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By Arthur N. Prior on NA/NA/NA

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1


Text 103: (
Read Russell on Ag. And Dem.
)
1

A.N. Prior



Suppose we write

Π
xFx

for

“For any
cogitabile

x, Fx”

Σ
xFx

for

“For some
cogitabile

x, Fx”

∀
xFx

for

“For any real thing x, Fx”

∃
xFx

“


“ some “ “ “, Fx”

& define the latter
two thus:
--

∀
xFx

= Πx (Rx →

Fx)

∃
x
Fx

= Σx

(Rx & Fx)


& Π

and Σ have Lemmon’s rules.

We then have


Π
xFx ├
∀
xFx

∃
xFx ├ ΣxFx


ΠxFx ├ Fa

Fa ├ Σ
x
Fx


∀
x
Fx, Ra ├ Fa

Fa, Ra ├
∃
xFx


Now let’s go back to our arg.



(1)
Σx

(Fx & Rx)

∴

(2)

∃
xFx


And Ans
e
l
m gives an argument for (1), using


I
∃
xFx ├ ΣxFx

So by substitution



I
∃
x (
Fx & Rx) ├ Σx (Fx & Rx).


Note also that



∃
x
Fx = Σ
x

(Fx & Rx) = Σ
x (
Fx & Rx &

Rx) =
∃
x

(
Fx & Rx)

So



I
∃
x
Fx = I
∃
x (Fx & Rx).

So now we have



I
∃
xFx → I
∃
x

(Fx & Rx)




→ Σx

(Fx &

Rx)




→
∃
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 {…}.


2


(1)

I
∃
xFx → ΣxFx

(2)

∃
xFx = Σx (Fx & Rx).


So better drop one of them. I think there are reasons for dropping (2), but I’ll 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 “x is the only thing there is”.


∃
xFx


be (4) I
∃
x
∀
y

(y = x).


“
It is impossible that something is the only thing there is”

i.e.
“It is imaginable that there should be only one real object.”


Does this imply


(5) Σx
∀
y

(y = x)

i.e. “Some
cogitabile

is the only real object”

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. Can’t 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, can’t we
just say th
at & be done with it. Note that


I
∃
x
Fx ╪>
2

∃
xIFx

“It is imaginable that something (real) is perfect” doesn’t entail “there is some (real)
thing of which it is imaginable that it is perfect”.
And this is still more obvious with
more modest operators than
“I”.
For example “I think t
hat someone stole my pencil”

╪>

“there is someone of whom I think that
he

stole my pencil.” And “
Dickens made
up a story to the effect th
at

someone was called Mr. P”

╪>

“there was someone of
whom D made up the story that he was called Mr. P.” These are fallacies, but perhaps
we have an urge to connect them,
& then we check ourselves & say “well, there
wasn’t a
real

someone that D.
s
aid was called Mr. P., but there was an

imaginary
someone of whom he said that.”

But do we
have

to make this move?
Does “There was
an imaginary someone of whom D.
s
aid he was P” mean anything more than “D.
s
aid
that there was a real someone who was P.
–

there
wasn’t

someone who
was

P, there
wasn’t even anyone of whom D.
s
aid he was P, but D.
s
aid that someone

was P.”

Isn’
t
that
all we want
? {4}

Well, we can certainly get quite a long way with this apparatus.
Let’s 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 ‘exists’ he seems to say inconsistent things
about it. Sometimes he seems to accept



Fa├ Ra

(eg. Soc. is ill ├ Soc. exists)

& so



-
Ra ├
-
Fa

(Soc. doesn’t exist ├ Soc. isn’t ill).

But elsewhere he comments that “X is Y”
doesn’t

always imply “X is”. He gives as
an example “X is thought about”
–

“Mr. Pickwick is thought about” for example,



2

The symbol used is a
ri
ghtarrow with stroke through it.

We represent it here as ‘
╪>
’.

3


doesn’t imply “Mr. P. exists.”
Some medieval writers divide predicates into those that
imply or
presuppose existence & those that don’t.
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 “
a
exists”
is meaningless

or

we define it as exists Y (a

= Y)
. If the latter we do have



Fa ├ Ra


i.e.

Fa ├
∃
y (a = y)

But more than that, we have




├ Ra.

Indeed we could get this from



Fa ├ Ra.

Because
-
Fa is a case of Fa, so we would have



-
Fa ├ Ra. {5}

So we now have

2

Fa
∨

-
Fa

2 (2) Fa


A

2 (3) Ra


2, Aristotle

4 (4)
–
Fa


A

4 (5)

Ra


4, Aristotle


(6)

Ra


1, 2, 3, 4, 5,

∨
E.


But suppose that the X in out ‘X is Y’ is a description, i.e. suppose the thing is



The thing that F’s G
’s.

This =



∃
1xFx &
∀
x (Fx → Gx)
3


& implies



∃
1xFx


i.e. The thing that F’s exists.


What about “The thing that F’s
doesn’t

G.”?

Well this might mean


∃
1xFx &
∀
x (Fx →
-
Gx).

& this also implies


∃
1xFx.

But we don’t have├
∃
1xFx because we don’t have a as law


(
∃
1xFx & (
∀
x (Fx → Gx))
∨

(
∃
xFx &
∀
x (Fx →
-
Gx)).

Or “The thing that F’s doesn’t G” might mean
–
(
∃
1xFx &
∀
x (Fx → Gx)).

& this doesn’t imply


∃
1xFx.

Russell dists. bwt. Primary & secondary occurrences of “The thing
that F’s”.
Secondary occurrence if it falls within the scope of some operator like
--

.Also like “It
is believed that”. (
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
ranscribers’ note:
]

this seems to be the exactly one quantifier, and is written with
the 1 immediately below the

∃
.


4





Fa ├ Ra





(
-
F)a├ Ra



But
not





Fa ├ Ra



-
(Fa)├ Ra



In this case, ┤
Ra.


Strawson treats descriptions like R. treats names.
4



1,3 (8)
∃
x (
∀
y (Py → Hxy) & Hxe)


7, E I


1,2 (9)
∃
x (
∀
y (Py → 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 haven’t 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 “exists” 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)

∴

Some thinkable both is perfect &

really exists.

(4)

∴

Something that is perfect really exists.


In symbols we could use P, T & R for “is perfect”, “is thinkable” and “really
exists”, & give the proof as follows:
--



1 (1)
∀
x

(Px → Rx)



A


2 (2)
∃
x (Tx & Px)



A


3 (3) Ta & Pa




A


1 (4)
Pa → 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)
∃
x (Px & Rx)


7, E I


1, 2 (9)
∃
x (Px & Rx)


2, 3, 8 E E

{3}



To this argument the contention that “exists” isn’t 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,
“Lions exist” comes out as
∃
xLx

and “Unicorns do not exist” as
–
∃
xLx

. Here
the grammatical subject, “lions’ or “unicorns” has a predicate hidden in it (“is a



4

[Transcribers’ note:] the next few pag
es are not always consecutively
numbered and it’s not clear just how they relate to what has gone before. The
first
page is numbered {2}.


5


lion” or “is a unicorn”); this becomes the explicit predicate of the predicate
-
calculus translation, & no other predicate is nee
ded.
The “exists” bit has become
absorbed into the quantifier, & the operator
∃
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 “exists” to f
orm a predicate from a name,
and moreover from a name that has no predicate hidden in it? What about
“John Smith exists”? Well, perhaps “John Smith” does have a predicate
concealed in it, namely the predicate “
–

is John Smith”, i.e. “
–

is identical with
Jo
hn Smith.”
In that case, “John Smith exists” means “Something is John
Smith”, i.e.
∃
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 “Something is identical with
–
“, which does form a sentence when
attached to a name. We might in fact define the predicate “
–

(really) exists” by


Ra

= (Df.)
∃
x
(
x

=
a
);

Or perhaps more naturally by the separated form


Ra
= (Df.)
∃
x
(
a

=
x
),

i.e. “
a

exists” = “
a

is something”, i.e. “There is something that
a

is”, i.e.
“There is something the
a

is identical with”.



So perhaps there is a sense of “exists” in which it is a genuine
predicate, & in which “John Smith exists”, for example, is a genuine
predication. But what about “Mr. Pickwick
doesn’t

exist”?
This, on the view
under consideration, would have the form
–
Ra
, i.e.
–
∃
x

(
a

=
x
), with
a

for Mr.
Pickwick.
Unfortunately in Lemmon’s 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 “exists” 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 “ordinary formal machinery”

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
∃
x

means “For some x, real or
imaginary”, &

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 “real” compartment of
this intended universe.
And we cannot now deny the first premise on the
ground that (real) existence isn’t a predicate
–

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?


Anselm’s 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
–

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 “It is imaginable that P”,
the principle is


I
∃
xFx

→
∃
xFx

Not
e that on the right
-
hand side the
∃
x

still only means “For some real
-
or
-
imaginary
x
”
–

the principle doesn’t enable us to pass straight from “It is
imaginable that something real
-
or
-
imaginary is F” to “Something real is F”.

However, if the predicate “real” is {new page}
a conjunct of
F
, we do get
“Something real” on the right, & so the ontological argument goes through.
5


So perhaps it
is

dangerous to have “real” as a predicate. But if we
don’t, how do we say “Some things ar
e real & some things are not”?
One
suggestion is that we follow the current practice of putting existence into the
quantifiers, but have two sorts of quantifiers, say
∃
(
m
)
x

for “For some real or
imaginary
x
” and

∃
(
r
)
x

for “For some real
x
”



Σx
–

Rx



→ I

∃
x(
–
Rx)



→ I
Σ
x (
–

Rx & Rx).


I
∃
x
–

Rx



→ I
Σ
x (
–

Rx & Rx & Rx)



→ I
∃
x (
–

Rx & Rx).



I
∃
xPx

∴


ΣxPx


Πx (Px →
∃
y (x = y))


Πx (Px → (Px &
∃
y (x = y))

∴

Σx (Px &
∃
y (x = y))

∴

I
∃
x (Px &
∃
y (x = y)).


OK.



∃
xFx


[not=] I Σx (
∃
y (x = y) & Fx).


∃
xFx


= Σx (
∃
y (x = y) & Fx)


Σx
–

∃
y (x = y)

But

I
∃
x

–

∃
y (x = y)
.

→ I
∃
x
–

(x = x)

No.



∃
xFx

∃
x
∃
y (x = y & Fy).


→
∃
y (x = y)

[Σx Px &
∃
y (x = y) ←→
∃
y (Px & x = y).

?
Σx (Px &
∃
y (x = y) ←→ Σx
∃
y (Px & x = y)]





5

[Prior’s note, in the margin:] I
∃
x
∀
y (x = y) → Σx
∀
y (x = y)

7




∃
x
–

∃
y (x = y)




→
∃
y
(




[
∃
x
–

∃
x (x = y)




→
∃
y (x = y &
–

∃
x (x = y))




∃
y]


{document ends}




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