## Pragmatic Maxims • 4

Re: Peirce List Discussion • (1)(2)

I haven’t been able to do more than randomly sample the doings on the Peirce List for the last half year, being deeply immersed in other Peirce work that I hope to report on one of these days, but Jon Alan Schmidt’s remarks on the Pragmatic Maxim drew me back in for a bit.  He recited one of the places where Peirce declares the role of the Pragmatic Maxim in giving a rule to abduction, a point often missed by many of the most careful commentators.  There are aspects of the Pragmatic Maxim that come more naturally to engineers, workers in the applied sciences, therapeutic professions, and other practical categories than they do to specialists in spectator philosophies.  But I have lost track of that direction for the moment.  No doubt the occasion will arise again …

## Pragmatic Maxims • 3

Inquiry begins in doubt and aims for belief but the rush to get from doubt to belief and achieve mental peace can cause us to short the integrated circuits of inquiry that we need to compute better answers.

For one thing, we sometimes operate under the influence of fixed ideas and hidden assumptions that keep us from seeing the sense of fairly plain advice, and here I would simply recommend reading those versions of the Pragmatic Maxim again and again and trying to triangulate the points to which they point.

For another thing, not everything in logic is an argument.  A well-developed formal system will have:  (1) Primitives, the undefined terms that acquire meaning from their place in the whole system rather than from explicit definitions, (2) Definitions, that connect derived terms to primitives, (3) Axioms, propositions taken to be true for the sake of the theorems that can be derived from them by means of certain (4) Inference Rules.

But that’s just the formal underpinnings — there’s all sorts of informal heuristics, regulative principles, rules of thumb that go toward sustaining any system of significant practical use, and that’s where bits of practical advice like the Maxim in question come into play.

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## Pragmatic Maxims • 2

I tend to think more in relative terms than absolute terms, so I would not expect to find an absolute best formulation of any core principle in philosophy, science, or even math.  But taken relative to specific interpreters and objectives we frequently find that symbolic expressions of meaningful principles can be improved almost indefinitely.

## Pragmatic Maxims • 1

Here is a set of variations on the Pragmatic Maxim that I collected a number of years ago, along with some commentary of my own as I last left it.  As I understand them, they all say essentially the same thing, merely differing in emphasis, point of view, or rhetorical style as befit the moment’s audience or occasion.

## { Information = Comprehension × Extension } • Comment 6

Note. This is a placeholder, to be developed later.

Figure 2 shows the implication ordering of logical terms in the form of a lattice diagram.

Figure 2. Disjunctive Term u, Taken as Subject

### 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.

## { Information = Comprehension × Extension } • Comment 5

Let’s stay with Peirce’s example of abductive inference a little longer and try to clear up the more troublesome confusions that tend to arise.

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

Figure 1. Conjunctive Term z, Taken as Predicate

One thing needs to be stressed at this point.  It is important to recognize that the conjunctive term itself — namely, the syntactic string “spherical bright fragrant juicy tropical fruit” — is not an icon but a symbol.  It has its place in a formal system of symbols, for example, a propositional calculus, where it would normally be interpreted as a logical conjunction of six elementary propositions, denoting anything in the universe of discourse that has all six of the corresponding properties.  The symbol denotes objects that may be taken as icons of oranges by virtue of bearing those six properties.  But there are no objects denoted by the symbol that aren’t already oranges themselves.  Thus we observe a natural reduction in the denotation of the symbol, consisting in the absence of cases outside of oranges that have all the properties indicated.

The above analysis provides another way to understand the abductive inference that reasons from from the Fact $x \Rightarrow z$ and the Rule $y \Rightarrow z$ to the Case $x \Rightarrow y.$  The lack of any cases that are $z$ and not $y$ is expressed by the implication $z \Rightarrow y.$  Taking this together with the Rule $y \Rightarrow z$ gives the logical equivalence $y = z.$  But this reduces the Case $x \Rightarrow y$ to the Fact $x \Rightarrow z$ and so the Case is justified.

Viewed in the light of the above analysis, Peirce’s example of abductive reasoning exhibits an especially strong form of inference, almost deductive in character.  Do all abductive arguments take this form, or may there be weaker styles of abductive reasoning that enjoy their own levels of plausibility?  That must remain an open question at this point.

### 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.

## So Many Modes Of Mathematical Thought

So many modes of mathematical thought,
So many are learned, so few are taught.
There are streams that flow beneath the sea,
There are waves that crash upon the strand,
Lateral thoughts that spread and meander —
Who knows what springs run under the sand?

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