1
Text 103: (
Read Russell on Ag. And Dem.
)
1
A.N. Prior
Suppose we write
Î
xFx
for
âFor any
cogitabile
x, Fxâ
Î£
xFx
for
âFor some
cogitabile
x, Fxâ
â
xFx
for
âFor any real thing x, Fxâ
â
xFx
â
â some â â â, Fxâ
& define the latter
two thus:

â
xFx
= Î x (Rx â
Fx)
â
x
Fx
= Î£x
(Rx & Fx)
& Î
and Î£ have Lemmonâs rules.
We then have
Î
xFx â
â
xFx
â
xFx â Î£xFx
Î xFx â Fa
Fa â Î£
x
Fx
â
x
Fx, Ra â Fa
Fa, Ra â
â
xFx
Now letâs go back to our arg.
(1)
Î£x
(Fx & Rx)
â´
(2)
â
xFx
And Ans
e
l
m gives an argument for (1), using
I
â
xFx â Î£xFx
So by substitution
I
â
x (
Fx & Rx) â Î£x (Fx & Rx).
Note also that
â
x
Fx = Î£
x
(Fx & Rx) = Î£
x (
Fx & Rx &
Rx) =
â
x
(
Fx & Rx)
So
I
â
x
Fx = I
â
x (Fx & Rx).
So now we have
I
â
xFx â I
â
x
(Fx & Rx)
â Î£x
(Fx &
Rx)
â
â
xFx

an awful result. We get it from
two
principles
:

{2}
1
Edit
ed by Max Cresswell and Adriane
Rini
.
The original is kept in the Prior
collection at Bodleian Library, Oxford, Box 6.
The page numbers in the original has
been put in {â¦}.
2
(1)
I
â
xFx â Î£xFx
(2)
â
xFx = Î£x (Fx & Rx).
So better drop one of them. I think there are reasons for dropping (2), but Iâll put them
aside for the moment.
What I want to say now is that even apart from ont. arg. there
are reasons for being suspicious of (1).
Let Fx be âx is the only thing there isâ.
â
xFx
be (4) I
â
x
â
y
(y = x).
â
It is impossible that something is the only thing there isâ
i.e.
âIt is imaginable that there should be only one real object.â
Does this imply
(5) Î£x
â
y
(y = x)
i.e. âSome
cogitabile
is the only real objectâ
Surely not. (5) means that there is some imaginable state of affairs such that [there?]
in that state of affair
s
there is only one real object.
(5) means that there is an
im
aginable object such that the only real object
in the actual state of affairs
is that
object.
But the introduction of I
raises the larger question of whether we really need
these
cogitabilia
&
possibilia
at all. Canât we deal with the imaginable & the thinkable
& the possible just by having
ordinary
quantifiers & operators {3} like I.?
If we want
to say, for example, that it can be imagined that some real thing is perfect, canât we
just say th
at & be done with it. Note that
I
â
x
Fx âª>
2
â
xIFx
âIt is imaginable that something (real) is perfectâ doesnât entail âthere is some (real)
thing of which it is imaginable that it is perfectâ.
And this is still more obvious with
more modest operators than
âIâ.
For example âI think t
hat someone stole my pencilâ
âª>
âthere is someone of whom I think that
he
stole my pencil.â And â
Dickens made
up a story to the effect th
at
someone was called Mr. Pâ
âª>
âthere was someone of
whom D made up the story that he was called Mr. P.â These are fallacies, but perhaps
we have an urge to connect them,
& then we check ourselves & say âwell, there
wasnât a
real
someone that D.
s
aid was called Mr. P., but there was an
imaginary
someone of whom he said that.â
But do we
have
to make this move?
Does âThere was
an imaginary someone of whom D.
s
aid he was Pâ mean anything more than âD.
s
aid
that there was a real someone who was P.
â
there
wasnât
someone who
was
P, there
wasnât even anyone of whom D.
s
aid he was P, but D.
s
aid that someone
was P.â
Isnâ
t
that
all we want
? {4}
Well, we can certainly get quite a long way with this apparatus.
Letâs look at some old questions. I want to sum all this up by
an historica
l puzzle & a
modern answer to it.
When Aristotle uses the predicate âexistsâ he seems to say inconsistent things
about it. Sometimes he seems to accept
Faâ Ra
(eg. Soc. is ill â Soc. exists)
& so

Ra â

Fa
(Soc. doesnât exist â Soc. isnât ill).
But elsewhere he comments that âX is Yâ
doesnât
always imply âX isâ. He gives as
an example âX is thought aboutâ
â
âMr. Pickwick is thought aboutâ for example,
2
The symbol used is a
ri
ghtarrow with stroke through it.
We represent it here as â
âª>
â.
3
doesnât imply âMr. P. exists.â
Some medieval writers divide predicates into those that
imply or
presuppose existence & those that donât.
What would Russell say? Or
Lemmon. Well, the first point is that our X is Y might be of 2 forms. Either X is a
log.prop.name, or it is a description. If it is a log.prop.name, either we say â
a
existsâ
is meaningless
or
we define it as exists Y (a
= Y)
. If the latter we do have
Fa â Ra
i.e.
Fa â
â
y (a = y)
But more than that, we have
â Ra.
Indeed we could get this from
Fa â Ra.
Because

Fa is a case of Fa, so we would have

Fa â Ra. {5}
So we now have
2
Fa
â¨

Fa
2 (2) Fa
A
2 (3) Ra
2, Aristotle
4 (4)
â
Fa
A
4 (5)
Ra
4, Aristotle
(6)
Ra
1, 2, 3, 4, 5,
â¨
E.
But suppose that the X in out âX is Yâ is a description, i.e. suppose the thing is
The thing that Fâs G
âs.
This =
â
1xFx &
â
x (Fx â Gx)
3
& implies
â
1xFx
i.e. The thing that Fâs exists.
What about âThe thing that Fâs
doesnât
G.â?
Well this might mean
â
1xFx &
â
x (Fx â

Gx).
& this also implies
â
1xFx.
But we donât haveâ
â
1xFx because we donât have a as law
(
â
1xFx & (
â
x (Fx â Gx))
â¨
(
â
xFx &
â
x (Fx â

Gx)).
Or âThe thing that Fâs doesnât Gâ might mean
â
(
â
1xFx &
â
x (Fx â Gx)).
& this doesnât imply
â
1xFx.
Russell dists. bwt. Primary & secondary occurrences of âThe thing
that Fâsâ.
Secondary occurrence if it falls within the scope of some operator like

.Also like âIt
is believed thatâ. (
Extensional
or
intensional
complexity). {6}
Sometimes
[?]
distn. bwt. int. & ext. complexity if we allow empty names.

Fa might
be

(Fa) or (

F)a.
3
[T
ranscribersâ note:
]
this seems to be the exactly one quantifier, and is written with
the 1 immediately below the
â
.
4
Fa â Ra
(

F)aâ Ra
But
not
Fa â Ra

(Fa)â Ra
In this case, â¤
Ra.
Strawson treats descriptions like R. treats names.
4
1,3 (8)
â
x (
â
y (Py â Hxy) & Hxe)
7, E I
1,2 (9)
â
x (
â
y (Py â Hxy) & Hxe)
2, 3, 8, E E
The usual move, confronted by this argument is to say that the first
premiss is false, existence not being a perfection because it is not even a
predicate.
Well, here we havenât made it a predicate, but a subject; but that is
almost certainly even worse, &
there is a slight modification of the above
argument in which âexistsâ does figure as a predicate. This modification
makes the argument run as follows:

(1)
Whatever is perfect really exists
(2)
Some thinkable[?] is perfect
(3)
â´
Some thinkable both is perfect &
really exists.
(4)
â´
Something that is perfect really exists.
In symbols we could use P, T & R for âis perfectâ, âis thinkableâ and âreally
existsâ, & give the proof as follows:

1 (1)
â
x
(Px â Rx)
A
2 (2)
â
x (Tx & Px)
A
3 (3) Ta & Pa
A
1 (4)
Pa â Ra
1, U E
3 (5) Pa
3, & E
1, 3 (6) Ra
4, 5, MPP
1, 3 (7) Pa & Ra
5, 6, & I
1, 3 (8)
â
x (Px & Rx)
7, E I
1, 2 (9)
â
x (Px & Rx)
2, 3, 8 E E
{3}
To this argument the contention that âexistsâ isnât a predicate would certainly
be releva
nt, if it were true. But is it?
It is certainly true that many ordinary
assertions & denials of existence can be put into predicate calculus symbolism
without using anything corresponding to our
R
. To use the stock examples,
âLions existâ comes out as
â
xLx
and âUnicorns do not existâ as
â
â
xLx
. Here
the grammatical subject, âlionsâ or âunicornsâ has a predicate hidden in it (âis a
4
[Transcribersâ note:] the next few pag
es are not always consecutively
numbered and itâs not clear just how they relate to what has gone before. The
first
page is numbered {2}.
5
lionâ or âis a unicornâ); this becomes the explicit predicate of the predicate

calculus translation, & no other predicate is nee
ded.
The âexistsâ bit has become
absorbed into the quantifier, & the operator
â
x
( )
x
is not a predicate, i.e.
something that forms a sentence from a name, but is something that forms a
sentence from a predicate.
But do we not sometimes use âexistsâ to f
orm a predicate from a name,
and moreover from a name that has no predicate hidden in it? What about
âJohn Smith existsâ? Well, perhaps âJohn Smithâ does have a predicate
concealed in it, namely the predicate â
â
is John Smithâ, i.e. â
â
is identical with
Jo
hn Smith.â
In that case, âJohn Smith existsâ means âSomething is John
Smithâ, i.e.
â
x
(
x
=
a
), where
a
is John Smith.
Now, however, there
is
in the
symbolic version {6}
[sic]
something that corresponds to a genuine predicate,
namely âSomething is identical with
â
â, which does form a sentence when
attached to a name. We might in fact define the predicate â
â
(really) existsâ by
Ra
= (Df.)
â
x
(
x
=
a
);
Or perhaps more naturally by the separated form
Ra
= (Df.)
â
x
(
a
=
x
),
i.e. â
a
existsâ = â
a
is somethingâ, i.e. âThere is something that
a
isâ, i.e.
âThere is something the
a
is identical withâ.
So perhaps there is a sense of âexistsâ in which it is a genuine
predicate, & in which âJohn Smith existsâ, for example, is a genuine
predication. But what about âMr. Pickwick
doesnât
existâ?
This, on the view
under consideration, would have the form
â
Ra
, i.e.
â
â
x
(
a
=
x
), with
a
for Mr.
Pickwick.
Unfortunately in Lemmonâs system, &
in most normal systems of
predicate calculus, this has contradictory consequences.
Without going into
the formalities, the argument is that Mr. Pickwick
is
after all Mr. Pickwick (by
the Law of Identity) so there
is
an
x
such that Mr. Pickwich is
x
.
Given
the
ordinary formal machinery, then, this definition {7}
will not do for that sense
of âexistsâ in which we want to say that some individuals exist & others (e.g.
Mr. Pickwick) do not.
The answer to this may be to modify the âordinary formal machineryâ
in some way; & various modifications of it have in fact been suggested.
Let us
stay with it for a while, however & try another tack.
Let us keep the existing
formalism but re

interpret it, so that
â
x
means âFor some x, real or
imaginaryâ, &
then let us say that some of these real

or

imaginary objects are
real, i.e. have the predicate
R
which we can leave undefined; & others of these
real

or

imaginary objects lack this predicate.
What becomes of the ontological
argument then?
It still seems t
o demonstrate that a real

or

imaginary object
that has all perfections is in fact real, i.e. falls into the ârealâ compartment of
this intended universe.
And we cannot now deny the first premise on the
ground that (real) existence isnât a predicate
â
in th
e system we are working,
with it is.
But what about the other premise, the one that asserts that some
thinkable has all perfections,
{8}
including existence?
Anselmâs ground for asserting this premiss
was that even the man who
denies the existence of a p
erfect being, if this denial is not mere verbiage,
must have found it possible to imagine such a being
â
to entertain the thought
that something is perfect, even if only to deny it.
And to entertain that thought,
& think it through, is to entertain the tho
ught that something is perfect &
6
really exists (for nothing can be perfect without that).
So the argument by
which the second premiss is arrived at is this: It can be thought that something
is perfect & real;
so something real

or

imaginary (something thought of)
is
perfect & real.
And the principle of it is: If it is imaginable that something is
F
, then some imaginable thing
is
F
.
If we write IP for âIt is imaginable that Pâ,
the principle is
I
â
xFx
â
â
xFx
Not
e that on the right

hand side the
â
x
still only means âFor some real

or

imaginary
x
â
â
the principle doesnât enable us to pass straight from âIt is
imaginable that something real

or

imaginary is Fâ to âSomething real is Fâ.
However, if the predicate ârealâ is {new page}
a conjunct of
F
, we do get
âSomething realâ on the right, & so the ontological argument goes through.
5
So perhaps it
is
dangerous to have ârealâ as a predicate. But if we
donât, how do we say âSome things ar
e real & some things are notâ?
One
suggestion is that we follow the current practice of putting existence into the
quantifiers, but have two sorts of quantifiers, say
â
(
m
)
x
for âFor some real or
imaginary
x
â and
â
(
r
)
x
for âFor some real
x
â
Î£x
â
Rx
â I
â
x(
â
Rx)
â I
Î£
x (
â
Rx & Rx).
I
â
x
â
Rx
â I
Î£
x (
â
Rx & Rx & Rx)
â I
â
x (
â
Rx & Rx).
I
â
xPx
â´
Î£xPx
Î x (Px â
â
y (x = y))
Î x (Px â (Px &
â
y (x = y))
â´
Î£x (Px &
â
y (x = y))
â´
I
â
x (Px &
â
y (x = y)).
OK.
â
xFx
[not=] I Î£x (
â
y (x = y) & Fx).
â
xFx
= Î£x (
â
y (x = y) & Fx)
Î£x
â
â
y (x = y)
But
I
â
x
â
â
y (x = y)
.
â I
â
x
â
(x = x)
No.
â
xFx
â
x
â
y (x = y & Fy).
â
â
y (x = y)
[Î£x Px &
â
y (x = y) ââ
â
y (Px & x = y).
?
Î£x (Px &
â
y (x = y) ââ Î£x
â
y (Px & x = y)]
5
[Priorâs note, in the margin:] I
â
x
â
y (x = y) â Î£x
â
y (x = y)
7
â
x
â
â
y (x = y)
â
â
y
(
[
â
x
â
â
x (x = y)
â
â
y (x = y &
â
â
x (x = y))
â
y]
{document ends}