❦ Pyramus & Thisbe ❦

Posted on November 12, 2014 by Jon Awbrey

It’s all about love
And the knots thereof
I have known beauty
I’ll bring it to you

Jon Awbrey • 12 Nov 2014

Posted in Anthem, Mantra, Verse | Tagged , , | Leave a comment

Frankl, My Dear • 11

Posted on November 10, 2014 by Jon Awbrey

Re: Dick Lipton & Ken Regan • (1) • (2)

Let’s take a moment from the differential analysis of the proposition in Example 1 to form a handy compendium of the results obtained so far.

Example 1


Venn Diagram PQR		 (1)

Enlargement Map \mathrm{E}(pqr) of the Conjunction pqr


Venn Diagram Frankl Figure 3		 (3)

Table 4.0 PQR Enlargement Map

Tacit Extension \boldsymbol\varepsilon(pqr) of the Conjunction pqr


Venn Diagram Frankl Figure 4		 (4)

Table 4.0 PQR Tacit Extension

Difference Map \mathrm{D}(pqr) of the Conjunction pqr


Venn Diagram Frankl Figure 5		 (5)

Table 4.0 PQR Difference Map

To be continued …

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Posted in Boolean Algebra, Boolean Functions, Computational Complexity, Differential Logic, Frankl Conjecture, Logic, Logical Graphs, Mathematics, Péter Frankl | Tagged , , , , , , , , | 10 Comments

Continuity, Generality, Infinity, Law, Synechism • 1

Posted on November 9, 2014 by Jon Awbrey

The concept of continuity Peirce highlights in his synechism is a logical principle somewhat more general than the concepts of either mathematical or physical continua.

Peirce’s concept of continuity is better understood as a concept of lawful regularity or parametric variation.  As such, it is basic to the coherence and utility of science, whether classical, relativistic, quantum mechanical, or any conceivable future science deserving the name.  (As Aristotle already knew.)

Perhaps the most pervasive examples of this brand of continuity in physics are the “correspondence principles” describing the relationship between classical and contemporary paradigms.

The importance of lawful regularities and parametric variations is not diminished one bit in passing from continuous mathematics to discrete mathematics, nor from theory to application.

It would be good to keep in mind two further points of information, the missing of which seems to lie at the root of many recent disputes on the Peirce List.

  • It is necessary to distinguish the mathematical concepts of continuity and infinity from the question of their physical realization.  The mathematical concepts retain their practical utility for modeling empirical phenomena quite independently of the (meta-)physical question of whether those continua and cardinalities are literally realized in the physical universe.  That is equally true of any other domain or level of phenomena — chemical, biological, mental, social, or whatever.
  • As far as the mathematical concept goes, continuity is relative to topology.  Thus what counts as a continuous function or transformation between spaces is relative to the topology under which those spaces are considered and the same spaces may be considered under many different topologies.  What topology makes the most sense in a given application is an abductive question.
Posted in Abduction, Aristotle, C.S. Peirce, Cardinality, Constraint, Continua, Continuity, Discreteness, Discretion, Epistemology, Generality, Infinity, Knowledge, Logic, Logic of Science, Mathematical Models, Mathematics, Natural Law, Physics, Quanta, Quantum Mechanics, Synechism, Topology | Tagged , , , , , , , , , , , , , , , , , , , , , , | 10 Comments

Frankl, My Dear • 10

Posted on November 4, 2014 by Jon Awbrey

Re: Dick Lipton & Ken Regan • (1) • (2)


Venn Diagram Frankl Figure 5		 (5)

Figure 5 shows the 14 terms of the difference map \mathrm{D}f as arcs, arrows, or directed edges in the venn diagram of the original proposition f(p, q, r) = pqr. The arcs of \mathrm{E}f are directed into the cell where f is true from each of the other cells. The arcs of \boldsymbol\varepsilon f are directed from the cell where f is true into each of the other cells.

The expansion of \mathrm{D}f computed in the previous post is shown again below with the terms arranged by number of positive differential features, from lowest to highest.

\begin{array}{*{4}{l}} \multicolumn{4}{l}{\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =} \\[10pt] & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & \texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} \cdot \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & \texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} \end{array}
 
\begin{array}{*{4}{l}} + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & \texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} \cdot \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} & + & \texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} \cdot \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} & + & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \end{array}

To be continued …

Resources

Posted in Boolean Algebra, Boolean Functions, Computational Complexity, Differential Logic, Frankl Conjecture, Logic, Logical Graphs, Mathematics, Péter Frankl | Tagged , , , , , , , , | 11 Comments

Frankl, My Dear • 9

Posted on November 3, 2014 by Jon Awbrey

“It doesn’t matter what one does,” the Man Without Qualities said to himself, shrugging his shoulders. “In a tangle of forces like this it doesn’t make a scrap of difference.” He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.

Robert Musil • The Man Without Qualities

Re: Dick Lipton & Ken Regan • (1) • (2)

We continue with the differential analysis of the proposition in Example 1.

Example 1


Venn Diagram PQR		 (1)

The difference operator \mathrm{D} is defined as the difference \mathrm{E} - \boldsymbol\varepsilon between the enlargement operator \mathrm{E} and the tacit extension operator \boldsymbol\varepsilon.

The difference map \mathrm{D}f is the result of applying the difference operator \mathrm{D} to the function f. When the sense is clear, we may refer to \mathrm{D}f simply as the difference of f.

In boolean spaces there is no difference between the sum (+) and the difference (-) so the difference operator \mathrm{D} is equally well expressed as the exclusive disjunction or symmetric difference \mathrm{E} + \boldsymbol\varepsilon. In this case the difference map \mathrm{D}f can be computed according to the formula \mathrm{D}f = (\mathrm{E} + \boldsymbol\varepsilon)f = \mathrm{E}f + \boldsymbol\varepsilon f.

The action of \mathrm{D} on our present example, f(p, q, r) = pqr, can be computed from the data on hand according to the following prescription.

The enlargement map \mathrm{E}f, computed in Post 5 and graphed in Post 6, is shown again here:

\mathrm{E}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =

\begin{smallmatrix} & p q r \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q \texttt{(} r \texttt{)} \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r & + & p \texttt{(} q \texttt{)} r \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & + & p \texttt{(} q \texttt{)(} r \texttt{)} \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r \\[4pt] + & \texttt{(} p \texttt{)} q r \,\cdot\, \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)} q \texttt{(} r \texttt{)} \,\cdot\, \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r & + & \texttt{(} p \texttt{)(} q \texttt{)} r \,\cdot\, \mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \,\cdot\, \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r \end{smallmatrix}

The tacit extension \boldsymbol\varepsilon f, computed in Post 7 and graphed in Post 8, is shown again here:

\boldsymbol\varepsilon f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =

\begin{array}{*{8}{l}} & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \\[4pt] + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \end{array}

The difference map \mathrm{D}f is the sum of the enlargement map \mathrm{E}f and the tacit extension \boldsymbol\varepsilon f.

Here we adopt a paradigm of computation for \mathrm{D}f that aids not only in organizing the stages of the work but also in highlighting the diverse facets of logical meaning that may be read off the result.

The terms of the enlargement map \mathrm{E}f are obtained from the table below by multiplying the base factor at the head of each column by the differential factor that appears beneath it in the body of the table.

Enlargement Map • Conjunction PQR

The terms of the tacit extension \boldsymbol\varepsilon f are obtained from the next table below by multiplying the base factor at the head of the first column by each of the differential factors that appear beneath it in the body of the table.

Tacit Extension • Conjunction PQR

Finally, the terms of the difference map \mathrm{D}f are obtained by overlaying the displays for \mathrm{E}f and \boldsymbol\varepsilon f and taking their boolean sum entry by entry.

Difference Map • Conjunction PQR

Notice that the “loop” or “no change” term p q r \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} cancels out, leaving 14 terms in the end.

To be continued …

Resources

Posted in Boolean Algebra, Boolean Functions, Computational Complexity, Differential Logic, Frankl Conjecture, Logic, Logical Graphs, Mathematics, Péter Frankl | Tagged , , , , , , , , | 13 Comments

Frankl, My Dear • 8

Posted on October 30, 2014 by Jon Awbrey

Re: Dick Lipton & Ken Regan • (1) • (2)


Venn Diagram Frankl Figure 4		 (4)

Figure 4 shows the eight terms of the tacit extension \boldsymbol\varepsilon f as arcs, arrows, or directed edges in the venn diagram of the original proposition f(p, q, r) = pqr. Each term of the tacit extension \boldsymbol\varepsilon f corresponds to an arc that starts from the cell where f is true and ends in one of the eight cells of the venn diagram.

For ease of reference, here is the expansion of \boldsymbol\varepsilon f from the previous post:

\boldsymbol\varepsilon f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =

\begin{array}{*{8}{l}} & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \\[4pt] + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \end{array}

Two examples suffice to convey the general idea of the extended venn diagram:

  • The term pqr \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} is shown as a looped arc starting in the cell where pqr is true and returning back to it. The differential factor \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} corresponds to the fact that the arc crosses no logical feature boundaries from its source to its target.
  • The term pqr \cdot \mathrm{d}p \; \mathrm{d}q \, \mathrm{d}r is shown as an arc going from the cell where pqr is true to the cell where \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} is true. The differential factor \mathrm{d}p \; \mathrm{d}q \, \mathrm{d}r corresponds to the fact that the arc crosses all three logical feature boundaries from its source to its target.

To be continued …

Resources

Posted in Boolean Algebra, Boolean Functions, Computational Complexity, Differential Logic, Frankl Conjecture, Logic, Logical Graphs, Mathematics, Péter Frankl | Tagged , , , , , , , , | 11 Comments

Frankl, My Dear • 7

Posted on October 29, 2014 by Jon Awbrey

Re: Dick Lipton & Ken Regan • (1) • (2)

We continue with the differential analysis of the proposition in Example 1.

Example 1


Venn Diagram PQR		 (1)

A proposition defined on one universe of discourse has natural extensions to larger universes of discourse. As a matter of course in a given context of discussion, some of these extensions come to be taken for granted as the most natural extensions to make in passing from one universe to the next and they tend to be assumed automatically, by default, in the absence of explicit notice to the contrary. These are the tacit extensions that apply in that context.

Differential logic, at the first order of analysis, treats extensions from boolean spaces of type \mathbb{B}^k to enlarged boolean spaces of type \mathbb{B}^k \times \mathbb{D}^k. In this setting \mathbb{B} \cong \mathbb{D} \cong \{ 0, 1 \} but we use different letters merely to distinguish base and differential features.

In our present example, the tacit extension \boldsymbol\varepsilon f of f is the boolean function \boldsymbol\varepsilon f : \mathbb{B}^3 \times \mathbb{D}^3 \to \mathbb{B} defined by the following equation:

\boldsymbol\varepsilon f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) ~=~ f(p, q, r).

The boolean expansion of \boldsymbol\varepsilon f takes the following form:

\boldsymbol\varepsilon f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =

\begin{array}{*{8}{l}} & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \\[4pt] + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \end{array}

In other words, \boldsymbol\varepsilon f is simply f on the base variables p, q, r extended by a tautology — commonly known as a “Don’t Care” condition — on the differential variables \mathrm{d}p, \mathrm{d}q, \mathrm{d}r.

To be continued …

Resources

Posted in Boolean Algebra, Boolean Functions, Computational Complexity, Differential Logic, Frankl Conjecture, Logic, Logical Graphs, Mathematics, Péter Frankl | Tagged , , , , , , , , | 12 Comments

Frankl, My Dear • 6

Posted on October 27, 2014 by Jon Awbrey

Re: Dick Lipton & Ken Regan • (1) • (2)


Venn Diagram Frankl Figure 3		 (3)

Figure 3 shows the eight terms of the enlarged proposition \mathrm{E}f as arcs, arrows, or directed edges in the venn diagram of the original proposition f(p, q, r) = pqr. Each term of the enlargement \mathrm{E}f corresponds to an arc into the cell where f is true from one of the eight cells of the venn diagram.

For ease of reference, here is the expansion of \mathrm{E}f from the previous post:

\mathrm{E}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =

\begin{smallmatrix} & p q r \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q \texttt{(} r \texttt{)} \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r & + & p \texttt{(} q \texttt{)} r \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & + & p \texttt{(} q \texttt{)(} r \texttt{)} \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r \\[4pt] + & \texttt{(} p \texttt{)} q r \,\cdot\, \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)} q \texttt{(} r \texttt{)} \,\cdot\, \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r & + & \texttt{(} p \texttt{)(} q \texttt{)} r \,\cdot\, \mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \,\cdot\, \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r \end{smallmatrix}

Two examples suffice to convey the general idea of the enlarged venn diagram:

  • The term p q r \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} is shown as a looped arc starting in the cell where p q r is true and returning back to it. The differential factor \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} corresponds to the fact that the arc crosses no logical feature boundaries from its source to its target.
  • The term \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r is shown as an arc from the cell where \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} is true to the cell where p q r is true. The differential factor \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r corresponds to the fact that the arc crosses all three logical feature boundaries from its source to its target.

To be continued …

Resources

Posted in Boolean Algebra, Boolean Functions, Computational Complexity, Differential Logic, Frankl Conjecture, Logic, Logical Graphs, Mathematics, Péter Frankl | Tagged , , , , , , , , | 11 Comments

Frankl, My Dear • 5

Posted on October 23, 2014 by Jon Awbrey

Re: Dick Lipton & Ken Regan • (1) • (2)

Putting all thought of the Frankl Conjecture out of our minds for the moment, let’s return to the proposition in Example 1 and work through its differential analysis from scratch.

Example 1


Venn Diagram PQR		 (1)

Consider the proposition p \land q \land r, in boolean terms the function f : \mathbb{B}^3 \to \mathbb{B} such that f(p, q, r) = pqr, as illustrated by the venn diagram in Figure 1.

The enlargement \mathrm{E}f of f is the boolean function \mathrm{E}f : \mathbb{B}^3 \times \mathbb{D}^3 \to \mathbb{B} defined by the following equation:

\mathrm{E}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) ~=~ f(p + \mathrm{d}p, q + \mathrm{d}q, r + \mathrm{d}r).

Given that f is the boolean product of its three arguments, \mathrm{E}f may be written as follows:

\mathrm{E}f ~=~ (p + \mathrm{d}p)(q + \mathrm{d}q)(r + \mathrm{d}r).

Difficulties of notation in differential logic are greatly eased by introducing the family of minimal negation operators on finite numbers of boolean variables. For our immediate purposes we need only the minimal negation operators on one and two variables.

  • The minimal negation operator on one variable x is notated with monospace parentheses as \texttt{(} x \texttt{)} and is simply another notation for the logical negation \lnot x.
  • The minimal negation operator on two variables x, y is notated with monospace delimiters as \texttt{(} x \texttt{,} y \texttt{)} and is simply another notation for the exclusive disjunction x + y.

In this notation the previous expression for \mathrm{E}f takes the following form:

\mathrm{E}f ~=~ \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)(} q \texttt{,} \mathrm{d}q \texttt{)(} r \texttt{,} \mathrm{d}r \texttt{)}.

A canonical form for \mathrm{E}f may be derived by means of a boolean expansion, in effect, a case analysis that evaluates \mathrm{E}f at each triple of values for the base variables p, q, r and forms the disjunction of the partial evaluations. Each term of the boolean expansion corresponds to a cell of the venn diagram and is formed by multiplying the value of that cell by a coefficient that amounts to the value of \mathrm{E}f on that cell.

For example, in the case pqr where all three base variables are true, the corresponding coefficient is computed as follows:

\begin{array}{lll} \mathrm{E}f(1, 1, 1, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) & = & (1 + \mathrm{d}p)(1 + \mathrm{d}q)(1 + \mathrm{d}r) \\[4pt] & = & \lnot\mathrm{d}p \land \lnot\mathrm{d}q \land \lnot\mathrm{d}r \\[4pt] & = & \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} \end{array}

Collecting the cases yields the boolean expansion of \mathrm{E}f via the following computation:

Step 1

\mathrm{E}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =

\begin{smallmatrix} & p q r \cdot f(1 + \mathrm{d}p, 1 + \mathrm{d}q, 1 + \mathrm{d}r) & + & p q \tilde{r} \cdot f(1 + \mathrm{d}p, 1 + \mathrm{d}q, 0 + \mathrm{d}r) & + & p \tilde{q} r \cdot f(1 + \mathrm{d}p, 0 + \mathrm{d}q, 1 + \mathrm{d}r) & + & p \tilde{q} \tilde{r} \cdot f(1 + \mathrm{d}p, 0 + \mathrm{d}q, 0 + \mathrm{d}q) \\[4pt] + & \tilde{p} q r \cdot f(0 + \mathrm{d}p, 1 + \mathrm{d}q, 1 + \mathrm{d}r) & + & \tilde{p} q \tilde{r} \cdot f(0 + \mathrm{d}p, 1 + \mathrm{d}q, 0 + \mathrm{d}r) & + & \tilde{p} \tilde{q} r \cdot f(0 + \mathrm{d}p, 0 + \mathrm{d}q, 1 + \mathrm{d}r) & + & \tilde{p} \tilde{q} \tilde{r} \cdot f(0 + \mathrm{d}p, 0 + \mathrm{d}q, 0 + \mathrm{d}q) \end{smallmatrix}

Step 2

\mathrm{E}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =

\begin{smallmatrix} & p q r \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q \texttt{(} r \texttt{)} \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r & + & p \texttt{(} q \texttt{)} r \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & + & p \texttt{(} q \texttt{)(} r \texttt{)} \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r \\[4pt] + & \texttt{(} p \texttt{)} q r \,\cdot\, \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)} q \texttt{(} r \texttt{)} \,\cdot\, \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r & + & \texttt{(} p \texttt{)(} q \texttt{)} r \,\cdot\, \mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \,\cdot\, \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r \end{smallmatrix}

To be continued …

Resources

Posted in Boolean Algebra, Boolean Functions, Computational Complexity, Differential Logic, Frankl Conjecture, Logic, Logical Graphs, Mathematics, Péter Frankl | Tagged , , , , , , , , | 13 Comments

Semiotic Theory Of Information • 6

Posted on October 14, 2014 by Jon Awbrey

2014 Oct 14

Through the 1970s I gradually recovered from my early traumas with Fortran, and with the aid of more symbol-friendly programming languages like Lisp and Pascal began to play around again with implementing simple forms of graphical calculi inspired by Peirce and Spencer-Brown in the form of programs that carried out the requisite transformations in automatic or guided interactive fashions.

Experiments like these, moving from scribblings on paper to algorithms and data structures in electronic media, brought about a transformation in my perspective on symbolic logic and other semiotic processes.  The shift was very gradual over the decade that followed, but I think it began with thinking of computer memory as very like a sheet of paper, a tabula rasa, or a Sheet of Assertion as Peirce dubbed the unmarked state in his systems of logical graphs.

Thinking this way naturally brings out the system-theoretic aspects of semiosis in general and logic in particular.  One begins with sign relations as subsets of cartesian products {O \times S \times I}, where {O, S, I} are sets of objects, signs, and interpretant signs, respectively, and over time one begins to see dynamic systems in place of those sets.  Then one day in the mid 1980s I distinctly remember flashing on the fact that the graph-theoretic data structures I and my programs were manipulating in memory were actually diagrams in Peirce’s sense.

With that pre-ramble, here is a bit of background from (Awbrey & Awbrey, 1990) that describes our system-theoretic approach to agents with a capacity for learning and reasoning.

The State Space Approach to Intelligent Systems

The common definition of a system as a list of variables is useful but not absolute.  It characterizes the system only as measured from a particular frame of observation.  The property of a system that we really care about is its state space, or a representation showing the possible states of a system.  When considering systems that exhibit complex sets of properties, such as being able to transform information and to act with intelligent purpose, it becomes more difficult to specify in advance the exact nature of the state space, or even whether a space exists that satisfies a given set of specifications.  Therefore we do not consider the state space of intelligent systems to be given absolutely, as the subset of some predefined space (like {\mathbb{R}^{n}}), but defined provisionally, subject to a list of constraints and subject to revision.

When considering intelligent behavior, we are most interested in the state of information that an interpreter system has about an object system, and this information has its expression in a system of signs.  The characterizations sign, object, interpreter are not so much exclusive categories of entities as roles that systems may play within the so-called sign relation.  Although interested in the nature and relations of these systems in themselves, whatever we can say about them takes place within the domain of signs.  As we use the words, sign is the general category, while symbol is a particular type of sign.  From the pragmatic point of view, almost all the actual work of computation is involved with transformations between expressions in the symbolic domain.

(Awbrey & Awbrey, 1990)

To be continued …

Posted in C.S. Peirce, Information, Information Theory, Inquiry, Inquiry Driven Systems, Logic, Logic of Relatives, Mathematics, Peirce, Peirce List, Pragmatic Information, Pragmatism, Relation Theory, Semiotic Information, Semiotics, Sign Relations, Triadic Relations, Triadicity | Tagged , , , , , , , , , , , , , , , , , | 13 Comments