Cf: Peirce's Categories • 21
http://inquiryintoinquiry.com/2020/06/25/peirces-categories-%e2%80%a2-21/
Robert, All ...
Let us 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, as indicated in the following from a standard textbook.
<QUOTE>
A set Γ of sentences is called a “theory”. A theory is said to be “closed” iff every consequence of Γ belongs to Γ. A
set Δ of sentences is said to be a “set of axioms for” a theory Γ iff Γ 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 Γ' of all consequences of Γ is
the unique closed theory which has Γ as a set of axioms.
(Chang and Keisler, p. 12).
</QUOTE>
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 to reserve 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.
(p. 12) https://oeis.org/wiki/User:Jon_Awbrey/Mathematical_Notes#MOD._Note_14
Resources
=========
• Precursors Of Category Theory
https://oeis.org/wiki/Precursors_Of_Category_Theory
• Survey of Precursors Of Category Theory
https://inquiryintoinquiry.com/2015/05/15/survey-of-precursors-of-category-theory-%e2%80%a2-1/
Regards,
Jon
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