Peirce’s 1903 Lowell Lectures • Comment 8

Re: Peirce List DiscussionJA

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.

Venn Diagram Two Variables {P Q}

(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.

Venn Diagram Two Variables P

(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.

Venn Diagram Two Variables Q

(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.

Venn Diagram Two Variables P (Q)

(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.

Venn Diagram Two Variables (P (Q)) 1

(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.

Venn Diagram Two Variables (P (Q)) 2

(6)

Reference

This entry was posted in C.S. Peirce, Diagrammatic Reasoning, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Peirce, Peirce List, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Visualization and tagged , , , , , , , , , , , , . Bookmark the permalink.

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