steven e. boËr

8/14/2019 Steven e. Bor

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STEVEN E. BOR

ON THE MULTIPLE RELATION THEORY OF JUDGMENT

0. INTRODUCTION

The locus classicus of the multiple relation theory of judgment is, ofcourse, the work of Bertrand Russell in the period from 1906 to 1919.1

Motivated by a desire to excise propositions from his ontology, Russell

attempted during this period to analyse the so-called propositional attitudesin terms of a family of judgment relations which would tie the thinkingsubject directly to the individuals, properties and relations judged aboutand which would permit a notion of judgmental truth (defined in termsof judgmental facts) to supplant the traditional notion of propositionaltruth. It is generally conceded that Russells original attempts were unsuc-cessful; indeed, so serious were their shortcomings that Russell himselfeventually abandoned the whole project. Since then, the multiple relationapproach has come to be regarded as a mere historical curiosity. Whilethe fate of those early versions is no doubt well-deserved, the lesson to belearned from their failure is not that multiple relation strategies per se areunworkable but that the particular blend of ingredients favored by Russellwas inadequate for their implementation.

The aim of this paper is to show how, by developing apparatus thathas roots in Russells own early work, one can vindicate a version (callit MRTJ) of the multiple relation theory of judgment by formally redu-cing it to a plausible representationalist theory. Section 1 briefly sketchesthe ideas behind MRTJ and presents some adequacy conditions on anyreductive vindication of such a theory. Section 2 presents the formalitiesof MRTJ: its base language, underlying logic, and some of its character-istic axioms. Section 3 adduces some general considerations about mentalrepresentation and, against the same formal background, discusses somesalient axioms of a representationalist theory I that treats thought as inner

speech in ones mental language. Section 4 provides the crucial definitionsand axioms which, when added to I to obtain the theory I+, function asbridge principles allowing the formal reduction of MRTJ to I+. Finally,Section 5 argues that this reduction constitutes a vindication inasmuch as

Erkenntnis 56: 181214, 2002. 2002 Kluwer Academic Publishers. Printed in the Netherlands.


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182 STEVEN E. BOR

MRTJ, reconstructed within I+, meets the adequacy conditions laid downin Section 1.

1. ADEQUACY CONDITIONS

Before becoming immersed in the technicalities of its axiomatization, itwill be useful to say just enough about MRTJ to enable us to articu-late certain adequacy conditions on our program of reductive vindication.As presented here, MRTJ is modelled after Russells final version of histheory, according to which the relata of judgment relations include notonly the objects of judgment (i.e., the individuals, properties, and re-lations that the judgment is intuitively about) but also a logical formthat somehow encodes the pattern in which these constituents are judged

to be arranged, thereby determining the judgments truth condition.2

Thus, e.g., an atomic belief ascription of the superficial form (1) is to beanalysed3 along the lines of (2), in which f is the yet-to-be-explainedlogical form of an n-ary predication and Bel is a yet-to-be-explained(n + 3)ary relation of the appropriate logical type for relating items havingthe respective logical types ofA, f, R, a1, . . . , an:

(1) A believes that R(a1, . . . , an)

(2) Bel(A,f ,R,a1, . . . , an).

(In what follows, belief will be our paradigm of a judgment relation.) Nowsuppose as will transpire in the next section that MRTJ is set out asa formalized theory in which the general notions of belief, logical form,and determining a truth condition are taken as primitive, along with somenotation for describing the logical complexity of the logical forms ofparticular beliefs. A reduction of MRTJ to another theory whose primitivesare taken as antecedently understood will not be adequate for the purposeof vindicating the formers notions of logical forms, multiple relations andthe like unless the reduced version of MRTJ provides plausible analysesof beliefs of arbitrary complexity. One test of the plausibility of such ana-lyses is their ability to accommodate and explain the intuitive validity or

invalidity of certain inferences about peoples beliefs. It is, e.g., notoriousthat the Substitutivity of Identity seems to fail in inferences about psycho-logical attitudes like belief. The reduced theory should offer some accountof this which is consistent with MRTJs commitment to the Substitutivity


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ON THE MULTIPLE RELATION THEORY OF JUDGMENT 183

of Identity. In sum, then, we have (C1) as an adequacy condition on thereduced version of MRTJ:

(C1) It should give a plausible analysis of A believes that p andThe belief that p is true/false for all grammatically admiss-ible replacements of p, not just for atomic ones like (1). Inparticular, this analysis should shed light upon alleged failuresof the Substitutivity of Identity.

Moreover, because MRTJ eschews propositions in favor of a multiplicity oftyped belief relations among logical forms and various non-propositionalobjects of belief, the reduced version of MRTJ must additionally meet theadequacy conditions (C2)(C5):

(C2) It should reveal what is common to the members of the infinitely

large family of differently typed belief relations (i.e., that invirtue of which they are belief-relations).

(C3) It should incorporate a precise account of the nature of theposited logical forms.

(C4) It should explain how the obtaining of a belief relation bringsthe objects of the belief into relation with a logical form soconceived; and this explanation should account for the role oflogical forms in fixing the logical structure of a belief ex-plaining, e.g., (a) how a specific order is thereby imposed onthe objects of the belief and (b) why certain possible orderingsmake sense but others do not.

(C5) It should neither appeal to propositions nor use locutions thatmight seem to presuppose propositions for their interpreta-tion such as, e.g., undefined predicates that take nominalizedsentences (as opposed to names of sentences) as arguments.

While Russells versions are set up to comply nominally with (C5), itis a matter of record that none ever came close to satisfying (C1)(C4).This is not surprising, since everything clearly depends upon what is said,on the one hand, about the nature of the posited logical forms and, on the

other hand, about the way in which belief (even when false) supposedlyunites its objects with logical forms so conceived. But both of thesetopics ultimately remain mysterious in Russells writings. Let us see ifwe can dispel the mystery.


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184 STEVEN E. BOR

2. THE FORMALITIES OF MRTJ

2.1. The Base Language and Underlying Logic

L MRTJ, we shall assume, is articulated in a quantificational language Lembodying a simple type theory in which the basic type is i (the type ofindividuals) and in which t1, . . . , t n counts as a type if t1, . . . , t n do.4

(Type indices appear as right superscripts on terms.) L contains variablesof every type and a finite selection (borrowed from English) of primitivenames for the individuals, properties and relations that serve as objects of

judgment. (We use name as a synonym for constant. When t = i, wewill also call a name of type t a predicate of type t. The variables andnames of a given type constitute the terms of that type.) Two sorts of com-plex terms are permitted. When t11 , . . . ,

tnn (n 1) are variables and is

a formula in which none of1, . . . , n occurs bound, L counts -abstracts

of the form [t1

1 . . . tn

n : ] as complex terms of type t1, . . . , t n. Andwhere t is a variable occurring free but nowhere bound in the formula ,L counts a definite description of the form ( t) as a complex termof type t. For future reference, let us call a formula pure iff it contains noprimitive names.

The underlying logic treats the sentential connectives (, &, ,, ) in classical fashion and subjects -abstracts to the familiar prin-ciples of -conversion. Identity is also treated classically: subject to theusual restrictions on free variables, every instance of t = t ( (//)) is an axiom. However, in light ofLs syntactic treatment ofdefinite descriptions as terms, will be so restricted as to ensure a freelogic for such terms ( being defined as usual via ). In other words,

where is a term containing no definite descriptions save those whosenonemptiness is assured by our axioms, (/) may be inferred from(); but the conditional (/) (/) may be inferred from() for any term substitutable for both and , provided that theformula is atomic. This departure from Russell as regards the treatmentof definite descriptions is purely for technical convenience, and nothingessential to the results obtained below depends upon it.

To the foregoing apparatus, we add the axiom scheme (A1), whichformalizes the plausible (albeit metaphysically optional) thesis that if Fand G are the indeed one and the same n-ary relation, then F cannotrequire its j th term to be identical with an object a while G requires its

j th term to be identical with a distinct object b:5


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(A1) If and are formulas, t11 , . . . , tjj , . . . ,

tnn are variables,

tj

and tj are terms distinct from each oft11 , . . . , tjj , . . . ,

tnn and

foreign to both and , then every instance of the following is

an axiom (1 j n):[

t11 . . .

tjj . . .

tnn : j = & ]

= [t11 . . .

tjj . . .

tnn : j = & ] = .

2.2. Some Primitive Vocabulary and Axioms of MRTJ

Since Russell never undertakes an axiomatic presentation of his multiplerelation theory, it is not clear what primitives or axioms he would havefavored for its articulation. For the purposes of our reconstruction, we shallassume that MRTJ is formulated in the languageLMRTJ obtained by addingto L the four families of primitive predicates specified in (P1)(P4):

(P1) the (n + 2)-ary predicates Beli,t1 ,...,tn,i,i,i,i,t1 ,...,tn (n 0);

(P2) the binary predicates Determinest1,...,tn,i,i,i,i,t1 ,...,tn,i(n 0);

(P3) the unary predicates LogicalFormt1,...tn,i,i,i,i (n 0);and

(P4) for each pure formula whose free variables (if any) are

t11 , . . . ,

tkk and each variable

i foreign to , the quatern-ary predicate [1 . . . k : ], of type t1, . . . t k, i, i, i, i,which is counted as syntactically simple (hence as not contain-ing occurrences oft11 , . . . ,

tkk ).

The underlined terms mentioned in (P4) serve as LMRTJs official namesfor particular logical forms. The reason for including the seeminglyredundant variable i will emerge later when we consider wholly gen-eral judgments. The informal interpretation of these special names (andtheir alphabetical variants) may be conveyed by the following sampleglosses: [Gixi di : G(x)] is to be read as the logical form of a

(first-order) unary predication; [Gixi di : G(x)] is to be read asthe logical form of the negation of a (first-order) unary predication;

[Fi,i

xi1x

i2G

i,i

yi1y

i2d

i

: F (x1, x2) G(y1, y2)] is to be read as the lo-gical form of the disjunction of two (possibly unrelated, first-order) unarypredications; and [di : (xi )(x = x)] is to be read as the logical formof the (first-order) universal quantification of a predication of self-identity.


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186 STEVEN E. BOR

We cannot undertake here to provide a full axiomatic development ofMRTJ. For our purposes it will be enough to concentrate upon certain keyaxiom schemes that might plausibly be proposed to govern the vocabulary

introduced in (P1)(P4). Let us begin with the connections between beliefrelations and logical forms. Since belief is supposed to essentially involvea relation to a logical form, [I] is an obvious candidate for axiomatic status:

[I] Beli,t1,...,tn,i,i,i,i,t1,...,tn(i , t1,...,tn,i,i,i,i, t11 , . . . , tnn )

LogicalForm().

(Axioms peculiar to MRTJ will be indexed with bracketed roman numer-als; logical axioms and the additional axioms ofI and I+ will be numberedas (A1), (A2), etc.)

The intuitive logic of de re belief ascriptions might be expected tosupply more candidates for axiomhood. For example, although we pre-

sumably do not want MRTJ to rule out someones having two implicitlycontradictory beliefs regarding the same object say, one under the lo-gical form [Gixi di : G(x)] and the other under [Gixi di : G(x)] we might well be averse to the prospect of someones having anexplicitly contradictory belief about an object under the logical form[Gixi di : G(x) & G(x)]. For the sake of argument, let us legislate thisaversion into the axiom scheme [II]:

[II] Beli,t1,...,tn,i,i,i,i,t1 ,...,tn(i0, [t11 . . .

tnn

i : & ],

t11 , . . . ,

tnn ).

Notoriously, peoples beliefs are not closed under logical consequence,

but we might at least expect some of their beliefs to respect the simplevalid inferences that characterize the logical constants. Thus, e.g., a con-

junctive de re belief should be accompanied by de re beliefs matching theconjuncts (though not always conversely). An appropriate axiom schemefor this kind of conjunctive belief would thus be [III]:

[III] Beli,t1,...,tn,i,i,i,i,t1,...,tn(0, [1 . . . n : & ],1, . . . , n)

{Beli,tg1 ,...,tgn ,i,i,i,i,tg1 ,...,tgn (0, [g1 . . . gn : ],g1 , . . . , gn ) &

Beli,tj1 ,...,tjn ,i,i,i,i,tj1 ,...,tjn (0, [j1 . . . jn : ],

j1 , . . . , jn )},

where (i) g1 , . . . , gn are those of the variables 1, . . . , n free in ,and g1 , . . . , gn are the corresponding terms from among 1, . . . , n; (ii)


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j1 , . . . , jn are those of the variables 1, . . . , n free in , and j1 , . . . , jnare the corresponding terms from among 1, . . . , n. Again, under the as-sumptions (i) and (ii), we might lay down [IV], which (roughly speaking)

requires that when objects are believed to satisfy neither of two conditions,they should be separately believed not to satisfy each of those conditions:

[IV] Beli,t1,...,tn,i,i,i,i,t1,...,tn(0, [1 . . . n : ( )],1, . . . , n)

{Beli,tg1 ,...,tgn ,i,i,i,i,tg1 ,...,tgn (0, [g1 . . . gn : ],g1 , . . . , gn ) &

Beli,tj1 ,...,tjn ,i,i,i,i,tj1 ,...,tjn (0, [j1 . . . jn : ],j1 , . . . , jn )},

Similarly, we might stipulate [V], which says (again roughly) that believ-

ing objects to satisfy some condition rules out believing that condition tobe unsatisfied:

[V] Beli,t1,...,tn,i,i,i,i,t1,...,tn(0, [1 . . . n : ], 1, . . . , n)

Beli,i,i,i,i(0, [ : (1) . . . (n)]).

No doubt numerous other principles of this ilk could be justified for beliefsof other basic logical forms, but having a complete list of them is notimportant for present purposes.

As for the nature of logical forms themselves, we may lay down at leastthis much at the outset. A logical form f is a relational entity that de-termines a particular ontological structure Rf, the latter being a (possiblyunexemplified) formal relation between ordinary properties, relations, andindividuals. The chief constraint on the identity of the structure Rf derivesfrom the role that a logical form is supposed to play vis--vis the condi-tions for someones believing truly/falsely under it: viz., a case of believingunder f is a case of believing truly/falsely under f iff the entities believed-about are/are not related by Rf. Accordingly, the logical form f must bedistinct from the determined structure Rf. For in a case of believing falselyunder f, the entities believed-about are of course not related by Rf. Yeteven ifRf is not exemplified by anything at all, f must still be the logicalform under which the subject believes, where this naturally suggests thatf (itself a relational entity) must somehow be exemplified in the belief.

By this we do notmean that there is an extra entity, the belief, which is arelatum off, but merely that any case of believing under f must involvefs actually relating some items. (As it stands, MRTJ offers no account offs relata; in our proposed reduction, we shall identify them with certain


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MRTJ, of course, treats belief ascriptions transparently. Some (e.g.,Salmon (1986) and Soames (1987a, b)) have argued on Russells behalfthat so-called attitudinal opacity is an illusion not a syntactic/semantic

datum but merely a pragmatic phenomenon hence that there are noreal substitution-failures in attitudinal contexts to contend with in the firstplace. Be that as it may, MRTJ does at least have a way of handling ap-parent substitution-failures involving definite descriptions, despite treatingthem syntactically as terms. Consider, e.g., the inference from (3ab) to(3c):

(3)a. George IV believes that Scott = the author ofWaverly.

b. The author ofWaverly = the author ofMarmion.

c. George IV believes that Scott = the author ofMarmion.

Because it can make fine-grained distinctions of logical form, MRTJ can

in principle formalize this inference in any of the four ways (4)(7):

(4)a. Beli,i,i,i,i,i,i,i,i (GeorgeI V, [xi yi di : x = y],

Scott, (yi )AuthorOf(y, Waverly)).

b. (yi )AuthorOf(y, Waverly) = (yi )AuthorOf(y,Marmion).

c. Beli,i,i,i,i,i,i,i,i (GeorgeI V, [xi yi di : x = y],

Scott, (yi )AuthorOf(y,Marmion)).

(5)a. Beli,i,i,i,i,i,i,i,i (GeorgeI V, [xi Fidi : x = (zi)F z],

Scott, [yi : AuthorOf(y, Waverly)]).

b. (xi )AuthorOf(x, Waverly) = (xi )AuthorOf(x,Marmion).

c. Beli,i,i,i,i,i,i,i,i (GeorgeI V, [xi Fidi : x = (zi )F z],Scott, [yi : AuthorOf(y,Marmion)]).

(6)a. Beli,i,i,i,i,i,i,i,i (GeorgeI V, [xi Fidi : x = (zi)F z],

Scott, [yi : AuthorOf(y, Waverly)]).

b. (xi )AuthorOf(x, Waverly) = (xi )AuthorOf(x,Marmion).

c. Beli,i,i,i,i,i,i,i,i (GeorgeI V, [xi yi di : x = y],

Scott, (yi )AuthorOf(y,Marmion)).

(7)a. Beli,i,i,i,i,i,i,i,i (GeorgeI V, [xi yi di : x = y],

Scott, (yi )AuthorOf(y, Waverly)).

b. (yi )AuthorOf(y, Waverly) = (yi )AuthorOf(y,Marmion).

c. Beli,i,i,i,i,i,i,i,i (GeorgeI V, [xi Fidi : x = (zi )F z],

Scott, [yi : AuthorOf(y,Marmion)]).


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190 STEVEN E. BOR

Now (4), which is univocal with respect to Bel and the logical formassignment in premisses and conclusion, not only is a plausible analysis ofthe transparent reading of (3) but also is formally valid, being an instance

of the Substitutivity of Identity. This leaves (5)(7) as candidates for rep-resenting the wholly or partly opaque readings of (3) on which it is invalid.Each of (5)(7) either equivocates on Bel and the logical form assignmentor changes the subject by making the statements about properties insteadof (or in addition to) individuals. None of (5)(7) is valid solely in virtue ofour logical axioms, although this leaves it open that a deeper understandingof what it is to have a belief under a logical form might lead us to acceptone of them. (5), which involves no equivocation, looks promising as arendering of the fully opaque construal of (3) and is intuitively invalid.Here the intuition is that if George IV thinks of Scott as uniquely authoringWaverly, then the mere fact that Scott uniquely authored both Waverlyand Marmion offers no assurance that George IV also thinks of Scott as

uniquely authoring Marmion. (7) likewise seems invalid. The fact that thetwo novels had the same author and that George IV has an identity-beliefthat happens to be about Scott and the author of Waverly is insufficientgrounds for concluding that George IV thinks of Scott as the author of

Marmion (or as of any other kind, for that matter). The intuitive statusof (6) is less clear and will be left to our reduction to settle.

2.3. Truth in MRTJ

Where f is a logical form, let us call the ontological structure it determinesthe value off and introduce for it the following abbreviatory notation:

(D1) Where is a term of type t1, . . . t n, i, i, i, i:|| =def(

t1,...,tn,I)Determines(,).

We have seen that the value |f| of a logical form f is intuitively con-nected with the truth conditions for beliefs under f. Can this intuitiveconnection be made explicit in MRTJ itself? As regards the truth-valuesof belief facts, it is easy to define typed counterparts of truly believesand falsely believes. The basic idea is simple: one bears the multiplerelation of truly/falsely believing to such-and-such items under a givenlogical form f iff (i) one bears to those items under f the multiple relationof believing and (ii) the value of f does/does not relate those items (inthe indicated order). Formally, this amounts to taking (D2) and (D3) as

definition schemes:

(D2) TrBeli,t1 ,...,tn,i,i,i,i,t1 ,...tn =def[i t1,...,tn,i,i,i,it11 . . .

tnn :

Beli,t1,...,tn,i,i,i,i,t1,...,tn(,,1, . . . , n) & ||(1, . . . , n, )].6


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(D3) FlsBeli,t1,...,tn,i,i,i,i,t1,...tn =def[it1,...,tn,i,i,i,it11 . . .

tnn :

Beli,t1,...,tn,i,i,i,i,t1,...,tn(,,1, . . . , n) &||(1, . . . , n, )].

Thus, e.g., (8) will be paraphrased in MRTJ by (9), which is provablyequivalent in MRTJ to (10):

(8) Othello truly believes that Desdemona loves Cassio.

(9) TrBeli,i,i,i,i,i,i,i,i,i,i,i,i (Othello, [Fi,ixi yi di : F(x,y)],Loves, Desdemona, Cassio).

(10) Beli,i,i,i,i,i,i,i,i,i,i,i,i (Othello, [Fi,ixi yi di : F(x,y)],Loves, Desdemona, Cassio) & Loves(Desdemona, Cassio).

Clearly, however, the definition schemes (D2) and (D3) will not sup-ply what Russell wanted of MRTJ, viz. (stratified) notions of truth andfalsehood for beliefs that will serve in place of the traditional notions ofpropositional truth and falsehood that he jettisoned along with the propos-itions themselves. The obvious problem is that infinitely many truths andfalsehoods may go unbelieved, so that there will not be enough surrogatebelief facts to go around (hence not enough real cases of believing-trulyor believing-falsely for (D2) and (D3) to take up the slack). However,even where there is no fact that someone believes, say, that Ra b, thereis still the corresponding relation between a thinker x and a logical form fthat consists in (i) xs bearing the appropriate multiple relation to R, a, b

underf

and (ii)f

s being such-and-such a logical formh

. Takingh

to be[Fi,iyi1yi2d

i : F (y1, y2)], this relation between thinker and logical formwould be the complex relation Bh:R,a,b defined by (D4):

(D4) Bh:R,a,b =def[xi fi,i,i,i,i,i,i,i,i,i,i,i,i :f = [Fi,iyi1y

i2d

i : F (y1, y2)] &

Beli,i,i,i,i,i,i,i,i,i,i,i,i (x,f,Ri,i, a , b )].

In sentences of the sort The belief that a bears R to b is true, we can nowthink of (11) as para-phraseable by (12), to which the relation Bh:R,a,b isassigned as referent:

(11) the belief that a bears R to b.

(12) beliefs under the logical form of a binary predication involvingR, a, b respectively.


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192 STEVEN E. BOR

In general, then, for any logical form gt1,...,tn,i,i,i,i and entities e1, . . . ,en of respective types t1, . . . , tn, we will have as a belief-surrogate thecomplex relation Bg:e1,...,en defined by (D5):

(D5) Bg:e1,...,en =def[xi ft1,...,tn,i,i,i,i: f = g &

Beli,t1,...,tn,i,i,i,i,t1,...,tn(x,f,e1, . . . , en)].

To complete the picture, we need a suitable family of typed truth-predicates for such belief-surrogates. What immediately suggests itself isan appeal to the notion of the value of the logical form relating the objectsof the belief. Tailored to fit (D5), a suitable definition scheme would be(D6):

(D6) For any term of type t1, . . . , t n, i, i, i, i and terms

t11 , . . . ,

tnn :

Truei,t1 ,...,tn,i,i,i,i(B:1,...,n )

=def(i )||(1, . . . , n,),

with Falsei,t1 ,...,tn,i being defined as Truei,t1,...,tn,i. In otherwords, a belief-surrogate Bg:e1,...,en is true" just in case the value of gactually relates e1, . . . , en to some object w. Trivially, we will have astheorems all instances of the corresponding truth scheme (T2), and fromthe instance of (T2) for the particular belief-surrogate Bh:R,a,b discussedabove, theorem (T3) will follow:

(T2) True(B[1...n:]:1,...,n ) (i)([

t11 . . .

tnn

i : ]

(1, . . . , n,)).

(T3) True(Bh:R,a,b) R(a, b).

Using this apparatus, we can translate a vernacularism like (13) by itsperspicuous counterpart (14), which, however unwieldy, will at least havethe virtue of being demonstrably equivalent to (15):

(13) The belief that Desdemona loves Cassio is true. [I.e.: Beliefsunder the logical form of a binary predication involving loving,Desdemona, and Cassio respectively are true.]

(14) Truei,i,i,i,i,i,i,i,i ([xi fi,i,i,i,i,i,i,i :f = [Fi,iyi1y

i2d

i : F (y1, y2)] &

Beli,i,i,i,i,i,i,i,i,i,i,i,i,i (x,f,Loves, Desdemona, Cassio)]).


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(15) Loves(Desdemona, Cassio).

These results, together with the fact that MRTJ treats the logical com-

plexity of a beliefs content as a function of its logical form, ensure thatthe foregoing account of truth for belief-surrogates applies not just to theatomic ones but to the molecular ones as well.

3. THE THEORY I

Our concern is to vindicate MRTJ, not the various doctrines entangledwith Russells own versions of the multiple relation theory (e.g., his Prin-ciple of Acquaintance and his sense-datum phenomenalism). In particular,we carry no brief for his curious view that the mind somehow directly

arranges the (nonmental) objects of judgment in accordance with a cer-tain logical form. Instead, we look for vindication in the direction of arepresentational theory of mind (RTM) according to which what gets ma-nipulated in thought are not the nonmental objects of thought per se butmental representations of those objects. Specifically, we shall consider aformal theory I that embraces RTM in the specific form of the so-calledlanguage of thought hypothesis (LOT), according to which (at the levelof representation-types) the minds stock of representations is regardedas forming a language-like system (the thinkers Mentalese) for whichan appropriate syntax and semantics could in principle be provided. Weshall be assuming rather than arguing for the plausibility of LOT here; forarguments see Fodor (1975, 1987) and Maloney (1989).

Ironically, Russells first, very brief and tentative sketch of his theory(Russell 1906/08) did involve mental representatives of objects in addi-tion to the objects themselves; and after officially renouncing the multiplerelation approach (Russell 1919) he immediately embraced a new viewof belief as a relation to certain mental representations constructed frommental images standing for objects, properties, and relations. Perhaps whatprevented Russell from exploiting his latent affinity for mental represent-ations in the interest of his multiple relation theory of judgment was hisfailure to get beyond the unprofitable traditional talk of mental imageryto some version of LOT. In any event, let us see whether, availing ourselvesof LOT, we can do better on behalf of MRTJ.

As loyal type-theorists, we assume that languages of thought have atype-structure like that ofL. Since we will both be talking in English aboutthe extended language LI ofI and using LI in turn to talk about thesyntax and semantics of Mentalese, it will be useful to adopt the following


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The following vocabulary (minus type indices and with obviousor bracketed interpretations) is employed for discussing the syntaxand semantics of a thinkers Mentalese sentences and the thinkers

psychological relations to those sentences:VarType(, t,M ); NameType(, t,M); TermType(, t,M);Sentence(,M );

Expression(,M ); SimpleIn(,M ); FreeIn(,);OccursIn(,); (1 . . . n/1 . . . n);

DesIn(, t,M) [i.e., designates t in M];Distinct(1, 2, . . . , m) [i.e., &1i=j m(i = j )];SynRel() [i.e., is a syntactic relation among Mentaleseexpressions]; Accepts(,).

In axiom, theorem and definition schemes where t is employed as aschematic letter whose replacements are particular type indices of our

formalism, its primed counterpart t is to be understood as a schematicletter whose replacements are the boldfaced indices for the correspondingMentalese types. (Thus, e.g., if t is replaced by i, i in an instance ofa scheme also containing t, then t must be replaced by i, i in thatinstance.) The metalinguistic expression t must not be confused with theobject language variable t.

In the case ofLIs syntactic vocabulary, the corresponding axiomschemes of I are predictable. There will be, e.g., an axiom schemeto guarantee that [1 . . . m : ] will count as a term of typet1, . . . ,tm in M if is a sentence and 1, . . . , m are variables of re-spective types t1, . . . ,tm in M; another axiom scheme to guarantee that if[1 . . . m : ] is a term of type t1, . . . ,tk, i in M with foreignto , then [1 . . . m : ] is a term of type t1, . . . ,tk, i, i, i, i inM; and so on. Since these axiom schemes are as uninteresting as they arepredictable, there is no point to enumerating and formalizing them here.

For present purposes there is no need for assumptions about the make-up or semantics of anyones non-logical Mentalese vocabulary. It will,however, be important to specify the designation of pure Mentalese -abstracts i.e., terms of the sort [v1 . . . vk : S] in which S contains noprimitive Mentalese names of any type. So I will require a correspondingsemantic axiom scheme. To formulate this axiom scheme (and others tocome), we introduce below the useful notion of the shadow of a pureformula.

Suppose [1, . . . , k; k+1, . . . , n] is a pure formula ofL. Then,for any distinct boldfaced letters 1, . . . , k, k+1, . . . , n, we say thatan expression is the shadow of [1, . . . , k ; k+1, . . . , n] un-der [1, . . . , k; k+1, . . . , n] iff: results from [1, . . . , k;


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196 STEVEN E. BOR

k+1, . . . , n] by first replacing 1, . . . , k, k+1, . . . , n with 1, . . . , k,k+1, . . . , n and then replacing any remaining non-boldfaced sym-bols with their boldface counterparts from LIs special vocabulary

for Mentalese. (For example: (v4)(v1(v4) & v2(v3)) would be theshadow of (x)(F(x) & G(y)) under [v1, v2, v3; v4] and(v1)(v2)v2(v1) would be the shadow of (x)(F)F(x) under [ ;v1, v2].)

Where is the shadow of the pure formula [t11 , . . . , tkk ;

tk+1k+1, . . . ,

tnn ] under [1, . . . , k; k+1, . . . , n] and is a bold-

faced letter foreign to , I has the axiom scheme (A2) for the semanticpredicate DesIn of type i, t1, . . . , t k, i:

(A2) (Distinct(1, . . . , n, ) & VarType(1, t1,M) & &

VarType(n, tn,M) & VarType(, i,M) &

OccursIn(, )) DesIn([1. . . k : ], [

t11 . . .

tkk

i :[1, . . . , k; k+1, . . . , n]], M).

Thus, since f(x) is the shadow of F(x), under [f, x; ], it followsfrom (A2) that iff, x, d are distinct variables of respective types i, i andi in ys Mentalese, (d foreign to f(x)), then ys Mentalese name [fxd:f(x)] designates the type i, i , i relation [Fixi di : F(x)].

Given the adoption of LOT, it is natural to model the having of beliefs asthe acceptance of certain sentences of ones Mentalese. Since, however,most of a persons beliefs at any moment are merely tacit, the correspond-ing notion of acceptance must not require occurrent tokening of Mentalesesentences. Accordingly, let us gloss Accepts(x, S) as x is disposed asif x inwardly and assertively tokens S,8 where to token S a property is to produce something exemplifying it. I will of course contain axiomscharacterizing this relation. The following, admittedly incomplete list ofaxioms (derived, with modifications, from Loar (1981: 72)) may serve toconvey the flavor of what would be involved:

(A3.1) (Sentence(S1,Mx ) & Sentence(S2,Mx )) Accepts(x, S1 &S2).

(A3.2) (Sentence(S,M

x) & Accepts(x, S) Accepts(x, S).

(A3.3) (Sentence(S1,Mx ) & Sentence(S2,Mx ) & Accepts(x, S1 &S2)) (Accepts(x, S1) & Accepts(x, S2)).


17/34


18/34

198 STEVEN E. BOR

The import of this complicated definition scheme is best appreciated byway of a simple example. Since f(x) is the shadow of the pure formulaFi(xi ) under [f, x; ], we have the definition (16) as an instance

of (D7):

(16) [Fixi di : F(x)] =def [i,i,iyi zdi : = [Fixi di : F(x)]& (w){NameType(w, i, i, i,My ) & DesIn(w, ,My ) &[rsi u: (f)(x)(d)(Distinct(f, x, d) & VarType(f, i,Ms ) &VarType(x, i,Ms ) & VarType(d, i,Ms ) & OccursIn(d, f(x))& r = [fxd : f(x)]) & u = r)]wyz}].

From (16) we learn that what MRTJ calls [Fixi di : F(x)] alias thelogical form of a (first-order) unary predication may be identified witha certain complex relation R between a structure , person y, expressionz ofMy , and arbitrary individual d. In effect, Ryzd obtains just in case:(a) is the structure [Fixi di : F(x)]; (b) for some distinct variables f,x, d of respective My-types i, i, i, the term [fxd : f(x)] designates thestructure in My , and (c) z is the term [fxd : f(x)] ofMy .

According to MRTJ, [Fixi di : F(x)] determines the ontological

structure [Fixi di : F(x)]. But by (16), [Fixi di : F(x)] contains[Fixi di : F(x)] in the sense of being a relation that requires its first termto be that very structure. This suggests a general definition of determiningvia containing, which is spelled out in (D8) with the aid of the predicateSynRel expressing the higher-order property of being a syntactic relationamong expressions of a persons Mentalese:

(D8) Determinest1,...,tk ,i,i,i,i,t1 ,...,tk,i =def[Ft1,...,tk ,i,i,i,iGt1,...,tk ,i: (Ki,i,i){SynReli,i,i(K)& F = [xt1,...,tk ,iyi zdi : x = G & (w){NameType(w,t1, . . . , t

k, i, My ) & DesIn(w, x, My ) & Kwyz}]}].

For present purposes, it is not important exactly how SynRel is ultimatelycharacterized. All we need assume here is that it has been so axiomatizedin I+ as to yield as a theorem any formula in which SynRel is applied toan instance of the schematic -term used in (D7); in other words, we shallassume that (T4) is a theorem scheme:

(T4) SynRel([rsi u: (1).. .(n)()(Distinct(1, . . . , n) &VarType(1,t1,Ms ) & . . . & VarTypen,tn,Ms ) &VarType(, i,Ms ) & OccursIn(,) &r = [1 . . . k : ]) & u = r)]).


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(T4) ensures that instances of this schematic -term count as expressing asyntactic condition on expressions of a persons Mentalese. Courtesy of(A1) and the definitions (D7) and (D8), the axiom scheme [VII] of MRTJ

is now a theorem scheme ofI

+

.

11

With Determines defined, we can now equate what MRTJ calls be-ing a logical form of a given type t1, . . . , t k, i, i, i, i, with being acertain relation of type t1, . . . , t k, i, i, i, i that determines relations ofthe corresponding lower type t1, . . . , t k, i. This is recorded in (D9):

(D9) LogicalFormt1 ,...,tk ,i,i,i,i =def[xt1,...,tk ,i,i,i,i:(yt1 ,...,tk,i)Determines(x,y)].

In light of (A1), (D8), and (D9), MRTJs axiom schemes [VIII] and [VI]are theorem schemes ofI+.Having embraced LOT, we shall naturally wish to speak not only about

a logical form determining a structure but also about its being a logicalform of something viz., someones Mentalese sentence. While it wouldbe possible to define a relation FormOf that would apply to any Men-talese sentence, our project here requires only a more restricted versionthat applies to pure Mentalese sentences, i.e., those all of whose simpleterms are variables. Accordingly, we schematically define FormOf as thefollowing complex relation:

(D10) FormOft1,...,tk ,i,i,i,i,i,i =def[ft1,...,tk ,i,i,i,iSxi:LogicalForm(f ) & Sentence(S,Mx ) &(v1) . . . (vk)(d){(Distinct(v1, . . . , vk, d) &VarType(v1, t

1,Mx ) & . . . & VarType(vk , t

k,Mx ) &

VarType(d, i,Mx ) & (v0)(FreeIn(v0, S) (v0 =v1 v0 = vk)) & OccursIn(d, S) &(r)(tx )((NameType(r,t,Mx ) & SimpleIn(r,Mx )) OccursIn(r, S)) & f (|f|, x, [v1 . . . vk d : S], x)}].

In other words, FormOf is that relation between a logical form f, sentence

S, and agent x consisting in Ss being a pure sentence ofMx containingfree variables v1, . . . , vk such that, for some Mx-type i variable d foreign toS, f relates its value to x and to the corresponding term [v1 . . . vkd : S]ofMx.


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200 STEVEN E. BOR

(D10) enables us to prove every instance of the important scheme(T5), in which is the shadow of the pure formula [t11 , . . . ,

tkk ;

tk+1k+1, . . . ,

tnn ] under [1, . . . , k; k+1, . . . , n]:

(T5) (Distinct(1, . . . , n) & VarType(1, t1,M) & . . . &VarType(n, t

n,M))

FormOf([t11 . . .

tkk

i : [1, . . . , k; k+1, . . . , n]],

, i ).

Thus, e.g., (G(v) being the shadow of Gi(xi ) under [G, v; ])we can prove that where G and v are free variables of respective Ma-types i and i, the logical form [Gixi di : G(x)] is a logical form ofthe elementary Ma-sentence G(v). This is recorded in theorem (T6):

(T6) (ai)(G)(v){{Distinct(G, v) & VarType(G, i,Ma ) &VarType(v, i,Ma )} FormOf(([G

ixi di : G(x)],G(v), a)}.

The axiom scheme (A2) that I+ inherits from I specifies the designataof Mentalese counterparts of relational names of the form [1 . . . k :]. But in order to complete the connection of MRTJs vocabulary forlogical forms with Is vocabulary for their mental representations, weneed to specify the designata of the Mentalese counterparts of relationalnames of the form [1 . . . k : ]. Intuitively, these Mentalese coun-terparts are supposed to be names of logical forms. Accordingly, wesuppose that, where is the shadow of the pure formula [t11 , . . . ,

tkk ;

tk+1k+1, . . . ,

tnn ] under [1, . . . , k; k+1, . . . , n] and is is a bold-

faced letter foreign to , the instances of scheme (A4) are axioms ofI+:

(A4) (Distinct(1, . . . , n, ) & VarType(1, t1,M) & . . . &VarType(n, t

n,M) & VarType(, i,M) &

OccursIn(, )) DesIn([1 . . . k : ],[1 . . . k : [1, . . . , k; k+1, . . . , n]],M).

12

Having said in I+ what sort of relation MRTJs logical forms are andwhat it is for one of them to be a logical form of a (pure) Mentalesesentence, we must now specify the nature of the multiple relation ofbelief in which logical forms are to figure as terms. Suppose, for the sakeof illustration, that [Fi,iyi1y

i2d

i : F (y1, y2)] is the logical form invoked

for Desdemona loves Cassio and that (17) is MRTJs analysis of (18):

(17) Beli,i,i,i,i,i,i,i,i,i,i (Othello, [Fi,iyi1yi2d

i : F (y1, y2)],Loves, Desdmona, Cassio).


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(18) Othello believes that Desdemona loves Cassio.13

The idea about Bel we want to articulate in I+ is that the truth of (18)

requires something like the following: (i) that Othello assertively tokens inhis language of thought some sentence S comprising names of loving, Des-demona, and Cassio; and (ii) that these names are syntactically arranged inS in exactly the way dictated by [Fi,iyi1y

i2d

i : F (y1, y2)], (so that S willbe a relational sentence of Othellos Mentalese whose binary predicate,first argument, and second argument are respectively the aforementionednames of loving, Desdemona, and Cassio). This would be straightforwardwere it not for a difficulty that arises in trying to generalize the account tocover nested beliefs.

A purported vindication of MRTJ must surely take seriously its guidingidea that belief is a multiple relation involving thinker, logical form,and the various entities thought about. If so, however, then that guiding

idea should be applied across the board, to any proffered analysis of beliefascriptions in Mentalese as well as in English! (This is part of the point of(C5), which bars appeal to any unanalysed Mentalese analogue of a that-clause construction.) Now MRTJ analyses an iterated belief ascription like(19) by something of the form (20):

(19) Iago believes that Othello believes that Desdemona lovesCassio.

(20) Bel1(Iago, f1, Bel2, Othello, f2, Loves, Desdemona, Cassio).

In (20), however, there are two logical forms to contend with f1, whichdetermines how Bel1 relates Iago to all the other constituents, and f2,which determines how Iago thinks of Bel2 as relating Othello, loving,Desdemona and Cassio! In other words, logical forms must be capableof figuring among the objects of a belief as well.

The difficulty facing the attempt to generalize the account to cover (20)is this: it is not enough merely that Iago should accept a Mentalese sentencewhose structure is dictated by f1 and whose terms respectively designatebelieving, Othello, loving, Desdemona, Cassio, and a certain logical formf2 (for Othellos belief). Rather, Iago must represent the logical form ofOthellos alleged belief in some canonical way that reveals to Iago theconditions under which Othellos belief would be true, i.e., puts Iago in

a position to appreciate those truth conditions (if he considers the matter,is smart enough, etc.). The need for such canonical representation of thelogical forms of others beliefs is intrinsic to any view of thinking as a kindof inner speech. This is most easily seen when we pretend that the inner


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202 STEVEN E. BOR

speech in question occurs in a natural language like English. Imagine, e.g.,a situation like the following. Iago, peering through a telescope, dimly seesa man a crouched behind a bush spying on a woman b who is embracing

a man c. Iago, thinking in approved multiple-relation jargon, then says tohimself (21) while mentally ostending a, b, c in connection with the tokensof he, her, and him:

(21) He has a beliefof the binary first-order relational sortabout herand him.

But unbeknown to Iago a = Othello, b = Desdemona and c = Cassio. Insuch a situation, a wholly transparent construal of (19) would presumablybe true, and the logical form of Othellos alleged belief (viz., binary first-order relational predication) is recoverable from what Iago has said tohimself. On the other hand, suppose Iago had instead said to himself (22):

(18) He has a beliefof his favorite sort about her and him.

This is plainly not enough to make true even a wholly transparent con-strual of (19), precisely because the logical form of Othellos alleged beliefwould no longer be recoverable from what Iago had said to himself.

In light of the foregoing, we need to ensure that when a logical form[1 . . . k : ] occurs among the objects of a persons belief, the per-sons mental name for it is canonical in the sense of being a Mentalese[1 . . . k : ]. Accordingly, we define by cases the ternary predicateCNO (read: . . . is a canonical name of . . . in . . . s Mentalese). For thecases in which t is a type of the sort t1, . . . t k, i, i, i, i, we providedefinition scheme (D11):

(D11) CNOi,t,i =def[wx

t yi : {LogicalForm(x) &(r)(S){Sentence(S,My ) & (v1)(v2)(v3)(d){Distinct(v1, v2, v3, d) & VarType(v1, t1, . . . , t

k , i,My ) &

VarType(v2, i,My ) & VarType(v3, i,My ) & VarType(d, i,My )& (v0)(FreeIn(v0, S) (v0 = v1 v0 = v2 v0 =v3)) & OccursIn(d, S) & (n)(ty )(NameType(n,t,My ) OccursIn(n, S)) & r = [v1v2v3d : S] &

DesIn(r, |x|, My ) & w = r}} {LogicalForm(x) &DesIn(w, x,My )}].

For types t not of the sort t1, . . . t k , i, i, i, i, canonical naming maybe equated with ordinary designation, as in (D12), since no type t entitycould be a logical form:

(D12) CNOi,t,i =def[wx

t yi : DesIn(w, xt

,My )].


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At last we are in a position to define the family of predicatesBeli,t1,...,tn,i,i,i,i,t1,...,tn. Despite the proliferation of Bel-relations towhich MRTJ is committed, it is nevertheless possible to capture them all

inI

+

by means of the definition scheme (D13):(D13) Beli,t1,...,tn,i,i,i,i,t1,...,tn =def [xi ft1,...,tn,i,i,i,iy

t11 . . . y

tnn :

(S)(v1) . . . (vn){Sentence(S,Mx) & VarType(v1, t1,Mx )

& . . . & VarType(vn, tn,Mx ) & Distinct(v1, . . . , vn) &(w)(FreeIn(w, S) (w = v1 w = vn) &FormOf(f, S, x) & (b1) . . . (bn){NameType(b1, t

1,Mx )

& . . . {NameType(bn, tn,Mx ) & CNO(b1, y1, x) & . . . &

CNO(bn, yn, x) & Accepts(x, S(b1 . . . bn/v1 . . . vn))}}].(n 0)

What (D13) tells us is that an (n + 2)-ary Bel-relation of given type

is the relation between an agent, a logical form, and entities y1, . . . yn(of corresponding types) that consists in the agents being disposed asone who inwardly and assertively tokens the (closed) Mentalese sentencewhich results from uniformly substituting names b1, . . . bn for the freevariables of an n-ary open Mentalese sentence having the given logicalform, where each bi is a canonical name ofyi in the agents Mentalese. Thedescription of b1, . . . , bn is couched in terms of canonical names for thesake of generality: we want MRTJ to apply to iterated belief ascriptions,its analysis of which requires logical forms to appear among the entitiesy1, . . . yn. However, in cases not involving nested belief predicates hencewhere none of y1, . . . yn is a logical form the description ofb1, . . . , bnsimply amounts to the requirement that b1, . . . , bn respectively designatey1, . . . yn in agents Mentalese. So, in the non-nested cases, to have a beliefabout y1, . . . yn under a logical form f is to accept the substitution-instanceS(b1 . . . bn/v1 . . . vn) of a pure sentence S of ones Mentalese which issuch that (i) f is a form of S and (ii) b1, . . . , bn respectively designatey1, . . . yn in ones Mentalese. In the formulation of (D13) the Mentalesenames b1, . . . , bn, unlike the Mentalese variables v1, . . . , vn they replace,are not required to be distinct from one another, for (like Russell) weare dealing with relational rather than notional belief, hence allowingthat identities among the objects of a belief may not tracked by identitiesamong their Mentalese names.

The unity of the family of Bel-relations is thus more intimate than

that of a group of relations that merely obey the same or similar laws:members of the family of Bel-relations are all structurally alike in theway depicted in (D13). It is this structural likeness, together with theirincorporation of the relation Accepts, which makes them all beliefrelations


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204 STEVEN E. BOR

though of course this cannot be said in I+ itself but only shown bythe definitional status of (D13)s instances.

Given (D10) and (D13), MRTJs axiom scheme [I] becomes a theorem

scheme ofI

+

. Indeed, in light of (D7) and (D13), all the remaining axiomschemes of MRTJ also become theorem schemes of I+: [II] via (A3.1);[III] via (A3.3); [IV] via (A3.5); and [V] via (A3.7). Furthermore, it isnow possible to say what it is for a type-indexed expression ofMx to bea belief predicate (a Bel) or a believes-truly predicate (a TrBel)of that Mentalese type two notions that will loom large in the next sec-tion. To simplify the formulation, let us take Vocab(vt11 , . . . , v

tnn , d

i, ui , yt11 ,. . . ytnn , S;Mx ) to say that v1, . . . , vn, d, u, y1, . . . yn are distinct variables ofrespective Mx-types t1, . . . ,tn, i, i, t1, . . . ,tn and that S is a pure formulaofMx in which the free variables are exactly v1, . . .vn but in which d doesnot occur. Now consider the schematic formula (23), in which x, B, andt1, . . . , t1are the only free variables:

(23) (v1) . . . (vn)(d)(u)(y1) . . . (yn)(S){Vocab(vt11 , . . . , v

tnn ,

di , ui , yt11 , . . .y

tnn , S; Mx ) Accepts(x,

(u) Bi,t1,...,tn,i,i,i,i,t1,...,tn(u, [v1 . . . vnd : S & S],y1, . . . , yn))}.

(23) attributes to x acceptance of Mentalese sentences that mimic instancesof the axiom scheme [II] of MRTJ. It seems clear that, whatever the plaus-ible axiom schemes for Bel turn out to be in MRTJ (including ones thatmight be added to relate belief to desire and intention), there will be corres-ponding open formulas ofI analogous to (23) for attributing acceptance

of instances of those schemes. Since x, B, and t

1, . . . , t

n are theonly free variables involved, let R(x, B,t1, . . . ,tn) denote the conjunc-tion of all such open acceptance-attribution formulas of I. Then what itis for a type-indexed expression Bi,t1,...,tn,i,i,i,i,t1,...,tn ofMx to be an(n + 2)-ary belief predicate of that Mentalese type can be equated withxs accepting (i.e., being disposed as one who assertively tokens) all therelevant axioms. In other words, we can lay down (D14) as a definitionscheme in I+:

(D14) BeliefPredn =def [btx x: (tx1) . . . (t

xn)(t = i, t1, . . . ,tn, i,

i, i, i, t1, . . . ,tn & (e)(Expression(e, Mx ) & b = et

& (d)(tx0 )(Expression(d, Mx ) & e = dt0 )) & R(x, e,

t1, . . . ,tn)))]. (n > 0)

Given (D14), we can define in I+ what it is for an expression pof Mx to be a believes-truly predicate of suitable type. This is ac-


25/34


complished by adoption of the definition scheme (D15), in which t =i, t1, . . . ,tn, i, i, i, i,t1, . . . ,tn:

(D15) TrueBeliefPredn =def[ptx x: (tx1 ).. .(t

xn)(e){t = t

&Expression(e,Mx ) & p = et & (d)(tx0 )Expression(d, Mx ) & e = dt0 ) & (b)BeliefPredn(b, t, x) &(b)(v1) . . . (vn)(d)(a)(y1) . . . (yn)(S)

{(BeliefPredn(b,t, x) & Vocab(vt11 , . . . , v

tnn , d

i , ui , yt11 , . . .ytnn , S,

M)) Accepts(x, (y1).. .(yn)([u : p(u,[v1 . . . vnd : S], y1, . . .yn)] = [u : b(u, [v1 . . . vnd : S],y1, . . . , yn) & S(y1, . . . , yn/v1. . .vn)]))}}].

The idea (near enough) is that a believes-truly predicate of xs Men-talese is one for which x accepts a Mentalese analogue of (D2), MRTJsdefinition of Trbel in terms of Bel. If we pretend that x thinks in re-

gimented English, we could put the idea by saying that is a Mentalesebelieves-truly predicate for x just in case x accepts every instance of(24):

(24) For any y1, . . . , yn: being a u such that (u, [1 . . . k : ],y1, . . . , yn) = being a u such that u has a belief about y1, . . . , ynunder [1 . . . k : ] where (y1, . . . , yn/1. . . k).

In particular (and continuing the inner English pretense), it follows thatif and are respectively Mentalese belief and true-belief predicates forx, then x accepts every instance of (25):

(25) For any y1, . . . , yn: being a u such that (u, [1. . . k : ],y1, . . . , yn) = being a u such that (u, [1 . . . k : ],y1, . . . , yn) where (y1, . . . , yn/1. . . k).

This is recorded in theorem scheme (T7), where t = i, t1, . . . ,tn, i, i,i, i, t, . . . ,tn:

(T7) (xi )(b)(v1).. .(vn)(d)(u)(y1).. .(yn)(S)(p){{TrueBeliefPredn(p, t

, x) & BeliefPredn(b, t, x) &

Vocab(vt11 , . . . v

tnn , d

i, ui , yt11 , . . . , ytnn , S; Mx )}

Accepts(x, (y1) . . . (yn)([u : p(u, [v1 . . . vnd : S],y1, . . . , yn)] = [u : b(u, [v1 . . . vnd : S], y1, . . . yn) &S(y1, . . . , yn/v1 . . . vn)]))}.

This completes the apparatus needed for the reduction of MRTJ to I+.Assuming (as we have) the plausibility of LOT in general and ofI in par-ticular, the status of this reduction as a vindication of MRTJ depends upon


26/34

206 STEVEN E. BOR

whether the reduced version of MRTJ satisfies the adequacy conditions(C1)(C5) laid down in Section 1. We shall now argue that these conditionshave indeed been satisfied.

5. VINDICATION AND THE ADEQUACY CONDITIONS

Since we have neither appealed to propositions nor made use of any un-defined predicates of a language that take nominalized sentences of thatlanguage as arguments, condition (C5) has clearly been met. Moreover,the definition schemes (D7)(D13) provide a precise ontological accountof the logical forms posited by MRTJ and reveal (via the systematicconnection with the single relation Acceptsi,i) what is structurallycommon to the members of the family of differently typed belief relationsin virtue of which they are belief relations; so (C2) and (C3) are satisfied.

What remains to be shown is that (C1) and (C4) are met as well.Let us begin with (C1). There are two issues here: viz., (a) whether

MRTJ, understood via the reduction to I+, can provide analyses of Abelieves that p and The belief that p is true/false for all grammat-ically admissible replacements of p; and (b) whether the profferedanalyses are plausible. The answer to question (a) is clearly affirmat-ive. Since logical forms themselves can be of arbitrary truth-functionaland/or quantificational complexity, and since according to MRTJ all ofa judgments structure derives from its logical form, it should be obvi-ous that MRTJ is not restricted merely to atomic judgments but canprovide analyses of belief ascriptions with content-clauses of any degreeof complexity.14 Question (b) is not so easily settled. Even assuming, as wehave, the acceptability of the general framework of LOT, we can only arguefrom representative examples in which the analyses of belief ascriptionsprovided by the reduced version of MRTJ can be seen to plausible.

Let us begin with (18), an ascription to Othello of the atomic beliefthat Desdemona loves Cassio. MRTJs analysis of (18) is (26):

(26) Beli,i,i,i,i,i,i,i,i,i, (Othello, [Fi,ixi yi di : F(x ,y )], Loves,Desdemona, Cassio).

In I+, (26) will be provably equivalent to (27):

(27) (b1)(b2)(b3){(NameType(b1, i,i,MOthello) &

NameType(b2, i,MOthello) & NameType(b3, i,MOthello)) &DesIn(b1,Loves,MOthello) & DesIn(b2,Desdemona,MOthello)& DesIn(b3, Cassio,MOthello)& Accepts(Othello, b1(b2, b3))}.


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According to I+, then, (18)/(26) obtains iff Othello is disposed as one whoinwardly and assertively tokens in his language of thought a relationalsentence whose binary predicate, first argument, and second argument

respectively designate loving, Desdemona, and Cassio.Consider next (28), which (suppressing the type-indices) is analysed byMRTJ as (29):

(28) A believes that either R(a1, . . . , an) or S(b1, . . . , bk).

(29) Bel(A;[F x1 . . . xnGy1 . . . yk d

i : F (x1, . . . , xn) G(y1, . . . , yk)];R, a1, . . . , an, S , b1, . . . , bk).

According to I+, (29) obtains just in case A accepts a Mentalese sentenceof the sort R(a1, . . . , an) S(b1, . . . , bk) in which (i) the Mentalese

names R and S respectively designate the relations R and S and (ii) theMentalese names a1, . . . , an, b1, . . . bk respectively designate the indi-viduals a1, . . . , an, b1, . . . , bk. No peculiar logical object is required tocorrespond to the disjunction sign itself, whether in English or in AsMentalese. The same results holds, mutatis mutandis, for belief ascriptionswith content-clauses involving other connectives, quantifiers, etc.

In our examples so far, the dummy variable in logical form specifica-tions has merely been along for the ride. However, when the content-clausein a belief ascription is wholly general, this otherwise inert element finallycomes into play. Thus, e.g., (30) is analyzed as (31):

(30) A believes that everything has properties.

(31) Beli,i,i,i,i (A; [di : (i)(Fi)F()]).

Now by (D7) the logical form [di : (i)(Fi)F()] is the relation

[xiyi zdi : x = [di : ()(F)F()] &(w){DesIn(w, x,My ) & [rsi u : (v)(F)(d)((VarType(d, i,Ms ) & VarType(v, i,Ms ) &VarType(F, i,Ms ) & r = [d : (v)(F)F(v)]) &u = r}]wyz].

So (31) is ultimately equivalent to (32):

(32) (v)(F){VarType(v, i,MOthello) & VarType(F, i,MOthello) &Accepts(Othello, (v)(F)F(v))}.


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208 STEVEN E. BOR

A wholly general belief is thus a binary relation between a thinker anda general logical form f, i.e., one which is a logical form for certainwholly general sentences of the thinkers Mentalese; and to be so related

to f is inwardly and assertively to token a sentence of that form (or at leastto be disposed as if one did so).The reduced version of MRTJ easily handles iterated belief ascriptions

like (33):

(33) Iago believes that Othello believes that Desdemona lovesCassio.

Where the two occurrences of believes in (33) are translated as predicatesBelt1 and Belt2 of appropriate (and, of course, distinct) types t1 and t2,we have the analysis (34) of (33):

(34) Belt1 (Iago, [Ht2 zi Gi,i,i,i,iJi,ixi yi di : H (z,G,J,x ,y )],

Belt2 , Othello, [Fi,ixi yi di : F(x,y)], Loves, Desdemona,Cassio).

Skipping the derivation and stating the result less formally, (34) obtainsjust in case (35) does:

(35) Iago [is disposed as if he] assertively tokens a certain sentencein MIago of the form b(o, f, l, d, c) in which (i) b, o, f, l, d,and c are names inMiago respectively designating Belt2 , Othello,[Fi,ixi yi di : F(x,y)], loving, Desdemona, Cassio; and (ii) fhas the form [v1v2v3d : v1(v2, v3) for distinct variables v1,v2, v3, and d of respective MIago-types i, i, i, i, and i.

By our definitions and axioms, Iagos belief is true (i.e., Iago truly be-lieves that Othello believes that Desdemona loves Cassio) iff, in additionto (35), (36) also obtains:

(36) [Ht2 zi Gi,i,i,i,iJi,ixi yi di : H (z,G,J,x ,y )](Belt2 ,Othello, [Fi,ixi yi di : F(x ,y )], Loves, Desdemona, Cassio).

But (36) is equivalent to (26), which we unpacked above. So, in the end,Iagos belief is true iff (35) obtains andOthello accepts in his language ofthought a relational sentence whose binary predicate, first argument, andsecond argument respectively designate loving, Desdemona, and Cassio.

Moreover, independently of any assumptions about what belief factsthere are, we are now entitled to assert (37):

(37) The beliefthat Othello believes that Desdemona loves Cassio istrue iff Othello believes that Desdemona loves Cassio.


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For MRTJs translation of (37) is (38):

(38) True([ui f : f =[Ht2 zi Gi,i,i,i,iJi,ixi yi di : H (z,G,J,x ,y )] & Belt1 (u, f,

Belt2 , Othello, [Fi,ixi yi di : F(x,y)], Loves, Desdemona,

Cassio)]) Belt2 (Othello, [Fi,ixi yi di : F(x,y)], Loves,Desdemona, Cassio).

And (38) follows from the corresponding instance of theorem scheme (T2)in which g is the logical form

[Ht2 zi Gi,i,i,i,iJi,ixi yi di : H (z,G,J,x ,y )]

and e1 e6 are respectively the relation Belt2 , Othello, the logical form[Fi,ixi yi di : F(x,y)], the relation Loves, and the individuals Desde-

mona and Cassio.By forcing Mentalese to incorporate the kind of multiple rela-

tion treatment of belief ascriptions found in MRTJ, our account of(33)/(34) requires Iagos Mentalese representation of the logical form[Fi,ixi yi di : F(x,y)] of Othellos belief to be of the canonical sort[v1v2v3d : v1(v2, v3)] This satisfies our original demand that Iago mustrepresent the logical form of Othellos alleged belief in some canonicalway that reveals (to Iago) the conditions under which Othellos beliefwould be true. For suppose that in Iagos Mentalese O, L, D, and C arerespectively names for, Othello, loving, Desdemona, and Cassio. Supposealso that Iagos Mentalese contains predicates B and T (both of Mentalesetype i, i,i,i,i,i,i,i,i, i,i,i,i) for believing and truly-believing re-spectively. Then it follows from (T7) that Iago accepts (39):

(39) (y1)(y2)(y3)([u : T(u, [v1v2v3d : v1(v2, v3)], y1, y2, y3)]= [u : B(u, [v1, v2, v3d : v1(v2, v3)], y1, y2, y3) &y1(y2, y3)]).

Although we cannot presume upon Iagos logical acumen, he is nonethe-less now in a position to reason validly from (39) to (40), and hence to(41):

(40) [u : T(u, [v1v2v3d : v1(v2, v3)], L, D, C)] =

[u : B(u, [v1v2v3d : v1(v2, v3)], L, D, C) & L(D, C)]

(41) T(O, [v1v2v3d : v1(v2, v3)], L, D, C)] B(O, [v1v2v3d : v1(v2, v3)], L, D, C) & L(D, C)]


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210 STEVEN E. BOR

By requiring Iago to employ a canonical representation of the sort[v1v2v3d : v1(v2, v3)] we ensure that he is primed to exploit theinferential connections between his three Mentalese sentences T(O,

[v1v2v3d : v1(v2, v3)], L, D, C), (B(O, [v1v2v3d : v1(v2, v3)], L, D,C) and L(D, C) hence to appreciate what he is attributing to Othello even if for some reason he fails to do so.

To handle alleged failures of Substitutivity of Identity in natural lan-guage arguments like (3), MRTJ appealed to the alternative formalizations(5)(7) of (3). I+ provides interpretations for (5)(7) that show how thisappeal sheds genuine light on the subject. The argument (5) is invalidbecause, despite the identity of the author of Waverly with the author of

Marmion, the fact that George IV accepts a Mentalese equation a =(x)fx for some name a of Scott and predicate f designating Waverly-authorship cannot guarantee that George IV accepts a Mentalese equationb = (x)gx for some for some name b of Scott and predicate g desig-

nating Marmion-authorship. For the distinctness of these two authorshipproperties requires the distinctness offand g. Similarly, the argument (7)is invalid because, despite the identity of the author of Waverly with theauthor ofMarmion, the fact that George IV accepts a Mentalese equationa = b for two names a and b of Scott/the author ofWaverly cannot guar-antee that George IV accepts a Mentalese equation c = (x)gx for somefor some name c of Scott and predicate g designating Marmion-authorship.For there is no guarantee that George IV has any such a predicate g in hisMentalese!

As it happens, the argument (6) will go through provided (as seemsplausible) that a full axiomatization ofI would yield (42) as a semantic

theorem scheme:(42) (yi )(Fi)(f)(x){(NameType(f, t, My ) & VarType(x, t,

My ) & DesIn(f, F, My)) (zt)(DesIn((x)f(x), z, My) z= (xt)F(x))}.

For if George IV accepts a Mentalese equation a = (x)fx for somename a of Scott and predicate f designating Waverly-authorship, then by(42) the ingredient name (x)fx will designate the author ofWaverly; sothe identity of the author of Waverly with the author of Marmion wouldguarantee that George IV accepts a Mentalese equation a = b for twonames a and b of the author ofMarmion.

Finally, we turn to (C4). Given our explanation of how the obtainingof a belief relation brings various constituents into relation with a logicalform, do we face either (a) a residual problem of accounting for the specificorder thereby imposed on these constituents or (b) a residual problem of


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accounting for the fact that certain possible orderings make sense butothers do not? Following Griffen (1985), we may call (a) the DirectionProblem and (b) the Nonsense Problem.

It should be clear that MRTJ, as reduced toI

+

, does not suffer from theDirection Problem. So long as believes is translated by an appropriatelytyped predicate, the ingredient type theory prevents any permutation ofarguments of unlike type. Moreover, the residual difference within a typebetween, say,

(43) Beli,i,i,i,i,i,i,i,i,i (Othello, [Fi,ixi yi di : F(x ,y )], Loves,Desdemona, Cassio)

and

(44) Beli,i,i,i,i,i,i,i,i,i (Othello, [Fi,ixi yi di : F(x ,y )], Loves,Cassio, Desdemona)

is reproduced at the level of Othellos Mentalese in terms of the differencebetween (i) a sentence a1(a2, a3) whose three terms respectively des-ignate the loving relation, Desdemona, and Cassio and (ii) an isomorphicsentence b1(b2, b3) whose three terms respectively designate the lovingrelation, Cassio, and Desdemona. Likewise, no problems are occasionedby differences in the alphabetical order of variables in the canonical namesof logical forms. Consider, e.g., the differences between (43), (45), and(46):

(45) Beli,i,i,i,i,i,i,i,i,i (Othello, [Fi,iyi xi di : F(x ,y )], Loves,

Desdemona, Cassio)

(46) Beli,i,i,i,i,i,i,i,i,i (Othello, [Fi,ixi yi di : F(y ,x )], Loves,Desdemona, Cassio)

In I+, (45) and (46) are both equivalent to (44). And of course both (47)and (48) are equivalent in I+ to (43):

(47) Beli,i,i,i,i,i,i,i,i,i (Othello, [Fi,iyi xi di : F(x ,y )], Loves,Cassio, Desdemona)

(48) Beli,i,i,i,i,i,i,i,i,i (Othello, [Fi,ixi yi di : F(y ,x )], Loves,

Cassio, Desdemona)

Nor does MRTJ, as reduced to I+, confront any Nonsense Problem. Forlogical forms and belief relations, as these are defined in I+, do provide


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212 STEVEN E. BOR

the requisite restrictions on meaningful combinations in the form of type-constraints on free variables in certain of the thinkers Mentalese sentencesinto which the Mentalese names of the various objects of the judgment are

to be inserted.

15

Provided the logical types are respected, every sentenceof the form

(49) Beli,t1,...,tn,i,i,i,i,t1,...,tn(i0, ([t11 . . .

tnn

i : ],

t11 , . . . ,

tnn )

receives via the reduction an interpretation that not only is intelligible butis so we have urged plausible as well.

In light of our results, it may be hoped that multiple relation theories ofjudgment will at least no longer be glibly dismissed as unworkable odditiesbut will instead be accorded the respect they deserve. For whatever theultimate fate of multiple relation theories may be, we have seen how to

elaborate at least one of them in a way that renders it just as viable as manyof the more standard theories that have paraded through the literaturesince Russells day.

ACKNOWLEDGEMENT

I am indebted to the referees for several valuable suggestions for improvingthe original manuscript.

NOTES

1 Although Russell tentatively proposed a multiple relation theory in (19061908), hisfirst publicly endorsed versions appear in (1910) and (1910/11). He revised the theory in(1912) and further modified it in his 1913 book manuscript (posthumously published asRussell 1992). Russell was still propounding a multiple relation theory in (1918/19), butofficially abandoned it in (1919).2 Our concentration on the final version of Russells theory is not meant to suggestthe impossibility of vindicating any of his earlier versions, in which the belief relationsthemselves are supposed to provide the pattern. These earlier versions have their ownattractions (and problems) and may well be independently defensible (see Jubien (to ap-pear)). The main reason for ignoring them here is that the technical apparatus employed inthe reduction of MRTJ to I+ requires isolating the structure-determining factors as discrete

elements. Doing so is trivial in MRTJ but not in a version that hides these factors insidethe belief relations unless, that is, the corresponding belief predicates are formalized insuch a way that the resulting theory becomes a mere notational variant on MRTJ. Insofaras these other versions are intended to be substantive alternatives to MRTJ, their defensewould require different strategies, but limitations of space preclude pursuing them here.


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3 In the spirit of Russells ideal language philosophy, talk of MRTJ analysing a nat-ural language sentence S via a formula should be understood as meaning that is thetranslation ofS into the (allegedly more perspicuous) language of MRTJ. No commitmentis undertaken here as to what role, if any, MRTJ might play in a theoretical account of

natural language syntax.4 Types themselves may be thought of as symbols that categorize both linguistic andextralinguistic items. So we could regard the letter i in the text as naming itself andthink of the set of types as being the smallest set T such that i T and t1, . . . , t n Tfor every t1, . . . , tn T (n 1).5 (A1), of course, entails the numerical distinctness of at least some necessarily coextens-ive relations.6 Since || will be an (n + 1)ary relation in which the (n + 1)th argument-place is adummy position, we have put in that slot just for definiteness. Alternatively, we couldexistentially quantify over it.7 Mentalese types, like the types ofL, may be thought of as symbols in this case,Mentalese symbols. This is important, since I needs to quantify univocally over thesesymbols in syntactic axioms like

(w)(x)(tx1 ){TermType(w, t1, Mx ) (!e(Expression(e, Mx ) & w = et1

& (d)(tx2 )(Expression(d, Mx ) & e = dt2 )},

which requires a term ofa given Mentalese type to be the result of indexing a unique (un-indexed) Mentalese expression with that very type. We exploit this treatment of Mentalesetypes in (D14) and (D15) of Section 4.8 For an interesting and persuasive account of tacit belief and being disposed as if ,see 2.3 of Crimmins (1992). Crimmins, however, is no fan of the language of thoughthypothesis.9 Of the principles listed, only this one runs afoul of the fact that limitations of memorywill put an upper bound on the complexity of Mentalese sentences that x can process,for S(a//b) may be more complex than S, where the latter is already at xs limit. To keepthe principle with its present consequent, an appropriate clause should be added to its

antecedent. Alternatively, we could weaken the consequent to Accepts(x, S(a//b).10 With its nesting of-abstracts, the formulation of the relation in question might seem tobe needlessly complex. There is a purely technical reason for this prolixity, but limitationsof space preclude explaining the details here.11 Once the salient definitions and axioms of I+ are in place, the derivation of MRTJsaxioms is trivial and hence will not be spelled out here.12 There is no circle here, for definition scheme (D7) appeals merely to the designation of[1, . . . k : ], identified by (A2) as

[1 . . . k : [1, . . . , k ; k+1, . . . , n]],

in order to define the logical form

[1 . . . k : [1, . . . , k ; k+1, . . . , n]],

which is then invoked by (A4) as designatum for [1 . . . k : ]. The status of

(A4) as a bridge principle would perhaps be clearer if its consequent had been writtenin biconditional form as

(x){DesIn([1. . . k : ], x, My ) x =[1. . . k : [1, . . . , k; k+1, . . . , n]]}.


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214 STEVEN E. BOR

13 As is customary, we pretend that the ingredient proper names designate real individuals.14 For ease of exposition, we have restricted attention to replacements of p whose logicalforms could be formalized in the base language L. But it should be clear that the addition,e.g., of tense and modal operators would be straightforward, so that L could be brought

ever closer to full-blown (albeit regimented) English.15 Of course, some sentences are called nonsensical for reasons that have nothing to dowith their type-theoretical well-formedness. Thus, e.g., if numbers are entities of type iand having a beard is a property of type i, type restrictions cannot rule out such Ryleancategory mistakes as The number 6 has a beard. Such residual oddities, however, areperhaps best viewed not as literal nonsense but merely as gross absurdities particularlyblatant necessary falsehoods.

REFERENCES

Crimmins, M.: 1992, Talk About Beliefs, Bradford/MIT Press, Cambridge, MA.

Fodor, J.: 1975, The Language of Thought, Thomas Y. Crowell Company, New York.Fodor, J.: 1987, Psychosemantics, MIT Press, Cambridge, MA.Griffin, N.: 1985, Russells Multiple Relation Theory of Judgment, Philosophical Studies

47, 213247.Jubien, M.: to appear, Propositions and the Objects of Thought, Philosophical Studies.Maloney, J. C.: 1989, The Mundane Matter of the Mental Language, Cambridge University

Press, Cambridge.Richard, M.: 1990, Propositional Attitudes: An Essay on Thoughts and How We Ascribe

Them, Cambridge University Press, Cambridge.Russell, B.: 19061908, The Nature of Truth, Proceedings of the Aristotelian Society 7,

2849.Russell, B.: 1910, On the Nature of Truth and Falsehood, in his Philosophical Essays,

Simon and Schuster, New York.Russell, B.: 1910/1911, Knowledge by Acquaintance and Knowledge by Description,

Proceedings of the Aristotelian Society 11, 108128.Russell, B.: 1912, The Problems of Philosophy, Oxford University Press, Oxford.Russell, B.: 1918/1919, The Philosophy of Logical Atomism, Monist28, 495527.Russell, B.: 1919, On Propositions: What They Are and How They Mean, Proceedings

of the Aristotelian Society, Supplementary Volume 2, pp. 143.Russell, B.: 1992, E. R. Eames and K. Blackwell (eds), Theory of Knowledge: The 1913

Manuscript, Routledge, New York.Russell, B. and A. N. Whitehead: 1910, Principia Mathematica, Vol. I, 2nd edn, Cambridge

University Press, Cambridge.Salmon, N.: 1986, Freges Puzzle, MIT Press, Cambridge, MA.Soames, S.: 1987a, Direct Reference, Propositional Attitudes and Semantic Content,

Philosophical Topics 15, 4487.Soames, S.: 1987b, Substitutivity, in J. J. Thomson (ed.), Essays in Honor of Richard

Cartwright, MIT Press, Cambridge, MA.

Department of PhilosophyThe Ohio State UniversityColumbus, Ohio 43210, USA

steven e. boËr

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