Doubt, Uncertainty, Dispersion, Entropy : 1

Re: Stephen Rose • The Second Law of Thermodynamics

Just a note to anchor a series of recurring thoughts that come to mind in relation to a Peirce List discussion of entropy etc., but I won’t have much to say on the bio-chemico-physico-thermo-dynamic side of things, so I’ll spin this off under a separate heading.  My interest in this topic arises mainly from my long-time work on inquiry driven systems (1)(2)(3)(4)(5)(6), where understanding the intertwined measures of uncertainty and information is critical to comprehending the dynamics of inquiry.

In a famous passage, Peirce says that inquiry begins with the “irritation of doubt” and ends when the irritation is soothed.  Here we find the same compound of affective and cognitive ingredients that we find in Aristotle’s original recipe for the sign relation.

When we view inquiry as a process taking place in a system the first thing we have to ask is what are the properties or variables that we need to consider in describing the state of the system at any given time.  Taking a Peircean perspective on a system capable of undergoing anything like an inquiry process, we are led to ask what are the conditions for the possibility of a system having “states of uncertainty” and “states of information” as state variables.

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Peirce’s 1870 “Logic Of Relatives” • Intermezzo

This brings me to the end of the notes on Peirce’s 1870 Logic of Relatives that I began posting to the web in various online discussion groups a dozen years ago. Apart from that there are only the scattered notes and bits of discussion with others that I’ve archived on the discussion page of the collateral InterSciWiki article.

I rushed through my last few comments a little too hastily, giving no more than sketches of proofs for Peirce’s logical formulas, and I won’t be reasonably well convinced of them until I examine a few more concrete examples and develop one or two independent lines of proof. So I have that much unfinished business to do before moving on to the rest of Peirce’s paper.

But I’ll take a few days to catch my breath, rummage through those old notes of mine to see if they hide any hints worth salvaging, and then start fresh, raveling out the rest of Peirce’s clues to the maze of logical relatives.

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Peirce’s 1870 “Logic Of Relatives” • Comment 12.5

The equation (\mathit{s}^\mathit{l})^\mathrm{w} = \mathit{s}^{\mathit{l}\mathrm{w}} can be verified by establishing the corresponding equation in matrices:

(\mathsf{S}^\mathsf{L})^\mathsf{W} ~=~ \mathsf{S}^{\mathsf{L}\mathsf{W}}

If \mathsf{A} and \mathsf{B} are two 1-dimensional matrices over the same index set X then \mathsf{A} = \mathsf{B} if and only if \mathsf{A}_x = \mathsf{B}_x for every x \in X.  Thus, a routine way to check the validity of (\mathsf{S}^\mathsf{L})^\mathsf{W} = \mathsf{S}^{\mathsf{L}\mathsf{W}} is to check whether the following equation holds for arbitrary x \in X.

((\mathsf{S}^\mathsf{L})^\mathsf{W})_x ~=~ (\mathsf{S}^{\mathsf{L}\mathsf{W}})_x

Taking both ends toward the middle, we proceed as follows:

\begin{array}{*{7}{l}}  ((\mathsf{S}^\mathsf{L})^\mathsf{W})_x  & = & \displaystyle  \prod_{p \in X} (\mathsf{S}^\mathsf{L})_{xp}^{\mathsf{W}_p}  & = & \displaystyle  \prod_{p \in X} (\prod_{q \in X} \mathsf{S}_{xq}^{\mathsf{L}_{qp}})^{\mathsf{W}_p}  & = & \displaystyle  \prod_{p \in X} \prod_{q \in X} \mathsf{S}_{xq}^{\mathsf{L}_{qp}\mathsf{W}_p}  \\[36px]  (\mathsf{S}^{\mathsf{L}\mathsf{W}})_x  & = & \displaystyle  \prod_{q \in X} \mathsf{S}_{xq}^{(\mathsf{L}\mathsf{W})_q}  & = & \displaystyle  \prod_{q \in X} \mathsf{S}_{xq}^{\sum_{p \in X} \mathsf{L}_{qp} \mathsf{W}_p}  & = & \displaystyle  \prod_{q \in X} \prod_{p \in X} \mathsf{S}_{xq}^{\mathsf{L}_{qp} \mathsf{W}_p}  \end{array}

The products commute, so the equation holds.  In essence, the matrix identity turns on the fact that the law of exponents (a^b)^c = a^{bc} in ordinary arithmetic holds when the values a, b, c are restricted to the boolean domain \mathbb{B} = \{ 0, 1 \}.  Interpreted as a logical statement, the law of exponents (a^b)^c = a^{bc} amounts to a theorem of propositional calculus that is otherwise expressed in the following ways:

\begin{matrix}  (a \Leftarrow b) \Leftarrow c & = & a \Leftarrow b \land c  \\[8pt]  c \Rightarrow (b \Rightarrow a) & = & c \land b \Rightarrow a  \end{matrix}

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Peirce’s 1870 “Logic Of Relatives” • Comment 12.4

Peirce next considers a pair of compound involutions, stating an equation between them that is analogous to a law of exponents in ordinary arithmetic, namely, (a^b)^c = a^{bc}.

Then (\mathit{s}^\mathit{l})^\mathrm{w} will denote whatever stands to every woman in the relation of servant of every lover of hers;  and \mathit{s}^{(\mathit{l}\mathrm{w})} will denote whatever is a servant of everything that is lover of a woman.  So that

(\mathit{s}^\mathit{l})^\mathrm{w} ~=~ \mathit{s}^{(\mathit{l}\mathrm{w})}.

(Peirce, CP 3.77)

Articulating the compound relative term \mathit{s}^{(\mathit{l}\mathrm{w})} in set-theoretic terms is fairly immediate:

\displaystyle \mathit{s}^{(\mathit{l}\mathrm{w})} ~=~ \bigcap_{x \in LW} \mathrm{proj}_1 (S \star x) ~=~ \bigcap_{x \in LW} S \cdot x

On the other hand, translating the compound relative term (\mathit{s}^\mathit{l})^\mathrm{w} into a set-theoretic equivalent is less immediate, the hang-up being that we have yet to define the case of logical involution that raises a dyadic relative term to the power of a dyadic relative term.  As a result, it looks easier to proceed through the matrix representation, drawing once again on the inspection of a concrete example.

Example 7

\begin{array}{*{15}{c}}  X & = & \{ & a, & b, & c, & d, & e, & f, & g, & h, & i\ & \}  \\[6pt]  L & = & \{ & b\!:\!a, & b\!:\!c, & c\!:\!b, & c\!:\!d, & e\!:\!d, & e\!:\!e, & e\!:\!f, & g\!:\!f, & g\!:\!h, & h\!:\!g, & h\!:\!i & \}  \\[6pt]  S & = & \{ & b\!:\!a, & b\!:\!c, & d\!:\!c, & d\!:\!d, & d\!:\!e, & f\!:\!e, & f\!:\!f, & f\!:\!g, & h\!:\!g, & h\!:\!i\ & \}  \end{array}


LOR 1870 Figure 56
(56)

There is a “servant of every lover of” link between u and v if and only if u \cdot S ~\supseteq~ L \cdot v.  But the vacuous inclusions, that is, the cases where L \cdot v = \varnothing, have the effect of adding non-intuitive links to the mix.

The computational requirements are evidently met by the following formula:

\displaystyle (\mathsf{S}^\mathsf{L})_{xy} ~=~ \prod_{p \in X} \mathsf{S}_{xp}^{\mathsf{L}_{py}}

In other words, (\mathsf{S}^\mathsf{L})_{xy} = 0 if and only if there exists a p \in X such that \mathsf{S}_{xp} = 0 and \mathsf{L}_{py} = 1.

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Peirce’s 1870 “Logic Of Relatives” • Comment 12.3

We now have two ways of computing a logical involution that raises a dyadic relative term to the power of a monadic absolute term, for example, \mathit{l}^\mathrm{w} for “lover of every woman”.

The first method operates in the medium of set theory, expressing the denotation of the term \mathit{l}^\mathrm{w} as the intersection of a set of relational applications:

\displaystyle \mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} L \cdot x

The second method operates in the matrix representation, expressing the value of the matrix \mathsf{L}^\mathsf{W} with respect to an argument u as a product of coefficient powers:

\displaystyle (\mathsf{L}^\mathsf{W})_u ~=~ \prod_{v \in X} \mathsf{L}_{uv}^{\mathsf{W}_v}

Abstract formulas like these are more easily grasped with the aid of a concrete example and a picture of the relations involved.

Example 6

Consider a universe of discourse X that is subject to the following data:

\begin{array}{*{15}{c}}  X & = & \{ & a, & b, & c, & d, & e, & f, & g, & h, & i & \}  \\[6pt]  W & = & \{ & d, & f & \}  \\[6pt]  L & = & \{ & b\!:\!a, & b\!:\!c, & c\!:\!b, & c\!:\!d, & e\!:\!d, & e\!:\!e, & e\!:\!f, & g\!:\!f, & g\!:\!h, & h\!:\!g, & h\!:\!i & \}  \end{array}

Figure 55 shows the placement of W within X and the placement of L within X \times X.


LOR 1870 Figure 55
(55)

To highlight the role of W more clearly, the Figure represents the absolute term ``\mathrm{w}" by means of the relative term ``\mathrm{w},\!" that conveys the same information.

Computing the denotation of \mathit{l}^\mathrm{w} by way of the set-theoretic formula, we can show our work as follows:

\displaystyle \mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} L \cdot x ~=~ L \cdot d ~\cap~ L \cdot f ~=~ \{ c, e \} \cap \{ e, g \} ~=~ \{ e \}

With the above Figure in mind, we can visualize the computation of \textstyle (\mathsf{L}^\mathsf{W})_u = \prod_{v \in X} \mathsf{L}_{uv}^{\mathsf{W}_v} as follows:

  1. Pick a specific u in the bottom row of the Figure.
  2. Pan across the elements v in the middle row of the Figure.
  3. If u links to v then \mathsf{L}_{uv} = 1, otherwise \mathsf{L}_{uv} = 0.
  4. If v in the middle row links to v in the top row then \mathsf{W}_v = 1, otherwise \mathsf{W}_v = 0.
  5. Compute the value \mathsf{L}_{uv}^{\mathsf{W}_v} = (\mathsf{L}_{uv} \Leftarrow \mathsf{W}_v) for each v in the middle row.
  6. If any of the values \mathsf{L}_{uv}^{\mathsf{W}_v} is 0 then the product \textstyle \prod_{v \in X} \mathsf{L}_{uv}^{\mathsf{W}_v} is 0, otherwise it is 1.

As a general observation, we know that the value of (\mathsf{L}^\mathsf{W})_u goes to 0 just as soon as we find a v \in X such that \mathsf{L}_{uv} = 0 and \mathsf{W}_v = 1, in other words, such that (u, v) \notin L but v \in W.  If there is no such v then (\mathsf{L}^\mathsf{W})_u = 1.

Running through the program for each u \in X, the only case that produces a non-zero result is (\mathsf{L}^\mathsf{W})_e = 1.  That portion of the work can be sketched as follows:

\displaystyle (\mathsf{L}^\mathsf{W})_e ~=~ \prod_{v \in X} \mathsf{L}_{ev}^{\mathsf{W}_v} ~=~ 0^0 \cdot 0^0 \cdot 0^0 \cdot 1^1 \cdot 1^0 \cdot 1^1 \cdot 0^0 \cdot 0^0 \cdot 0^0 ~=~ 1

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Peirce’s 1870 “Logic Of Relatives” • Comment 12.2

Let us make a few preliminary observations about the operation of logical involution, as Peirce introduces it here:

I shall take involution in such a sense that x^y will denote everything which is an x for every individual of y.  Thus \mathit{l}^\mathrm{w} will be a lover of every woman.

(Peirce, CP 3.77)

In ordinary arithmetic the involution x^y, or the exponentiation of x to the power y, is the repeated application of the multiplier x for as many times as there are ones making up the exponent y.

In analogous fashion, the logical involution \mathit{l}^\mathrm{w} is the repeated application of the term \mathit{l} for as many times as there are individuals under the term \mathrm{w}.  According to Peirce’s interpretive rules, the repeated applications of the base term \mathit{l} are distributed across the individuals of the exponent term \mathrm{w}.  In particular, the base term \mathit{l} is not applied successively in the manner that would give something like “a lover of a lover of … a lover of a woman”.

For example, suppose that a universe of discourse numbers among its elements just three women, \mathrm{W}^{\prime}, \mathrm{W}^{\prime\prime}, \mathrm{W}^{\prime\prime\prime}.  This could be expressed in Peirce’s notation by writing:

\mathrm{w} ~=~ \mathrm{W}^{\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime\prime}

Under these circumstances the following equation would hold:

\mathit{l}^\mathrm{w} ~=~ \mathit{l}^{(\mathrm{W}^{\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime\prime})} ~=~ (\mathit{l}\mathrm{W}^{\prime}), (\mathit{l}\mathrm{W}^{\prime\prime}), (\mathit{l}\mathrm{W}^{\prime\prime\prime})

This says that a lover of every woman in the given universe of discourse is a lover of \mathrm{W}^{\prime} that is a lover of \mathrm{W}^{\prime\prime} that is a lover of \mathrm{W}^{\prime\prime\prime}.  In other words, a lover of every woman in this context is a lover of \mathrm{W}^{\prime} and a lover of \mathrm{W}^{\prime\prime} and a lover of \mathrm{W}^{\prime\prime\prime}.

The denotation of the term \mathit{l}^\mathrm{w} is a subset of X that can be obtained as follows:  For each flag of the form L \star x with x \in W, collect the elements \mathrm{proj}_1 (L \star x) that appear as the first components of these ordered pairs, and then take the intersection of all these subsets.  Putting it all together:

\displaystyle \mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} \mathrm{proj}_1 (L \star x) ~=~ \bigcap_{x \in W} L \cdot x

It is very instructive to examine the matrix representation of \mathit{l}^\mathrm{w} at this point, not the least because it effectively dispels the mystery of the name involution.  First, let us make the following observation.  To say that j is a lover of every woman is to say that j loves k if k is a woman.  This can be rendered in symbols as follows:

j ~\text{loves}~ k ~\Leftarrow~ k ~\text{is a woman}

Reading the formula \mathit{l}^\mathrm{w} as “j loves k if k is a woman” highlights the operation of converse implication inherent in it, and this in turn reveals the analogy between implication and involution that accounts for the aptness of the latter name.

The operations defined by the formulas   x^y = z   and   (x\!\Leftarrow\!y) = z   for x, y, z in the boolean domain \mathbb{B} = \{ 0, 1 \} are tabulated as follows:

\begin{array}{ccc}  x^y & = & z \\  \hline  0^0 & = & 1 \\  0^1 & = & 0 \\  1^0 & = & 1 \\  1^1 & = & 1  \end{array}  \qquad\qquad\qquad  \begin{array}{ccc}  x\!\Leftarrow\!y & = & z \\  \hline  0\!\Leftarrow\!0 & = & 1 \\  0\!\Leftarrow\!1 & = & 0 \\  1\!\Leftarrow\!0 & = & 1 \\  1\!\Leftarrow\!1 & = & 1  \end{array}

It is clear that these operations are isomorphic, amounting to the same operation of type \mathbb{B} \times \mathbb{B} \to \mathbb{B}.  All that remains is to see how this operation on coefficient values in \mathbb{B} induces the corresponding operations on sets and terms.

The term \mathit{l}^\mathrm{w} determines a selection of individuals from the universe of discourse X that may be computed by means of the corresponding operation on coefficient matrices.  If the terms \mathit{l} and \mathrm{w} are represented by the matrices \mathsf{L} = \mathrm{Mat}(\mathit{l}) and \mathsf{W} = \mathrm{Mat}(\mathrm{w}), respectively, then the operation on terms that produces the term \mathit{l}^\mathrm{w} must be represented by a corresponding operation on matrices, say, \mathsf{L}^\mathsf{W} = \mathrm{Mat}(\mathit{l})^{\mathrm{Mat}(\mathrm{w})}, that produces the matrix \mathrm{Mat}(\mathit{l}^\mathrm{w}).  In other words, the involution operation on matrices must be defined in such a way that the following equations hold:

\mathsf{L}^\mathsf{W} ~=~ \mathrm{Mat}(\mathit{l})^{\mathrm{Mat}(\mathrm{w})} ~=~ \mathrm{Mat}(\mathit{l}^\mathrm{w})

The fact that \mathit{l}^\mathrm{w} denotes the elements of a subset of X means that the matrix \mathsf{L}^\mathsf{W} is a 1-dimensional array of coefficients in \mathbb{B} that is indexed by the elements of X.  The value of the matrix \mathsf{L}^\mathsf{W} at the index u \in X is written (\mathsf{L}^\mathsf{W})_u and computed as follows:

\displaystyle (\mathsf{L}^\mathsf{W})_u ~=~ \prod_{v \in X} \mathsf{L}_{uv}^{\mathsf{W}_v}

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Peirce’s 1870 “Logic Of Relatives” • Comment 12.1

To get a better sense of why the above formulas mean what they do, and to prepare the ground for understanding more complex relational expressions, it will help to assemble the following materials and definitions:

X is a set singled out in a particular discussion as the universe of discourse.

W \subseteq X is the monadic relation, or set, whose elements fall under the absolute term \mathrm{w} = \text{woman}. The elements of W are referred to as the denotation or the extension of the term \mathrm{w}.

L \subseteq X \times X is the dyadic relation associated with the relative term \mathit{l} = \text{lover of}\,\underline{~~~~}.

S \subseteq X \times X is the dyadic relation associated with the relative term \mathit{s} = \text{servant of}\,\underline{~~~~}.

\mathsf{W} = (\mathsf{W}_x) = \mathrm{Mat}(W) = \mathrm{Mat}(\mathrm{w}) is the 1-dimensional matrix representation of the set W and the term \mathrm{w}.

\mathsf{L} = (\mathsf{L}_{xy}) = \mathrm{Mat}(L) = \mathrm{Mat}(\mathit{l}) is the 2-dimensional matrix representation of the relation L and the relative term \mathit{l}.

\mathsf{S} = (\mathsf{S}_{xy}) = \mathrm{Mat}(S) = \mathrm{Mat}(\mathit{s}) is the 2-dimensional matrix representation of the relation S and the relative term \mathit{s}.

Recalling a few definitions, the local flags of the relation L are given as follows:

\begin{array}{lll}  u \star L  & = & L_{u\,@\,1}  \\[6pt]  & = & \{ (u, x) \in L \}  \\[6pt]  & = & \text{the ordered pairs in}~ L ~\text{that have}~ u ~\text{in the 1st place}.  \\[9pt]  L \star v  & = & L_{v\,@\,2}  \\[6pt]  & = & \{ (x, v) \in L \}  \\[6pt]  & = & \text{the ordered pairs in}~ L ~\text{that have}~ v ~\text{in the 2nd place}.  \end{array}

The applications of the relation L are defined as follows:

\begin{array}{lll}  u \cdot L  & = & \mathrm{proj}_2 (u \star L)  \\[6pt]  & = & \{ x \in X : (u, x) \in L \}  \\[6pt]  & = & \text{loved by}~ u.  \\[9pt]  L \cdot v  & = & \mathrm{proj}_1 (L \star v)  \\[6pt]  & = & \{ x \in X : (x, v) \in L \}  \\[6pt]  & = & \text{lover of}~ v.  \end{array}

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