Why Grice cared an iota -- as did Peano; or, the implicatures of "the"

Speranza

Grice mentions a list of formal devices at the beginning of "Logic and Conversation" (now WoW). He was being jocular for the rest of the essay is NOT about the formal devices and he rarely comes back to them in the remaining of the lectures.

Among the formal devices are:

a one-place truth-functor: negation -- the tilde -- "not"

two-place truth-functors or connectives: the symbols for "and", "or" and "if"

quantifiers: existential and universal: "all" and "some (at least one)"

and finally

the iota operator: "the".

So it was OBVIOUS that Grice was in a big campaign, for he is wanting to say that the logical form reflects the semantics, and that if there is a felt divergence -- a divergence felt by some, that is -- both formalists or 'modernists' like Carnap and Russell, or informalists or neotraditionalists like STRAWSON) they are WRONG.

He started lecturing on these things at Oxford. I wonder what his students made of his notes. Harvard was perhaps different, in that it was the faculty who attended the lectures!

Anyway, the 'iota operator' has a long history and can be traced back to Peano. I suppose he chose 'iota' because it occurs in Greek phrases with the root 'id-' (I won't say 'idiot') that mean, plainly 'individual'. Only that it's not indidivual or singleton we are talking about but "the" individual or "the" singleton. Russell thought that "the" was very important in mathematics, as it was, I think he believed, the basis for the analysis of 'first'. Peano possibly had seen this too.

Now, R. B. Jones helps clarify the status of the iota. It should be pointed out that in "Presupposition and Conversational Implicature", as reprinted originally in P. Cole, for Academic Press, there is a footnote crediting Hans Sluga.

Grice would often rely on logicians or professional logicians -- he was a philosopher -- for this stuff. Sluga's comments are important when it comes to the logical form involving the iota operator. But Grice remains a Russellian and thus thinks that the iota operator can be ELIMINATED, as they say, by way of a tripartite analysis:

Thus, where "K" represents the predicate "present king of France" and "B" represents the predicate "bald" we have:

∃x[∀y(Ky ≡ y=x) & Bx]

where the 'iota operator' notably does not occur.

Grice was interested in the negation of the above, for STRAWSON has claimed, contra Russell, that the negation of the above, "The present king of France is NOT bald" is neither true nor false but involved what Quine labelled a true-value gap.

Grice is able to demonstrate, via the logical form, the alleged ambiguity here:

∃x[∀y(Ky ≡ y=x) & ~Bx]
~∃x[∀y(Ky ≡ y=x) & Bx]

In the above, the 'definiteness' of "the" (_cum_ uniqueness) is part of the common ground (Grice used extra square brackets for that). But in the second, the 'tilde' of 'not' has widest scope, and thus, no implicature towards the 'definiteness' of "the" (_cum_ existence and uniqueness) is required.

"The king of France ain't bald: France has been a monarchy for some time now, and THAT man you are referring to is De Gaulle, the "president" of France, not the king.

(De Gaulle was elected as the 18th President of France, until his resignation in 1969 -- and Strawson knew it).

But 'iota' is still important as an abbreviatory device. On top of that Elbourne in his book, "Definite descriptions" (Oxford) notes that Grice takes for granted too easily that his tripartite analysis is Russell's ONLY way to regard 'the'. (Elbourne makes a passing reference to Grice's "Vacuous Names" where the use of subscripts would be used instead of square brackets to indicate mere syntactic scope indicator).

In Grice's story then, the affirmative "The King of France is bald" ENTAILS the existence of the king; while the negative "The King of France is NOT bald" merely IMPLICATES it (and is thus cancellable).

When listing the devices, Grice then has 'the iota', and R. B. Jones writes about it:

Jones writes:

"In Church's Simple Type Theory, a really neat simplification of Russell's Theory of types, iota is just a (higher order) function (of type (i,(o,I)), i.e. a function which takes a property and returns a value which might have that property."

Jones goes on:

"The axiom of choice is optional, if you don't include it in the logic then iota is indefinite description, i.e. "an", but when you add the axiom of choice (in the prescribed way) it becomes definite description (" the")."

That's a great remark. I'm surprised philosophers like Grice never cared much for indefinite descriptions. I suppose he was just stuck with the existential quantifier which he takes care to rephrase in the vernacular as "some (at least one)".

Now take the "one" in "some (at least one"). In a way, it resembles, "an", or "a". In others it doesn't. But in the ways it resembles it, I would suppose it is IDENTICAL with it, etymologically.

One does not think much about this way, but when one says

"I saw a chicken in the garden".

The idea is that

"I saw ONE chicken in the garden."

i.e.

"I saw SOME chicken in the garden".

I think G. J. Warnock deals with this in "Metaphysics and Logic" (a lovely essay), where he allows for

"I saw some chickens in the garden"

to be a truthful report of

"I saw one chicken in the garden".

The "at least" is of course prone for implicature. Consider:

A: How old is the baby?
B: He is at least five months old.

versus

B: He is at most five months old.

In a book by Bert Bultinck on numeral meanings alla Grice, the specificity of the reading of "five months old" as "at least five months old and AT MOST five months old" come into consideration.

Jones goes on:

"It's more common of course, to use Hilbert's "epsilon" for choice"

Indeed, but there are indeed cross-references with Donnellan on this. I have commented on the epsilon at the Grice Club -- I hope the search engine does retrieve it!

Jones goes on:

"and of course you can tweak Church's system to use iota just for indefinite and epsilon for definite descriptions."

I see.

I wonder where the 'epsilon' came from. I mean, I can see that Peano chose 'iota' because it does relate to words in Greek beginning with "id-" which would feature the iota. But then I also should care a lambda why the lambda calculus is named after that particular Greek letter.

Jones concludes:

"As to italics, ... the systems I work with don't usually support italics!"

Well, when Grice re-labelled Donnellan's referential/attributive distinction in the use of 'the' in "Vacuous Names" he does not use iota but find it more entertaining to use FULL CAPITALS:

THE F

for the identificatory use -- and the italics

the F

for the non-identificatory.

I think that's pretty neat, but I grant, you have to import italics which means someone has to export them for you!