Dynamic Meinongian logic for descriptions
Minao Kukita
August 1, 2014

Abstract
The aim of this article is to present a formal language and a semantics
for it which reflect two ideas expressed but not formally implemented in
Graham Priest’s Towards Non-Being. One is that a description can beevaluated in diﬀerent ways depending on the speaker’s intention, so that
one and the same statement can be true on one evaluation and be false
on the other. The other idea is that two diﬀerent kinds of linguistic activities are involved in fiction, as well as in mathematics: characterisingan object and inferring about the object thereby characterised. The language defined here is designed so as to conform to the ideas as naturally
as possible. The resulting language and its semantics is Meinongian inthat it assign every term a denotation, but also Russellian in that it treat
existing objects as specially privileged.

1

Introduction

There are two radically opposing kinds of approaches to determining the meanings of those sentences which contain descriptions (description sentences hereafter): Meinongian and Russellian. In a Meinongian approach, a description istreated as a term which refers to an objects, just as a name does. However,since some descriptions, say “the present King of France,” seem not to refer toany existent object, it should be possible for descriptions (or terms in general)
to refer to non-existent objects. We will call such objects “non-beings.” Nonbeings are just another kind of genuine objects, capable of having a variety ofproperties though they lack the property of existence. On the other hand, in a1

Russellian approach, descriptions are not integral grammatical units in themselves and have no referents at all. Therefore we need not worry about whom
(what) “the present King of France” refers to. Thus, these two approaches standin a sharp contrast as for fundamental semantic principles.1
For Russell, all genuine individual objects are existent. For Meinong, therange of genuine objects includes abstract entities like numbers, fictional objects
like Sharlock Holmes, or even logically impossible objects like a circle that is
not round, in addition to concrete existent objects.
An obvious advantage of Meinongianians is that they can save the intuition
that a sentence like “Sharlock Holmes is a detective” is true. This is due tothe “Characterisation Principle (Postulate)” (CP), which Meinongians will in
general embrace. Meinongians think that fictional objects, non-existent objects
can be identified by characterisation. For example, Sharlock Holmes is identified by all the properties that Doyle attributed to Holmes in his novels. Suchdescriptions as “the present King of France” attributes the property of being
the present King of France to the object it refers to. So descriptions are alsocharacterisation. The Characterisation Principle states that(CP) Non-existent objects have those properties which they are characterised as having, and those which follows from them.
In virtue of this, we can attribute various properties to non-beings.
However, CP has serious diﬃculties as well. For example, consider the referent of the description “the circle that is not round.” It follows from CP thatthe object is round and not round. Since Russell pointed out this diﬃculty,
several attempts have been made to resolve it. Among them, Terence Parsons[7] has been perhaps most prominent. His idea was to discriminate those descriptions which refer from those which do not. For this purpose, he proposedseveral criteria for a description to refer. However such criteria cannot help
being arbitrary and highly artificial.
1 Yet another view concerning description sentences is Strawson’s, which treat any sen-tence containing an empty description as being neither true or false (Strawson [13]). Thoughseemingly quite opposing, Strawson’s and Russell’s views are similar in that they treat all thesentences containing empty descriptions as not false, no matter what the sentences say. So
we can apply our method to both of the theories. See §3.

2

In Towards Non-Being (TNB, hereafter), Priest builds a semantics based on
Meinongian theory of objects, which he calls noneism following Richard Routley(Routley [9]), and thereby resolves the diﬃculties of CP, as well as some puzzles
in traditional epistemic logics. His approach does not only resolve puzzles, butalso is a natural analysis of our practice of speaking about fictional objects ormathematical objects. Moreover the semantics has a merit of being able to treat
every intentional verbs in a single apparatus, instead of treating exclusively“know” or “believe.” On the other hand, TNB’s approach does not seem todeal well with typically Russellian cases like “The present King of France isbald.” Yet, Priest suggests informally how to deal with those cases. In shortthe idea is to shift the methods of evaluation of description sentences according
to the speaker’s intention. It is the aim of this paper to formally implement
this idea. At the same time, we also take into consideration another suggestion
by Prist that two diﬀerent kinds of activities are involved in fiction, as well as
mathematics: characterising an object and discussing the object characterised.

2

TNB’s approach to description sentences

In this section, we will see how descriptions are dealt with in TNB, how their
meanings are determined, and compare its merits and demerits with those of
the Russelian approach.

2.1

The world semantics

First we will review the semantic framework of TNB. 2 The semantics is avariation of possible world semantics with a constant domain, where there areimpossible and open worlds in addition to usual possible worlds. This is hence
called world semantics. Impossible worlds are those worlds where some of logical truth might not hold. Open worlds are those worlds which might not be
closed under logical consequences. Namely, an open world w are such that there
might be propositions A1 , A2 , . . . , An , B such that B is a logical consequence ofA1 , A2 , . . . , An , A1 , A2 , . . . , An are true in w, but B is not. The actual world @2 The following is a fairly simplified version of the semantics of TNB, omitting some aspects

irrelevant to the current issue.

3

is among the possible ones. We will denote possible, impossible and open worlds
by P, I, and O respectively. The worlds in P ∪ I are called closed worlds, written C. C ∪ O is written W. Given an assignment s of an object to each of thevariables, we write w ⊩s A to express that A is true at w under the assignments.3
The domain of the objects is common to all the worlds, denoted by D. D
contains both existent and non-existent objects in a world. There is no essential
distinction between these objects. “Exists” is no more or no less an ordinary
predicate than predicates like “is red” or “ is a human” are. Existent (nonexistent) objects may diﬀer from world to world. Holmes does not exist in ouractual world but he does in some other worlds. This is why we need not variabledomains.
One purpose of TNB is to provide a semantics for verbs expressing propositional attitudes like “believes that A.” This kind of verbs are called intentional
operators. Let a be a term, Ψ an intentional operator and A a formula. Thenthe formula aΨA stands for the sentence “a Ψs that A.” In the following, weneed not consider the subject a, and we will omit it and write “ΨA” and so on.
In TNB, intentional operators are treated as a kind of necessity modal operators. That is, relative to each intentional operator Ψ, there is an accessibility
relation RΨ such thatw ⊩s ΨA ⇐⇒ for all w′ ∈ W if wRΨ w′ then w′ ⊩s A.
The worlds accessible from w by RΨ are the ones where every proposition toward
which the subject has the attitude Ψ in w is true. For example, if Ψ stands forbelieving, the worlds accessible from w by RΨ are the worlds where whatever
the subject believes in w is true.
This is standard in traditionla epistemic logic.4 What is unique in TNB is theintroduction of impossible and closed worlds so as to block some of problematic
inferences such as Logical Omniscience etc.5 But we are not concerned withthem here.
3 s is needed because A may contain free variables.

When it is the case, its truth can not

be decided unless we have assigned an object to each variable as its denotation.
4 For standard semantics in epistemic logic, see, for example, Hintikka [3] or Meyer and vander Hoek [6].
5 A similar approach can be found in Hintikka [4].

4

2.2

The Meinongian account of descriptions

What is most characteristic in the Meinongian account of descriptions is that,
unlike the Russelian account, it regards a description as a well-formed term neverfailing to refer to an object in the domain. It is natural to ask the following
questions:
1. What do seemingly empty descriptions such as “the gold mountain” referto?
2. How is the Characterisation Principle satisfied?
3. Is it possible to avoid the paradox of CP?
Let’s see the answers to these questions in TNB.
As for the question 1. A description describes a certain condition and thereby
picks up an object which satisfies the condition. If it is intended that only one
object satisfies the condition, the description is called definite and otherwise
indefinite. In a formal language, the conditions is expressed in the form oflogical formula. The object denoted by a description is that which satisfies theformula. Therefore, a description can be constructed by designating a formulaand a variable to be satisfied. Following TNB notation, we will write ιxA (orεxA) for the definite (or indefinite) description determined by a formula A anda variable x. If we are indiﬀerent to the definite/indefinite distinction, we will
write ξxA instead. We will call A in ξxA the characterisation or characterisingproperty of the description.
TNB assigns an object to every description. The assigned object is called thedenotation of the description. If the characterisation is satisfiable in the actual
world, then it has as its denotation an object that satisfies the characterisationin the actual world. Therefore, each description ξxA has a selection function
ϕξxA : P(D) → D. Let S = {d ∈ D : @ ⊩s(x7→d) A}, where for any objects
d1 , d2 , . . . , dn ∈ D and any variablesx1 , x2 , . . . , xn , s(x1 7→ d1 , . . . , xn 7→ dn ) is
the assignment defined bys(x1 7→ d1 , . . . , xn 7→ dn )(v) =


 d

i

if v = xi (1 ≤ i ≤ n)

 s(v) otherwise,

where xi ̸= xj for every 0 ≤ i, j ≤ n such that i ̸= j. In other words, S is theset of objects that satisfy the characterisation in the actual world. When S is
5

not empty, it is postulated that ϕξxA (S) ∈ S. In this paper, we will call suchdescritpions realistic. When S is empty, ϕξxA (S) is not restricted, and can be
any term in D. we will call such descritpion idealistic.
A realistic description denotes an object that satisfies the characterisation in
the actual world, and the truth of a sentence containing the description generally
depends on actual properties of the denoted object. For example, the sentence“The present president of France is bald” is about the present president of France
in the actual world and may be true or false depending on the actual fact about
the object. On the other hand, although a idealistic description does denotes acertain object, it does not satisfy the characterisation in the actual world. It is
the objects which the speaker indends to satisfy the characterisation.
It is natural to question what kind of objects idealistic objects like the present
King of France are. Of course they are not physical objects. Nor are they mental,
for if so, it wiould be entirely impossible for two persons to talk about one andthe same non-being. But we can talk about the same Sharlock Holmes or Zeus
with others.6
As for the question 2. In relation to descriptions, the Characterisation Principle may be interpreted as stating that for any variable x and any formulaA|= A[x := ξxA],
where A[x := ξxA] stands for the result of substituting ξxA for free occurrencesof x in A. Of course this is not always the case. For example, let A be thepredicate “x is a gold mountain.” Then CP requires that something be a gold
mountain in this actual world.7 But there is nothing that is actually a gold
mountain.
Priest regards this as an inappropriate way to interpret CP concerning descriptions. To say something about a characterised object is related to an intentional propositional attitude of representing a certain situation. When someone
says “The gold mountain is a gold mountain” one is not stating it as a fact
about this world, but a fact about the situation (or the possible world) that one
6 See TNB chapter 5 for the identity of non-beings.
7 The logical consequence relation |= is defined as the truth preservation in the actual world

for every interpretation. Therefore |= A ⇐⇒ for any variable assignment s, @ ⊩s A holds inevery model.

6

is representing using the description “the gold mountain.” If we use Φ for theintentional operator “represents,” CP actually requires that for any x and A|= ΦA[x := ξxA].8
Interpreted this way, the constraint causes no trouble in the semantics of TNB
for this says only that A[x := ξxA] is true in every world where everything thatthe speaker represents is true.
This account applies to other kinds of characterisation than descriptions.
For example, Sharlock Holmes has as its characterisation the whole body of thedeclarative sentences in the Holmes series written by Doyle. By CP, Holmes satisfies the characterisation (being detective, living in the Baker Street etc.). Thisis, however, not the case in the actual world but in the world where everything
Doyle represented in the Holmes series holds true.
Thus TNB makes CP completely general without any exception, by relativising it to the worlds connected to the actual one by the intentional operator“represents.”
As for the question 3. If CP holds true without exceptions, it appears thatthe paradoxes mentioned above are inevitable. It indeed is, in a sense. Take thedescription “a circle that is not a circle” as an example. Let P x be a predicate“x is a circle.” Then “a circle that is not a circle” is written εx(P x ∧ ¬P x). Letτ be the description. By CP |= Φ(P x ∧ ¬P x)[x := τ ], namely |= Φ(P τ ∧ ¬P τ ).
This means that in any interpretation, @ ⊩ Φ(P τ ∧ ¬P τ ) where @ is the actual
world and Φ is the intentional operator for “represents.” Then for any worldw ∈ W, if @RΦ w then w ⊩ P τ ∧ ¬P τ . Therefore, both P τ and ¬P τ holdtrue in any world that is Φ-accessible from the actual world @. But it onlymeans that such worlds are impossible worlds (therefore not the actuall worldin particular). It does not matter if any contradiction arises in an impossible
world.

2.3

Types of descriptions

However, this analysis does not seem to apply to every description sentence.
Consider, for example, the sentence “The planet which is nearer to the sun
8 Recall we decided to forget about who has that propositional attitude.

7

Speaker’s intention

Referent(1)

To represent the reality

Existent(2)

To represent the reality

Non-existent(3)

To represent a fictional situation

Existent(4)

To represent a fictional situation

Non-existent

Table 1: Four types of descriptions

than Mercury causes its perihelion shift” uttered by an astronomer. Call thesentence S. According to the above approach, S is interpreted as speaking about
the worlds represented by the astronomer in which there is a planet nearer tothe sun than Mercury. If the worlds are represented as obeying ordinary lawsof physics, the planet will have an eﬀect on the orbit of Mercury in any of thoseworlds. So S will be true in such worlds. Is this, however, correct analysis of theutterance? The astronomer would not have represented the planet as something
existing in his imagination but as something existing in this actual world. So
the sentence should not interpreted as stating something about the worlds inwhich his representation is realized, but stating something about this actual
world. In this case, the description sentence should not be relativised to therepresented (non-actual) worlds.
Another kind of cases to which the above approach does not apply are theones where the speaker uses description intending to represent a fictinal object
but there exists an object satisfying the characterisation. For example, supposethat someone uses the descriptions “the present emperor of Japan” not knowingthat there is one in Japan. According to TNB, “the present emperor of Japan”
has as its referent the actual present emperor of Japan. On the other hand, theobject intended by the description has properties such as being imagined by the
speaker, which the actual emperor does not have. This seems problematic.
When we use descriptions, there seem to be two kinds of cases: on the one
hand, the speaker intends to represent some fictional situation, while, on theother hand, the speaker intends to refer to some actual object. In addition,
descriptions may or may not have their referents in the actual world. Then wecan classify the cases where a description is used as shown in the table 1.

8

Russellian

Meinongian

(1)

✓

✓

(2)

✓

×

(3)

×

×

(4)

×

✓

Table 2: Russell versus Meinong

Russellian approaches intend to deal well with the cases (1) and (2), and notwith (3) and (4). On the other hand, Meinongian approaches are supposed todeal with the cases (1) and (4), and not with (2) and (3), as shown in the table
2.
Priest does not give a detailed account for the case (2), but says that it
is possible that we use a description which no existent object fits, intending
to apply it to the actual world. He suggested that we have diﬀerent kindsof representing. Namely, representing something as fictional, and representing
something as real. In the latter case, the associated accessibility relation will
be reflexive, i.e., relate the the actual world to itself. Then @ ⊩ ΦA implies@ ⊩ A. Therefore, @ ̸⊩ A implies @ ̸⊩ ΦA.
As for the case (3), Priest, metioning the possibility that a certain existentobject may satify by chance characterising conditions of a fictional character,and hence the writer of the fiction may be talking about the object contrary to
his intention, says as follows:
But when I tell a work of fiction, I deliberately intend to exclude
this possibility. Thus, we should perhaps rule out this world as a
candidate for satisfying it. It therefore makes sense to suppose thatthe appropriate intentional state involved in representing things inthis case, Φ′ , is diﬀerent from that, Φ, in which I intend the stroy tobe veridical. We may take aΦ′ A to be something like: ‘a represents
A as holding non-actually (in the matter at hand)’. (Priest [8] p.
124)
In contrast to the case (2), this time the associated accessibility relation should
be modified so that it will not relate the actual world to itself. Moreover, the9referent of the description should be diﬀerent from the usual one. This is because
the imagined object may have properties which any actual object does not, saybeing imagined by the author at such and such a moment and so on.
What this means is that (syntactically) the same description can refer todiﬀerent objects depending on the speaker’s intention or the situation in which it
is used. This is intuitively obvious. If someone is talking about Dr. Strangelove
and says “the president of the United States,” then he refers to the faint-hearted
man with eye glasses in the film whose part Peter Sellers played, and not to
Barack Obama.
In TNB, diﬀerent ways in which the speaker represents the object may bereflected by diﬀerent models. Here we will provide an alternative approach toaddress the problem of the speaker’s intention.

3

Dynamic Meinongian logic

In TNB, Priest makes another important point, though he does not formally
implement it. He thinks that our activities concerning fiction and mathematics
are very much alike, and that for both of them, there are two distinct kinds ofactivities: ‘specifying a characterization’ on the one hand, and ‘figuring out what
follows from it’ on the other (Priest [8] p. 148). Yet when thinking about fiction,
one naturally thinks of the former, while when thinking about mathematics,
one thinks of the latter. And this explains why fiction and mathematics arerecognized as totally diﬀerent kinds of activities.
The former activity, i.e., the activity of specifying a characterisation, is not
explicitly dealt with in Priest [8] except for the characterisation by descriptions.
Here we try to define the notion of characterisation in general, in the way that it
will cover descriptions as well. Then we make up a semantics for characterisation
which reflects the intuition (pointed out in the previous section) that the wayof evaluating a description (or a name of a characterised object) can diﬀer
according to the speaker’s intention.
In the following, V stands for the set of variables, C for that of constants, S
for that of formulas in the first-order language. In addition, we write S∗ for theset of all the finite sequence on S. For simplicity, we consider a language without

10

function symbols. For A ∈ S, F V (A) is the set of free variables occuring in A.
We fix a interpretation I for this language.
We extend the language with following expressions.
Definition 3.1 (Characterising declaration; discourse). For any formula A andvariable x, ⟨A, x⟩ is called a characterising declaration. Beside usual well-formed
formulas, CA is also a wwf if C is a characterising declaration and A is a wwf.
Let C1 , C2 , . . . , Cn be characterising declarations and A be a formula. Then wecall C1 C2 . . . Cn A a discourse.
A characterising decration is not evaluated on its own. It functions by affecting the evaluation of the formula that comes after it in a discourse. In thissense it looks like a propositional operator such as □.
We shall give a semantics for this language as follows. We denote by D thedomain of the objects.
Definition 3.2 (Characterising context). A function from V to S∗ is called acharacterising context. Given a characterising context χ : V → S∗ , a variable
x and a sequence of formulas s, χ(x7→s) stands for the characterising context
defined by
χ(x7→s)(y) = sif x = y χ(y) otherwise.
Definition 3.3 (denotations of variables). For any variable x, ϕx : P (D) →D ∪ {⊥} is a function such that ϕx (M ) ∈ M if M is not empty, and ϕx (∅) = ⊥,where ⊥ ∈/ D.
AFor any formula A and assignment s : V → D, define σs,I
: V → P(D) by
Aσs,I(x) = {d ∈ D : I |=s(x7→d) A}(x ∈ V).

χ
Furthermore, for a given characterising context χ, define σs,I
: V → P(D) by
χ
σs,I
(x) =

∩

σsA (x)(x ∈ V).

A∈χ(x)

Then a denoting function [[·]]χs,I : V → D ∪ {⊥} is defined by[[x]]χs,I = ϕx (σsχ (x))
11(x ∈ V).

We call [[x]]χs,I the denotation of x (relative to the interpretation I, the assignment s and the characterising context χ). The subscript I may be omitted if
no confusion arises.
We extend the evaluation of formulas so as to deal with the cases in whichsome of the variables have as their value ⊥ ∈/ D. Let s be a function from V toD ∪ {⊥}. For any atomic formula P x1 . . . xn , if s(xi ) = ⊥ for some 1 ≤ i ≤ n,then Is (P x1 . . . xn ) = ⊥. Otherwise, Is (P x1 . . . xn ) is either true or false as
usual. For a complex formula A, if Is (B) = ⊥ for a subformula B of A, thenIs (A) = ⊥. Otherwise Is (A) is true or false as usual. We can see Is as a partial
function from S to {true, f alse}, as opposed to a total function that Is usuallyis.
Definition 3.4 (local logics). Any subset ⊢ of S∗ × S∗ is called a local logic onS.
A local logic is a logic in some particular piece of fiction or a theory. It may
seem natural to impose some constraints on local logics, such as reflexivity or
monotonicity. You can do so and in many cases you should. However, we canthink of a story where monotonicity, say, does not hold. Therefore we leave
local logics without any constraint.
Definition 3.5 (evaluation, Meinongian and Russellian). Given a local logic ⊢and an assignment s : V → D of variables. Let x ⊆fin V. Then an evaluation⊩sx is defined for any characterising context χ and formula A by ∪x∈x χ(x) ⊢ A if x ̸= ∅sχ ⊩x A ⇐⇒ I |= χ Aotherwise.
[[·]]sAn evaluation is called Meinongian if x ̸= ∅, and Russellian otherwise.
Intuitively, what χ ⊩sx A states is that, when we regard the set of variables x
as names for Meinongian objects and/or “objectives,” we can infer A from theircharacterising properties according to the local logic ⊢. If x is empty, it says thatthe denotations of the free variables in A satisfy A under the interpretation I.
Thus, in the latter case, the variables are dealt with like Russelian descriptions.
When any of the variables has ⊥ as its denotation, A is false, which conforms to
the Russellian treatment of description of which the denotation does not exist.
12Note that we can treat (definite) descriptions and fictional proper names inthe same manner. They are both variables associated with a particular set of
formulas. Therefore “Holmes” and “Pegasus” are not constants. We reserve
constatns as names for existent objects. Therefore, we cannot characterise thethings that exist.
We have not yet given an account on how characterisation will be carriedout. An act of characterisation will be carried out by using a characterisingdeclaration. A characterising declaration ⟨A, x⟩ has the eﬀect of altering a given
characterising context χ. To be more specific, ⟨A, x⟩ will add A to the set ofcharacterising properties of x. Formally, this is defined as follows.
Definition 3.6 (dynamic evaluation). Consider characterising declarations Ci =⟨Ai , xi ⟩ (1 ≤ i ≤ n) and a characterising context χ. For a variable assignments and a finite set of variables x, the dynamic evaluation χ ⊩∗s
x C1 C2 . . . Cn A ofC1 C2 . . . Cn A is defined as follows:s(1) if n = 0, χ ⊩∗s
x A ⇐⇒ χ ⊩x A, and′∗s(x1 7→[[x1 ]]χs )′(2) if n ≥ 1, χ ⊩∗s
x C1 C2 . . . Cn A ⇐⇒ χ ⊩x

C2 . . . Cn A,′

where χ = χ(x1 7→⟨A1 ,χ(x1 )⟩) ．
As mentioned above, Priest distinguishes two kinds of linguistic activities
concerning fiction and mathematics, namely, the activity of designating thecharacterisation and that of explicating what follows from the characterisation.
Each Ci in the discourse C1 C2 . . . Cn A corresponds to the act of designating
the characterisation, while A to the conclusion from the characterisation. As
to fiction, the former corresponds to the creation of a work of fiction, while thelatter to the discussion of the fiction (particularly, argumentation about what is
entailed from it). As to mathematics, the former corresponds to postulation of
axioms, while the latter to statement of theorems that follow from the axioms.
This approach can be applied not only to fiction and mathematics, but also
to cases where one uses a description intending to represent the reality. For
example, some 19th century astronomers thought that there was a planet nearer
to the sun than Mercury, and it causes its perihelion shift. They often called
this imagined planet “Vulcan.” They did not make up the story as a piece offiction. They intended the name to refer to some existing object. Therefore the13statement “Vulcan causes Mercury’s perihelion shift” is false. This intuition isreflected in our system as follows:
Let C1 = ⟨x is a planet, x⟩,
C2 = ⟨x is nearer to the sun than Mercury, x⟩,
A = x causes Mercury’s perihelion shift, andχ(z) = ∅, for any variable z.
Then the Russellian evaluation χ ⊩s∅ C1 C2 A is false, while the Meinongianevaluation χ ⊩sx C1 C2 A is true under a proper local logic ⊢.
Another kind of cases to consider are the ones in which the speaker intends
to represent a fictional situation but the characterisation happens to hold true ofsome existent object. Then the object will have unintended properties, and thesemantics of TNB allows us to infer such properties about that object. On theother hand, in our system you can signify your intention by indicating variables
that are meant to refer to Meinongian objects. Then you cannot infer anythingother than those that can be deduced from the characterising properties. Thissituation might seem strange, but it is in fact common when we turn our eye tothe practice of mathematics, as we will see in §3.1

3.1

Consideration

Two opposing intuitions concerning description sentences led to two quite different semantic accounts of them, namely, Meinongian and Russellian. The
diﬀerence lies in whether we regard a description as refering to something (notnecessarily existent) or being merely quantificational and not referential.9 Our
everyday discourse involves instances which lend support to either of the two
intuitions, and so, over a hundred years, researchers have tended to appeal tosome of them and expressed preference for one theory over the other. However,neither of them alone seems to be able to deal with all the instances where
9 In this sense, Strawson’s view is similar to the Meinongian view. Meanwhile, he thoughtthat the cases where the descriptions fail to refer are anomalies, and the sentences containing
such descriptions cannot be true. In this respect, Strawson is more similar to Russell, whothought that in such cases the sentences are always false, than to Meinong, who thought thatsuch sentences can be either true or false.

14

descriptions are used. If so, all we can say is that there are some cases where
Russell scores over Meinong, and there are other cases where Meinong beats
Russell, as shown in the table 2. Therefore, here we did not try to judge which
theory is better and instead build a system which accomodate these two opposite
way to evaluate description sentences.
One merit of our system is that it reflects two distinct kind of linguisitic activities involved in fiction as well as mathematics. Namely characterising an object(writing a work of fiction or postulating axioms), and drawing conclusions aboutfrom the characterisation (making a guess about fictional situations, or deducing theorems from a set of axioms). In a discourse C1 C2 . . . Cn A, C1 C2 . . . Cncorresponds to the former, and A to the latter.
This analysis will be of some interest for the philosophers of mathematics.
We observed in the previous section that we may use the Meinongian evaluation
for descriptions for which there are existent objects that satisfy their characterising properties. In such a case, we refrain from ascribing any property thatis not deduced from the characterising properties. This is what happens whena mathematician make a new axiomatic system abstracted from some existenttheories. For example, when Dedekind came up with the theory of lattice, he
has in mind a lattice of submodules of a module over a ring. Such a lattice has
more properties than are deduced from the lattice axioms.10 Such generalization is important because it allows us to convey any result obtained from theaxioms to wider range of mathematical structures such as Boolean algebras andso on.11
Our system has another merit of not allowing arbitrary conclusions to be
derived from axioms which no mathematical structrures can satisfy. For even forsuch axiomatic systems, we have a Meinongian object which satisfy the axioms,
and so we do not have to be worried about vacuity, which has concerned manymathematical structuralists.12

10 Among others, it has the property of being modular.
11 For the history of lattice theory, see Schlimm [10].
12 Cf. Hellman [2].

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4Conclusion

In this article, we presented a formal system which reflects the two intuition
expressed in Priest [8]: One is that a description can be evaluated in diﬀerent
ways depending on the speaker’s intention. The other idea is that two diﬀerent
kinds of linguistic activities are involved in fiction and in mathematics.
In our everyday discourse we can find instances of description sentences
which favour or challenge either of the two method of evaluation: Meinongianand Russellian. We saw that in the resulting system, we can carry out bothRusselian and Meinongian evaluations of description sentence, so that we canmake up for the defects of these methods. Furthermore, our system is provide
a plausible model for mathematical activities.

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[8] G. Priest. Towards Non-Being: The Logic and Metaphysics of Intentionality. Oxford University Press, Oxford, 2005. 邦訳『存在しないものに向かって』，久木田水生，藤川直也訳，解説．勁草書房，2011 年．.

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