Peirce's 1870 “Logic of Relatives” • Selection 3.3
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Comment on ❝The Signs of Inclusion, Equality, Etc.❞

Peirce's use of a square bracket \([t]\) to indicate a mapping from logical terms to numbers provides a basis for the computation of frequencies, probabilities, and other statistical measures constructed from them, thus affording a “principle of correspondence” between probability theory and its limiting case in the forms of logic.

This brings us once again to the relativity of contingency and necessity, as one way of approaching necessity is through the avenue of probability, describing necessity as a probability of \(1,\) but the whole apparatus of probability theory only figures in if it is cast against the backdrop of probability space axioms, the reference class of distributions, and the sample space we cannot help but abduce on the scene of observations.

Aye, there's the snake eyes. And with them we can see there is always an irreducible quantum of facticity to all our necessities. More plainly spoken, it takes a fairly complex conceptual infrastructure just to begin speaking of probabilities, and that setting can only be set up by means of abductive, fallible, hypothetical, and inherently risky mental acts.

Pragmatic thinking is the logic of abduction, which is another way of saying it addresses the question: What may be hoped? We have to face the possibility it may be just as impossible to speak of absolute identity with any hope of making practical philosophical sense as it is to speak of absolute simultaneity with any hope of making operational physical sense.

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Peirce's 1870 “Logic of Relatives” • Selection 3.2
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❝§3. Application of the Algebraic Signs to Logic❞

❝The Signs of Inclusion, Equality, Etc.❞

❝But not only do the significations of \(=\) and \(<\) here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations. Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes.

❝So, to write \(5 < 7\) is to say that \(5\) is part of \(7,\) just as to write \(\mathrm{f} < \mathrm{m}\) is to say that Frenchmen are part of men. Indeed, if \(\mathrm{f} < \mathrm{m},\) then the number of Frenchmen is less than the number of men, and if \(\mathrm{v} = \mathrm{p},\) then the number of Vice-Presidents is equal to the number of Presidents of the Senate; so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities.❞

(Peirce, CP 3.66)

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Peirce's 1870 “Logic of Relatives” • Selection 3.1
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❝§3. Application of the Algebraic Signs to Logic❞

❝The Signs of Inclusion, Equality, Etc.❞

❝I shall follow Boole in taking the sign of equality to signify identity. Thus, if \(\mathrm{v}\) denotes the Vice-President of the United States, and \(\mathrm{p}\) the President of the Senate of the United States,

\[\mathrm{v}=\mathrm{p}\]

❝means that every Vice-President of the United States is President of the Senate, and every President of the United States Senate is Vice-President.

❝The sign “less than” is to be so taken that

\[\mathrm{f}<\mathrm{m}\]

❝means that every Frenchman is a man, but there are men besides Frenchmen. Drobisch has used this sign in the same sense. It will follow from these significations of \(=\) and \(<\) that the sign \(-\!\!\!<\) (or \(\leqq,\) “as small as”) will mean “is”. Thus,

\[\mathrm{f}-\!\!\!<\mathrm{m}\]

❝means “every Frenchman is a man”, without saying whether there are any other men or not. So,

\[\mathit{m}-\!\!\!<\mathit{l}\]

❝will mean that every mother of anything is a lover of the same thing; although this interpretation in some degree anticipates a convention to be made further on. These significations of \(=\) and \(<\) plainly conform to the indispensable conditions.

❝Upon the transitive character of these relations the syllogism depends, for by virtue of it, from \(\mathrm{f}-\!\!\!<\mathrm{m}\) and \(\mathrm{m}-\!\!\!<\mathrm{a}\) we can infer that \(\mathrm{f}-\!\!\!<\mathrm{a},\) that is, from every Frenchman being a man and every man being an animal, that every Frenchman is an animal.❞

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Peirce's 1870 “Logic of Relatives” • Selection 2.2
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Peirce's remarks at CP 3.65 are so replete with remarkable ideas, some of them so taken for granted in mathematical discourse as usually to escape explicit mention, others so suggestive of things to come in a future remote from his time of writing, and yet so smoothly slipped into the stream of thought it's all too easy to overlook their significance — that all I can do to highlight their impact is to dress them up in different words, whose main advantage is being more jarring to the mind's sensibilities.

• Peirce's mapping of letters to numbers, or logical terms to mathematical quantities, is the very core of what quantification theory is all about, definitely more to the point than the mere “innovation” of using distinctive symbols for the so-called quantifiers.

• The mapping of logical terms to numerical measures, to express it in current language, would probably be recognizable as a “morphism” or “functor” from a logical domain to a quantitative co-domain.

• Peirce follows the mathematician's usual practice of making the status of being an individual or being a universal relative to a discourse in progress.

• Peirce takes the plural denotation of terms for granted. Indeed. what's the number of a term for, if it could not vary apart from being one or nil?

• Peirce takes the individual objects of a particular universe of discourse in a generative way, as opposed to a totalizing way, and thus its contingent individuals afford us with a basis for talking freely about collections, constructions, properties, qualities, subsets, and higher types built on them.

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Peirce's 1870 “Logic of Relatives” • Selection 2.1
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❝§3. Application of the Algebraic Signs to Logic❞

❝Numbers Corresponding to Letters❞

❝I propose to use the term “universe” to denote that class of individuals about which alone the whole discourse is understood to run. The universe, therefore, in this sense, as in Mr. De Morgan's, is different on different occasions. In this sense, moreover, discourse may run upon something which is not a subjective part of the universe; for instance, upon the qualities or collections of the individuals it contains.

❝I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual. Thus in a universe of perfect men \((\mathrm{men}),\) the number of “tooth of” would be 32. The number of a relative with two correlates would be the average number of things so related to a pair of individuals; and so on for relatives of higher numbers of correlates. I propose to denote the number of a logical term by enclosing the term in square brackets, thus, \([t].\)❞

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Peirce's 1870 “Logic of Relatives” • Selection 1.2
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❝The conjugative term involves the conception of third, the relative that of second or other, the absolute term simply considers an object. No fourth class of terms exists involving the conception of fourth, because when that of third is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship. Whether this reason for the fact that there is no fourth class of terms fundamentally different from the third is satisfactory of not, the fact itself is made perfectly evident by the study of the logic of relatives.❞

One thing that strikes me about the above passage is a pattern of argument I can recognize as invoking a closure principle. This is a figure of reasoning Peirce uses in three other places: his discussion of continuous predicates, his definition of a sign relation, and his formulation of the pragmatic maxim itself.

One might also call attention to the following two statements:

❝Now logical terms are of three grand classes.❞

❝No fourth class of terms exists involving the conception of fourth, because when that of third is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship.❞

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Peirce's 1870 “Logic of Relatives” • Selection 1.1
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We pick up Peirce's text at the following point.

❝§3. Application of the Algebraic Signs to Logic❞

❝Use of the Letters❞

❝The letters of the alphabet will denote logical signs.

❝Now logical terms are of three grand classes.

❝The first embraces those whose logical form involves only the conception of quality, and which therefore represent a thing simply as “a ──”. These discriminate objects in the most rudimentary way, which does not involve any consciousness of discrimination. They regard an object as it is in itself as such (quale); for example, as horse, tree, or man. These are absolute terms.

❝The second class embraces terms whose logical form involves the conception of relation, and which require the addition of another term to complete the denotation. These discriminate objects with a distinct consciousness of discrimination. They regard an object as over against another, that is as relative; as father of, lover of, or servant of. These are simple relative terms.

❝The third class embraces terms whose logical form involves the conception of bringing things into relation, and which require the addition of more than one term to complete the denotation. They discriminate not only with consciousness of discrimination, but with consciousness of its origin. They regard an object as medium or third between two others, that is as conjugative; as giver of ── to ──, or buyer of ── for ── from ──. These may be termed conjugative terms.❞

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Peirce's 1870 “Logic of Relatives” • Preliminaries 5
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Individual terms are taken to denote individual entities falling under a general term. Peirce uses upper case Roman letters for individual terms, for example, the individual horses \(\mathrm{H}, \mathrm{H}^{\prime}, \mathrm{H}^{\prime\prime}\) falling under the general term \(\mathrm{h}\) for horse.

The path to understanding Peirce's system and its wider implications for logic can be smoothed by paraphrasing his notations in a variety of contemporary mathematical formalisms, while preserving the semantics as much as possible. Remaining faithful to Peirce's orthography while adding parallel sets of stylistic conventions will, however, demand close attention to typography-in-context.

Current style sheets for mathematical texts specify italics for mathematical variables, with upper case letters for sets and lower case letters for individuals. So we need to keep an eye out for the difference between the individual \(\mathrm{X}\) of the genus \(\mathrm{x}\) and the element \(x\) of the set \(X\) as we pass between the two styles of text.

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Peirce's 1870 “Logic of Relatives” • Preliminaries 4
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Conjugative Terms (Higher Adic Relatives)
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The Table displays the single-letter abbreviations and their verbal equivalents for the “conjugative terms” (or “higher adic relative terms”) used in Peirce's examples of logical formulas. Peirce used a distinctive typeface for the abbreviations of higher adic relative terms, rendered here as LaTeX “mathfrak”, Fraktur, or Gothic.

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Peirce's 1870 “Logic of Relatives” • Preliminaries 3
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Simple Relative Terms (Dyadic Relatives)
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The Table displays the single-letter abbreviations and their verbal equivalents for the “simple relative terms” (or “dyadic relative terms”) used in Peirce's examples of logical formulas. Peirce used a distinctive typeface for the abbreviations of dyadic relative terms, rendered here as LaTeX “mathit” or Italics.

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