Threshold to Grice

---- By JLS
--------- For the GC

---- WE ARE DISCUSSING GRICE's sympathy (and indeed that of most of us) for 'enough' ("Enough said" -- but cfr. "Enough implicated" -- "The man who implicated too much". "The addressee who read too much between the lines," etc. "Enough", Kramer notes, "is the quintessentially analog word", and everybody gotta love it. It's a threshold word, too. As when we infer that if Joan Rivers says she is 65 that's all she is. 'Nuff said, 'nuff implicated.

The idea of a threshold is due to Altham in Keenan. He starts with the idea that the digital /\x ('all' -- qua conjunction) and \/x ('some' -- qua disjunction) are hardly part ('if not parcel' he adds) of the most analogous of our conversations. "Too much alcohol?" "All the alcohol?". Nonsense.

As the notes, the English words

"every", "any", and "none"

can be qualified by certain adverbs, so that we may say, for instance,

"nearly every", "scarcely any", and "almost none". Consider

(1) Amost every man has a penis. (Indeed, by default, most are born with one).

This is logically equivalent to

(2) Few men do not have a penis.

which in turn is equivalent to

(3) Not many men do not have a penis.

"There is," Altham notes, "indeed, a pretty close correspondence between two sets of words as follows"

always ever often seldom sometimes never
every any many few some none

where the terms in the upper line interrelate in the same way as do those on
the lower. Consider:

(4) I have a few balls.

is equivalent to

(5) I do not have many balls.

Similarly,

(6) I seldom die.

is equivalent to

(7) I do not often die.

Further quantifiers are discernible in English, unless the eyes deceives
one. As well as "few", "many", and "nearly all", we have

"very few", "very many", and "very nearly all",

and yet more result from reiterated prefixing of "very".

In their representation in a formal syntax, all the foregoing expressions
coume out as what Altham calls an

(I,I)-quantifier

i.e. a quantifier which binds one variable in one formula. A formal syntax, together with appropriate semantics, which gives an appropriate treatment to all these is a significant generalisation of classical quantificational methods on the pattern of ordinary logic. The further generalisation to a

(I, k)-quantifier

- binding one variable in an ordered k-triple of formulae, gives a further
increase in power. Thus, consider

(8) There are exactly as many members in the Grice Club as you care to have them.

We have here a (1,care)-quantifier. It seems significant that we can build up additional quanitifers in much the same way as we can (1,1)-quantifiers. For instance, from

"more than"

we can to to

"many more than"

and

"very many more than". We have such expressions as

"nearly as many as" and "almost as few as" -- and so on. As to their truth-conditional semantics, one thing that is clear about the truth conditions of

(9) There are many members of the Grice Club.

is that that it implicates, but does not say:

(10) It is not the case that there is only one member of the Grice Club.

It also seems that, in general, how many members there need to be for there to be
"many members" depends on the size of the envisaged domain of discourse, in this case, 'The Grice Club'. E.g. in

(11). There are many chairmen in the Grice Club.

the domain of discourse would probably be the electorate of the constituency of the Grice Club. This domain is smaller than the one envisaged in

(13) There are many female members in the Grice Club.

and consequently the number of female members in the Grice Club have to be for there to be many chairmen in the Grice Club is smaller than the number there have
to be for there to be many female members in the aforementioned club (to wit: the Grice Club).

This suggests the use of a numerical method in providing the appropriate
truth-conditional semantics, by selecting a number. Let's use

"n"

which is such that it refers to

the LEAST number of things (or persons)

there have to be with a certain property, e.g. "the Grice Club", for there to be "many" things (or persons) with that property (or club), an important constraint being, of course, that it better be that

n > 1.

(but cf. Groucho Marx -- and the one-member set). Now, "n" varies with the domain of discourse, and its value relative to numbers associated with other quantifiers should be correct. Thus, the quantifier

"a few"

is given a truth-conditional semantics in a way similar to those for

"many", in terms of

the aforementioned LEAST number of things (or persons)

that must have some property (such as a club) if there are to be "a few" things with that property (i.e. belonging to that club, or having a penis). If such numbers are termed

THRESHOLD-NUMBERS,

the essential condition is that the threshold-number associated with "a few"
should be smaller than that associated with "many".

This method -- the Grice Projector -- can be used also in the case of the quantifiers compounded with "very" (archaically, 'verily', i.e. truly). Thus, the
threshold-number associated with

"very many", or 'verily many,' if you must

will be

n + m,

with m positive, if n is the threshold-number for "many". Analogously,
if "k" is the threshold for

"a few",

the threshold for "a very few" will be

k - l.

(read "k minus 1")

Emphatic repetitions, on occasion, of "very" can be coped with similarly, and, also, the multiplicity of threshold-numbers is reduced by the possibility of defining
some quantifiers in terms of others.

Thus:

"nearly all" is "not many not".

Now consider

(13) Thre are many persons which both have a penis and are members of the Grice Club.

This is one in which the quantifier is not "sortal", as Altham calls it, and is thus logically equivalent to

(14) There are many persons which are both members of the Grice Club and have a penis.

In contrast,

(15) Many members of the Grice Club have a penis.

involves a queer "sortal quantifier", and is not equivalent to

(16) Many penis-holders are members of the Grice Club. (nonsequitur).

In (15), the quantifier's range is restricted to the set of the members of the Grice Club. Thus, the set of such members becomes the domain of discourse whose size determines an appropriate threshold number. Consequently, since the set of members of the Grice Club may not, by chance, have even nearly the same cardinal
number as the set of persons with penises, the threshold-number determined by one may be different from that determined by the other, as in:

(17) Many specialists in Grice are not members of the Grice Club.

(18) Many specialists in blog officers are members of the Grice Club.

It seems that (17) is true and (18) false: there are "more" specialists in Grice
officers than members of the Grice Club (we assume). I.e. the threshold-number for "many specialists in Grice" is correspondingly, we hope, larger than that for "many holders of a penis". Now, consider the conditional,

(19) If many members of the Grice Club own a convertible, and there are at least as many holders of a penis as owners of a convertible, many owners of a convertible are members of the Grice Club.

To prove its validity, let us take a segment of (19), viz.

(20) There are at least as many members of the Grice Clb as owners of a convertible.

How do we represent that formally? Altham suggests it be done by a

(I,k)-quantifier

and as such the method would involve the application of a proceudre which enables sortal quantifiers to be replaced by more complex quantifiers which are *not* sortal. Thus, it is clear that the sortal

(21) Most penis-holders are men.

is logically equivalent to

(22) There are more things which are both penis-holders and men than there are things which are penis-holders and not men.

which involves a non-sortal (1,2)-quantifier. Similarly we may think that

(23) Many penis-holders are men if not far from half the penis-holders are men.

In this case, we could give, as logically equivalent to:

(24) There are at least nearly as many specialists in Grice who are (honorary) members of the Grice Club as there are penis-holders who are not men.

where (24) is not sortal. This transcription renders patent the lack of
equivalence between the pair above, for, as anyone can see, the latter emerges as

(25) There are at lest nearly as many members of the Grice Club who are specialists in Grice as there are penis-holders who are not men.

-- which trades, alas, on the pragmatically ambiguous expression, 'to hold'.