Pleonetetic Implicatures

By J. L. Speranza,

for the Grice Club







In "Logic and Conversation", Grice proposes some counterparts to the logical devices of the logicians:

& and
v or
-> if
(x) all
(Ex) some
(ix) the


--- At this point, Toulmin interrupted, "What about 'most'"?

"Most what?", Grice asked.

"Most xs. Or Few xs, for that matter"

"Your point?"

"Surely we can use them in inferences"

"I would be inclined to suggest," Grice said, "that what we mean by 'many' may well end up depending on what I'd call a 'threshold': 'many people read the Times', 'many people read Kant's Critique of Practical Reason'. What counts as 'many', while logic, is pleoretetic".

---









"You should keep using 'pleonetetic. It _is_ a useful
word, but its usage too narrow to merit entrance
in the OED" -- JulieFields@oed.oup.co.uk, to J. L. S.


"Logicians bore me: Grice primarily.
It's all goaty words with them: never
non-logical goats, like "most" or "few""

Toulmin, Uses of Argument.




The logic of "many", "few" and "nearly all"

What a good thing is to have a contact in Cambridge, even for an Oxonian
like me!

J. E. J. Altham. Jimmy Altham was Sidgwick Lecturer in Philosophy until July
1999: he later became part-time member of the Faculty, and a Fellow of Gonville
and Caius College. He was a student at Trinity College, Cambridge, where he
was taught by Casimir Lewy.

In 1965 he was elected a Fellow and Director of
Studies in Philosophy at Caius.

His PhD dissertation 'Assertion, Command,
Obligation' investigated the logical powers of linguistic force indicators.
He continued to study logic, and published "The Logic of Plurality" (London:
Methuen, 1971), which is about such quantifiers as


'many','few' and 'nearly all'.


- whereas Grice, as we'll recall, had only considered "all" and "some (at least one)" in his "Logic & Conversation", repr. in Studies in the Way of Words.

Although he has continued to work and publish in philosophical logic, the
most recent example being 'Reporting Indexicals' in Philosophical Logic
edited by Timothy Smiley (Oxford: Clarendon Press, 1998), ethics is now his
principal area of study. This grew out of one topic of his dissertation, and
resulted first in 'Evaluation & Speech' in "Morality & Moral Reasoning",
edited by John Casey (London: Methuen, 1971).

This was an essay in post-Gricean neo-emotivism.

Hoorah! Trust Jimmy to be one of us! H. Paul Grice being the English Oxford
philosopher usually associated with "emotivism" after his "Studies in the
Way of Words". For a defense of emotivism, see M. Nantongo.

He co-ordinates the Ethics teaching for the Faculty of Philosophy, and gives
several courses in the subject. These include

'The Ends of Action',
'Responsibility',
'Rights' and

Things Julian O'Dea think they don't exist.


'Punishment'.

He recently introduced 'Green Values' a course in environmental ethics. A
paper 'Ethics of Risk' (Proceedings of the Aristotelian Society, 1983), was
reprinted in The Philosopher's Annual for that year. It argued for a
quasi-contractualist account of the extent to which

it is legitimate to impose risks on others,

and against utilitarian and natural rights theories. He contributed an essay
'Reflection & Confidence' to "World, Mind & Ethics (Cambridge, CUP 1995), a
collection of essays on the work of

that Oxonian apostate.

Professor Bernard A O Williams, which he also edited with Ross Harrison. His
essay casts doubt on Williams's claim that

reflection can destroy knowledge.

on some people.

Jimmy Altham has taught for other Faculties and Departments. He devised and
was the first to give the course on professional ethics and intellectual
property for the Computer Sciences Tripos, and he has contributed to the
Teaching programmes of the Diploma in Public Health, History and Philosophy
of Science, Social and Political Sciences, Classics, and History. He is to
become a Q.A.A. Subject Reviewer in Philosophy, and he is a member of the
Philosophy Benchmarking Group for Q.A.A. Jimmy Altham's current research is
in Environmental Ethics. An essay 'Quality of Life, Environment & Markets'
is forthcoming. This endorses a conception of efficiency drawn from E. F.
Schumacher, according to which

efficiency is determined by the amount of well being
yielded by each unit of resource.

He is working on a theory of property which attempts to combine Humean
elements with an environmentalist perspective. He is trying to understand
conceptions of

non-anthropocentric value,

but has not yet succeeded. He recently gave a lecture on

'Human Interests & Natural Value' for the Master of Studies in
Interdisciplinary Design for the Built Environment, March 2000, and a talk
'Towards an Environmental Theory of Property' to the Philosophy Society,
University of Kent, November 1999. Other recent talks include 'Taking
account of natural value' to the consultation on the Future of Farming &
Public Aspirations, at St George's House, Windsor Castle, January 2000., and
a lecture on environmental ethics in the Bio-Ethics programme for trainee
teachers of Biology, Homerton College, February 2000. Other interests
include membership of the Ethics Committee of the East Anglian Ambulance
Trust. His concern with the environment finds more practical expression
through his membership of the
Henry Doubleday Research Association,

which studies organic gardening; the Permaculture Association, which
furthers design for sustainable living; Cambridge Local Exchange Trading
System, a kind of 'alternative economy' whose currency is the Cam, and the
Woodland Trust.

He is also developing a garden to provide a habitat
for wildlife and food for his family.

He enjoys music and has been a member of the Faculty Board of Music, and is
organiser, Cambridge Piano Weekend (recital and masterclasses with Bernard
Roberts), April 2000; he rides a small motor-bike, and has recently taken a
course in Buddhist meditation.

For the use of "pleonetetic" cf.

1. Gyula Klima: Approaching Natural Language Via Mediaeval Logic
... a cue from Montague, is intended to cope with the troubles caused by
'pleonetetic'
determiners and common noun phrases of natural languages in general. [8 ...
http://www.fordham.edu/gsas/phil/klima/NLN.htm -

2. REVIEW of the Geach festschrift,
"Peter Geach: Philosophical Encounters", ed. by H. A. Lewis.
London: Kluver, 1991.
http://www.nb.vse.cz/kfil/win/logpoint/93-3/REVIEWS.htm -
... of the quantifier "most" ("most As are B"), considers some interesting
examples with
this quantifier, and gives some principles for monadic pleonotetic logic. ...
J.E.J. Altham (in 'Plural & Pleonetetic Quantification') investigates
relational quantifiers, such as "there are more As than Bs", "nearly every A
is B", and "many As are B". He also deals with pleonetetic logic, that is
the logic of the quantifier "most" ("most As are B"), considers some
interesting examples with this quantifier, and gives some principles for
monadic pleonotetic logic.
http://www.nb.vse.cz/kfil/win/logpoint/93-3/REVIEWS.htm -

3. Essay by E P Brandon on "Deductive Logical Competence"
http://www.uwichill.edu.bb/bnccde/epb/NGLC.HTM


MVAL [valid] Most As are B, all B are C.
Ergo, most As are C.

MINVAL [invalid] Most As are B, most Bs are C.
Ergo, most As are C.


UMMOST [invalid] Most As are B, all Cs are A.
Ergo, some C are B.

"An interesting and unusual feature of the investigations reported here is
that principles involving the plurative or pleonetetic (Geach's terms [P T
Geach, Reason & Argument, Oxford: Blackwell, 1976]) quantifier "most" have
often been used. This quantifier is not normally studied in elementary
formal logic, even though its meaning is perhaps closer to what we often
intend in using plurals or the universal quantifier "all" than how that
universal quantifier is itself construed in formal logic (cf. Hodges [W
Hogdes, Logic. Harmondsworth: Penguin, 1977], p. 196)."

This essay suggests that J E J Altham originally favoured the term
"plurative" in his book "The Logic of Plurarity", of 1971, and that
"pleonetetic" was indeed first used by P T Geach 5 years later, in his book
"Reason & Argument" of 1976. The Geach festschfirt cited above, published in
1991, to which Altham contributed, suggests that Altham thus adopted the
term after Geach).


"Sortal Quantification",

in E. A. Keenan, "Formal Semantics of Natural Language". Cambridge, UP., 46ff.

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 owns a car.

This is logically equivalent to

2. Few men do not own a car.

which in turn is equivalent to

3. Not many men do not own a car.

There is, 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, e.g.

4. I have a few books.

is equivalent to

5. I do not have many books.

Similarly,

6. I seldom go to London.

is equivalent to

7. I do not often go to London.

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 I call

(I,I)-quantifiers

i.e. quantifiers which bind 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

(I, k)-quantifiers

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

8. There are exactly as many Apostles as there are days of Xmas.

We have here a (1,2)-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 As.

is that

10. It is not the case that there is only one A.

It also seems that, in general, how many As there need to be for there to be
"many As" depends on the size of the envisaged domain of discourse. E.g. in

11. There are many communists in this constituency.

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

12. There are many communists in England.

and consequently the number of communist there have to be for there to be
many communists in this constituency is smaller than the number there have
to be for there to be many communists in England. This suggests the use of a numerical method in providing the appropriate truth-conditional semantics, by selecting a number

"n"

which is

the LEAST number of things

there have to be with a certain property A for there to be "many" things
with that property, an important constraint being, of course, that

n > 1.

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 LEAST number of things

that must have some property if there are to be "a few" things with that
property. 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 can be used
also in the case of the quantifiers compounded with "very". Thus, the
threshold-number associated with

"very many"

will be

n + m,

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

"a few",

the threshold for "a very few" will be

k - l.

Repetitions 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 things which are both A & B.

This is one in which the quantifier is not "sortal", as I call it, and is
logically equivalent to

14. There are many things which are both B & A.

In contrast,

15. Many As are Bs.

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

16. Many As are Bs.

In (15), the quantifier's range is restricted to the set of As: thus the set
of As becomes the domain of discourse whose size determines an appropriate
threshold number. Consequently, since the set of As may NOT have even nearly the same cardinal number as the set of Bs, the threshold-number determined by one may be
different from that determined by the other, as in 17 vs. 18:

17. Many specialists in Old Norse are university officers.
18. Many university officers are specialists in Old Norse.

It seems that (17) is true and (18) false: there are "more" university
officers than specialists in Old Norse, the threshold-number for "many
university officers" is correspondingly larger than that for "many
specialists in Old Norse". Or consider the conditional,

19. If many professional men own French motorcars, and
there are at least as many professional men as owners
of French motorcars, many owners of French motorcars
are professional men.

Now, take a segment of (19), viz.

20. There are at least as many professional men as owners
of French motorcars.

How do we represent that formally? I suggest this be done by the what I call

(I,k)-quantifiers.

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

21. Most As are Bs.

is logically equivalent to

22. There are more things which are both A and B
than there are things which are A and not B.

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

23. Many As are B if not far from half the As are B.

In this case, we could give, as logically equivalent to (17)

24. There are at least nearly as many specialists in Old Norse
who are university officers as there are specialists in
Old Norse who are not university officers.

where (24) is not sortal. This transcription renders patent the lack of
equivalence between (17) and (18) above, for, as anyone can see, (18) emerges as

25. There are at lest nearly as many university officers
who are specialists in old Norse as there are university
officers who are not specialists in Old Norse.