Rigidity of Predicates as Rigidity of Descriptions
(A Proposal Framed within a Hyperintensional Semantics)
Jiří Raclavský
Abstract: Unlike rigidity of singular terms, there seems to be no consensus as regards rigidity of predicates
(general terms). The aim of this paper is to establish the rigid / non-rigid distinction on exact concepts, viz.
the concept of reference as framed within a certain (hyperintensional) semantic theory. Consequently,
a rigorous definition of a rigid designator of an individual (which is applicable to proper names and individual
descriptions) is possible. The definition is straightforwardly adaptable to the definition of a rigid designator of
a class of individuals (applicable to common predicates), etc. Thus not only generality, but also inner unity of
the theory of rigidity can be achieved. Another important feature of the present proposal is its stress on
modality, since the modal stability of reference is the exclusive aspect leading to the division of expressions
into rigid and non-rigid.
Keywords: rigidity; descriptions; general terms; reference.
1. Introduction
The notion of rigid designator introduced by Saul A. Kripke in his seminal lectures Naming
and Necessity (cf. Kripke 1980) is one of the most discussed notions of the recent philosophy
of language. As a rigid designator it is considered a singular term (a proper name or
individual description) which refers to one and the same individual under all
circumstances, i.e. in all possible worlds.1 Proper names such as ‘X’ are rigid designators; their
reference is modally stable and unique. Common descriptions such as ‘the president of
U.S.A.’ are non-rigid designators because their reference is modally variable.2 Descriptions
such as ‘the only individual identical with X’ are rigid, their reference is stable. The notion
of rigid designator thus apparently interconnects reference of expressions with modality.
1
The definition is usually supplemented by the condition ‘in all worlds in which the individual exist’. I
am not going to examine this additional condition (which seems to me questionable) and I will ignore it
throughout the whole paper.
2
Expressions which do not refer to anything, non-designators, will not be investigated in this paper.
1
Recently, various theoreticians applied the rigid / non-rigid distinction even to
general terms (I will usually utilize the neutral term predicates).3 A bit surprisingly, no
consensus has been achieved – although there seems to be consensus concerning the
rigidity of singular terms. There are roughly four basic standpoints (the list is not
exhaustive):
a. natural kind terms are rigid, while artificial kind terms are non-rigid;
b. all general terms are rigid;
c. not only general terms, (but) all other designators are rigid, too;
d. application of the notion of rigidity to general terms does not have a good sense.
It is a disputable matter whether Kripke is a direct exponent of the position b. when
he repeatedly indicated that, for instance, ‘(is a) tiger’ is a rigid designator (cf. Kripke 1980).4
Kripke mentions no predicates which are now often thought to be non-rigid designators.
Modally conditioned changes of extensions of artificial kind terms, e.g. ‘(is a) pencil’, are
usually used to demonstrate that these predicates are, unlike natural kind terms such as ‘(is
a) tiger’, non-rigid. Nathan Salmon (cf. his 1982) was perhaps the first theoretician to
consider artificial terms as non-rigid and he still holds this position (e.g. 2003, 2005). The
view a. is supported also by Michael Devitt (2005: 146) and many others. Although the
extension of a predicate such as ‘(is a) tiger’ changes depending on circumstances, it is often
said that ‘(is a) tiger’ still signifies the kind (property) “tiger”.5
3
Literature on rigidity of general terms, especially the natural kind ones, is vast. One should recall here
at least the seminal contribution by Hillary Putnam 1975 or the recent book Beebee, Sabbarton-Leary 2010.
4
When recording a predicate, ‘(is a) …’ is used to distinguish the predicate from the corresponding
‘kind-description’. For instance, the description ‘X’s most favourite colour’ picks out the property (say) “pink”,
while the predicate ‘(is an) X’s most favourite colour’ picks out the singleton containing “pink”. (Martí and
Martinéz-Fernandéz 2010 do not distinguish predicate from the corresponding kind-description and that
seems to press them to puzzles related to possible predicative occurrences of kind-descriptions.)
5
Theoreticians differ not only in terminology, but mainly in conceiving things. Many say that
a predicate denotes a kind. Some say that a predicate expresses a (structured) property but it designates
a kind. Some other theoreticians, intensional logicians among them, say that a predicate means a property. It
is also difficult to decide what entities kinds, as perhaps distinct from properties, are (properties can be aptly
modelled as certain function from possible worlds). The present author inclines rather to the (Churchian)
view that a predicate expresses a (structured) concept of a property, whereas the property is the denotatum of
the predicate (cf. below for details). Anyway, the rather metaphysical notion of kind will not be employed.
2
It has been sometimes maintained that the difference between natural and artificial
kind terms is a bit debatable matter.6 For ‘(is a) pencil’ surely signifies the property “pencil”
just as ‘(is a) tiger’ signifies the property “tiger”. When questioning their rigidity, Stephen
P. Schwartz then writes that an application of the distinction rigid / non-rigid to general
terms does not have a good sense (Schwartz 2002: 275) and Scott Soames writes that the
distinction should be restricted to singular terms only (Soames 2002: 263).
But even a stronger challenge to the notion of rigidity than the view d. has already
been advocated. For instance, David Kaplan writes, ‘almost all single words other than
particles seem to me to be rigid designators’ (Kaplan 1973: 518n31). Thus not only proper
names and natural kind terms, but also artificial terms and all one-word descriptions would
be rigid designators. One can find even a bit more radical attitude, namely that of Pavel
Tichý who argued that entirely all designators are rigid (Tichý 1994: 30, 1986: 255). Without
mentioning Kaplan or Tichý, the view like c. is often condemned as trivializing the notion of
rigid designator (cf., e.g., Salmon 2003: 480-481).
Discrepancies and ambiguities as regards applicability of the rigid / non-rigid
distinction give rise to the hypothesis that the distinction is not based on clear concepts. To
avoid this, the present paper aims to treat the distinction with help of exact concepts and
criteria. Moreover, the distinction is intended to be generally applicable to whatever kind of
expressions (proper names, descriptions, predicates, sentences). It should be applicable in
a non-trivial way, unequivocally classifying expressions into rigid or non-rigid ones.
The key idea of this paper is rooted in the fact that rigidity is a semantic property of
expressions because it is best defined in terms of reference of expressions, which is
a semantic matter. In order to put the rigid / non-rigid distinction on rigorous basis it is
thus inevitable to enlighten what the semantics of expressions is, i.e. how one explicates
their meaning, denotation, and reference. Of course, a certain well-elaborated semantic
theory has to be adopted and I am going to take an advantage of Pavel Tichý’s
hyperintensional semantic theory (Tichý 1988, 2004).7 The intuitive notion of reference will
be then identified with a certain exactly explicated concept of this theory.
6
According to a common view, natural kind terms enjoy a specific status among general terms.
A persuasive critical discussion of this view can be found in (Wikforss 2010). In the present paper, no special
attention to natural kind terms is made; they are treated semantically on a par with other general terms.
7
There are several reasons for this choice (cf. below).
3
Here is the plan of the present paper. I begin with a semantics of singular terms,
disclosing thus the semantic theory I use. The explication of semantic notions follows. I
utilize the explication for an exact definition of the concept of rigid designator applicable to
singular terms. Then I insert a short discussion of Tichý’s reasons for the thesis that all
designators are rigid which stems from certain Kripke’s claims. I proceed to the semantics
of general terms and then formulate exact definition of the concept of rigid designator
applicable to general terms.
2. Reference, denotation and meaning of singular terms
On the natural view, the referent of the proper name ‘X’ is the individual X which is
named by the expression ‘X’. The semantics of individual descriptions is, of course,
different. In sentences such as ‘D is an F’ (where ‘F’ is a predicate applicable to individuals),
the description ‘D’ serves as a pointer at the individual, say X, which actually fits the
description. Many theoreticians rightly claim that the verification of the sentence consists
in taking X, described by the description, and checking whether it is an F. Some authors
then seem to maintain that the meaning of ‘D’ is just the individual X.
The last statement is attractive but questionable.8 For the semantics of the description
‘the president of U.S.A.’ cannot be identified with the semantics of the proper name ‘Barack
Obama’; in other words, Obama cannot be the meaning of the description. Had
circumstances (possible worlds) been different, quite a distinct individual, say Bill Clinton,
would fit the description. The modal variability shows in which the semantics of
descriptions differs from the semantics of proper names.
It is then not difficult to explicate the semantics of descriptions in terms of functions.
Since a description points at one individual on certain circumstances, yet it points at
another individual on distinct circumstances, it seems plausible to construe the meaning of
the description as a function which associates circumstances (arguments of the function)
with individuals (the values of the function). Thus the meaning of the description is
8
Note that it does not follow from the view how to verify sentences such as ‘D is an F’.
4
explicated as an intension which is a function from possible worlds (and, I will assume with
Tichý, moments of time) to individuals.9
The just sketched idea is the core of intensional semantics (celebrated mainly by
Montagovians) which has been demonstrated to be fruitful also for explication of meaning
of many other expressions. For instance, a sentence signifies not one of the truth-values
(for surely there are more sentential meanings than just two, viz. T and F), but certain
proposition, i.e. an intension having truth-values as its values. A common monadic predicate
signifies a property of individuals, i.e. an intension having classes of individuals as its values.
Etc.
Before I raise certain doubts about such intensional semantics, I am going to expose
some other reasons for its adoption. These are prompted by certain intensional transitives.
Consider for example the sentence ‘X seeks the Fountain of Youth’. If affirmed, the sentence
does not inform us that X is related to certain individual. Since on the contrary assumption
− namely that X stands in a relation to certain individual − one is entitled to carry out an
existential import leading to the sentence according to which there exists an individual
which is the Fountain of Youth (viz. ‘The Fountain of Youth exists’). Yet the input sentence
can be true, while the output sentence would not, thus the inference is intuitively invalid.
Intensional semantics correctly dismisses the view that X is related to an individual and so
treats the just discussed existential import as unwarranted.
For another example consider the sentence ‘X contemplates the Fountain of Youth’;
moreover, suppose that it is true. The purpose of a description (as occurring in many
sentences) is to pick out an individual which actually fits the description. Nevertheless, this
is not the case in the sentence we just discuss: ‘the Fountain of Youth’ does not serve here
for singling out a definite individual.10 If the sentence is true, the object of X’s attitude
cannot be an individual but an object O, which is explicated by intensional semantics as an
intension.
In Tichý’s semantics (going back to late 1960s, cf. Tichý 2004), the individual at which
a description contingently points at is called the referent of the description in a given
9
Tichý called such intensions individual offices (or roles). They are nearly the same entities as individual
concepts considered, e.g., by R. Stalnaker. An extensive defence of the notion of individual office can be found
in Tichý 2004.
10
I do not deny that there is a reading of the sentence according to which X is related to a certain
individual, i.e. that the purpose of the description is to refer to that individual. On this reading, however, the
sentence could not be actually true. Nevertheless, we assume just the opposite.
5
possible world W and moment of time T. The intension having the individuals as its values is
called the denotatum of the description.11 The referent (in W at T) of a proper name is
identified with its denotatum. (Denotation and reference of other kinds of expressions are
explained quite analogous way. For example, some predicates denote − as well as refer to −
just classes of objects, while many common predicates denote intensions having classes of
objects as values.)
It has been already found that intensional semantics is incapable to explain
hyperintensional contexts such as those involved in ‘X believes that 2+3=5’. Intensional
semantics cannot figure out why there are differences in meaning of sentences which are
equivalent due to logical or mathematical transformations. In other words, intensional
semantics renders the intuitively invalid inference from ‘X believes that 2+3=5’ and ‘5=√25’
to ‘X believes that 2+3=√25’ as valid. Now having ‘X calculates two plus three’ as an example,
it cannot be explained to which object an agent X which calculates 2+3 is related. It cannot
be the number 5, which is the result of that calculation; for obvious reasons, it cannot be the
English expression ‘two plus three’.
Moreover, intensional semantics leaves unclear what binds functions and objects
denoted by sub-expressions of an expression into a complex, structured whole which is
characteristic of the (intuitive) expression’s meaning. In other words, not only the meaning
of the sentence ‘X is an F’ is not a truth-value, a possible-world proposition is its meaning
neither. For a proposition is only a class of worlds (world-time couples), thus it embodies no
trace of the individual and the property denoted by the sub-expressions of ‘X is an F’. On the
natural view, however, the meaning of the sentence is a structured complex which
combines, in a certain unique way, X and F. Hence the genuine semantics of natural
language expressions is to be a hyperintensional one.
The hyperintensional semantics I adopt is the well-elaborated semantic theory of Pavel
Tichý, see esp. his 2004, 1988. According to his theory, the meaning of an expression is socalled construction (for exact specification and defence of constructions see Tichý 1988).
Constructions are extra-linguistic abstract entities which are structured (the way no settheoretical object is); they have algorithmic nature. Constructions construct objects. An
11
Unlike Montague and many other intensional semanticists, Tichý maintained that an intension is the
denotatum of description not only in ‘opaque’ but also in ‘transparent’ contexts (this is why Tichý called his
semantic system transparent intensional logic). This means that Tichý sustains our intuitive idea that the
meaning of the description is stable, insensitive to those contexts.
6
intension (or a non-intension) is constructible by infinitely many equivalent – yet
non-identical – constructions.12 In other words, hyperintensional semantics provides more
fine-grained individuations of meanings than intensional semantics.
In Tichý’s semantics the meaning of an individual description ‘D’ is explicated as
a construction of the intension denoted by ‘D’. The complexity of the construction
corresponds to the syntactical complexity of ‘D’. The intension constructed by the
construction has individuals as its values; the individuals are the referents of ‘D’ in the
respective worlds and times.13 Proper names express (mean) simple constructions of
individuals. The simplicity of these constructions corresponds to the logical
unstructuredness of proper names.
3. Language and semantic notions
As mentioned above, the term ‘rigid designator’ is a semantic predicate. It is beyond
doubt that an expression can be a rigid designator in one language while it can have quite
distinct meaning (if any) in another language, being then a non-rigid designator (or even
a non-designator). All semantic predicates and semantic concepts are to be relativized to
language. To define the meaning of such predicate, i.e. to provide a definition of the
respective semantic concept, it is inevitable to elucidate how language is modelled.
Unfortunately, there is no consensus as regards the explication of language, no
particular explication suggested up to now has been acknowledged as the genuine one.
Some theoreticians say that language is a social and perhaps also a normative device
enabling us to communicate. However, it is not exactly clear what kind of entity
a normative object of this sort can be. Many logicians and philosophers of language rather
assume that language can be modelled as a class of expression-meaning pairs. In other
12
Russell’s propositional functions are akin to Tichý’s constructions. For example, they both are
stratified in orders due to the Vicious Circle Principle. However, the crucial difference between them consists
in that Tichý’s constructions construct objects (e.g., functions-mappings), while Russell’s functions do not
(moreover, Russell appreciated no functions-mappings in his ‘no-class theory’ framework). Recall also that
Russell neglected modal variability, while Tichý did not.
13
According to Tichý, constructions expressed by descriptions can occur in larger constructions either
directly, or applied to Ws and Ts. This makes possible to model certain intriguing phenomena discussed in the
first part of this section. Another remark: the above consideration can be easily generalized for the case of
descriptions of other object than individuals, say numbers.
7
words, language (in the synchronic sense) is fittingly explicable as a function from
expressions to meanings. If meanings are explicated as constructions, language is then
explicated as a function from expressions to (k-order) constructions, it is a (k-order) code Lk
(Tichý 1988: 228). Which means that coding, expressive means of language of (say) English
are modelled just this way.14
The semantic scheme adopted in this paper is thus:
expression N
|
N expresses in Lk:
meaning of N in Lk
|
N denotes in Lk:
denotatum of N in Lk
|
= construction
= non-intension / intension
N refers in Lk in W at T to:
referent of N in Lk in W at T
= non-intension or the value of intension in W at T
All just mentioned semantic notions can be rigorously explicated, one can define the
respective concepts. Here, I am only going to indicate how to proceed since the formal
definitions are rather easy. Since a language (code) Lk is a function from expressions to
(k-order) constructions, its application to an argument, i.e. an expression N, leads to
a (k-order) construction Ck which is the meaning of N in Lk. The denotatum of an expression N
in Lk is an object (of type ξ) which is constructed by the construction Ck, which is the
meaning of N in Lk; thus the denotatum can be obtained by executing Ck. As regards the
reference, one can distinguish the reference of empirical expressions (i.e. those denoting
intensions, e.g. ‘the Pope’, ‘(is a) tiger’, ‘It rains in Paris’) and the reference of non-empirical
expressions (i.e. those denoting non-intensions, e.g. ‘and’, ‘four’). In the former case, the
intension constructed by Ck is applied to W and T in order to get the value of the intension,
i.e. the referent of the expression N in that W at T in Lk. In the latter case, the referent of N
(in W at T in Lk) is just its denotatum. Of course, one can define concept of the referent
which is uniformly applicable to expressions of both sorts.15
14
For more details of the proposal, see Author 2009 (the chapter IV.5 Language and Selected Semantic
Notions).
15
The explication of semantic concepts just described has a valuable benefit that it does not lead to
semantic paradoxes (cf. the chapter III.2 Paradoxes of Reference in Author 2009.)
8
4. Rigidity of singular terms
We have just seen that semantic notions are rigorously explicable. Their explications
can then be utilized for the definition of the concept of rigidity. It enables us to give the
concept of rigidity – which is a semantic one – into relationships to other, more
fundamental semantic concepts. Recall that the notion of rigidity was originally construed
just this way: a singular term was classified as rigid or non-rigid in correspondence to the
modal (in)variability of its reference.
We have just three basic semantic concepts at our disposal (viz. meaning, denotation,
and reference) which can be deployed in the definition of the concept of rigidity. To select
the right one, I am first going to exclude the concept of meaning. Given a language (in the
synchronic sense), the linkage between an expression and its meaning is fixed by semantic
convention, thus it is modally stable. Rigidity defined as invariability of meaning of
a singular term would then be ascribable to all (meaningful) singular terms, i.e. each
singular term would be a rigid designator. Thus the distinction would come out trivial.
An almost analogous result would be achieved by defining rigidity as invariability of
denotation. The difference would be minimal, only terms such as ‘3÷0’, which are
meaningful, would be classified as non-rigid designators (or rather non-designators)
because ‘3÷0’ lacks a denotatum, it denotes nothing at all. All other singular terms would
come out rigid. Defining rigidity on the basis of denotation, which is stable for all singular
terms, therefore also trivializes the rigid / non-rigid distinction.
To construe the distinction as non-trivial, some singular terms have to be evaluated as
non-rigid. Such demand is fulfilled by the definition of rigidity which is based on
invariability of reference. Any proper name has a stable, fixed denotation and its referent is
identical with its denotatum, thus its reference is also invariable. On the other hand, any
common (individual) description has a stable denotation but a variable reference. (Of
course, the reference of a description such as ‘the only individual which is identical with X’,
which denotes a constant intension, is invariable.) Thus to define rigidity of singular terms
on the basis of reference leads to the results, i.e. classification of singular terms, which are
widely accepted.
9
Here is the exact definition:16
an expression n is, in Lk, a rigid designator of an individual =df there exists an individual x
such that in every possible world w and moment of time t, x is the referent of n in Lk
The concept of rigid designator of an individual defined just this way is construed in
the so-called total sense. This means that if an expression is (say) a predicate, it is evaluated
as a non-rigid designator of an individual. One might regard this as not intuitive enough
and require another concept of rigid designator of an individual which is defined in
a manner avoiding this: expressions which are in principle incapable to refer to individuals
(e.g. predicates, connectives, sentences, etc.) would not be evaluated as rigid or non-rigid
designators of an individual. To define such concept in the partial sense is possible, but it is
technically rather complicated thus I omit the definition here (see Author 2009: 219).
One might perhaps think of yet another concept of rigid designator of an individual.
That concept would apply even to a singular term which does not refer to certain individual
in some possible world and moment of time, yet it refers to one and the same individual
under all other circumstances.17 Again, the definition is possible, thought a bit complicated
(see Author 2009: 219).
The definition presented in this section can be easily adapted in order to define the
concept of rigid designator of a class of individuals, the concept of rigid designator of
a property of individuals, the concept of rigid designator of a class of properties of
individuals, etc. Before I show that, I am going to discuss one special topic.
5. Rigidity and counter-factual truth-conditions
16
I consider definitions to be certain deduction rules of Tichý’s system of deduction (cf., e.g., Tichý 1986).
When writing an expression standing for construction, I use various shortcut conventions following analogous
conventions
in
Tichý
1988.
The
rigorous
definition
verbally
expressed
above
is
just:
[RigidDesignatorOfIndividual n L ] ≡ ∃x ∀w ∀t [x = [ReferentIn wt n L ]] (Author 2009: 220). Both sides of the
ι
k
k
definition construct (if they both construct, depending on a particular valuation, anything at all) one and the
same truth-value. The construction (or concept) RigidDesignatorOfIndividual constructs the relation which
divides expressions of a given language to rigid or non-rigid due to their reference to one and the same
individual under all circumstances. (Analogously below.)
17
Such designator would be a ‘persistent’, as distinct from ‘obstinate’, designator (cf. Salmon 1982: 34).
10
As indicated above, Tichý maintained that ‘All designators, without exception, are
rigid.’ (Tichý 1994: 30). Tichý supported this claim by making consequences of Kripke’s
claim that a singular term ‘E’, which is a part of a sentence such as ‘E is an F’ (where ‘F’ is
a suitable predicate), is a rigid designator if the truth-conditions of the sentence are
invariable with respect to counter-factual situations (Kripke 1980: 7, 9).
To illustrate it, let a common description ‘D’ be our concrete example of ‘E’ and the
individual X be the actual referent of ‘D’. Then the truth-condition of ‘D is an F’ in the actual
possible world (and the present moment of time) consists in that X has the property F.
Under different circumstances, when ‘D’ does not refer to X, the truth of the sentence is
dependent on the fact whether the other individual (if any) which fits the description ‘D’
has the property F. This is the case when the truth-condition ‘vary wildly’ from those truthconditions which are peculiar to rigidity (Kripke 1980: 7). On the other hand, if a proper
name ‘X’ is our concrete example of ‘E’, there are no such changes of truth-conditions − ‘X is
an F’ has only one and the same truth-condition, viz. that the individual X has the property
F.
Now consider the passive form of the sentence discussed already above: ‘The Fountain
of Youth is contemplated by X’.18 The sentence is true, if the object O which is signified by
the description ‘the Fountain of Youth’ has the property in question. Under counter-factual
circumstances, however, it has the very same truth-condition. Thus the description is,
according to Kripke’s own criterion for rigidity, a rigid designator.
One might perhaps object that the example does not exactly match Kripke’s
assumption that ‘the Fountain of Youth’ signifies an individual. Nevertheless, Kripke’s own
example with predicates such as ‘(is a) tiger’ defies the assumption in the same way. Since
such predicates are construed as rigid designators, it seems hold that one can ascertain it by
inspection of truth-conditions of sentences such as ‘Tiger(hood) is contemplated by X’. In
other words, the rigidity is based not on invariability of extensions of the predicates (i.e.
their reference) but on invariability of properties denoted by such predicates. In the case of
individual descriptions: their reference (which is typically variable) is ignored and their
rigidity is based on their denotation (which is invariable). The conclusion that all
designators are rigid is then unavoidable. Since this construal trivializes the distinction
rigid / non-rigid, I have decided to base the distinction rather on reference.
18
My example is radically different from that by Tichý, yet it perfectly fits his conclusion (cf. Tichý 1994:
30).
11
6. Reference, denotation and meaning of general terms
Dubiousness whether this or that general term is a rigid or non-rigid designator is
partly caused by uncertainty as regards the adopted semantics of general terms. The aim of
this section is to preclude this uncertainty.
If we begin with general terms such as ‘{X,Y}’ (i.e. ‘(is) identical with X or Y’), it is clear
that they denote classes of individuals; their referents are naturally construed as identical
with their denotata. Note that such terms have semantics similar to proper names: they
refer to the objects explicitly mentioned by them.
In the case of more common predicates such as ‘(is a) tiger’ or ‘(is a) tiger or lion’ it is
often maintained that they signify properties of individuals, not classes of individuals. The
semantics of such predicates is thus analogous to the semantics of individual descriptions:
under certain circumstances, a predicate points at certain class of individuals, while under
distinct circumstances, it points at distinct class. Thus a predicate of this type denotes an
intension which has classes of individuals as its values. In Tichýan semantic theory, the
classes are called referents of such predicates (in given worlds and times), while the
intensions are called denotata of such predicates.
Intensions having classes of individuals as their values serve as explications of (the
intuitive notions of) properties of individuals.19 Note that such explication is sufficiently
materially adequate. If property had been explicated as a mere class of individuals, then an
individual would instantiate given property by logical necessity – for there would be no
possibility not to instantiate it (every class is individuated by its membership). Such
explication would be obviously wrong because individuals acquire or lose (common)
properties by contingent chance; i.e. the intuitive notion of property has a modal feature.
A property is also something which has – dependently on circumstances – this or that
extension. Intensional logic fittingly accommodates also this feature.20
19
For a detailed defence of such explication of properties see Author 2011 (the chapter 3. Intensional
Explication of Properties).
20
One can go perhaps further and try to capture the intuition that properties have a structure. In Tichý’s
framework, constructions of intensions having classes of individuals as their values would fill the bill, but I do
not enter this idea here. Properties are thus modelled simply as structureless conditions (cf. also the preceding
footnote).
12
In hyperintensional semantics I adopt, the meaning of the predicate ‘(is a) tiger’ or ‘(is
a) tiger or lion’ is a (structured) construction which constructs the property denoted by the
predicate.
It is surely obvious how to adapt these considerations for cases of n-ary predicates
which denote n-ary relations between individuals (etc.). Moreover, meaning, denotation
and reference of ‘higher-order’ predicates is explicable an analogous way. Thus one covers
frequently discussed predicates such as ‘(is a) species of a big feline predator’ (or ‘(is an) X’s
most favourite property of individuals’), the denotatum of which is a property of properties
of individuals, namely a property actually instantiated by the property “tiger”.
Remember that common predicates are in fact descriptions of certain sort (realize
that this is not ‘descriptivism’): they do not directly denote but rather refer to this or that
class of objects.21 This fact enables us to generalize claims about semantic features of
singular terms to the case of general terms.
7. Rigidity of general terms
In order to keep the theory of rigidity homogenous and to preserve non-triviality of
the rigid / non-rigid distinction we are lead to construe rigidity of general terms in the
same way as the rigidity of singular terms. Thus it will be also defined on the basis of modal
invariability of reference. (Let us restrict our attention in this section only to general terms
denoting properties of individuals.)
The definition of the respective concept is a straightforward adaptation of the
definition stated above in the section 4.:22
an expression n is, in Lk, a rigid designator of a class of individuals =df there exists a class of
individuals s such that in every possible world w and moment of time t, s is the referent
of n in Lk
21
On the other hand, both descriptions and predicates are akin to proper names in that they name
objects they denote. (Of course, the similarity is amplified if the description and predicates in questions are
simple expressions.)
22
[RigidDesignatorOfClassOfIndividuals n Lk] ≡ ∃s ∀w ∀t [s = [ReferentOfIn(οι)wt n Lk]].
13
The applicability of such concept of rigid designator of a class of individuals is clear.
Predicates such as ‘(is) identical with X or Y’ are rigid, while predicates such as ‘(is a) tiger’,
‘(is a) pencil’, ‘(is a) tiger or lion’ are non-rigid.23
Let us examine one possible objection to this proposal. In his 2007, Arthur Sullivan
mentioned a possibility to define rigidity of general terms as I do, i.e. as invariability of
extensions across the logical space. Sullivan immediately repudiated such proposal because
too many general terms, including natural kind ones, would then be sorted out as non-rigid.
In my opinion, however, an appeal to quantity is not quite decisive from a general scientific
point of view.24 I think that it would be better to argue in favour − or against − the present
theory by an appeal to its overall compactness and also its relations to other particular
explications of our whole conceptual scheme.
This can be illustrated on the example of Sullivan’s own theory. Unlike my proposal,
his proposal classifies natural kind terms such as ‘(is a) tiger’ as rigid. It is so for the reason
that Sullivan defines rigidity as unstructuredness of terms in questions. Sullivan realizes
that there are several problems with his proposal. For instance, he has to explain how the
apparently contingent linguistic fact about labelling “tiger” by ‘tiger’ (and not by, say,
‘black stripped feline predator’) is unproblematic as regards his explanation. However, it is
difficult to be convinced because contingent linguistic data concerning English run against
general philosophical demands. He has also persuaded us how his theory expands to
singular terms. This is complicated by the fact that there are possible unstructured labels
(‘names’) of descriptive contents and many such one-word descriptions are intuitively nonrigid (yet Sullivan’s theory says the opposite).
But I think that his theory must pay even a bigger price. Sullivan’s definition of
rigidity employs syntactical (only partly semantical) features of expressions; it lacks,
however, a connection to modality. But recall that modality was crucial for the very original
exposition of rigidity by Kripke and others. Contrary to Sullivan’s proposal, my approach
heavily accents modal profile of the semantic content of expressions: if a singular term /
predicate / sentence has a modally stable semantic content (viz. constant reference), it is
a rigid designator.
23
As noted already above, no contrast between natural and artificial kind terms is drawn (similarly as in,
e.g., LaPorte 2000: 305).
24
It is also not entirely persuasive to object that the present approach does not match original intuitions
of (say) Kripke. For intuitions can be confused or their product can be of low importance (cf., e.g., the section
5.).
14
Now let us define a slightly different concept of rigid designator of a class of
individuals. According to this concept, the class s, which is an extension of the property
denoted by the predicate, is always non-empty. This excludes, for instance, the predicate
‘(is) round and not round’ from rigid designators. The definition originates from the
preceding one by means of simple supplementation:25
an expression n is, in Lk, a rigid designator2 of a class of individuals =df there exists a class of
individuals s such that in every possible world w and moment of time t, s is the referent
of n in Lk, and there exists an individual x which is in s
The definition has an interesting philosophical feature because a designator of a class
of individuals which is rigid in this sense denotes a property which is naturally
characterized as essential.26 A property f is essential =df there exists an individual x such that
in every possible world w and moment of time t, x instantiates f.
To utilize the concept of essential property in the definition of rigidity for general
terms was suggested already by Scott Soames (2002: 249-263, 287-288). But Soames did not
take into account that properties such as “(is) as tall as X” are essential properties (for the
individual X has it under all circumstances, it cannot lack it), yet they have a variable ranges
of extensions (for the individual Y, or Z, has that property if it happens being as tall as X).
Properties of this sort were called and defined by Pavel Cmorej 1996 partly essential
properties. A property f is partly essential =df f is essential but not constant; a property f is
constant =df there exists s such that in every w and t, s is the extension of f.27
Of course, we do not wish partly essential properties to be denotata of rigid
designators of a class (or classes) of individuals. We intend to admit only those essential
properties which have stable ranges of their extensions, i.e. purely essential properties.
A property f is purely essential =df f is essential and constant. Thus the alternative definition
of the concept of rigid designator2 of a class of individuals is:28
25
[RigidDesignatorOfClassOfIndividuals2 n Lk] ≡ ∃s ∀w ∀t [ [s = [ReferentOfIn(οι)wt n Lk]] ∧ ∃x [s x] ].
26
Note that this fact does not commit us to any suspicious metaphysics. An example of an essential
property covered by the subsequent definition is identity, a property no well-constituted object can lack.
27
An extensive classification of properties is presented in Author 2011. Note that partly essential
properties are not so-called impure properties discussed by contemporary metaphysicians.
28
[RigidDesignatorOfClassOfIndividuals2 n Lk] ≡ ∃f [ [f = [DenotatumOfIn(((οι)τ)ω n Lk]] ∧ [PurelyEssential f] ].
15
an expression n is, in Lk, a rigid designator2 of a class of individuals =df there exists a property
f which is the denotatum of n in Lk and which is purely essential
8. Rigidity of other general terms
Predicates denoting properties of individuals are not the only ones. There exist also
predicates denoting properties of properties of individuals, properties of properties of
properties of individuals, etc., i.e. properties of the second, third, ..., n-th order.
To illustrate, a property such as “tiger” can posses a second-order property such as
“species of a big feline predator” or “(being) X’s most favourite property of individuals”.
The predicates for such higher-order properties are classified by many theoreticians as
non-rigid designators (cf., e.g., Martí 2004: 135, LaPorte 2000, 2006, Linsky 2006).
More accurately, these predicates are not rigid designators of a class of individuals
because they in principle do not refer to any class of individuals. (The same holds for all
predicates of such higher order.) However, we naturally want to classify such predicates as
rigid or non-rigid with respect to the reference they have.29 For instance, a possible referent
of the predicate ‘(is a) species of a big feline predator’ is a class of properties of individuals
(the class actually includes the property “tiger(hood)”); since the reference of the predicate
varies, we would like to classify it as a non-rigid designator of a class of properties of
individuals.
The definition of the respective concept is a straightforward adaptation of the first
definition given in the preceding section:30
an expression n is, in Lk, a rigid designator of a class of properties of individuals =df there exists
a class of properties of individuals u such that in every possible world w and moment of
time t, u is the referent of n in Lk
Thus ‘(is a) species of a big feline predator’ and also ‘(is an) X’s most favourite property of
individuals’ are non-rigid designators of a class of properties of individuals, while
29
This idea has been exposed already by Marián Zouhar (2009) who even suggested to base rigidity of
general terms on their reference in the Tichýan sense.
30
[RigidDesignatorOfClassOfPropertiesOfIndividuals n Lk] ≡ ∃u ∀w ∀t [u = [ReferentOfIn(ο(((οι)τ)ω))τ)ωwt n Lk]].
16
predicates such as ‘(is a) property identical with the property “tiger” ’ are rigid designators
of a class of properties of individuals.
It is evident that the concept of rigid designator defined above can be adapted even
for other kinds of expressions, e.g., for descriptions referring to properties of individuals
(an example of which is the apparently non-rigid description ‘X’s most favourite property of
individuals’). For another noteworthy kind of expressions consider rigid designators of
truth-values, especially sentences denoting analytically (constantly) true or analytically
(constantly) false propositions.
Summing up, if we define the concept of rigidity on the basis of the modal profile of
their reference, we get a rigid / non-rigid distinction which is applicable on the large
amount of natural language expressions. These expressions divide into two substantial
groups, (non-)rigid designators of an object (e.g. an individual, a property of individuals,
a truth-value, etc.) or a class of objects (e.g. a class of individuals, a class of properties of
individuals, etc.).
References
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31
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version
is
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17
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from
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32
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18