Griceian infinity and the null set -- Ø --

Speranza

Further to previous commentary.

Part of the charm of "Vacuous Names" is how it connects with some demons within Grice -- and perhaps Carnaap.

In "Reply to Richards", Grice speaks of the demon (or bête noire) of Extensionalism, which may be made to connecd vis–à–vis certain points by Jones re: the policy of having the null set do duty for more things that it should!

On p. 68, Grice refers to  "Extensionalism" as a

"position imbued with teh spirit of Nominalism [another demon], and dear both to those who feel that 'Because it is red' is no more informative as an anser to the question 'Why is an English mail-box called 'red'?' than would 'Because he is Paul Grice' is an answer to the question 'Why is that distinguished-looking philosopher called 'Paul Grice'?', AND also to those whe are particularly impressed by the power of set theory."

----

Grice goes on

"The picture which, I suspect, is liable to go along with Extensionalism is that of the world of
PARTICULARS as a DOMAIN stocked with innumerable tiny pellets, internally INdistinguishable
from one another, BUT distinguished by the groups within which they fall, by the 'clubs' to which they belong."

"And since the clubs are distinguished ONLY by their memberships, there can

ONLY BE ONE CLUB TO WHICH NOTHING BELONGS."




The null set --  ø --


----


Grice goes on:

"As one might have predicted from the outset, this leads to the trouble when it comes to the
accomodation of EXPLANATION within such a system."



"Explanation of the ACTUAL presence of a particular feature"
in a particular subject depends
CRUCIALLY on the possibility of
saying what WOULD be the
consequence of the presence of such
and such features in that subject,
regardless of whether the
features in question even DO appear
in that subject, or indeed in ANY subject."



"On the face of it, if one adopts an
extensionalist viewpoint, the
presence of a feature in some particular
will have to be RE-EXPRESSED in terms
of that particular's membership of a
certain set; but if we proceed along
those lines, there
THERE IS ONLY ONE EMPTY SET,
the potential consequences of the
possession of in fact UNexemplified
features would be
INVARIABLY THE SAME,
now matter how different in meaning
the expressions used to specify
such features would ordinarily be
judged to be."

:

"I can think of TWO ways in which
to avoid [this unacceptable conclusion
of extensionalism -- as per above]."

Both ways, Grice says, "seem to me to suffer from serious drawbacks"

He cares them to expound them in some detail-

FIRST EXTENSIONALIST MANOUVRE with the null set:

"The first shows some degree of analogy

with a move which, as a matter of history,

was made by empiricists in connection with simple and complex ideas.

In that region an idea would be redeemed

from a charge of failure to conform to

empiricist principles though not being

derived from experience of its instantiating

particulars (there being no such particulars)

if it could be exhibited as a complex

idea whose component simple ideas were so derived."



Grice goes on:

"Somewhat similarly, the first proposal

seeks to

RELIEVE CERTAIN VACUOUS PREDICATES

or general terms from the

embarrasing consequence of

DENOTING the empty set

by exploiting the

NON-VACUOUSNESS of OTHER predicates

or general terms which are constitutents

IN THE DEFINITION of the original vacuous terms."



Grice goes on:

"(alpha) Start with TWO vacuous predicates,

say (ALPHA-1) 'is married to a daughter

of an English queen and a pope' and

(ALPHA-2) 'is a climber on hands and knees of

29,000 foot mountain."

"(BETA) If alpha-1 and alpha-2 are vacuous,

then the following predicates are

satisfied by the empty set Ø"




---

Grice continues:

(BETA-1) 'is a set composed of

daughters of an English queen and pope', and

(BETA-2) 'is a set composed of climbers

on hands and knees of a 29,000 foot moutantain".

Grice proceeds with a third step:

"(GAMMA) Provided R1 and R2 are suitably

interpreted, the predicates

beta-1 and beta-2 may be trated as

CO-EXTENSIVE

respectively with the following

REVISED predicates

'gamma-1' 'stands in R1 to a sequence

composed of the sets 'married to', 'daughters',

'English queens' and 'popes''

and

'gamma-2, 'stands in relation R2 to a

sequence composed of the set 'climbers',

'29,000 foot mountains', and 'things

done on hands and knees'."


The fourth step he calls delta.


"DELTA. We may FINALLY correlate with

the two initial predicates alpha-1 and alpha-2,

respectively, the following sequences

derived from gamma-1 and gamma-2:

delta-1, the sequence composed of the

relation R1 (taken in EXTENSION), the

set 'married to', the set 'daughters', the

set 'English queens', and the set 'popes';

and

delta-2: the sequence composed of the

relation R2, the set

'climbers', the set '29,000 foot mountains',

and the set 'things done on hands and knees'".


Grice goes on:


"These sequences are clearly distinct, and the proposal

is that THEY, rather than the EMPTY SET, should be

used for determining, in some way yet to be

specified, the explanatory potentialities of the

vacuous predicates alpha-1 and alpha-2."


Grice writes against this proposal: "My chief complaint against this proposal is that

it involves YET another commission of what I regard as one

of the main MINIMALIST sins, that of imposing

IN ADVANCE a limitation on the character of

explanations."

---

Why? Well, 'for it implicitly recognises it

as a condition on the propriety of using

vacuous predicates in explanation"

--- should that be proved!

"that the terms in question should be

representable as being correlated with

a sequence of NON-EMPTY sets."



Grice notes: "This is a condition which, I suspect, might

not be met by every vacuous predicate. But

the possibility of representing an explanatory

term as being, in this way or that,

reducible to some favoured iterm or types of

items should be a BONUS which some theories achieve,

demonstrating their elegance, not a condition of

eligibility for a particular class of would-be

explanatory items."


What about a second proposal then?

"The second suggested way of avoiding the

unwanted consequence is perhaps more intuitive

thatn the first. It certainly seems simpler."


Grice goes on: "The admissibility of vacuous predicates

in explanations of possible but non-actual

phenomena (why they would happen if they did happen),

depends, it is suggested, on the availability

of acceptable non-trivial generalisations wherein

which the predicate in question specifies the

antecedent condition."

Clear enough.

Grice goes on: "And, we may add, a generalisation whose

acceptability would be unaffected by any

variation on the specification of its antecedent

condition, provided the substitute were

vacuous, would certainly be trivial."



Grice goes on: "Non-trivial generalisations of this sort

are certainly available, if (1) they are

derivable as special cases from other generalisations

involving less specific antecedent conditions, and

(2) these other generalisations are adequately

supported by further specifics shose antecedent

conditions are expressed by means of non-vacuous

predicates."



Grice concludes about this second proposal: "The explanatory opportunities for vacuous

predicates depend on their embodiment in

a SYSTEM".

However, there's the drawback.

Grice notes: "My doubt about this second suggestion relate

to the STEPS which woud be needed in order

to secure an adequately powerful system."



Grice goes on:

"I conjecture, but cannot demonstrate, that

the ONLY WAY to secure such a system

would be to confer SPECIAL ONTOLOGICAL

privilege upon the entities of physical science"


Grice goes on:

"together with the system which that

science provides. But now a problem arises:

the preferred entities seem NOT to

be observable"





Grice goes on:

"or, in so far as they ARE observable,

their observability seesm to be more a

matter of conventional decisions

to COUNT such and such occurrences"



Grice goes on: "AS observations than it is a matter of

FACT. It looks as if states of affairs

in the preferred scientific world NEED,

for credibility, support from the

vulgar world of ordinary observation reported in the language of common sense."

-

Grice goes on:

"But to give THAT support, the

judgements and the linguistic usage

of the VULGAR nneds to be endowed

with a certain authority , which

as a matter of history"

--- things CAN change; people learn.

"the kind of minimalists whom I know or

know of have NOT seemed anxious

to confer."



To conclude: "But even if there WERE anxious

to confer it, what would validate

the conferring, since ex hypothesi it is NOT

the vulgar world but the specialist

scientific world which enjoys

ontological privilege? (If this objection

is sound, the second suggestion, like the

first, takes something which when present

is an assert, bouns, or embellishment, namely

systematicity, and under philosophical pressure

converts it into a necessity)."

And so on.