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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.
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.
One of the questions that arises at this point, where we have a very small object domain '''O''' = {Falsity, Truth} and a very large sign domain '''S''' <u>≈</u> {rooted trees}, 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?
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'' ''I''<sub>1</sub> and ''I''<sub>2</sub>, and the ''algebraic initials'' ''J''<sub>1</sub> and ''J''<sub>2</sub>.
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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