[lit-ideas] Re: Griceian Numbers
- From: Jlsperanza@xxxxxxx
- To: lit-ideas@xxxxxxxxxxxxx
- Date: Sun, 17 Jun 2012 12:42:01 -0400 (EDT)
We are analysing the meaning of 'three' (as Wittgenstein uses it in 'The
Brown Book'). Ayer claims that 'three' is not a sense-datum ("Surely it would
be ridiculous to say, "I saw three reds today"). Yet, D. McEvoy claims
that "Witters" (as the name is sometimes abbreviated) is _onto something_.
To clarify, consider Grice's analysis of universal quantifiers in plural
domains.
Consider
(1) All dogs are barking
(2) All the dogs are barking
(3) All three dogs are barking.
Item (1) is fairly straightforward, being analyzed as follows.
"All dogs are pets . (1õ.)õ. 1õ. are barking Ïõ [(1õ.)õ.] Ï all 1 dogs
"x { D+[x] ²B[x] } Q1 "x { D+[x] ² Q(x) } lx1 B[x] are barking. l P0lQ1"x{…}
lx0 D+[x] all1. .dogs."
This reads the sentence as saying that every plural-set of dogs has the
following property – its members are barking. Only granting that "barking" (as
perhaps unlike "pissing") is distributive, this amounts to saying that
every dog is barking.
Now, item (2) is not so straightforward, since it appears to have a
double-determiner, involving in
particular a type-mismatch between ‘all’ and ‘the’. In order to resolve
this problem, Grice proposes an optionally-pronounced partitive ‘of’
interposed between ‘all’ and ‘the’, as in the following grammatical analysis:
All [of] the dogs are barking
for which the clearer logical form becomes:
(1õ.)õ. 1õ. are barking Ïõ [(1õ.)õ.] Ï all 1 õÏ [of] Ïõ Ï the dogs x {
D[x] ²B[x] } l Q1 "x { D[x] ² Q(x) } lx1 B[x]
.are barking. l P0lQ1"x{…} lx0 D[x] . all1. l y lx0 {x.y} mx D+[x]/of.l P0
mxP(x) lx0 D+[x] the. .dogs.
Thus, according to Grice's analysis, the sentence says that every
dog-entity "is" barking, which granting
distributivity is the same as every individual dog (in the relevant
domain) is barking. Note incidentally that the partitive use of ‘of’, can be
easily and categorially rendered alla Frege: type( of) = õÏ . of. = ly lx0
{x.y}
Thus, Grice concedes, ‘of’ converts a proper-noun phrase into a
common-noun phrase, pretty much reversing the effect of ‘the’. ("I owe most of
my
reflections on 'of' to my once tutor at Corpus Christie, R. W. Hardie, who
once challenged me with, "And what, if I may know, do you mean by 'of'?").
Item (3), "All three dogs are barking" is, perhaps, less straightforward.
First, a naïve analysis, which we should of course, rejects (alla Witters)
goes as follows.
all three dogs are barking. (1õ.)õ. 1õ.are barking Ïõ [(1õ.)õ.] Ï all 1
ÏõÏ Ï three dogs
"x { D[x] & 3[x] .²B[x] }l Q1 "x { D[x] & 3[x] .Q(x) } lx1 B[x] are
barking. P0lQ1"x{…} lx0 { D[x] & 3[x] }
. all1 .l P0 lx0 { P(x) & 3[x] } lx0 D+[x] three. .dogs.
This reads the sentence as saying that the members of every 3-membered set
of dogs are barking.
This is not a very plausible reading, unless we bring a lot of IMPLICATURE
to save it (as Stephen Yablo, a former student of Grice's once said,
"Implicatures happen").
The problem with the Implicature account is of course Witters, for he NEVER
cares to provide a conversational context to his odd utterances ("Pass me
three of those solid red bricks" -- does he mean, "at least three", "at
most three"? What is SHOWN in those utterances by one mason to another?)
A more plausible reading, of "All three dogs are barking" posits
unpronounced material as in the following.
all [of the] three dogs are barking . (1õ.)õ. 1õ. are barking Ïõ
[(1õ.)õ.] Ï all 1
õÏ [of] Ïõ Ï [the] ÏõÏ Ï three dogs x { x.mx{D[x]&3[x]} ² B[x] }
Q1"x{x.mx{D[x]&3[x]}²Q(x)} lx1 B[x]
.are barking l P0lQ1"x{…} lx0{x.mx{D[x]&3[x]}} all1. ylx0{x.y}
mx{D[x]&3[x]} of. l P0 mxP(x) lx0{D[x]&3[x]}
.the. l P0lx0{P(x)&3[x]} lx0D+[x] three. .dogs.
This reads (with G. Hardegree) the sentence as _saying_ (as, for once
Grice and Witters may agree), in effect, that there are exactly three dogs,
and
they are all barking. Whether there are further agreements it's not for
this one post to verify (or falsify for that matter). (
Cheers,
Speranza
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