## Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Selection 6

### Chapter 3. The Logic of Relatives (cont.)

#### §2. Relatives (concl.)

222.   Instead of considering the system of a relative as consisting of non-relative individuals, we may conceive of it as consisting of relative individuals.  Thus, since

$\begin{array}{*{11}{c}} \mathrm{A} & = & \mathrm{A:A} & + & \mathrm{A:B} & + & \mathrm{A:C} & + & \mathrm{A:D} & + & \text{etc.}, \end{array}$

we have

$\begin{array}{*{11}{c}} \mathrm{A:B} & = & \mathrm{(A:A):B} & + & \mathrm{(A:B):B} & + & \mathrm{(A:C):B} & + & \mathrm{(A:D):B} & + & \text{etc.} \end{array}$

But

$\begin{array}{*{11}{c}} \mathrm{B} & = & \mathrm{B:A} & + & \mathrm{B:B} & + & \mathrm{B:C} & + & \mathrm{B:D} & + & \text{etc.}; \end{array}$

so that

$\begin{array}{*{11}{c}} \mathrm{A:B} & = & \mathrm{A:(B:A)} & + & \mathrm{A:(B:B)} & + & \mathrm{A:(B:C)} & + & \mathrm{A:(B:D)} & + & \text{etc.} \end{array}$

### References

• Peirce, C.S. (1880), “On the Algebra of Logic”, American Journal of Mathematics 3, 15–57.  Collected Papers (CP 3.154–251), Chronological Edition (CE 4, 163–209).
• 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.  Volume 3 : Exact Logic, 1933.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Volume 4 (1879–1884), 1986.

## Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Selection 5

### Chapter 3. The Logic of Relatives (cont.)

#### §2. Relatives (cont.)

221.   From the definition of a simple term given in the last section, it follows that every simple relative is the negative of an individual term.  But while in non-relative logic negation only divides the universe into two parts, in relative logic the same operation divides the universe into $2^n$ parts, where $n$ is the number of objects in the system which the relative supposes;  thus,

$\begin{array}{*{5}{l}} \infty & = & \mathrm{A} & + & \overline{\mathrm{A}} \end{array}$

$\begin{array}{*{9}{l}} \infty & = & \mathrm{A:B} & + & \mathrm{\overline{A}:B} & + & \mathrm{A:\overline{B}} & + & \mathrm{\overline{A}:\overline{B}} \end{array}$

$\begin{array}{*{9}{l}} \infty & = & \mathrm{A:B:C} & + & \mathrm{\overline{A}:B:C} & + & \mathrm{A:\overline{B}:C} & + & \mathrm{A:B:\overline{C}} \\[4pt] & + & \mathrm{\overline{A}:\overline{B}:\overline{C}} & + & \mathrm{A:\overline{B}:\overline{C}} & + & \mathrm{\overline{A}:B:\overline{C}} & + & \mathrm{\overline{A}:\overline{B}:C}. \end{array}$

Here, we have

$\begin{array}{*{5}{l}} \mathrm{A} & = & \mathrm{A:B} & + & \mathrm{A:\overline{B}} \\[4pt] \mathrm{\overline{A}} & = & \mathrm{\overline{A}:B} & + & \mathrm{\overline{A}:\overline{B}} \end{array}$

$\begin{array}{*{5}{l}} \mathrm{A:B} & = & \mathrm{A:B:C} & + & \mathrm{A:B:\overline{C}} \\[4pt] \mathrm{A:\overline{B}} & = & \mathrm{A:\overline{B}:C} & + & \mathrm{A:\overline{B}:\overline{C}} \\[4pt] \mathrm{\overline{A}:B} & = & \mathrm{\overline{A}:B:C} & + & \mathrm{\overline{A}:B:\overline{C}} \\[4pt] \mathrm{\overline{A}:\overline{B}} & = & \mathrm{\overline{A}:\overline{B}:C} & + & \mathrm{\overline{A}:\overline{B}:\overline{C}}. \end{array}$

It will be seen that a term which is individual when considered as $n$-fold is not so when considered as more than $n$-fold;  but an $n$-fold term when made $(m + n)$-fold, is individual as to $n$ members of the system, and indefinite as to $m$ members.

### References

• Peirce, C.S. (1880), “On the Algebra of Logic”, American Journal of Mathematics 3, 15–57.  Collected Papers (CP 3.154–251), Chronological Edition (CE 4, 163–209).
• 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.  Volume 3 : Exact Logic, 1933.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Volume 4 (1879–1884), 1986.

## Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Selection 4

### Chapter 3. The Logic of Relatives (cont.)

#### §2. Relatives (cont.)

220.   Every relative, like every term of singular reference, is general;  its definition describes a system in general terms;  and, as general, it may be conceived either as a logical sum of individual relatives, or as a logical product of simple relatives.  An individual relative refers to a system all the members of which are individual.  The expressions

$\begin{array}{lll} (\mathrm{A : B}) & \qquad & (\mathrm{A : B : C}) \end{array}$

may denote individual relatives.  Taking dual individual relatives, for instance, we may arrange them all in an infinite block, thus,

$\begin{array}{*{11}{c}} \mathrm{A:A}&&\mathrm{A:B}&&\mathrm{A:C}&&\mathrm{A:D}&&\mathrm{A:E}&&\text{etc.} \\[4pt] \mathrm{B:A}&&\mathrm{B:B}&&\mathrm{B:C}&&\mathrm{B:D}&&\mathrm{B:E}&&\text{etc.} \\[4pt] \mathrm{C:A}&&\mathrm{C:B}&&\mathrm{C:C}&&\mathrm{C:D}&&\mathrm{C:E}&&\text{etc.} \\[4pt] \mathrm{D:A}&&\mathrm{D:B}&&\mathrm{D:C}&&\mathrm{D:D}&&\mathrm{D:E}&&\text{etc.} \\[4pt] \mathrm{E:A}&&\mathrm{E:B}&&\mathrm{E:C}&&\mathrm{E:D}&&\mathrm{E:E}&&\text{etc.} \\[4pt] \text{etc.}&& \text{etc.}&& \text{etc.}&& \text{etc.}&& \text{etc.}&& \text{etc.} \end{array}$

In the same way, triple individual relatives may be arranged in a cube, and so forth.  The logical sum of all the relatives in this infinite block will be the relative universe, $\infty,$ where

$x \,-\!\!\!< \infty,$

whatever dual relative $x$ may be.  It is needless to distinguish the dual universe, the triple universe, etc., because, by adding a perfectly indefinite additional member to the system, a dual relative may be converted into a triple relative, etc.  Thus, for lover of a woman, we may write lover of a woman coexisting with anything.  In the same way, a term of single reference is equivalent to a relative with an indefinite correlate;  thus, woman is equivalent to woman coexisting with anything.  Thus, we shall have

$\begin{array}{*{13}{c}} \mathrm{A} & = & \mathrm{A:A} & + & \mathrm{A:B} & + & \mathrm{A:C} & + & \mathrm{A:D} & + & \mathrm{A:E} & + & \text{etc.} \end{array}$

$\begin{array}{*{11}{c}} \mathrm{A:B} & = & \mathrm{A:B:A} & + & \mathrm{A:B:B} & + & \mathrm{A:B:C} & + & \mathrm{A:B:D} & + & \text{etc.} \end{array}$

### References

• Peirce, C.S. (1880), “On the Algebra of Logic”, American Journal of Mathematics 3, 15–57.  Collected Papers (CP 3.154–251), Chronological Edition (CE 4, 163–209).
• 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.  Volume 3 : Exact Logic, 1933.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Volume 4 (1879–1884), 1986.

## Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Selection 3

### Chapter 3. The Logic of Relatives (cont.)

#### §2. Relatives

218.   A relative is term whose definition describes what sort of a system of objects that is whose first member (which is termed the relate) is denoted by the term;  and names for the other members of the system (which are termed the correlates) are usually appended to limit the denotation still further.  In these systems the order of the members is essential;  so that $(\mathrm{A}, \mathrm{B}, \mathrm{C})$ and $(\mathrm{A}, \mathrm{C}, \mathrm{B})$ are different systems.  As an example of a relative, take ‘buyer of ── for ── from ── ’;  we may append to this three correlates, thus, ‘buyer of every horse of a certain description in the market for a good price from its owner’.

219.   A relative of only one correlate, so that the system it supposes is a pair, may be called a dual relative;  a relative of more than one correlate may be called plural;  A non-relative term may be called a term of singular reference.

### References

• Peirce, C.S. (1880), “On the Algebra of Logic”, American Journal of Mathematics 3, 15–57.  Collected Papers (CP 3.154–251), Chronological Edition (CE 4, 163–209).
• 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.  Volume 3 : Exact Logic, 1933.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Volume 4 (1879–1884), 1986.

## Six Ways of Looking at a Triadic Relation ⌬ 1

A triadic relation has $3! = 6$ converses, six grammatically and rhetorically different ways of representing what is logically the same information.  Peirce illustrates the situation as follows, with six variations on the theme of giving.

So in a triadic fact, say, for example

$A ~\mathrm{gives}~ B ~\mathrm{to}~ C$

we make no distinction in the ordinary logic of relations between the subject nominative, the direct object, and the indirect object.  We say that the proposition has three logical subjects.  We regard it as a mere affair of English grammar that there are six ways of expressing this:

$\begin{array}{lll} A ~\mathrm{gives}~ B ~\mathrm{to}~ C & \qquad & A ~\mathrm{benefits}~ C ~\mathrm{with}~ B \\ B ~\mathrm{enriches}~ C ~\mathrm{at~expense~of}~ A & \qquad & C ~\mathrm{receives}~ B ~\mathrm{from}~ A \\ C ~\mathrm{thanks}~ A ~\mathrm{for}~ B & \qquad & B ~\mathrm{leaves}~ A ~\mathrm{for}~ C \end{array}$

These six sentences express one and the same indivisible phenomenon.

### References

• Peirce, C.S., “The Categories Defended”, Harvard Lectures on Pragmatism : Lecture 3 (MS 308, delivered on 9 April 1903).  Published in Collected Papers (CP 5.66–81, 88–92, in part), Harvard Lectures (HL 167–188), Essential Peirce : Volume 2 (EP 2, 160–178).
• 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.  Volume 5 : Pragmatism and Pragmaticism, 1934.  (Cited as CP).
• Peirce, C.S., Pragmatism as a Principle and Method of Right Thinking : The 1903 Harvard Lectures on Pragmatism, Patricia Ann Turrisi (ed.), State University of New York Press, Albany, NY, 1997.  (Cited as HL).
• Peirce, C.S., The Essential Peirce : Selected Philosophical Writings, Volume 2 (1893–1913), Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1998.  (Cited as EP 2).

## Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Selection 2

### Chapter 3. The Logic of Relatives (cont.)

#### §1. Individual and Simple Terms (concl.)

216.   Just as in mathematics we speak of infinitesimals and infinites, which are fictitious limits of continuous quantity, and every statement involving these expressions has its interpretation in the doctrine of limits, so in logic we may define an individual, $\mathrm{A},$ as such a term that

$\mathrm{A} \,\overline{-\!\!\!<}\, 0,$

but such that if

$x < \mathrm{A}$

then

$x \,-\!\!\!< 0.$

And in the same way, we may define the simple, $\alpha,$ as such a term that

$\infty \,\overline{-\!\!\!<}\, \alpha,$

but such that if

$\alpha < x$

then

$\infty \,-\!\!\!< x.$

The individual and the simple, as here defined, are ideal limits, and every statement about either is to be interpreted by the doctrine of limits.

217.   Every term may be conceived as a limitless logical sum of individuals, or as a limitless logical product of simples;  thus,

$\begin{array}{lll} a & = & \mathrm{A}_1 + \mathrm{A}_2 + \mathrm{A}_3 + \mathrm{A}_4 + \mathrm{A}_5 + \text{etc.} \\[8pt] \overline{a} & = & \overline{\mathrm{A}}_1 \times \overline{\mathrm{A}}_2 \times \overline{\mathrm{A}}_3 \times \overline{\mathrm{A}}_4 \times \overline{\mathrm{A}}_5 \times \text{etc.} \end{array}$

It will be seen that a simple is the negative of an individual.

### References

• Peirce, C.S. (1880), “On the Algebra of Logic”, American Journal of Mathematics 3, 15–57.  Collected Papers (CP 3.154–251), Chronological Edition (CE 4, 163–209).
• 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.  Volume 3 : Exact Logic, 1933.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Volume 4 (1879–1884), 1986.

## Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Selection 1

### Chapter 3. The Logic of Relatives

#### §1. Individual and Simple Terms

214.   Just as we had to begin the study of Logical Addition and Multiplication by considering $\infty$ and $0,$ terms which might have been introduced under the Algebra of the Copula, being defined in terms of the copula only, without the use of $+$ or $\times,$ but which had not been there introduced, because they had no application there, so we have to begin the study of relatives by considering the doctrine of individuals and simples,— a doctrine which makes use only of the conceptions of non-relative logic, but which is wholly without use in that part of the subject, while it is the very foundation of the conception of a relative, and the basis of the method of working with the algebra of relatives.

215.   The germ of the correct theory of individuals and simples is to be found in Kant’s Critic of the Pure Reason, “Appendix to the Transcendental Dialectic,” where he lays it down as a regulative principle, that, if

$\begin{array}{lll} a \,-\!\!\!< b & ~ & b \,\overline{-\!\!\!<}\, a, \end{array}$

then it is always possible to find a term $x,$ that

$\begin{array}{lll} a \,-\!\!\!< x & ~ & x \,-\!\!\!< b \\[8pt] x \,\overline{-\!\!\!<}\, a & ~ & b \,\overline{-\!\!\!<}\, x. \end{array}$

Kant’s distinction of regulative and constitutive principles is unsound, but this law of continuity, as he calls it, must be accepted as a fact.  The proof of it, which I have given elsewhere, depends on the continuity of space, time, and the intensities of the qualities which enter into the definition of any term.  If, for instance, we say that Europe, Asia, Africa, and North America are continents, but not all the continents, there remains over only South America.  But we may distinguish between South America as it now exists and South America in former geological times;  we may, therefore, take $x$ as including Europe, Asia, Africa, North America, and South America as it exists now, and every $x$ is a continent, but not every continent is $x.$

### References

• Peirce, C.S. (1880), “On the Algebra of Logic”, American Journal of Mathematics 3, 15–57.  Collected Papers (CP 3.154–251), Chronological Edition (CE 4, 163–209).
• 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.  Volume 3 : Exact Logic, 1933.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Volume 4 (1879–1884), 1986.