## { Information = Comprehension × Extension } Revisited • Comment 1

At this point in his inventory of scientific reasoning, Peirce is relating the nature of inference, information, and inquiry to the character of the signs mediating the process in question, a process he is presently describing as symbolization.

In the interest of clarity let’s draw from Peirce’s account a couple of quick sketches, designed to show how the examples he gives of conjunctive terms and disjunctive terms might look if they were cast within a lattice-theoretic frame.

Let’s examine Peirce’s example of a conjunctive term — spherical, bright, fragrant, juicy, tropical fruit — within a lattice framework.  We have these six terms:

$\begin{array}{lll} t_1 & = & \mathrm{spherical} \\ t_2 & = & \mathrm{bright} \\ t_3 & = & \mathrm{fragrant} \\ t_4 & = & \mathrm{juicy} \\ t_5 & = & \mathrm{tropical} \\ t_6 & = & \mathrm{fruit} \end{array}$

Suppose $z$ is the logical conjunction of the above six terms:

$\begin{array}{lll} z & = & t_1 \cdot t_2 \cdot t_3 \cdot t_4 \cdot t_5 \cdot t_6 \end{array}$

What on earth could Peirce mean by saying that such a term is not a true symbol or that it is of no use whatever?

In particular, consider the following statement:

If it occurs in the predicate and something is said to be a spherical bright fragrant juicy tropical fruit, since there is nothing which is all this which is not an orange, we may say that this is an orange at once.

In other words, if something $x$ is said to be $z$ then we may guess fairly surely $x$ is really an orange, in short, $x$ has all the additional features otherwise summed up quite succinctly in the much more constrained term $y,$ where $y$ means an orange.

Figure 1 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 1. Conjunctive Term z, Taken as Predicate

What Peirce is saying about $z$ not being a genuinely useful symbol can be explained in terms of the gap between the logical conjunction $z,$ in lattice terms, the greatest lower bound (glb) of the conjoined terms, $z = \mathrm{glb} \{ t_1, t_2, t_3, t_4, t_5, t_6 \},$ and what we might regard as the natural conjunction or natural glb of these terms, namely, $y,$ an orange.  That is to say, there is an extra measure of constraint that goes into forming the natural kinds lattice from the free lattice that logic and set theory would otherwise impose.  The local manifestations of this global information are meted out over the structure of the natural lattice by just such abductive gaps as the one between $z$ and $y.$

### Reference

• Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.

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