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| | ===Commentary Note 12.1=== | | ===Commentary Note 12.1=== |
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| − | Let us make a few preliminary observations about the operation of ''logical involution'', as Peirce introduces it here:
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| − | {| align="center" cellspacing="6" width="90%" <!--QUOTE-->
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| − | <p>I shall take involution in such a sense that <math>x^y\!</math> will denote everything which is an <math>x\!</math> for every individual of <math>y.\!</math></p>
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| − | <p>Thus</p>
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| − | | align="center" | <math>\mathit{l}^\mathrm{w}\!</math>
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| − | <p>will be a lover of every woman.</p>
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| − | <p>(Peirce, CP 3.77).</p>
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| − | |}
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| − | In ordinary arithmetic the ''involution'' <math>x^y,\!</math> or the ''exponentiation'' of <math>x\!</math> to the power of <math>y,\!</math> is the repeated application of the multiplier <math>x\!</math> for as many times as there are ones making up the exponent <math>y.\!</math>
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| − | In analogous fashion, the logical involution <math>\mathit{l}^\mathrm{w}\!</math> is the repeated application of the term <math>\mathit{l}\!</math> for as many times as there are individuals under the term <math>\mathrm{w}.\!</math> According to Peirce's interpretive rules, the repeated applications of the base term <math>\mathit{l}\!</math> are distributed across the individuals of the exponent term <math>\mathrm{w}.\!</math> In particular, the base term <math>\mathit{l}\!</math> is not applied successively in the manner that would give something like "a lover of a lover of … a lover of a woman".
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| − | For example, suppose that a universe of discourse numbers among its contents just three women, <math>\mathrm{W}^{\prime}, \mathrm{W}^{\prime\prime}, \mathrm{W}^{\prime\prime\prime}.</math> This could be expressed in Peirce's notation by writing:
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| − | {| align="center" cellspacing="6" width="90%"
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| − | | <math>\mathrm{w} ~=~ \mathrm{W}^{\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime} ~+\!\!,~ \mathrm{W}^{\prime\prime\prime}</math>
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| − | Under these circumstances the following equation would hold:
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| − | {| align="center" cellspacing="6" width="90%"
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| − | | <math>\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}).</math>
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| − | This says that a lover of every woman in the given universe of discourse is a lover of <math>\mathrm{W}^{\prime}</math> that is a lover of <math>\mathrm{W}^{\prime\prime}</math> that is a lover of <math>\mathrm{W}^{\prime\prime\prime}.</math> In other words, a lover of every woman in this context is a lover of <math>\mathrm{W}^{\prime}</math> and a lover of <math>\mathrm{W}^{\prime\prime}</math> and a lover of <math>\mathrm{W}^{\prime\prime\prime}.</math>
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| − | 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:
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| − | {| align="center" cellspacing="6" width="90%"
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| − | | height="40" | <math>X\!</math> is a set singled out in a particular discussion as the ''universe of discourse''.
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| − | | height="40" | <math>W \subseteq X</math> is the 1-adic relation, or set, whose elements fall under the absolute term <math>\mathrm{w} = \text{woman}.\!</math> The elements of <math>W\!</math> are sometimes referred to as the ''denotation'' or the set-theoretic ''extension'' of the term <math>\mathrm{w}.\!</math>
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| − | |-
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| − | | height="40" | <math>L \subseteq X \times X\!</math> is the 2-adic relation associated with the relative term <math>\mathit{l} = \text{lover of}\,\underline{~~~~}.</math>
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| − | | height="40" | <math>S \subseteq X \times X\!</math> is the 2-adic relation associated with the relative term <math>\mathit{s} = \text{servant of}\,\underline{~~~~}.</math>
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| − | |}
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| − | {| align="center" cellspacing="6" width="90%"
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| − | | height="40" | <math>\mathfrak{W} = (\mathfrak{W}_x) = \operatorname{Mat}(W) = \operatorname{Mat}(\mathrm{w})</math> is the 1-dimensional matrix representation of the set <math>W\!</math> and the term <math>\mathrm{w}.\!</math>
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| − | |-
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| − | | height="40" | <math>\mathfrak{L} = (\mathfrak{L}_{xy}) = \operatorname{Mat}(L) = \operatorname{Mat}(\mathit{l})</math> is the 2-dimensional matrix representation of the relation <math>L\!</math> and the relative term <math>\mathit{l}.\!</math>
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| − | |-
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| − | | height="40" | <math>\mathfrak{S} = (\mathfrak{S}_{xy}) = \operatorname{Mat}(S) = \operatorname{Mat}(\mathit{s})</math> is the 2-dimensional matrix representaion of the relation <math>S\!</math> and the relative term <math>\mathit{s}.\!</math>
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| − | |}
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| − | Recalling a few definitions, the ''local flags'' of the relation <math>L\!</math> are given as follows:
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| − | {| align="center" cellspacing="6" width="90%"
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| − | <math>\begin{array}{lll}
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| − | u \star L
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| − | & = &
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| − | L_{u \,\text{at}\, 1}
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| − | \\[6pt]
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| − | & = & \{ (u, x) \in L \}
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| − | \\[6pt]
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| − | & = &
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| − | \text{the ordered pairs in}~ L ~\text{that have}~ u ~\text{in the 1st place}.
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| − | \\[9pt]
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| − | L \star v
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| − | & = &
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| − | L_{v \,\text{at}\, 2}
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| − | \\[6pt]
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| − | & = &
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| − | \{ (x, v) \in L \}
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| − | \\[6pt]
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| − | & = &
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| − | \text{the ordered pairs in}~ L ~\text{that have}~ v ~\text{in the 2nd place}.
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| − | \end{array}</math>
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| − | |}
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| − | The ''applications'' of the relation <math>L\!</math> are defined as follows:
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| − | {| align="center" cellspacing="6" width="90%"
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| − | <math>\begin{array}{lll}
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| − | u \cdot L
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| − | & = &
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| − | \operatorname{proj}_2 (u \star L)
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| − | \\[6pt]
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| − | & = &
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| − | \{ x \in X : (u, x) \in L \}
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| − | \\[6pt]
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| − | & = &
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| − | \text{loved by}~ u.
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| − | \\[9pt]
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| − | L \cdot v
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| − | & = &
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| − | \operatorname{proj}_1 (L \star v)
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| − | \\[6pt]
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| − | & = &
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| − | \{ x \in X : (x, v) \in L \}
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| − | \\[6pt]
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| − | & = &
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| − | \text{lover of}~ v.
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| − | \end{array}</math>
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| − | |}
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| − | The denotation of the term <math>\mathit{l}^\mathrm{w}\!</math> is a subset of <math>X\!</math> that can be obtained as follows: For each flag of the form <math>L \star x</math> with <math>x \in W,</math> collect the elements <math>\operatorname{proj}_1 (L \star x)</math> that appear as the first components of these ordered pairs, and then take the intersection of all these subsets. Putting it all together:
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| − | {| align="center" cellspacing="6" width="90%"
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| − | | <math>\mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} \operatorname{proj}_1 (L \star x) ~=~ \bigcap_{x \in W} L \cdot x</math>
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| − | It is very instructive to examine the matrix representation of <math>\mathit{l}^\mathrm{w}\!</math> 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 <math>\mathrm{J}\!</math> is a lover of every woman is to say that <math>\mathrm{J}\!</math> loves <math>\mathrm{K}\!</math> if <math>\mathrm{K}\!</math> is a woman. This can be rendered in symbols as follows:
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| − | {| align="center" cellspacing="6" width="90%"
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| − | | <math>\mathrm{J} ~\text{loves}~ \mathrm{K} ~\Leftarrow~ \mathrm{K} ~\text{is a woman}</math>
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| − | |}
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| − | Interpreting the formula <math>\mathit{l}^\mathrm{w}\!</math> as <math>\mathrm{J} ~\text{loves}~ \mathrm{K} ~\Leftarrow~ \mathrm{K} ~\text{is a woman}</math> highlights the form of the 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.
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| − | The operations of the forms <math>x^y = z\!</math> and <math>(x\!\Leftarrow\!y) = z</math> for <math>x, y, z \in \mathbb{B} = \{ 0, 1 \}</math> are tabulated below:
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| − | {| align="center" cellspacing="6" width="90%"
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| − | <math>
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| − | \begin{matrix}
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| − | 0^0 & = & 1
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| − | 0^1 & = & 0
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| − | \\
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| − | 1^0 & = & 1
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| − | \\
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| − | 1^1 & = & 1
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| − | \end{matrix}
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| − | \qquad\qquad\qquad
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| − | \begin{matrix}
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| − | 0\!\Leftarrow\!0 & = & 1
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| − | \\
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| − | 0\!\Leftarrow\!1 & = & 0
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| − | \\
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| − | 1\!\Leftarrow\!0 & = & 1
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| − | \\
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| − | 1\!\Leftarrow\!1 & = & 1
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| − | \end{matrix}
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| − | </math>
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| − | |}
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| − | It is clear that these operations are isomorphic, amounting to the same operation of type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math> All that remains is to see how this operation on coefficient values in <math>\mathbb{B}</math> induces the corresponding operations on sets and terms.
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| − | The term <math>\mathit{l}^\mathrm{w}\!</math> determines a selection of individuals from the universe of discourse <math>X\!</math> that may be computed by means of the corresponding operation on coefficient matrices. If the terms <math>\mathit{l}\!</math> and <math>\mathrm{w}\!</math> are represented by the matrices <math>\mathfrak{L} = \operatorname{Mat}(\mathit{l})</math> and <math>\mathfrak{W} = \operatorname{Mat}(\mathrm{w}),</math> respectively, then the operation on terms that produces the term <math>\mathit{l}^\mathrm{w}\!</math> must be represented by a corresponding operation on matrices, say, <math>\mathfrak{L}^\mathfrak{W} = \operatorname{Mat}(\mathit{l})^{\operatorname{Mat}(\mathrm{w})},</math> that produces the matrix <math>\operatorname{Mat}(\mathit{l}^\mathrm{w}).</math> In other words, the involution operation on matrices must be defined in such a way that the following equations hold:
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| − | {| align="center" cellspacing="6" width="90%"
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| − | | height="60" | <math>\mathfrak{L}^\mathfrak{W} ~=~ \operatorname{Mat}(\mathit{l})^{\operatorname{Mat}(\mathrm{w})} ~=~ \operatorname{Mat}(\mathit{l}^\mathrm{w})</math>
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| − | |}
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| − | Because <math>\mathit{l}^\mathrm{w}\!</math> denotes the elements of a subset of <math>X\!</math> the matrix <math>\mathfrak{L}^\mathfrak{W}</math> is a 1-dimensional array of coefficients in <math>\mathbb{B}</math> that is indexed by the elements of <math>X.\!</math> The value of the matrix <math>\mathfrak{L}^\mathfrak{W}</math> at the index <math>a \in X</math> is written <math>(\mathfrak{L}^\mathfrak{W})_a</math> and computed as follows:
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| − | {| align="center" cellspacing="6" width="90%"
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| − | | height="60" | <math>(\mathfrak{L}^\mathfrak{W})_a ~=~ \prod_{x \in X} \mathfrak{L}_{ax}^{\mathfrak{W}_{x}}</math>
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| − | |}
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| | <pre> | | <pre> |
