Peirce’s 1903 Lowell Lectures • Comment 9

JFS:
In 1911, Peirce clarified [the] issues by using two distinct terms:  ‘the universe’ and ‘a sheet of paper’.  The sheet is no longer identified with the universe, and there is no reason why one couldn’t or shouldn’t shade a blank area of a sheet.

Extracting the moral John Sowa draws from Peirce, there is a difference between being a universe of discourse and representing a universe of discourse.

• On the one hand we have an initial universe of discourse $X.$  This provides the basis for a prospective object domain $O$ to be constructed out of its elements as our description of the universe develops.
• On the other hand we have the various systems of signs that we use to represent aspects of the universe of discourse $X.$  These go to make up whatever sign domain $S$ and interpretant domain $I$ are needed for the ongoing discussion and inquiry.

With logic as formal semiotics and semiotics as the study of triadic sign relations, properly understanding how Peirce’s graphical symbol systems manage to represent universes of discourse requires us to consider the larger contexts of triadic sign relations in which they play their role.

Peirce’s 1903 Lowell Lectures • Comment 8

Many aspects of Peirce’s alpha graphs can be clarified by seeing how they relate to the corresponding Venn diagrams.

In particular, there is a series of diagrams in this vein that I’ve found to be very illuminating with regard to understanding the properties of logical implications or material conditionals, under whatever name or notation they may be invoked.

Figure 1 shows the frame of a Venn diagram for two features, predicates, propositions, properties, qualities, variables, or whatever they may be called, signified by the letters ${}^{\backprime\backprime} p {}^{\prime\prime}$ and ${}^{\backprime\backprime} q {}^{\prime\prime},$ respectively.  The rectangular area represents a set or space $X,$ usually called the universe of discourse, though viewed from the angle of Peircean semiotics it is really just the ground level of a more complex object domain $O$ to be built on its base.

 (1)

The circular area marked ${}^{\backprime\backprime} p {}^{\prime\prime}$ represents the subset of $X$ that has the property $p.$  Figure 2 shows this area shaded blue.  We may think of the shading in the diagram as indicating the corresponding subset of the universe, in other words, associating a distinctive value with it.

 (2)

The circular area marked ${}^{\backprime\backprime} q {}^{\prime\prime}$ represents the subset of $X$ that has the property $q.$  Figure 3 shows this area shaded blue.  We may think of the shading in the diagram as indicating the corresponding subset of the universe, in other words, associating a distinctive value with it.

 (3)

The crescent-shaped area shaded blue in Figure 4 represents the subset of $X$ that has the property $p$ but not the property $q.$  We may think of this as the region where ${}^{\backprime\backprime} p ~\text{without}~ q{}^{\prime\prime}$ is true.  Further, we may interpret either the propositional form ${}^{\backprime\backprime} p \texttt{(} q \texttt{)} {}^{\prime\prime}$ or the corresponding logical graph as indicating the same subset of the universe as the shading in the Venn diagram.

 (4)

The shaded area in Figure 5 represents the subset of $X$ that constitutes the set-theoretic complement of the subset represented in Figure 4.  We may think of this as the region where ${}^{\backprime\backprime} \text{not}~ p ~\text{without}~ q {}^{\prime\prime}$ is true.  Finally, we may interpret either the propositional form ${}^{\backprime\backprime} \texttt{(} p \texttt{(} q \texttt{))} {}^{\prime\prime}$ or the corresponding logical graph as indicating the same subset of the universe as the shading in the Venn diagram.

 (5)

So far we are simply describing different regions of the universe $X$ based on the coordinate frame mapped out by the properties $p$ and $q.$  This amounts to the functional interpretation of the Venn diagrams, propositional formulas, and corresponding logical graphs, each one associating a subset of $X$ with a distinctive logical value, say “true” or “1” or “looky here”, it doesn’t really matter so long as we know the subset it indicates.

But the same Venn diagrams, propositional forms, and logical graphs may be interpreted another way, as bearing information about constraints on the structure of the universe as a whole, specifying what sorts of things, that is, what combinations of properties $p$ and $q$ have or have not existence in it.  This marks an interpretive transition from the functional interpretation to the relational interpretation of all these styles of signs.

In my mind’s eye I see the rectangular space of the Venn diagram as a soap film suspended in a wire frame, with two circles of thread for the properties $p$ and $q,$ and various regions of soap film tinted with the indicative color.  I see the transformation from Figure 5 to Figure 6 as occurring when a pin pops the untinted space of the first and the region collapses to give the arrangement of extant regions in the final diagram.  This is the sort of diagram we usually draw to indicate a subset relation, in this case showing the set $P$ where $p$ is true being a subset of the set $Q$ where $q$ is true.

 (6)

Peirce’s 1903 Lowell Lectures • Comment 7

I’ll go ahead and copy out the first part of the article on Logical Implication, as I find I am still pleased with all I was able to say in such a short space.

Logical Implication

The concept of logical implication encompasses a specific logical function, a specific logical relation, and the various symbols that are used to denote this function and this relation.  In order to define the specific function, relation, and symbols in question it is first necessary to establish a few ideas about the connections among them.

Close approximations to the concept of logical implication are expressed in ordinary language by means of linguistic forms like the following:

$\begin{array}{l} p ~\text{implies}~ q. \\[6pt] \text{if}~ p ~\text{then}~ q. \end{array}$

Here ${}^{\backprime\backprime} p {}^{\prime\prime}$ and ${}^{\backprime\backprime} q {}^{\prime\prime}$ are propositional variables that stand for any propositions in a given language.  In a statement of the form ${}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime},$ the first term, $p,$ is called the antecedent and the second term, $q,$ is called the consequent, while the statement as a whole is called either the conditional or the consequence.  Assuming that the conditional statement is true, then the truth of the antecedent is a sufficient condition for the truth of the consequent, while the truth of the consequent is a necessary condition for the truth of the antecedent.

Note.  Many writers draw a technical distinction between the form ${}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime}$ and the form ${}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime}.$  In this usage, writing ${}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime}$ asserts the existence of a certain relation between the logical value of $p$ and the logical value of $q,$ whereas writing ${}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime}$ merely forms a compound statement whose logical value is a function of the logical values of $p$ and $q.$  This will be discussed in detail below.

Peirce’s 1903 Lowell Lectures • Comment 6

Re: Peirce List Discussions • (1)(2)(3)

The concept of logical implication or material conditional — its names are legion — is ever a source of logical bedevilment.  It has proved no different in the discussion of Peirce’s logical graphs however much potential they bear for resolving the confusion if given the chance.

Here’s a link to an article on Logical Implication I originally wrote for Wikipedia where I attempted to synthesize the more coherent perspectives on the subject I had learned from various communities of usage over the years.

The last time I looked at the Wikipedia relic it had devolved into the usual run-of-the-mill obscurities of nominalism so let’s not go there.  The Wikiversity edition still looks okay.

I’ve been meaning for years now to add the graphical representations — maybe these recent discussions will nudge me to do that …

The Difference That Makes A Difference That Peirce Makes : 19

Peirce uses the word “formal” in a sense that gives it normative force.  It is this sense in which he defines logic as formal semiotic.

But taking “formal” in a normative sense creates difficulties for John Sowa’s thesis of a “sharp distinction between ‘formal logic’, which is part of mathematics, from logic as a normative science”.

I don’t think it means that “formal logic” is “formal formal semiotic”, much less a part of mathematics.  It simply means that logic is inherently formal (= normative) and adding the adjective “formal” is redundant.

Selections from C.S. Peirce, “Carnegie Application” (1902)

No. 12. On the Definition of Logic

Logic will here be defined as formal semiotic.  A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time.  Namely, a sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C.  It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic.  I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has virtually been quite generally held, though not generally recognized.  (NEM 4, 20–21).

Selection from C.S. Peirce, “Ground, Object, and Interpretant” (c. 1897)

Logic, in its general sense, is, as I believe I have shown, only another name for semiotic (σημειωτική), the quasi-necessary, or formal, doctrine of signs.  By describing the doctrine as “quasi-necessary”, or formal, I mean that we observe the characters of such signs as we know, and from such an observation, by a process which I will not object to naming Abstraction, we are led to statements, eminently fallible, and therefore in one sense by no means necessary, as to what must be the characters of all signs used by a “scientific” intelligence, that is to say, by an intelligence capable of learning by experience.  As to that process of abstraction, it is itself a sort of observation.  (CP 2.227).

Peirce’s 1903 Lowell Lectures • Comment 5

The World of Ā

The relationship between classical and non-classical logic is a topic that comes up from time to time in these discussions.  Inquiry into it tends to take a different tack when guided by Peirce’s placement of logic as a normative science within the more general study of triadic sign relations.  Here’s a brief comment I posted on the Peirce List with links to previous comments on the Foundations Of Math List when the subject came up there.

On the relationship between the classical laws of logic and their easement, relaxation, stay, or violation in the qualities of generality and vagueness, see the quotes and remarks I posted to the Foundations Of Mathematics List.

Peirce’s 1903 Lowell Lectures • Comment 4

A couple of comments in response to questions about the relationship between Spencer Brown’s Laws of Form and the broader scope of Peirce’s Logical Graphs.

George Spencer Brown mentions Charles S. Peirce and also Christine Ladd-Franklin in the chapter notes, appendices, and references of his Laws of Form.  Just scanning very quickly, I find references on pages 90, 111, and 136 in my copy.

Almost in spite of its extremely elegant style, Laws of Form did succeed in reviving a visual way of looking at logic that Peirce had pioneered but that few other logicians took up with any success in the intervening years.  It drew out and clarified a number of insights into the mathematical forms and methods of logic that Peirce had the depth of vision to peer into but did not always have the opportunity to develop as far as possible.