Alan McMichael; Ed Zalta -- An Alternative Theory of Nonexistent Objects

8/9/2019 Alan McMichael; Ed Zalta -- An Alternative Theory of Nonexistent Objects

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ALANMcMICHAELANDEDZALTA

AN ALTERNATIVE THEORY OF

NONEXISTENT OBJECTS

Many of us share certain intuitions regarding the truth-values of sentences

about nonexistent objects. For example, we believe “Pegasus is a winged

horse” is true, whereas “Pegasus is a minotaur” is false. Theories which

renounce nonexistent objects, attributing them no being at all, have

difficulty accounting for these intuitions.

Consider a theory apparently held by Russell. According to that theory,

any ordinary name, on a given occasion of its use, is short for a definite

description. Thus, “Pegasus is a winged horse,” on a given occasion, is

short for “The F is a winged horse,” for some predicate F.’ But by

Russell’s Theory of Definite Descriptions, “The F is a winged horse” is

true just in case there exists a unique thing satisfying F and it is a

winged horse. Since there exist no winged horses, the second sentence is

false. And by Russell’s theory, the falsity of this sentence implies the

falsity of the original sentence - contradicting out intuitions.

To avoid this problem, others have suggested that “Pegasus is a winged

horse” is a truth about a winged horse existing in another possible world,

and that it can be true of that other world even though Pegasushas neither

existence nor being here.2 However, this suggestion runs into difficulty with

another set of intuitions. It is evident that we do have intentional attitudes

toward non-existent objects. For example, Conan Doyle created Sherlock

Holmes. Some Greeks imagined Pegasus.Children dream about

Rumpelstiltskin. Sentences expressing these attitudes are true of

this

world. The above suggestion gives us no clue how to interpret such truths.

In fact, it seems to preclude uniform treatment: “Pegasus is a winged

horse” is a truth about a winged horse existing in another possible world,

but “Some Greeks imagined Pegasus” is a truth about no winged horses at

all (since it is true of this world, where there are no winged horses).

Meinongian theories which countenance nonexistent objects by

supposing them to have being in this world may yield a more satisfying

explanation of these data. One such theory, developed by Terence Parsons,

treats nonexistent objects on a par with existent objects.3 Parsons

Journal of Phibsophical Logic 9 (1980) 297-313.

0022-3611/80/0093-0297$01.70.

Copyr ight 0 1980 by D. Reidel Publ ishing Co., Dordrech t , Hol land and Boston, CI. .A.



298 ALANMcMICHAELANDEDZALTA

distinguishes two types of properties, nuclear and extranuclear. Objects are

correlated one-to-one with the sets of nuclear properties they have. A com-

prehension principle says that there is an object for every expressible set of

nuclear properties. Some of these objects have, while others lack, the extra-

nuclear property of existence. Parsons assignsone of the nonexistent

objects to “Pegasus” - the one which exemplifies just the nuclear proper-

ties attributed to the winged horse of Greek mythology. Pegasus,so

identified, exemplifies certain extranuclear properties like being imagined,

being thought about, being dreamed about, etc. Parsons’ theory is a power-

ful and insightful explanation of the data with which we’re concerned.

However, some philosophers may share another of our intuitions,

namely, that non-existent objects don’t have or exemplify their properties

in quite the same way that actual objects do. Parsons’ theory, with its

single domain of objects and single mode ofpredication, doesn’t capture

this intuition. But the theory discussed by W. Rapaport4 (which has its

roots in theories by Mally and Castaiieda) does justice to this intuition by

sharply distinguishing between existent and nonexistent objects, and by

introducing a second form of predication for the latter. Actual objects

have or exempZi& their properties, whereas non-actual objects may

exemplify some properties and include others. For example, Pinkerton

exemplifies detectivehood but does not include it. Holmes includes

detectivehood but does not exemplify it. And Holmes exemplifies being

thought about by Conan Doyle, but he does not include that property.

Notice that a fictional object includes the properties ascribed to it in the

work of fiction, but exemplifies certain intentional properties, such as

being imagined, being discussed, being more famous than any real detective,

and so forth.’

In an earlier paper by one of the authors, a language was produced in

which the distinctions between exemplification and inclusion, and between

the two kinds of objects, could be represented precisely.6 In addition, an

axiomatic theory of objects was developed there. The system presented here

incorporates several improvements on that earlier paper - we give a fuller

treatment of relations and we develop a semantics for the A-notation which

is used to form complex property expressions. The Relations axiom, which

appears here for the first time, is an important addition to the original

axiomatic theory. We expect that our system will serve as a prototype for a

larger and richer language into which we may translate significant portions

of English.



ALTERNATIVE THEORY OF NONEX ISTENT OBJECTS 299

I . THE LANGUAGE ZM

ZM is a second-order language with a number of special features: (1) There

are three sorts of object-terms, Meinongian object terms (x-variables and

a-constants), actual object terms (y-variables and b-constants), and neutral

object variables @I’S). (2) The language contains both simple and complex

predicate terms. Under any admissible interpretation, these denote

relations, nor sets. (3) There are two basic ways of forming atomic

predicative sentences. When n object terms, TV, . . . , r,, occur to the left

of a bracketed relation term, the resulting sentence says that the objects

exemplify the relation. When a single M-object term occurs to the right of

a bracketed property term, the resulting sentence says that the M-object

includes the property.

A. Syntax

1. Primitive Vocabulary

a. Singular terms

i. Meinongian object terms.

Constants: al, a2, a3, . . . .

Variables: x 1, ~2, ~3, . . . .

ii. Actual object terms.

Constants: br , bs, bs, . . . .

Variables: yI,y2,ys, . . . .

iii . Object variables: z,, z2, z3, . . . .

b. Simple predicate terms.

i. Constants: fl, z, E, . . . I2 and E (existence).

ii. Variables: p:, p$ pz, . . . .

NOTE: We wil l systematically misspell formulas of the form “r1r2[Z2] ”

m “7, =A 72”. =A is the relation in our system which most closely

resembles the identity relation of orthodox theories (namely, identity

among existing objects).

c. Connectives: &, v, -, +, E.

d. Existential quantifier: 3.

e. Purely formal identity sign: =.

f. Parentheses and brackets: ( , ), [ , 1.

g. Lambda: X.

2. Wffs



300

ALAN McMICHAEL AND ED ZALTA

a. Atomic: If CYs an n-place predicate term, and ri, . . . , 7, are

singular terms, then rl. . . rn[o] is a wff.

b. Atomic: If OLs a l-place predicate term, and r is a Meinongian object

term, then [cu]7 is a w ff.

c. Atomic: If CY, are both singular terms or both predicate terms with

the same number of places, then a = fl is a wff.

d. Molecular: If 9, J, are wffs, then so are @I J/, # v $, # + $,$ E $,

and-@.

e. Quantified’ If I$ s a wff, and a is any variable, then (a)# and ( ~cY)#

are wffs.

3. X-predicates: If 4 is a formula which has at least n free z-variables,

71,. * *,

r,, which has no inclusion subformulas “[cr]?‘, which has no

purely formal identities as subformulas, which has no predicate

quantifiers, and which contains no x’s, then Xri . , . XT,@ s an n-place

predicate.

z-variables are used to form X-predicates because neutral placeholders

are needed in complex predicate terms.

We will use the following defmed notation for asserting relations

among non-existents:

[R”]a,. .

.a, =a,,br [AZ 2112. . a,[R”]]al&

& [AZ ulzug.. . a,[R”]]uz &. . .

& [hzu1up. . a,-,z[R”]]u,.

There is an asymmetry between exemplification and inclusion. Objects may

exemplify (bear) relations, but they include only properties. Nevertheless,

there is a sense n which M-objects may “include” relations to one another,

and this is captured by the definition.

B. Semantics

An interpretation of ZM is to be any octuple, S, which meets the following

conditions:

1. The first member of S is a non-empty set M which will be called the

classof Meinongian objects.

2. The second member of S is a set A which will be called the class of

actual objects.



ALTERN ATIVE THEORY OF NONEXISTENT OBJECTS 301

3. The third member of S is a sequence of non-empty classes

Rl,&,R3.. . . Each

R,

is called the class of n-place relations.

RIuR2vR3u... must be closed under all the logical operations

specified in the seventh member of the interpretation, F.

(When ‘R” is followed only by subscripts, it will stand for classesof

relations.)

4. The fourth member of S is a function, extR, which maps each r E R,

to a class of ordered n-tuples of M u A. Intuitively, for n > 1, these sets

of n-tuples are the extensions of the relations among the objects which

exemplify

these relations. And for n = 1, the sets of objects serve as the

extensions of the properties among the objects which have or exemplify

these properties.

5. The fifth member of S is a function, extI lM, which maps each r E R 1

to the class of members ofM which include r. (Read ext[ l,&): the

extension of r among the M-objects which include it.)

6. The sixth member of S is a one-to-one function, f, which maps each

member of

M

to a subclassof

R 1 in

such a way that

f(x) = {r E R I Ix E ext[ I&}.

7. The seventh member of S is a class of logical functions, F, which has as

members PLUG (“plug”), CONV (“conversion”), REFL (“reflection”),

PROJ (“projection”), M-PROJ (“Meinongian projection”), A-PROJ

(“actual projection”), NEG (“negation”), and CONJ (“disjoint

conjunction”):’

(a) PLUGisafunctionfrom(R2uR3...)x(MuA)x~into

@lu&.

. . ) subject to the condition: eXtR(PLuG(R “, 0, i)) =

eXtR(PLUG(R”, 0, i)) = {bl,. . . , oiel, ol+l,. . . , on>1

(01,. . . ,

01-1909 Oi+ls * . . 3

0,) E eXtR(R”)}

Intuitively, PLUG (R”, o, i) is the n - 1 place relation which is the result

of plugging individual o into R” in its ith place.

For example,

PLUG(loves, Abelard, 2) = loving Abelard,

PLUG(loves, Heloise, 1) = being loved by Heloise.

(b) CONV is a function from (R2 u RP . . . ) x w x w into



302


(R2u Rs . .

. ) subject to the condition:

ext&O~W, i,i)) = {bl, . . . , OL~,O~,O~+~, . .

,of+,

01, oj+l,. . . , o,>l(ol, . . . ,o,) E extR(R”)}.

Intuitively, CONV(R”, i,j) is the n place relation which is the conver-

sion of the relation R” about its ith andith places.

For example,

CONV(loves, 1,2) = being loved by,

CONV(gives ( ) to, 1,3) = being given ( ) by.

(c) REFLisafunctionfrom(RauRg...)x ox winto

(RIu&.

. . ) subject to the condition:

extn(RJZFL(R”, i,i)) = ((01,. . . , oi,. . . , Oj-1,

oj+1,. * - ,

On)l(O~,...,Oi,...,Oj,..~,On>E

EextR(Rn)&oi = oi}

Intuitively, REFL(R”,

i,

i) is the n - 1 place relation which is the

reflection of R ” that links its ith and ith places.

For example,

REFL(kil1, 1,2) = committing suicide,

REFL(shaves, 1,2) = shaving oneself.

(d) PROJisafunctionfrom(R~uRg...)x~into(RRuRR2...)

subject to the condition:

extR(PROJ(R”,i)) = ((or,. . . , o~-~,o~+~, . . . ,o,)I

(30)((01, . . . , oiwl, o, o~+~, . . , 0,) E extR(Rn))}.

Intuitively, PROJ(R”, i) is the n - 1 place relation which is the ith

projection of R”.

For example,

PROJ(loves, 2) = loving someone,

PROJ(loves, 1) = being loved by someone.

(e) M-PROJisafunctionfrom(RzuR~...)x~into(R1uRa...)




ALTER NATIVE THEORY OF NONEXISTENT OBJECTS 303

extR(M-PROJ(R”, i)) = {(or,. . . , oiml, ~r+~, . . . , o”)l.

(30)(oEM&(ol,. . . ,o~-~,o,o~+~, . . . ,o”EextR(R”))).

Intuitively, M-PROJ (R", ) is the n - 1 place relation which is the ith

projection of R" with respect to the Meinongian objects.

For example,

M-PROJ (thinking about, 2) = thinking about some

Meinongian object.

(f) A-PROJisafunctionfrom(R2uRg...)x~into(RIuR2...)


extR(A-PROJ(R”,i)) = {(or,. . . , of-l,of+l,. . . , o”>l

(30)(0 EA & (0

I, . . . , of+, 0, of+l, . . . , 0,) E e%(R”))).

Intuitively, A-PROJ (R", ) is the n - 1 place relation which is the ith

projection of R" with respect to the actual objects.

For example,

A-PROJ (thinking about, 1) = being thought about

by some real thing,

A-PROJ (thinking about, 2) = thinking about some

real thing.

(g) NEGisafunctionfrom(R1uRz...)into(R1uRz...)


extR(NEG(R”)) = ((or,. . . ,o,,)l(or, . . . ,o”>

4 ext&“)}.

Intuitively, NEG (R") is the n-place relation which is the negation of

R".

For example,

NEG (red) = being non-red,

NEG (met) = not having met.

Q CONJisafunctionfrom(R1uR2...)x(RluR2...)into

(R,u RJ... ) subject to the condition:



304

ALAN McMlCHAEL AND ED ZALTA

extn(CONJ(Z2”,9”)) = {(or,. . . , on,oi, . . . , &)I

(01,

. . . ,o,) k ext,(R”) &

& <o’l, . . . ,c&> E extn(Y)}.

Intuitively, CONJ (R”, Sm) is the n + m place relation which is the

disjoint conjunction of R” and Sm.

For example,

CONJ (red, round) = the relation any two objects bear to

each other just in case the first is red

and the second is round

(Note that REFL(CONJ(red, round), 1,2) = the property of being red

and round.)

CONJ (being clever, being the mother of) =

= the three place relation any three objects

bear to one another just in case the first is

clever and the second is the mother of the third.

(Note that REFL(CONJ(being clever, being the mother of), 1,2)

= the relation any two objects bear to each other just in case the first

is the clever mother of the second. Also note that PROJ(R.PFL.(CONJ

(being clever, being the mother of, 1,2), 1) = the property of having a

clever mother.)s

8. The eighth member of S is a function Z which is the assignment function.

Z s defined on the terms and simple predicates of the language:

(a) If r is an M-object term, Z(r) EM.

(b) If r is an A-object term, Z(r) EA.

(c) If r is a z-variable, Z(T) EM u A.

(d) If a is an n-place simple predicate, Z(o) E R,.

Moreover,

extR(Z(=A)) = ((0, o>lo E A}, extn(Z(E )) = A.

C. Definition

of

“extended assignments ”

Given an interpretation S with assignment function Z, I* is the extended

assignments. Z* is defined on object terms, simple predicates, and

A-predicates, in accordance with the rules:




1. For any object term or simple predicate &Z*(a) = Z(a).

2. If a is a simple n-place predicate, and rl, . . . , rn are distinct z-variables,

Z+(hl . . . X7, r1 . . .7, [a]) = Z*(a).

3. If x71.. .

AT,@ s a A-predicate, and K is an object term

I* 07 1 . . . iTi_, ATi+1 . . . h,, (K /Ti))9 =

= PLUG(ZS(AT1

. . A?,$ ), z*(K), i)).

4. IfATl...

AT&J s a A-predicate,

zyx71.. . X7&1 ATjATf+1. . .

ATj-1 A7fAT j+l- - , A7n ) =

= CONv(Z’(kI . . . AT&J ), l-J).

5. If x71.. .

AT,$I is a A-predicate,

Z*(AT~ - e wA71a . . ATi- ATj+l s aA~n@(T~/Tj)) =

= REFL(Z*(Arl . . . AT&), i, j).

6. IfArl...

AT,@

s a A-predicate and T is a z-variable not occurring in @,

f(X71

. . . Aqel AT~+~. .

A~,(W#W1)) =

= PROJ(Z*(AT, . . .

AT,@),

).

7. IfArl...

AT,+ is a A-predicate and p is an M-object variable not

occurring in 9,

z*(x71

. . .

Ari-lAri+l

. . .

A~n(W401hN =

= M-PROJ(Z*(AT~ . . . AT&), i).

8. IfAT1... AT& is a A-predicate and 7) s an actual object variable not

occurring in $,

Z*(AT~. . . AqselAT~+~ . . A7,(317)$(7)/71)) =

= A-PROJ(Z*(Aq . . . AT,&), i).

9. If kl . . . AT&J s a A-predicate,

Z*(hl. . . AT, - #) = NEG(Z*(A~l . . . AT,@)).

10. IfArl.. .A?,@and Au,. . .

Au, JI are A-predicates and all the T’Sand

u’s are distinct,



D1

D2

03

04

D S

D6

D l

D S

D9

40


I*(&. . . XT&J~. . . Au& & 4)) =

= CONJ(Z* (Xrl

. . . AT”@), z*@u~. . . Au,JI)).

D. Recursive definition of “trues ”

If 01 s an n-place predicate and r1 . . .7, are any singular

term& then r1 . . . TV a] is trues iff (1(7,), . . . , Z(7,)) E

extR(Z+(a)).

If ar s a 1 place predicate term and T is an M-object term,

then [ar] 7 is trues iff Z*(a) E

f(Z(7)).

a = p is trues iff I*@) = I’@).

-@istruesifff$isnottrues.

(# & JI) is trues iff 4 and $ are both trues.

(@I $) is trues iff at least one of 4, $4 s trues.

(4 -f $) is trues iff at least one of - 4, $ is trues.

($J= $) is trues iff $ and JI are both trues or neither is

trues.

(~CY)I#Js trues iff + is trues’ for some S’ which is just like S

except that its assignment function may differ from the

assignment function of S in what it assigns o a.

(a)# is trues iff - (301) - (J s trues.

IL AXIOMS AND CONSEQU ENCES OF A THEORY OF OBJECTS

1. Non-existence: (x) -x [E ]

No M-objects have existence.

2. Existence: @)r[E ]

All actual objects have existence.

3. Objects: For any wff $ which does not contain x free, the universal

closure of the following is an axiom:

(3x)W)([P’lx = $4.



4.

5.

6.

7.


For every expressible set of properties, there is a Meinonglan object

which includes just those properties.

Some one might want to adopt the view that for eve)3rset of

properties, there is an Object which includes just those properties.

But we won’t commit ourselves to that view.

Relations: For any X-predicate hrr . . . XT,+, the universal closure

of the following is an axiom:

(21)..(Z”)

(

z1 .Z,[ATl..AT&J]4 ‘;I* .‘i

( 1171 7,

This is our “abstraction schema” for relations.

VqriabZes: (z)((-Jx)x = z v (3y)y = z) & (x)@z)z =x &

cv)(3z)z =Y.

Ident@ forMobjects: (x1)(x2)(x1 =x2 = @‘)([P’]x~ = [p’]x2)).

We deny that M-objects exist (the Nonexistence Axiom accomplishes

this). Nevertheless, we do quantify over them. Thus it might be said that

we admit Meinongian objects into our ontology. We do indeed count them

among the things there a7e.1’

Some M-objects are objects of imagination. But not all M-objects have

been imagined. This is fairly clear from the Objects Axiom, which can

generate infinitely many M-objects. For any expressible set of properties,

we obtain an M-object. For example, we get the existent golden mountain

as follows:

(3x)@)([p]x = p=E vp=G vp=M)

(instance of Objects Axiom)

where “G” stands for being golden and “M” stands for being a mountain.

By similar applications of this axiom, we can generate all the other non-

existent objects as well, for instance, the round square. Also, by letting

$J= rp f ~1, we obtain the empty M-object. It includes no properties.

That there is such an object may seem strange, but this shouldn’t constitute

an objection to the theory.

There is an important correlation which arises n this system, a



308


correlation between actual objects and certain M-objects. For each actual

object, there is an M-object which includes exactly the properties the

actual object exemplifies. When this happens, we call the M-object a

“blueprint” and the corresponding actual object its “ .Seincorrelate”

(Rapaport’s terminology). We introduce a defined notation for Sin-

correlation:

‘yscC” may not express a relation, since the notation it abbreviates cannot

be made into a A-predicate. This is so because the notation which is abbrevi-

ated contains an inclusion formula, violating a restriction on h-predicates.

By the Objects Axiom, we know that every actual object has at least one

M-object blueprint. We can also show that an actual object has at most one

blueprint. For suppose it had two distinct blueprints. By the Meinongian

Identity Axiom, there is a property which one blueprint includes and the

other doesn’t. By the definition of Sein-correlate, it would follow that the

actual object both has and doesn’t have this property.

Also, no two actual objects have the same blueprints. For suppose

distinct actual objects, b1 and b2, do. Then, b1 has iu z =A bl, whereas

bz lacks this property. So they can’t have the same blueprint.

Certain other M-objects could have been blueprints of actual objects.

Let us call these potential blueprints. They will prove extremely useful

when developing a modal version of this theory. Potential blueprints are

the principal bearers of possibility. For example, Jimmy Carter is possible

onty in the sense that his blueprint is possible. That is, his blueprint could

have, and indeed does have, a Seincorrelate. Pegasus s possible in the

primary sense, for in some world he has a Seincorrelate.”

III. APPLY ING THE SYS TEM TO FICTION

In this section of the paper, we shall put the system to work translating

English and motivate certain features of the language in the process.

Naturally we shalI concentrate on sentences containing names of fictional

characters, so that our translations wil l illustrate the uses of inclusion

predications and fictional object constants.

In our translations, we use the first letters of the names of fictional

characters to stand in for Meinongian object constants, the first letters of




the names of actual objects to stand in for actual object constants, and the

fmt letters of English predicates to stand in for predicate constants. The

intended interpretation of an M-object constant is that fictional object

which includes (1) all the properties expressed by predicates which are

attached to the corresponding English name in the work of fiction, and

(2) whatever properties normal readers infer from th~se ~

Preferred readings are given first.

1. Raskolnikov is a student.

(a) [S] r “Raskolnikov includes being a student.” (True, since this was

ascribed to him in the novel.)r3

(b) r]S] “Raskohrikov exemplif%s being a student.” (False. He never

attended any existing school.)

2. Dostoyevsky created Raskohrikov.

(a) &[C’J “Dosteyevsky bears the creation relation to Raskohrikov.”

(‘Due, since Dostoyevsky did create the fictional character

Raskolnikov.)

(b) fAzdz[Cj]r “Raskohlik ov includes the property of being created

by Dostoyevsky.” (False, since this was not ascribed to Raskolnikov

in the novel.)

3. Porphyry arrested Raskolnikov.

(a) [A]pr “Porphyry ‘includes’ the arresting relation to Raskolnikov,”

or “Raskolnikov includes the property of being arrested by

Porphyry, and Porphyry includes the property of arresting

Raskolnikov.” (‘Due. “[A]@’ is an abbreviation for [AZ pz [A]] r &

[AZ zr[A]]p. The conjunction is true because Raskolnikov and

Porphyry are attributed these properties in the novel.)

(b) pr[.4] “Porphyry bears the arresting relation to Raskolnikov.”

(False. No real arrest took place between these two characters.)

4. Smerdyakov hung a cat.

(a) [hz (3y)& [c3 & zy ]I Jl)J s “Smerdyakov includes the property of

having hung a cat .” (True).

(b) (3x)( [c]x & [H] se) “Some M-object which includes being a cat is

such that Smerdyakov ‘includes’ the relation of hanging to it,” or

“Some M-object which includes being a cat is such that it includes

the property of being hung by Smerdyakov, and Smerdyakov



310 ALAN McMICHAEL AND ED ZALTA

5.

includes the property of hanging it.” (This is true only if

Smerdyakov hung some cat which is a chmucter of the novel.)

(c) (3y)(~ [Cl & [AZ zy [H]] s) “For some actual cat, Smerdyakov

includes the property of hanging it.” (False).

(d) (3y)O, [Cl & sy [H’J) “Smerdyakov bears the hanging relation to

some actual cat.” (False).

Everyone that met Alyosha loved him.

(a) [AZ (v)@z [M] + yz [L])] a “Alyosha includes the property of being

loved by everyone that meets him.” (True, let’s suppose.)

(b) (x)( [M] xa + [L] xa) “Every M-object such that it includes meeting

Alyosha and Alyosha includes being met by it includes loving

Alyosha and AIyosha includes being loved by it.” This is true just

in caseevery character that meets Alyosha loves him. It is not

likely to be a good translation of a sentence of the novel, but could

be used to interpret a critical remark.)

6. Dmitri lived in Russia.

(a) [AZ zr[L]] d “Dmitri includes the property of living in Russia.”

(True, since this was ascribed to him in the novel.)

(b) dr[L] “Dmitri bears the living in relation to Russia.” (False, since

Dmltri is not among the real inhabitants of Russia.)

7. Dmitri lived in an actual country.

(3y)(j~ [q & [Xz zy [L ]] d) Note that this is true whereas 4(c) is

false. ‘Y’ names the actual country, Russia, in 6(a).

IV. PARADOXES

In certain naive versions of our Meinongian theory, paradoxes arise. For

example, if we may introduce the property of an object being its own Sein-

correlate and the property of an object not being its own Seincorrelate,

contradictions soon follow. r4 And if for every M-object x, there is a

property of being identical to x, we also get contradictions ’ One way in

which we might avoid such contradictions is by giving a restricted abstrac-

tion schema for relations, so we do not assert the existence of properties

corresponding to such predicates as “is its own Sein-correlate” and “is

identical to

ai”.

For example, we might use the schema:




where C#Iontains no formal identities or inclusions. In the present language,

we have chosen a less direct means. By restricting the formulas which may

appear in A-predicates, we are unable to denote any paradoxical propertiesr6

Although any solution to these paradoxes involves some compromises,

our solution does leave us with a rich variety of properties and relations.

The intuitive closure conditions are preserved: every property and relation

has a negation, every two properties or relations have a conjunction, all

relations have their appropriate projections, and so on. Many solutions fail

to preserve these conditions. For instance, one solution examined by

Rapaport has the result that some properties lack negations.”

The paradoxes have forced us to banish formal identities from A-

predicates. The loss s not grievous; we stil l have an identity relation which

is well-behaved on the actual objects.

Naive Meinongian theories employing one mode of predication also

engender contradictions. For suppose that for every set of objects, there is

a property which just those objects have, and that for every set of properties

there is an object which has just those properties. Then the set of properties

is of greater cardinality than the set of objects, and the set of objects is of

greater cardinality than the set of properties - which is absurd.

Yet such theories have an even more elementary difficulty. From them

we may prove,

“The existent golden mountain exists.” By “the existent

golden mountain” we mean the object which has just tbis set of properties:

{existence, being composed of gold, being a mountain}. On the theories in

question, this object has existence. Also we may prove, “Russell never

thought about the round square,” since the object which has just the

properties roundness and squareness does not have the property of being

thought about by Russell.

Parsons evades the contradictions by distinguishing nuclear and extra-

nuclear properties. For every set of objects, there is an extrunuclear

property which just those objects have, but it is only for every set of

nuclear properties that there is an object which has just those properties.

No contradictions arise. The difficulty concerning the existent golden

mountain is solved by asserting that existence is an extranuclear property,

and not a nuclear property. Hence there is no set of nuclear properties, no

object-forming set, which contains existence. Similarly, the property of



312


being thought about by Russell is extranuclear, not nuclear, and so does not

appear in an object-forming set.

The difficulty concerning the existent golden mountain is solved here by

asserting that existence is included, not exemplified, by the object which

also includes goldenness and mountainhood. Also, the property of being

thought about by Russell is exemplified, but not included, by the object

which includes exactly roundness and squareness. So possible advantages

of our theory are (a) it doesn’t exclude existence and being thought about

by Russell from object-forming sets, and (b) it doesn’t leave us with the

difficulty of trying to say what the “watered-down versions” of extra-

nuclear properties are, something which Parsons must face in order to

relate his domains of properties systematically.”

We concede that these advantages may not be telling. We are content to

have provided a viable theory based on the fundamental distinctions

between real and imaginary objects and between two kinds of predication.

Any final decision for or against the theory must be the outcome of future

discussion.

University of Massachusetts at Amherst

NOTES

* For Russe ll, the description might exist in thought alone, with no corresponding

English expression.

a See, for exam ple, David Le wis, ‘Truth in fiction’, Am erican Philosophical Quarterly

l&37-46.

3 ‘Prolegomenon to Meinongian se ma ntics’, Journal ofphilosophy 71(1974), 561-

580. ‘Nuclear and extranuclear properties, Meinong , and Leibniz’, Nou s 12 (1978)

137-151. Non existent Ob jects, (forthcoming), Yale Unive rsity Press A famil iarity

wi th the ideas in these works m ade our essay possible.

4 ‘Meinongian theories and a Russell ian paradox’, Nou s 12 (1978). 153-180.

5 Indeed, fictional objects exem plify ma ny “negative” properties as well, such as

being non-red and not being clever.

6 Ed Zalta, ‘Alternative Meinongian sem antics’, unpublished. In that paper, the basic

logic of inclusion wa s worked out - the several doma ins of quan tification, ext func

tions, correlation function s, and the logical function PLU G. This basic logical

apparatus resembles that of Parsons in Nonexistent Objects. The semantics for the

remaining logical functions wa s developed by McM ichael as a result of famiharity with

work by Quine (see Note 5) . The ReZations axiom replaces the property abstract ion

axiom of ‘Alternative Mem ongian S ema ntics’.

’ The logical function s are ultimately derived from Quine’s predicate operators in



ALTERNATIVE THEORY OF NONEX ISTENT OBJECTS 313

‘Variables explained away’ , Selected Logical Papers, Random House , New York, 1966,

pp. 227-235. Of course , the difference betwee n our function s and his operators is

enormous.

8 Other logical function s, such as DIS J (“ ‘disjunction”), BIC ON (“biconditional”),

and MC ON (“‘material conditional”), can be added if you think equivalent logical

function s can yield distinct properties when applied to the sam e argum ents.

9 W e use the symbol “@(a/r) ‘* to mean the fomula which results f rom @when every free

occurrence of r is replaced by an occurrence of K.

” Perhaps this is what R apaport means when he says that Meinongian objects are

actua l. (‘Meinongian theories. . . ‘ , p. 171).

I’ Str ictly speaking , Pegasus is not a potential blueprint. Since he includes only the

properties ascribed to him in my th, he is not comp lete ( i .e., for some properties, he

includes neither them nor their negations). How eve r, there are man y potential blue-

prints which include all the properties Pega sus includes, and each of them might have

had a S&correlate.

” This is essen tial ly the account of fictional charac ters given in Zalta’s ‘Alternative

Meinongian sema ntics’ . We presume that a more detailed account could be constructed

along the sam e l ines as Parsons’ acco unt iniVonexistenf Ob jects.

I3 W e obtain no other readings using h -predicates. For e xam ple, “[ t i z [S] ] r”

expresses the same proposit ion,as “ [S] r ” , and ‘ t[hz z[S]] ” expresses the same

proposition as ‘t[S] “. The first fol lows from clause 2 of the definition of “ ‘extended

assignm entg”, and the second fol lows from our intuition that express ions which are

equivalent by the Axiom of Relat ions express the same proposit ion.

” See Rapa port. ‘Meinongian Theo ries. . . ‘ , p. 172. The contradiction is worked out

in detai l in Zalta’s ‘Alternative Meinongian sem antics ’.

” For suppose there were. Since for any dist inct M-objects x, and x, , the property

of being identical to x L would be distinct from the property of being identical to x, ,

a one-to-one correspondence would exist between the set of M-objects and a subset

of the set of properties This leads to contradiction.

Consider the fol lowing instantiation of the Ob jects Axiom :

@)([p]x = (3x,)@ = A zz = x, & - [U 2 = x,]x,) ) .

From this we may deduce a contradiction:

[Azz = XIX = - [ t i t = XIX.

I6 W e have also barred property-quantifiers from A-predicates. Had we allowed them ,

we would need a more co mplicated set of logical operations.

I’ ‘Meinongian theories. . . ‘ , p. 173.

I* One reason he must relate them is so that he may claim that the sentence, “The

existen t golden m ountain is existen t”, ma y be read, ‘“The object which ha s nuclear

existen ce, nuclear goldenne ss, and nuclear mountainhood has nuclear existenc e.” So

he mu st say what nuclear e xistence is. His “wateringdown” axiom ( in Nonexistent

Objec ts) generates the “nuclear ve rsions” of extranuclear properties withou t really

tel ling us precisely wha t these nuclear versions are.

Alan McMichael; Ed Zalta -- An Alternative Theory of Nonexistent Objects

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