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

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

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

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

(Peirce, CP 3.77)

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

We come to the end of the “number of” examples that we noted at this point in the text.

NOF 4.5

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 the text at this point. Let us first recall the conventions we are using in this transcription, in particular, \mathit{1} for the italic 1 that signifies the dyadic identity relation and \mathfrak{1} for the “antique figure one” that Peirce defines as \mathit{1}_\infty = \text{something}.

CP 3 gives [\mathit{1}] = \mathfrak{1}, which I cannot make sense of. CE 2 gives the 1’s in different styles of italics, but reading the equation as [\mathit{1}] = 1, makes the best sense if the “1” on the right hand side is read as the numeral “1” that denotes the natural number 1, and not as the absolute term “1” that denotes the universe of discourse. In this reading, [\mathit{1}] is 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:

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

And so the “number of” mapping v : S \to \mathbb{R} has another one of the properties that would be required of an arrow S \to \mathbb{R}.

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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 ground this 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 4.4

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)

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 above by means of a case where independence fails. The details of that counterexample are summarized below.


LOR 1870 Figure 53
(54)

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)}.

To recall the derivation, the definition of conditional probability plus the independence condition yields \mathrm{P}(A|Z) = \displaystyle{\mathrm{P}(AZ) \over P(Z)} = \displaystyle{\mathrm{P}(A)\mathrm{P}(Z) \over \mathrm{P}(Z)}, in short, \mathrm{P}(A|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

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

NOF 4.4

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)

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

The example apparently assumes a universe of “things in general”, encompassing among other things the denotations of the absolute terms \mathrm{m} = \text{man} and \mathrm{b} = \text{black}. That suggests to me that we might well illustrate this case in relief, by returning to our earlier staging of Othello and seeing how well that universe of dramatic discourse observes the premiss that “men are just as apt to be black as things in general”.

Here are the relevant 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 is tantamount to this:  “Men are just as apt to be black as things in general are apt to be black”. In other words, men are a fair sample of things in general with respect to the factor of being black.

Should this hold, the consequence would be:

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

When [\mathrm{b}] is not zero, we obtain the result:

[\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{~~~~}. 

Consider the bigraph for the composition:

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

This is represented below in the equivalent form:

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


LOR 1870 Figure 53
(53)

Thus we observe one of the more factitious facts affecting this very special universe of discourse, namely:

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

This is equivalent to the implication \mathrm{b} \Rightarrow \mathrm{m} that Peirce would have written in the form \mathrm{b} ~-\!\!\!<~ \mathrm{m}.

That is enough to puncture any notion that \mathrm{b} and \mathrm{m} are statistically independent, but let us continue to develop the plot a bit more. 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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Peirce’s 1870 “Logic Of Relatives” • Comment 11.21

One more example and one more general observation and then we’ll be all caught up with our homework on Peirce’s “number of” function.

NOF 4.4

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)

The protasis, “men are just as apt to be black as things in general”, is elliptic in structure, and presents us with a potential ambiguity. If we had no further clue to its meaning, it might be read as either of the following:

  1. Men are just as apt to be black as things in general are apt to be black.
  2. Men are just as apt to be black as men are apt to be things in general.

The second interpretation, if grammatical, is pointless to state, since it equates a proper contingency with an absolute certainty. So I think it is safe to assume the following paraphrase of what Peirce intends:

  • Men are just as likely to be black as things in general are likely to be black.

Stated in terms of the conditional probability:

\mathrm{P}(\mathrm{b}|\mathrm{m}) ~=~ \mathrm{P}(\mathrm{b}).

From the definition of conditional probability:

\mathrm{P}(\mathrm{b}|\mathrm{m}) ~=~ \displaystyle{\mathrm{P}(\mathrm{b}\mathrm{m}) \over \mathrm{P}(\mathrm{m})}.

Equivalently:

\mathrm{P}(\mathrm{b}\mathrm{m}) ~=~ \mathrm{P}(\mathrm{b}|\mathrm{m})\mathrm{P}(\mathrm{m}).

Taking everything together, we obtain the following result:

\mathrm{P}(\mathrm{b}\mathrm{m}) ~=~ \mathrm{P}(\mathrm{b}|\mathrm{m})\mathrm{P}(\mathrm{m}) ~=~ \mathrm{P}(\mathrm{b})\mathrm{P}(\mathrm{m}).

This, of course, is the definition of independent events, as applied to the event of being Black and the event of being a Man. It seems to be the most likely guess that this is the meaning of Peirce’s statement about frequencies:

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

The terms of this equation can be normalized to produce the corresponding statement about probabilities:

\mathrm{P}(\mathrm{m}\mathrm{b}) ~=~ \mathrm{P}(\mathrm{m})\mathrm{P}(\mathrm{b}).

Let’s see if this checks out.

Let N be the number of things in general. In terms of Peirce’s “number of” function, then, we have the equation [\mathbf{1}] = N. On the assumption that \mathrm{m} and \mathrm{b} are associated with independent events, we obtain the following sequence of equations:

\begin{array}{lll}  [\mathrm{m,} \mathrm{b}] & = & \mathrm{P}(\mathrm{m}\mathrm{b}) N  \\[6pt]  & = & \mathrm{P}(\mathrm{m}) \mathrm{P}(\mathrm{b}) N  \\[6pt]  & = & \mathrm{P}(\mathrm{m}) [\mathrm{b}]  \\[6pt]  & = & [\mathrm{m,}] [\mathrm{b}].  \end{array}

As a result, we have to interpret [\mathrm{m,}] = “the average number of men per things in general” as \mathrm{P}(\mathrm{m}) = “the probability of a thing in general being a man”. This seems to make sense.

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