Relations & Their Relatives • 3

Posted on February 18, 2015 by Jon Awbrey

Here are two ways of looking at the divisibility relation, a dyadic relation of fundamental importance in number theory.

Table 1 shows the first few ordered pairs of the relation on positive integers corresponding to the relative term, “divisor of”.  Thus, the ordered pair {i\!:\!j} appears in the relation if and only if {i} divides {j}, for which the usual notation is {i|j}.

Elementary Relatives for the “Divisor Of” Relation

Table 2 shows the same information in the form of a logical matrix.  This has a coefficient of {1} in row {i} and column {j} when {i|j}, otherwise it has a coefficient of {0}.  (The zero entries have been omitted for ease of reading.)

Logical Matrix for the “Divisor Of” Relation

Just as matrices in linear algebra represent linear transformations, these logical arrays and matrices represent logical transformations.

Resources

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Relations & Their Relatives • 2

Posted on February 17, 2015 by Jon Awbrey

What is the relationship between “logical relatives” and “mathematical relations”?  The word relative used as a noun in logic is short for relative term — as such it refers to an item of language used to denote a formal object.

What kind of object is that?  The way things work in mathematics we are free to make up a formal object corresponding directly to the term, so long as we can form a consistent theory of it, but it’s probably easier and more practical in the long run to relate the relative term to the kinds of relations ordinarily treated in mathematics and universally applied in relational databases.

In those contexts a relation is just a set of ordered tuples and if you are a fan of strong typing like I am, such a set is always set in a specific setting, namely, it’s a subset of a specified cartesian product.

Peirce wrote k-tuples (x_1, x_2, \ldots, x_{k-1}, x_k) in the form x_1 : x_2 : \ldots : x_{k-1} : x_k and referred to them as elementary k-adic relatives.  He treated a collection of k-tuples as a logical aggregate or logical sum and often regarded them as being arranged in k-dimensional arrays.

Time for some concrete examples, which I will give in the next post.

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Relations & Their Relatives • 1

Posted on February 17, 2015 by Jon Awbrey

Sign relations are special cases of triadic relations in much the same way binary operations in mathematics are special cases of triadic relations.  It amounts to a minor complication that we participate in sign relations whenever we talk or think about anything else but it still makes sense to try and tease the separate issues apart as much as we possibly can.

As far as relations in general go, relative terms are often expressed by slotted frames like “brother of __”, “divisor of __”, and “sum of __ and __”.  Peirce referred to these kinds of incomplete expressions as rhemes or rhemata and Frege used the adjective ungesättigt or unsaturated to convey more or less the same idea.

Switching the focus to sign relations, it’s fair to ask what kinds of objects might be denoted by pieces of code like “brother of __”, “divisor of __”, and “sum of __ and __”.  And while we’re at it, what is this thing called denotation, anyway?

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

Posted on February 16, 2015 by Jon Awbrey

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.

Resources

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

Posted on February 15, 2015 by Jon Awbrey

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.

Resources

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

Posted on February 12, 2015 by Jon Awbrey

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.

Resources

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

Posted on February 11, 2015 by Jon Awbrey

Chapter 3. The Logic of Relatives (cont.)

§2. Relatives

218.   A relative is a 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.

Resources

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Six Ways of Looking at a Triadic Relation ⌬ 1

Posted on February 4, 2015 by Jon Awbrey

A triadic relation and its converses form a set of 3! = 6 triadic relations all together, six grammatically and rhetorically distinct 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:

Six Ways of Looking at a Triadic Relation

These six sentences express one and the same indivisible phenomenon.
(EP 2, 170–171).

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).

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

Posted on February 3, 2015 by Jon Awbrey

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.

Resources

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

Posted on February 1, 2015 by Jon Awbrey

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.

Resources

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