Inquiry Into Inquiry

Re: Operator Variables in Logical Graphs • 11

The rules given in the previous post for evaluating cactus graphs were given in purely formal terms, that is, by referring to the mathematical forms of cacti without mentioning their potential for logical meaning.  As it turns out, two ways of mapping cactus graphs to logical meanings are commonly found in practice.  These two mappings of mathematical structure to logical meaning are formally dual to each other and known as the Entitative and Existential interpretations respectively.  The following Table compares the entitative and existential interpretations of the primary cactus structures, from which the rest of their semantics can be derived.

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Re: Futures Of Logical Graphs • Themes and Variations

This post and the next wrap up the Themes and Variations section of my speculation on Futures of Logical Graphs.  I made an effort to “show my work”, reviewing the steps I took to arrive at the present perspective on logical graphs, whistling past the least productive of the blind alleys, cul‑de‑sacs, detours, and forking paths I explored along the way.  It can be useful to tell the story that way, partly because others may find things I missed down those roads, but it does call for a recap of the main ideas I would like readers to take away.

Partly through my reflection on Peirce’s use of operator variables I was led to what I called a “reflective extension of logical graphs”, amounting to a graphical formal language called the “cactus language” or “cactus syntax” after its principal graph-theoretic data structure.

The abstract syntax of cactus graphs can be interpreted for logical use in a couple of ways, both of which arise from generalizing the negation operator in a particular direction, treating as the controlled, moderated, or reflective negation operator of order 1 and adding another operator for each integer greater than 1.  The resulting family of operators is symbolized by bracketed argument lists of the forms and so on, where the number of places is the order of the reflective negation operator in question.

Two rules suffice for evaluating cactus graphs.

• The rule for evaluating a -node operator, corresponding to an expression of the form is as follows.

• The rule for evaluating a -lobe operator, corresponding to an expression of the form is as follows.

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Re: Operator Variables in Logical Graphs • 9

Let’s examine the Formal Operation Table for the third in our series of reflective forms to see if we can elicit the general pattern.

Alternatively, if we think in terms of the corresponding cactus graphs, writing for an unmarked node and for a terminal edge, we get the following Table.

Evidently, the rule is that denotes the value denoted by if and only if exactly one of the variables has the value denoted by otherwise denotes the value denoted by   Examining the whole series of reflective forms shows this to be the general rule.

• In the Entitative Interpretation where = false and = true,
  translates as “not just one of the is true”.

• In the Existential Interpretation where = true and = false,
  translates as “just one of the is not true”.

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The following Table will suffice to show how the “streamer‑cross” forms C.S. Peirce used in his essay on “Qualitative Logic” and Spencer Brown used in his Laws of Form, as they are extended through successive steps of controlled reflection, translate into syntactic strings and rooted cactus graphs.

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Re: Operator Variables in Logical Graphs • 7

A trick of discovery I learned by observing Peirce’s working methods, more than anything he wrote outright, might be put in the following words.

Take what is constant, Treat it as variable, See if anything remains the same.

The step of controlled reflection we took with the previous post can be repeated at will, as suggested by the following series of forms.

Written inline, we have the series and so on, whose general form is   With this move we have passed beyond the graph-theoretical form of rooted trees to what graph theorists know as rooted cacti.

I will discuss this cactus language and its logical interpretations next.

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Re: Operator Variables in Logical Graphs • 6

A funny thing just happened.  Let’s see if we can tell where.  We started with the algebraic expression where the operand suggests the contemplated absence or presence of an arbitrary arithmetic expression.  Next we contemplated the absence or presence of the operator in to be determined by the value of a newly introduced variable, say which is placed in a new slot of a newly extended operator form, as suggested by the following Figure.

What happened here is this.  Our contemplation of a constant operator as being potentially variable gave rise to the contemplation of a newly introduced but otherwise quite ordinary operand variable, albeit in a newly-fashioned formula.  In its interpretation for logic the newly formed operation may be viewed as an extension of ordinary negation, one in which the negation of the first variable is controlled by the value of the second variable.

We may regard this development as marking a form of controlled reflection, or a form of reflective control.  From here on out we’ll use the inline syntax to indicate the corresponding operation on two variables, whose formal operation table is given below.

• The Entitative Interpretation for which and
  calls this operation logical equality.
• The Existential Interpretation for which and
  calls this operation logical difference.

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Another tactic I tried by way of porting operator variables into Peirce’s logical graphs and Spencer Brown’s logical forms was to hollow out a leg of the latter’s crosses, gnomons, or markers, whatever you wish to call them, as shown below.

The initial idea I had in mind was the same as before, that the operator over would be counted as absent when evaluates to a space and present when evaluates to a cross.

However, much in the same way operators with a shade of negativity tend to be more generative than the purely positive brand, it turned out more useful to reverse the initial polarity of operation, letting the operator over be counted as absent when evaluates to a cross and present when evaluates to a space.

So that is the convention I’ll adopt from this point on.

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Re: Operator Variables in Logical Graphs • 4

We have encountered the question of how to extend our formal calculus to take account of operator variables.

In the days when I scribbled my logical graphs on the backs of computer punchcards, the first thing I tried was drawing big loopy script characters, placing some inside the loops of others.  Lower case alphas, betas, gammas, deltas, and so on worked best.  Graphs like that conveyed the idea that a character-shaped boundary drawn around an enclosed space can be viewed as absent or present depending on whether the formal value of the character in question is unmarked or marked.  The same idea can be conveyed by attaching characters directly to the edges of graphs.

For example, the next Figure shows how we might suggest an algebraic expression of the form where the absence or presence of the operator depends on the value of the algebraic expression the operator being absent whenever is unmarked and present whenever is marked.

It was clear from the outset that this sort of tactic would need a lot of work to become a usable calculus, especially when it came time to feed those punchcards back into the computer.

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Re: Operator Variables in Logical Graphs • 3

Last time we contemplated the penultimately simple algebraic expression as a name for a set of arithmetic expressions, specifically, taking the equal sign in the appropriate sense.

Then we asked the corresponding question about the operator   The above set of arithmetic expressions is what it means to contemplate the absence or presence of the arithmetic constant in the place of the operand in the algebraic expression   But what would it mean to contemplate the absence or presence of the operator in the algebraic expression

Evidently, a variation between the absence and the presence of the operator in the algebraic expression refers to a variation between the algebraic expression and the algebraic expression somewhat as pictured below.

But how shall we signify such variations in a coherent calculus?

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And if he is told that something is the way it is, then he thinks:  Well, it could probably just as easily be some other way.  So the sense of possibility might be defined outright as the capacity to think how everything could “just as easily” be, and to attach no more importance to what is than to what is not.

— Robert Musil • The Man Without Qualities

To get a clearer view of the relation between primary arithmetic and primary algebra consider the following extremely simple algebraic expression.

Here we see the variable name appearing as an operand name in the expression   In functional terms, is called an argument name but it’s best to avoid the potentially confusing connotations of the word argument here, since it also refers in logical discussions to a more or less specific pattern of reasoning.

In effect, the algebraic variable name indicates the contemplated absence or presence of any arithmetic expression taking its place in the surrounding template, which expression is proxied well enough by its formal value, and of which values we know but two.  Putting it all together, the algebraic expression varies between the following two choices.

The above set of arithmetic expressions is what it means to contemplate the absence or presence of the arithmetic constant in the place of the operand in the algebraic expression   But what would it mean to contemplate the absence or presence of the operator in the algebraic expression

That is the question I’ll take up next.

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