## Peirce’s Categories • 21

Dear Robert,

Let’s go back to a point where paths diverged in the yarrow wood and a lot of synchronicity was lost …

Variant understandings of words like axiom, definition, predicate, proposition, proof, relation, theory, and the like make mutual understanding difficult.  For example, when I mention Peirce’s definition of a sign, many people will bring to mind a long list of short statements Peirce made in describing the properties of signs, and when I refer to Peirce’s theory of signs, many people will bring to mind the entire corpus of Peirce’s writings on signs, so far as they know it, augmented perhaps with reliable reports of statements he may have made about signs.

There are fields of study where such expansive understandings of definitions and theories are the prevailing ones, perhaps the only feasible ones.  One example would be scriptural hermeneutics, where the full sense of a word’s meaning is determined by its use in every context where it occurs.  Thus the use of concordances to bring the diversity of contextual meanings into harmony.  We plow this field in a hermeneutic circle, according each bit of authoritative text equal priority, none privileged above the other, as if equidistant from a central point radiating a pervasive message.  It’s all you can do when there’s nothing but the text in view.

Curiously enough, the branch of mathematical logic known as model theory sets out with an equally expansive view, taking a maximally inclusive definition of theory as its initial point of departure and defining a theory as an arbitrary set of sentences from a formal language.  Naturally, logical and mathematical attention almost immediately shifts to more focused spheres of theory.

A set $\Gamma$ of sentences is called a theory.  A theory is said to be closed iff every consequence of $\Gamma$ belongs to $\Gamma.$  A set $\Delta$ of sentences is said to be a set of axioms for a theory $\Gamma$ iff $\Gamma$ and $\Delta$ have the same consequences.  A theory is called finitely axiomatizable iff it has a finite set of axioms.  Since we may form the conjunction of a finite set of axioms, a finitely axiomatizable theory actually always has a single axiom.  The set $\bar\Gamma$ of all consequences of $\Gamma$ is the unique closed theory which has $\Gamma$ as a set of axioms.  (Chang and Keisler, p. 12).

That’s all well and good as far as esoteric technical usage goes but outside those cloisters I would recommend using the word corpus when we want to talk about an arbitrary set of sentences or texts and reserving the word theory for those corpora possessing more differentiated and substantial anatomies than a mere hermeneutic blastula.

### Reference

• Chang, C.C., and Keisler, H.J. (1973), Model Theory, North-Holland, Amsterdam.

### Resources

cc: CyberneticsOntolog • Peirce List (1) (2) (3)Structural ModelingSystems Science

### 1 Response to Peirce’s Categories • 21

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