☯ Quantum Mechanics ☯


Particle ☯ Wave

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Why is there so much falsity in the world?

Because people prefer falsity to truth, illusion to reality.

Being the drift of my reflections on the plays I saw at Stratford this summer —
King Lear, King John, Man of La Mancha, Alice Through the Looking-Glass,
Crazy for You, Hay Fever.

The Beaux’ Stratagem • Masks, Madness, & Sonnets • Antony and Cleopatra

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Doubt, Uncertainty, Dispersion, Entropy : 2

Re: John Baez • Entropy and Information in Biological Systems

Re: To develop the concept of evolutionary games as “learning” processes in which information is gained over time.

My customary recommendation on this point is to look more deeply into the work of C.S. Peirce on the themes of evolution, inquiry, and their interaction.  Peirce stands out as one of the few pioneers in the study of scientific method who managed to avoid the dead-ends of naive deductivisim and naive inductivisim.  He developed Aristotle’s concept of abductive reasoning in a way that anticipated later insights into the dynamics of paradigm shifts.  A question worth exploring in this connection is whether abductive hypothesis formation is the analogue within scientific method of random mutation.

Posted in Biological Systems, Cybernetics, Dispersion, Doubt, Entropy, Evolution, Information, Information Theory, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Learning Theory, Peirce, Peirce List, Uncertainty | Tagged , , , , , , , , , , , , , , | Leave a comment

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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