## Animated Logical Graphs • 29

I invoked the general concepts of equivalence and distinction at this point in order to keep the wider backdrop of ideas in mind but since we’ve been focusing on boolean functions to coordinate the semantics of propositional calculi we can get a sense of the links between operations and relations by looking at their relationship in a boolean frame of reference.

Let $\mathbb{B} = \{ 0, 1 \}$ and $k$ a positive integer.  Then $\mathbb{B}^k$ is the set of $k$-tuples of elements of $\mathbb{B}.$

• A $k$-variable boolean function is a mapping $\mathbb{B}^k \to \mathbb{B}.$
• A $k$-place boolean relation is a subset of $\mathbb{B}^k.$

The correspondence between boolean functions and boolean relations may be articulated as follows:

• Any $k$-place relation $L,$ as a subset of $\mathbb{B}^k,$ has a corresponding indicator function (or characteristic function) $f_L : \mathbb{B}^k \to \mathbb{B}$ defined by the rule that $f_L (x) = 1$ if $x$ is in $L$ and $f_L (x) = 0$ if $x$ is not in $L.$
• Any $k$-variable function $f : \mathbb{B}^k \to \mathbb{B}$ is the indicator function of a $k$-place relation $L_f$ consisting of all the $x$ in $\mathbb{B}^k$ where $f(x) = 1.$  The set $L_f$ is called the fiber of $1$ or the pre-image of $1$ in $\mathbb{B}^k$ and is commonly notated as $f^{-1}(1).$

## Abduction, Deduction, Induction, Analogy, Inquiry : 26

Projects giving a central place to computation in scientific inquiry go back to Hobbes and Leibniz, at least, and then came Babbage and Peirce.  One of the first issues determining their subsequent development is the degree to which one identifies computation and deduction.  The next question concerns how many types of reasoning one counts as contributing to the logic of empirical science:

1. Is deduction alone sufficient?
2. Are deduction and induction irreducible to each other and sufficient in tandem?
3. Are there three irreducible types of inference:  abduction, deduction, induction?

## Animated Logical Graphs • 28

Re: Ontolog ForumJSJA

I will have to focus on other business for a couple of weeks — so just by way of reminding myself what we were talking about at this juncture where logical graphs and differential logic intersect, here’s my comment on R.J. Lipton and K.W. Regan’s blog post about Discrepancy Games and Sensitivity.

Just by way of a general observation, concepts like discrepancy, influence, sensitivity, etc. are differential in character, so I tend to think the proper grounds for approaching them more systematically will come from developing the logical analogue of differential geometry.

I took a few steps in this direction some years ago in connection with an effort to understand a certain class of intelligent systems as dynamical systems.  There’s a motley assortment of links here:

## Animated Logical Graphs • 27

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.

## Animated Logical Graphs • 26

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 ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime}$ in a particular direction, treating ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime}$ 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 ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime},$ ${}^{\backprime\backprime} \texttt{(} ~ \texttt{,} ~ \texttt{)} {}^{\prime\prime},$ ${}^{\backprime\backprime} \texttt{(} ~ \texttt{,} ~ \texttt{,} ~ \texttt{)} {}^{\prime\prime},$ 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 $k$-node operator, corresponding to an expression of the form ${}^{\backprime\backprime} x_1 x_2 \ldots x_{k-1} x_k {}^{\prime\prime},$ is as follows:

• The rule for evaluating a $k$-lobe operator, corresponding to an expression of the form ${}^{\backprime\backprime} \texttt{(} x_1 \texttt{,} x_2 \texttt{,} \ldots \texttt{,} x_{k-1} \texttt{,} x_k \texttt{)} {}^{\prime\prime},$ is as follows:

## Animated Logical Graphs • 25

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:

$\begin{array}{|*{3}{c}||c|} \multicolumn{4}{c}{\text{Formal Operation Table} ~ \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}} \\[4pt] \hline a & b & c & \texttt{(}a\texttt{,}b\texttt{,}c\texttt{)} \\ \hline\hline \texttt{Space}&\texttt{Space}&\texttt{Space}&\texttt{Cross} \\ \texttt{Space}&\texttt{Space}&\texttt{Cross}&\texttt{Space} \\ \texttt{Space}&\texttt{Cross}&\texttt{Space}&\texttt{Space} \\ \texttt{Space}&\texttt{Cross}&\texttt{Cross}&\texttt{Cross} \\ \hline \texttt{Cross}&\texttt{Space}&\texttt{Space}&\texttt{Space} \\ \texttt{Cross}&\texttt{Space}&\texttt{Cross}&\texttt{Cross} \\ \texttt{Cross}&\texttt{Cross}&\texttt{Space}&\texttt{Cross} \\ \texttt{Cross}&\texttt{Cross}&\texttt{Cross}&\texttt{Cross} \\ \hline \end{array}$

Or, thinking in terms of the corresponding cactus graphs, writing ${}^{\backprime\backprime} \texttt{o} {}^{\prime\prime}$ for a blank node and ${}^{\backprime\backprime} \texttt{|} {}^{\prime\prime}$ for a terminal edge, we get the following Table:

$\begin{array}{|*{3}{c}||c|} \multicolumn{4}{c}{\text{Formal Operation Table} ~ \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}} \\[4pt] \hline \quad a \quad & \quad b \quad & \quad c \quad & \texttt{(}a\texttt{,}b\texttt{,}c\texttt{)} \\ \hline\hline \texttt{o} & \texttt{o} & \texttt{o} & \texttt{|} \\ \texttt{o} & \texttt{o} & \texttt{|} & \texttt{o} \\ \texttt{o} & \texttt{|} & \texttt{o} & \texttt{o} \\ \texttt{o} & \texttt{|} & \texttt{|} & \texttt{|} \\ \hline \texttt{|} & \texttt{o} & \texttt{o} & \texttt{o} \\ \texttt{|} & \texttt{o} & \texttt{|} & \texttt{|} \\ \texttt{|} & \texttt{|} & \texttt{o} & \texttt{|} \\ \texttt{|} & \texttt{|} & \texttt{|} & \texttt{|} \\ \hline \end{array}$

Evidently, the rule is that ${}^{\backprime\backprime} \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)} {}^{\prime\prime}$ denotes the value denoted by ${}^{\backprime\backprime} \texttt{o} {}^{\prime\prime}$ if and only if exactly one of the variables $a, b, c$ has the value denoted by ${}^{\backprime\backprime} \texttt{|} {}^{\prime\prime},$ otherwise ${}^{\backprime\backprime} \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)} {}^{\prime\prime}$ denotes the value denoted by ${}^{\backprime\backprime} \texttt{|} {}^{\prime\prime}.$  Examining the whole series of reflective forms shows this is the general rule.

• In the Entitative Interpretation $(\mathrm{En}),$ where $\texttt{o}$ = false and $\texttt{|}$ = true, ${}^{\backprime\backprime} \texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)} {}^{\prime\prime}$ translates as “not just one of the $x_j$ is true”.
• In the Existential Interpretation $(\mathrm{Ex}),$ where $\texttt{o}$ = true and $\texttt{|}$ = false, ${}^{\backprime\backprime} \texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)} {}^{\prime\prime}$ translates as “just one of the $x_j$ is not true”.

## Animated Logical Graphs • 24

Boolean functions $f : \mathbb{B}^k \to \mathbb{B}$ 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 did run across an enlightening article about it just yesterday and I did once begin an exploration of 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.