undulating humming

Based on some links chrisamaphone tweeted a while back—and because UBC has decent (and English-language) libraries—I checked out Vidyabhusana's A History of Indian Logic. I've been reading (and tweeting) odd passages from it, usually just for amusement; the book was finished in 1920 (this copy seems to have been printed around 1971, though the paper is decaying so fast that I assumed it was printed in the 1920s).

The examples are not the typical ones found in European logic. The role of the default Aristotelian syllogism, headed by "All men [sic] are mortal" ("major premise"), seems to be played by a syllogism headed by "This hill is fiery" ("proposition"). The Indian syllogism (avayava) has five "members": a proposition, a reason, an "explanatory example", an "application of the example", and a restatement of the proposition (conclusion).

The five members may be fully set forth as follows:—
(i) Proposition—This hill is fiery.
(ii) Reason—Because it is smoky.
(iii) Example—Whatever is smoky is fiery, as a kitchen (homogeneous or affirmative).
(iv) Application—"So" is this hill (smoky)—(affirmative).
(v) Conclusion—Therefore this hill is fiery. (page 61)

Since the first and last members are the same, the syllogism looks like a four-place relation. The fourth member, "Application", also seems redundant, which gives a three-place relation. But the "explanatory example" really includes both a quantified implication (≈∀X. smoky(X) ⊃ fiery(X)) and an instantiation (a kitchen). So, boiling it down (on the kitchen fire):

(i) This hill is fiery.
(ii) Because this hill is smoky.
(iii)(a) Whatever is smoky is fiery—
(iii)(b) for example, a kitchen.

We can make this look like a judgment of the form

(i)conclusion ⇐ (ii)reason〈(iii)(a)quantified-implication, e.g. example〉

this hill is fiery ⇐ this hill is smoky〈whatever is smoky is fiery, e.g. a kitchen〉

Reading this right-to-left, this looks like the all-men* syllogism:

whatever is smoky is fiery
this hill is smoky
∴ this hill is fiery

…except that the example is omitted. This is interesting: the Indian syllogism, as an inference rule, has all the usual components of a combined ∀-elim/⊃-elim, plus an example. You can't use the rule unless you give an example of something that is both smoky and fiery; the set you're quantifying over can't be empty!

I don't want to overstate this; even from the fragments of the book I've read, it seems that the members of the syllogism can often be omitted. Moreover, a syllogism can use negative or heterogeneous reasoning: "whatever is not fiery is not smoky, as a lake", which doesn't incur a non-vacuousness obligation (or rather, you have to show that the complement is nonempty). But the default form of syllogism does seem to require giving an example.

This may have practical significance, at least if we trust the following anecdote (which I heard via my brother; who knows how many adventures it's had along the way):

A graduate student was defending their dissertation, which proved many interesting theorems about a particular topological structure—~manifolds satisfying properties P, Q, and R~. A random faculty member (not on the student's committee) showed up to the defense**. After the student's talk, the random faculty member asked, isn't your entire thesis vacuous for the following trivial reasons? Whereupon the random faculty member showed*** that

∀m. (P(m) ∧ Q(m) ∧ R(m)) ⊃ ⊥

And sadness descended.

In the version I heard, the student managed to rewrite their thesis so it stated theorems about manifolds that actually existed. But if only they'd been in the habit of finding suitable kitchens…

* #NotAllMen.
** This supposedly happened somewhere in the US.
*** With far more verbal scaffolding, of course.

Update: A few minutes after posting this, I found another passage, summarizing some writings of Māṇikya Nandi, who wrote "a standard work on Jaina logic" and who "seems to have lived about 800 A.D." He discussed the terms of a syllogism, summarized thusly (p. 191):

Example is superfluous.

The middle term and the major term are the parts of an inference, but the example is not. Nevertheless for the sake of explaining matters to men of small intellect, the example, nay, even the application and the conclusion, are admitted as parts of an inference.[Untranslated parentheticals omitted.]

That is rude, nonsense, unethical!