C.S. Peirce • Algebra of Logic ∫ Philosophy of Notation

[Jon Awbrey]

Cf: C.S. Peirce • Algebra of Logic ∫ Philosophy of Notation • 1
https://inquiryintoinquiry.com/2021/10/06/c-s-peirce-algebra-of-logic-%e2%88%ab-philosophy-of-notation-1/

All,

There's a few passages from Peirce I will need as background for the
inquiry I'm making into the iconic, indexical, and symbolic aspects of
Peirce's logical graphs and Spencer Brown's calculus of indications.
(As always, the formatting will be better on the above-linked blog post.)

Here's the first piece ...

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.
(3.359).

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. (3.360).

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. (3.361).

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. (3.362).

References
==========

• Peirce, C.S. (1885), “On the Algebra of Logic : A Contribution to the Philosophy of Notation”, American Journal of
Mathematics 7, 180–202.

• 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 (Published
Papers), 1933. CP 3.359–403.

• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana
University Press, Bloomington and Indianapolis, IN, 1981–. Volume 5 (1884–1886), 1993. Item 30, 162–190.

[Jon Awbrey]

Cf: C.S. Peirce • Algebra of Logic ∫ Philosophy of Notation • 2
http://inquiryintoinquiry.com/2021/10/10/c-s-peirce-algebra-of-logic-%e2%88%ab-philosophy-of-notation-2/

All,

Continuing the previous passage in preparation for discussing the

iconic, indexical, and symbolic aspects of Peirce's logical graphs
and Spencer Brown's calculus of indications.

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

§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.

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.

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).