Inquiry Into Inquiry

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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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 reflections on Peirce’s use of operator variables I was led to what I called the reflective extension of logical graphs, or what I now call the “cactus language”, after its principal graph-theoretic data structure.  This graphical formal language arises 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 parameter greater than 1.  This 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: Animated Logical Graphs • 23

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.

Or, thinking in terms of the corresponding cactus graphs, writing for a blank 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 is 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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Re: Ontolog Forum • Joseph Simpson

JS:

Today I found an interesting publication that might relate to the current discussion of Animated Logical Graphs (ALG).  Please see:

• Mathematician Solves Computer Science Conjecture in Two Pages

The “sensitivity” conjecture may be a topic that could be explored using ALG.  There appear to be many interesting connections between ALG and the sensitivity conjecture.  I am looking for an area where ALG application examples may be created and discussed.  My first attempt at an example, using the Augmented Model-Exchange Isomorphism (AMEI), raised a number of conceptual and structural application issues.  Which can be addressed as we move forward.  My plan is to continue the search for specific application areas.  I believe that finding domain specific applications will help me better understand the ALG material.

Take care, be good to yourself and have fun,

Joe

Dear Joe,

Boolean functions and different ways of contemplating their complexity are definitely the right ballpark, or at least the right planet, for field-testing logical graphs.

I don’t know much about the Boolean Sensitivity Conjecture but I happened to run across an enlightening article about it just yesterday and I did a while ago begin an exploration into what appears to be a related question, Péter Frankl’s “Union-Closed Sets Conjecture”.  See the resource pages linked below.

At any rate, now that we’ve entered the ballpark, or standard orbit, of boolean functions, I can skip a bit of dancing around and jump to the next blog post I have on deck.

Resources

• R.J. Lipton ∧∨ K.W. Regan
• The Easiest Impossible Problem
• Enriching the Frankl Conjecture
• Discrepancy Games and Sensitivity
• Scott Aaronson • Sensitivity Conjecture Resolved
• Frankl Conjecture • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12)

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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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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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A funny thing just happened.  Let’s see if we can tell where.  We started with the algebraic expression in which the operand suggests the contemplated absence or presence of any arithmetic expression or its value, then we contemplated the absence or presence of the operator in to be indicated by a cross or a space, respectively, for the value of a newly introduced variable, placed in a new slot of a newly extended operator form, as suggested by the following picture.

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 Space = False and Cross = True,
  calls this operation equivalence.
• The Existential Interpretation for which Space = True and Cross = False,
  calls this operation distinction.

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Another tactic I tried by way of porting operator variables into logical graphs and laws of form was to hollow out a leg of Spencer Brown’s crosses, gnomons, 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 this 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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We have encountered the question of how to extend our formal calculus to take account of operator variables.

In the days when I scribbled these things 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.  Graphics like these conveyed the idea that a character-shaped boundary drawn around another space can be viewed as absent or present depending on whether the formal value of the character is unmarked or marked.  The same idea can be conveyed by attaching characters directly to the edges of graphs.

Here is 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 obvious from the outset 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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