, as quickly as possible, in real time, as they come in, no matter how obscurely our quick and dirty expressions of the problematic situation might otherwise be.  Of course, very little of this can be apparent at the level of primary arithmetic, but I think it should become a little more obvious as we enter the primary algebra.

The answer to the puzzle of semiotic multiplicity appears to have something to do with the use of language in interacting with a complex external world.  The objective world throws its multiplicity of problems at us, and the first duty of language is to provide some expression of their structure, on the fly, as quickly as possible, in real time, as they come in, no matter how obscurely our quick and dirty expressions of the problematic situation might otherwise be.

I will now give a reference version of the CSP&ndash;GSB axioms for the abstract calculus that is formally recognizable in several senses as giving form to propositional logic.

The first order of business is to give the exact forms of the axioms that I use, devolving from Peirce's Logical Graphs via Spencer-Brown's ''Laws of Form'' (LOF).  In formal proofs, I will use a variation of the annotation scheme from LOF to mark each step of the proof according to which axiom, or ''initial'', is being invoked to justify the corresponding step of syntactic transformation, whether it applies to graphs or to strings.

 

 

The first order of business is to give the exact forms of the axioms that I use, devolving from Peirce's "Logical Graphs" via Spencer-Brown's ''Laws of Form'' (LOF).  In formal proofs, I will use a variation of the annotation scheme from LOF to mark each step of the proof according to which axiom, or ''initial'', is being invoked to justify the corresponding step of syntactic transformation, whether it applies to graphs or to strings.

The axioms are just four in number, and they come in a couple of flavors:  the ''arithmetic initials'' <math>I_1\!</math> and <math>I_2,\!</math> and the ''algebraic initials'' <math>J_1\!</math> and <math>J_2.\!</math>

The axioms are just four in number, and they come in a couple of flavors:  the ''arithmetic initials'' <math>I_1\!</math> and <math>I_2,\!</math> and the ''algebraic initials'' <math>J_1\!</math> and <math>J_2.\!</math>

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Notice that all of the axioms in this set have the form of equations.  This means that all of the inference steps they allow are reversible.  In the proof annotation scheme below, I will use a double bar "=====" to mark this fact, but I may at times leave it to the reader to pick which direction is the one required for applying the indicated axiom.

Notice that all of the axioms in this set have the form of equations.  This means that all of the inference steps they allow are reversible.  In the proof annotation scheme below, I will use a double bar <math>=\!=\!=\!=\!=</math> to mark this fact, but I may at times leave it to the reader to pick which direction is the one required for applying the indicated axiom.

==Frequently used theorems==

==Frequently used theorems==