Peirce's 1885 “Algebra of Logic”

[Jon Awbrey]

Peirce's 1885 “Algebra of Logic” • Selection 1.1
• https://inquiryintoinquiry.com/2024/03/24/peirces-1885-algebra-of-logic-selection-1/

All,

I'm laying down a few source materials
in preparation for a later discussion.

Selection from C.S. Peirce, “On the Algebra of Logic :
A Contribution to the Philosophy of Notation” (1885)

❝§1. Three Kinds Of Signs❞

❝Any character or proposition either concerns one subject,
two subjects, or a plurality of subjects. For example, one
particle has mass, two particles attract one another, a particle
revolves about the line joining two others. A fact concerning
two subjects is a dual character or relation; but a relation
which is a mere combination of two independent facts concerning
the two subjects may be called “degenerate”, just as two lines
are called a degenerate conic. In like manner a plural character
or conjoint relation is to be called degenerate if it is a mere
compound of dual characters.

❝A sign is in a conjoint relation to the thing denoted and to the mind.
If this triple relation is not of a degenerate species, the sign is
related to its object only in consequence of a mental association,
and depends upon a habit. Such signs are always abstract and general,
because habits are general rules to which the organism has become
subjected. They are, for the most part, conventional or arbitrary.
They include all general words, the main body of speech, and any
mode of conveying a judgment. For the sake of brevity I will call
them “tokens”. [Note. Peirce more frequently calls these “symbols”.]

Regards,

Jon

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[Jon Awbrey]

Peirce's 1885 “Algebra of Logic” • Selection 1.2
• https://inquiryintoinquiry.com/2024/03/24/peirces-1885-algebra-of-logic-selection-1/

❝On the Algebra of Logic❞
❝A Contribution to the Philosophy of Notation❞

❝§1. Three Kinds Of Signs❞ (cont.)

❝But if the triple relation between the sign, its object,
and the mind, is degenerate, then of the three pairs

sign object
sign mind
object mind

❝two at least are in dual relations which constitute the
triple relation. One of the connected pairs must consist
of the sign and its object, for if the sign were not related
to its object except by the mind thinking of them separately,
it would not fulfill the function of a sign at all. Supposing,
then, the relation of the sign to its object does not lie in
a mental association, there must be a direct dual relation of
the sign to its object independent of the mind using the sign.

❝In the second of the three cases just spoken of, this dual
relation is not degenerate, and the sign signifies its object
solely by virtue of being really connected with it. Of this
nature are all natural signs and physical symptoms. I call
such a sign an “index”, a pointing finger being the type of
this class.

❝The index asserts nothing; it only says “There!” It takes hold
of our eyes, as it were, and forcibly directs them to a particular
object, and there it stops. Demonstrative and relative pronouns are
nearly pure indices, because they denote things without describing
them; so are the letters on a geometrical diagram, and the subscript
numbers which in algebra distinguish one value from another without
saying what those values are.❞

Regards,

Jon

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[Jon Awbrey]

Peirce's 1885 “Algebra of Logic” • Selection 1.3

• https://inquiryintoinquiry.com/2024/03/24/peirces-1885-algebra-of-logic-selection-1/

❝On the Algebra of Logic❞
❝A Contribution to the Philosophy of Notation❞

❝§1. Three Kinds Of Signs❞ (cont.)

❝The third case is where the dual relation between the sign and its
object is degenerate and consists in a mere resemblance between them.

❝I call a sign which stands for something merely because it resembles it,
an icon. Icons are so completely substituted for their objects as hardly
to be distinguished from them. Such are the diagrams of geometry.

❝A diagram, indeed, so far as it has a general signification, is not
a pure icon; but in the middle part of our reasonings we forget that
abstractness in great measure, and the diagram is for us the very thing.

❝So in contemplating a painting, there is a moment when we lose consciousness
that it is not the thing, the distinction of the real and the copy disappears,
and it is for the moment a pure dream — not any particular existence, and yet
not general. At that moment we are contemplating an icon.❞

Regards,

Jon

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[Jon Awbrey]

Peirce's 1885 “Algebra of Logic” • Selection 2.1
• https://inquiryintoinquiry.com/2024/03/26/peirces-1885-algebra-of-logic-selection-2/

❝On the Algebra of Logic❞
❝A Contribution to the Philosophy of Notation❞

❝§1. Three Kinds Of Signs❞ (cont.)

❝I have taken pains to make my distinction of icons, indices, and tokens
clear, in order to enunciate this proposition: in a perfect system of
logical notation signs of these several kinds must all be employed.

❝Without tokens there would be no generality in the statements, for they
are the only general signs; and generality is essential to reasoning.

❝Take, for example, the circles by which Euler represents the relations of terms.
They well fulfill the function of icons, but their want of generality and their
incompetence to express propositions must have been felt by everybody who has
used them. Mr. Venn has, therefore, been led to add shading to them; and
this shading is a conventional sign of the nature of a token. In algebra,
the letters, both quantitative and functional, are of this nature.❞ (3.363).

Regards.

Jon

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[Jon Awbrey]

Peirce's 1885 “Algebra of Logic” • Selection 2.2

• https://inquiryintoinquiry.com/2024/03/26/peirces-1885-algebra-of-logic-selection-2/

❝On the Algebra of Logic❞
❝A Contribution to the Philosophy of Notation❞

❝§1. Three Kinds Of Signs❞ (cont.)

❝But tokens alone do not state what is the subject of discourse; and
this can, in fact, not be described in general terms; it can only be
indicated. The actual world cannot be distinguished from a world of
imagination by any description. Hence the need of pronouns and indices,
and the more complicated the subject the greater the need of them.

❝The introduction of indices into the algebra of logic is the greatest merit
of Mr. Mitchell's system. He writes F_1 to mean that the proposition F is
true of every object in the universe, and F_u to mean that the same is true
of some object. This distinction can only be made in some such way as this.
Indices are also required to show in what manner other signs are connected
together.❞ (3.363).

Regards.

Jon

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[Jon Awbrey]

Peirce's 1885 “Algebra of Logic” • Selection 2.3

• https://inquiryintoinquiry.com/2024/03/26/peirces-1885-algebra-of-logic-selection-2/

❝On the Algebra of Logic❞
❝A Contribution to the Philosophy of Notation❞

❝§1. Three Kinds Of Signs❞ (cont.)

❝With these two kinds of signs alone any proposition can be expressed;
but it cannot be reasoned upon, for reasoning consists in the observation
that where certain relations subsist certain others are found, and it
accordingly requires the exhibition of the relations reasoned with in
an icon.

❝It has long been a puzzle how it could be that, on the one hand,
mathematics is purely deductive in its nature, and draws its
conclusions apodictically, while on the other hand, it presents
as rich and apparently unending a series of surprising discoveries
as any observational science.

❝Various have been the attempts to solve the paradox by breaking down
one or other of these assertions, but without success. The truth,
however, appears to be that all deductive reasoning, even simple
syllogism, involves an element of observation; namely, deduction
consists in constructing an icon or diagram the relations of whose
parts shall present a complete analogy with those of the parts of
the object of reasoning, of experimenting upon this image in the
imagination, and of observing the result so as to discover unnoticed
and hidden relations among the parts.❞ (3.363).

Regards.

Jon

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[Jon Awbrey]

Peirce's 1885 “Algebra of Logic” • Selection 3
• https://inquiryintoinquiry.com/2024/03/30/peirces-1885-algebra-of-logic-selection-3/

❝On the Algebra of Logic❞
❝A Contribution to the Philosophy of Notation❞

❝§1. Three Kinds Of Signs❞ (cont.)

❝For instance, take the syllogistic formula,

• All M is P
S is M
∴ S is P.

❝This is really a diagram of the relations of S, M, and P.
The fact that the middle term occurs in the two premisses
is actually exhibited, and this must be done or the notation
will be of no value.

❝As for algebra, the very idea of the art is that it presents
formulæ which can be manipulated, and that by observing the effects
of such manipulation we find properties not to be otherwise discerned.
In such manipulation, we are guided by previous discoveries which are
embodied in general formulæ. These are patterns which we have the right
to imitate in our procedure, and are the icons par excellence of algebra.
The letters of applied algebra are usually tokens, but the x, y, z, etc.
of a general formula, such as

• (x + y)z = xz + yz,

❝are blanks to be filled up with tokens, they are indices of tokens.
Such a formula might, it is true, be replaced by an abstractly stated
rule (say that multiplication is distributive); but no application
could be made of such an abstract statement without translating it
into a sensible image.❞ (3.363).

Regards.

Jon

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[Jon Awbrey]

Peirce's 1885 “Algebra of Logic” • Selection 4
• https://inquiryintoinquiry.com/2024/04/01/peirces-1885-algebra-of-logic-selection-4/

❝On the Algebra of Logic❞
❝A Contribution to the Philosophy of Notation❞

❝§1. Three Kinds Of Signs❞ (concl.)

❝In this paper, I purpose to develop an algebra adequate to
the treatment of all problems of deductive logic, showing as
I proceed what kinds of signs have necessarily to be employed
at each stage of the development. I shall thus attain three
objects. The first is the extension of the power of logical
algebra over the whole of its proper realm. The second is the
illustration of principles which underlie all algebraic notation.
The third is the enumeration of the essentially different kinds
of necessary inference; for when the notation which suffices
for exhibiting one inference is found inadequate for explaining
another, it is clear that the latter involves an inferential
element not present to the former. Accordingly, the procedure
contemplated should result in a list of categories of reasoning,
the interest of which is not dependent upon the algebraic way
of considering the subject.

❝I shall not be able to perfect the algebra sufficiently to
give facile methods of reaching logical conclusions: I can
only give a method by which any legitimate conclusion may be
reached and any fallacious one avoided. But I cannot doubt
that others, if they will take up the subject, will succeed in
giving the notation a form in which it will be highly useful in
mathematical work. I even hope that what I have done may prove
a first step toward the resolution of one of the main problems of
logic, that of producing a method for the discovery of methods in
mathematics.❞ (3.364).

Regards,

Jon

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