Peirce’s 1870 “Logic of Relatives” • Intermezzo

Posted on June 18, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives”

Update • 10 April 2022

This brings me to the end of the notes on Peirce’s 1870 Logic of Relatives I began posting to the web in various discussion groups a dozen (now a score) years ago.  Apart from that there are only the assorted notes and bits of discussion with other people I archived on the InterSciWiki talk page linked here.

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.

References

  • Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9, 317–378, 26 January 1870.  Reprinted, Collected Papers 3.45–149, Chronological Edition 2, 359–429.  Online (1) (2) (3).
  • Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.
  • Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.

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

Posted on June 15, 2014 by Jon Awbrey

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.

Matrix Equation ((S^L)^W)_X = (S^(LW))_X

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 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

Posted on June 14, 2014 by Jon Awbrey

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

Peirce next considers a pair of compound involutions, stating an equation between them analogous to a law of exponents from 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

(s^ℓ)^w = s^(ℓw)

(Peirce, CP 3.77)

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

Denotation Equation s^(ℓw)

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

Involution Example 2

Consider a universe of discourse X subject to the following data.

\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}

Bigraph Involution S^L
\text{Figure 56. Bigraph Involution}~ \mathsf{S}^\mathsf{L}

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.

Matrix Computation S^L

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

Posted on June 12, 2014 by Jon Awbrey

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

We now have two ways of computing a logical involution raising 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 applies set-theoretic operations to the extensions of absolute and relative terms, expressing the denotation of the term \mathit{l}^\mathrm{w} as the intersection of a set of relational applications.

Denotation Equation L^W

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

Matrix Computation L^W

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

Involution Example 1

Consider a universe of discourse X 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.

Bigraph Involution L^W
\text{Figure 55. Bigraph Involution}~ \mathsf{L}^\mathsf{W}

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

Computing the denotation of \mathit{l}^\mathrm{w} by way of the class intersection formula, we can show our work as follows.

Class Intersection L^W

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 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 producing a non-zero result is (\mathsf{L}^\mathsf{W})_e = 1.  That portion of the work can be sketched as follows.

Matrix Coefficient L^W

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

Posted on June 11, 2014 by Jon Awbrey

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

Let us make a few preliminary observations about the operation of logical involution which Peirce introduces in the following words.

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}.  On 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 on the order of “a lover of a lover of … a lover of a woman”.

By way of example, suppose a universe of discourse numbers among its elements just three women, \mathrm{W}^{\prime}, \mathrm{W}^{\prime\prime}, \mathrm{W}^{\prime\prime\prime}.  In Peirce’s notation the fact may be written as follows.

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

In that case 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})

The equation says a lover of every woman in the aggregate \mathrm{W}^{\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime\prime} 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 the universe at hand 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 which may be obtained by the following procedure.  For each flag of the form L \star x with x \in W collect the subset \mathrm{proj}_1 (L \star x) of elements which appear as the first components of the pairs in L \star x and then take the intersection of all those subsets.  Putting it all together, we have the following equation.

Denotation Equation ℓ^w

It is 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, we make the following observation.  To say j is a lover of every woman is to say j loves k if k is a woman.  That 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 which accounts for the aptness of the latter name.

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

Involution ≅ Implication

It is clear the two operations are isomorphic, being effectively the same operation of type \mathbb{B} \times \mathbb{B} \to \mathbb{B}.  All that remains is to see how operations like these on values in \mathbb{B} induce the corresponding operations on sets and terms.

The term \mathit{l}^\mathrm{w} determines a selection of individuals from the universe of discourse X which may be computed via 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 which produces the term \mathit{l}^\mathrm{w} must be represented by a corresponding operation on matrices, \mathsf{L}^\mathsf{W} = \mathrm{Mat}(\mathit{l})^{\mathrm{Mat}(\mathrm{w})}, which produces the matrix \mathrm{Mat}(\mathit{l}^\mathrm{w}).  In short, the involution operation on matrices must be defined in such a way that the following equation holds.

Matrix Involution L^W

The fact that \mathit{l}^\mathrm{w} denotes individuals in a subset of X tells us its matrix representation \mathsf{L}^\mathsf{W} is a 1‑dimensional array of coefficients in \mathbb{B} 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.

Matrix Computation L^W

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

Posted on June 10, 2014 by Jon Awbrey

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

To get a better sense of why Peirce’s formulas in Selection 12 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 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}.

A few concepts from the article on Relation Theory, touched on again in Comment 11.7, will also be useful.

The local flags of the relation L are defined as follows.

\begin{array}{lll} u \star L & = & L_{u\,@\,1} \\[6pt] & = & \{ (u, x) \in L \} \\[6pt] & = & \text{ordered pairs in}~ L ~\text{with}~ u ~\text{in the 1st place}. \\[9pt] L \star v & = & L_{v\,@\,2} \\[6pt] & = & \{ (x, v) \in L \} \\[6pt] & = & \text{ordered pairs in}~ L ~\text{with}~ 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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Peirce’s 1870 “Logic of Relatives” • Selection 12

Posted on June 9, 2014 by Jon Awbrey

On to the next part of §3. Application of the Algebraic Signs to Logic.

Peirce’s 1870 “Logic of Relatives” • Selection 12

The Sign of Involution

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.  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

(s^ℓ)^w = s^(ℓw)

(Peirce, CP 3.77)

References

  • Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9, 317–378, 26 January 1870.  Reprinted, Collected Papers 3.45–149, Chronological Edition 2, 359–429.  Online (1) (2) (3).
  • Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.
  • Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.

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Peirce’s 1870 “Logic of Relatives” • Comment 11.24

Posted on June 8, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.24

We come to the last of Peirce’s observations about the “number of” function in CP 3.76.

NOF 4.4

It is to be observed that

[\mathit{1}] ~=~ 1.

Boole was the first to show this connection between logic and probabilities.  He was restricted, however, to absolute terms.  I do not remember having seen any extension of probability to relatives, except the ordinary theory of expectation.

Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it.

(Peirce, CP 3.76 and CE 2, 376)

There are problems with the printing of text at this point.  To recall the conventions used in this transcription, the italic figure ``\mathit{1}" denotes the dyadic identity relation \mathit{1} while the antique figure ``\mathfrak{1}" denotes what Peirce otherwise defines as \mathit{1}_\infty = \text{something}.

Collected Papers CP 3 gives [\mathit{1}] = \mathfrak{1}, which does not make sense.  Chronological Edition CE 2 gives the 1’s in different styles of italics but reading the equation as [\mathit{1}] = 1 makes better sense if the latter “1” is the numeral denoting the natural number 1 and not the absolute term “1” denoting the universe of discourse.  The quantity [\mathit{1}] is defined as the average number of things related by the identity relation \mathit{1} to one individual, and so it makes sense that [\mathit{1}] = 1 \in \mathbb{N}, where \mathbb{N} is the set of non-negative integers \{ 0, 1, 2, \ldots \}.

With respect to the relative term ``\mathit{1}" in the syntactic domain S and the number 1 in the non-negative integers \mathbb{N} \subset \mathbb{R}, we have the following.

v(\mathit{1}) ~=~ [\mathit{1}] ~=~ 1.

At long last, then, the “number of” mapping v : S \to \mathbb{R} has another one of the properties required of an arrow from logical terms in S to real numbers in \mathbb{R}.

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Peirce’s 1870 “Logic of Relatives” • Comment 11.23

Posted on June 5, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.23

Peirce’s description of logical conjunction and conditional probability via the logic of relatives and the mathematics of relations is critical to understanding the relationship between logic and measurement, in effect, the qualitative and quantitative aspects of inquiry.  To root that connection firmly in mind, I will try to sum up as succinctly as possible, in more current notation, the lesson we ought to take away from Peirce’s last “number of” example, since I know the account I have given so far may appear to have wandered widely.

NOF 4.3

So if men are just as apt to be black as things in general,

[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}],

where the difference between [\mathrm{m}] and [\mathrm{m,}] must not be overlooked.

(Peirce, CP 3.76)

Viewed in different lights the formula [\mathrm{m,}\mathrm{b}] = [\mathrm{m,}][\mathrm{b}] presents itself as an aimed arrow, fair sampling, or statistical independence condition.  The concept of independence was illustrated in the previous installment by means of a case where independence fails.  The details of that counterexample are summarized below.

Othello Product M,B,
\text{Figure 54. Bigraph Product}~ M,B,

The condition that “men are just as apt to be black as things in general” is expressed in terms of conditional probabilities as \mathrm{P}(\mathrm{b}|\mathrm{m}) = \mathrm{P}(\mathrm{b}), which means that the probability of the event \mathrm{b} given the event \mathrm{m} is equal to the unconditional probability of the event \mathrm{b}.

In the Othello example it is enough to observe that \mathrm{P}(\mathrm{b}|\mathrm{m}) = \tfrac{1}{4} while \mathrm{P}(\mathrm{b}) = \tfrac{1}{7} in order to recognize the bias or dependency of the sampling map.

The reduction of a conditional probability to an absolute probability, as \mathrm{P}(A|Z) = \mathrm{P}(A), is one of the ways we come to recognize the condition of independence, \mathrm{P}(AZ) = \mathrm{P}(A)P(Z), via the definition of conditional probability, \mathrm{P}(A|Z) = \displaystyle{\mathrm{P}(AZ) \over \mathrm{P}(Z)}.

By way of recalling the derivation, the definition of conditional probability plus the independence condition yields the following sequence of equations.

\mathrm{P}(A|Z) = \displaystyle{\mathrm{P}(AZ) \over P(Z)} = \displaystyle{\mathrm{P}(A)\mathrm{P}(Z) \over \mathrm{P}(Z)} = \mathrm{P}(A).

As Hamlet discovered, there’s a lot to be learned from turning a crank.

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Peirce’s 1870 “Logic of Relatives” • Comment 11.22

Posted on June 4, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.22

Let’s look at that last example from a different angle.

NOF 4.3

So if men are just as apt to be black as things in general,

[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}],

where the difference between [\mathrm{m}] and [\mathrm{m,}] must not be overlooked.

(Peirce, CP 3.76)

Viewed in various lights the formula [\mathrm{m,}\mathrm{b}] = [\mathrm{m,}][\mathrm{b}] presents itself as an aimed arrow, fair sampling, or stochastic independence condition.

Peirce’s example assumes a universe of things in general encompassing the denotations of the absolute terms \mathrm{m} = \text{man} and \mathrm{b} = \text{black}.  That allows us to illustrate the case in relief, by returning to our earlier staging of Othello and examining the premiss that “men are just as apt to be black as things in general” within the frame of that empirical if fictional universe of discourse.

We have the following data.

\begin{array}{*{15}{l}} \mathrm{b} & = & \mathrm{O} \\[6pt] \mathrm{m} & = & \mathrm{C} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[6pt] \mathbf{1} & = & \mathrm{B} & +\!\!, & \mathrm{C} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[12pt] \mathrm{b,} & = & \mathrm{O\!:\!O} \\[6pt] \mathrm{m,} & = & \mathrm{C\!:\!C} & +\!\!, & \mathrm{I\!:\!I} & +\!\!, & \mathrm{J\!:\!J} & +\!\!, & \mathrm{O\!:\!O} \\[6pt] \mathbf{1,} & = & \mathrm{B\!:\!B} & +\!\!, & \mathrm{C\!:\!C} & +\!\!, & \mathrm{D\!:\!D} & +\!\!, & \mathrm{E\!:\!E} & +\!\!, & \mathrm{I\!:\!I} & +\!\!, & \mathrm{J\!:\!J} & +\!\!, & \mathrm{O\!:\!O} \end{array}

The fair sampling condition amounts to saying men are just as likely to be black as things in general are likely to be black.  In other words, men are a fair sample of things in general with respect to the predicate of being black.

On that condition the following equation holds.

[\mathrm{m,}\mathrm{b}] ~=~ [\mathrm{m,}][\mathrm{b}].

Assuming [\mathrm{b}] is not zero, the next equation follows.

[\mathrm{m,}] ~=~ \displaystyle{[\mathrm{m,}\mathrm{b}] \over [\mathrm{b}]}.

As before, it is convenient to represent the absolute term \mathrm{b} = \text{black} by means of the corresponding idempotent term \mathrm{b,} = \text{black that is}\,\underline{~~~~}. 

Let is next consider the bigraph for the following relational product.

\mathrm{m,}\mathrm{b} ~=~ \text{man that is black}.

We may represent that in the following equivalent form.

\mathrm{m,}\mathrm{b,} ~=~ \text{man that is black that is}\,\underline{~~~~}.

Othello Product M,B,
\text{Figure 53. Bigraph Product}~ M,B,

The facts of the matter in the Othello case are such that the following formula holds.

\mathrm{m,}\mathrm{b} ~=~ \mathrm{b}.

And that in turn is equivalent to each of the following statements.

\begin{matrix} m \land b = b \\[6pt] \mathrm{b} \implies \mathrm{m} \\[6pt] \mathrm{b} ~-\!\!\!< \mathrm{m} \end{matrix}

Those last implications puncture any notion of statistical independence for \mathrm{b} and \mathrm{m} in the universe of discourse at hand but it will repay us to explore the details of the case a little further.  Putting all the general formulas and particular facts together, we arrive at the following summation of the situation in the Othello case.

If the fair sampling condition were true, it would have the following consequence.

\displaystyle [\mathrm{m,}] ~=~ {[\mathrm{m,}\mathrm{b}] \over [\mathrm{b}]} ~=~ {[\mathrm{b}] \over [\mathrm{b}]} ~=~ \mathfrak{1}.

On the contrary, we have the following fact.

\displaystyle [\mathrm{m,}] ~=~ {[\mathrm{m,}\mathbf{1}] \over [\mathbf{1}]} ~=~ {[\mathrm{m}] \over [\mathbf{1}]} ~=~ {4 \over 7}.

In sum, it is not the case in the Othello example that “men are just as apt to be black as things in general”.

Expressed in terms of probabilities:

\mathrm{P}(\mathrm{m}) = \displaystyle{4 \over 7}   and   \mathrm{P}(\mathrm{b}) = \displaystyle{1 \over 7}.

If these were independent terms, we would have:

\mathrm{P}(\mathrm{m}\mathrm{b}) = \displaystyle{4 \over 49}.

In point of fact, however, we have:

\mathrm{P}(\mathrm{m}\mathrm{b}) = \mathrm{P}(\mathrm{b}) = \displaystyle{1 \over 7}.

Another way to see it is to observe that:

\mathrm{P}(\mathrm{b}|\mathrm{m}) = \displaystyle{1 \over 4}   while   \mathrm{P}(\mathrm{b}) = \displaystyle{1 \over 7}.

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