MODAL
LOGIC WITH SUBJUNCTIVE MARKER:
A NEW PERSPECTIVE ON RIGID DESIGNATION
Helge Rückert
Department
of Philosophy
<heru0001@stud.uni-sb.de>
Abstract:
According to Kripke’s
Naming and Necessity the thesis
that proper names are synonymous with certain definite descriptions is false
because proper names are rigid designators whereas definite descriptions are
(in general) non-rigid designators. It is argued that standard modal logic,
the formal tool used by Kripke to deal with these
questions, does not reflect adequately the relations between the modal operators,
their scopes and the subjunctive mood. Then, by using K. Wehmeier’s modal logic with subjunctive marker (S5*), it is shown (against Kripke) that in one sense of rigidity (reflecting Kripke’s criterion of rigidity) proper names as well as definite
descriptions are rigid designators, but in another sense of rigidity neither
proper names nor definite descriptions are rigid designators (apart from definite
descriptions that have been called de facto rigid by Kripke and alike proper
names). Nevertheless Kripke is right in not accepting
the synonymy thesis: Proper names are not synonymous with definite descriptions.
But the correct reasons for this claim are not Kripke’s.
Extended abstract:
1. Kripke
on rigid designators
In his very influential Naming and Necessity Kripke introduced the
distinction between rigid and non-rigid designators. With respect to any possible
world rigid designators name always the same object whereas non-rigid designators
name different objects with respect to different possible worlds. That means:
Inside the scope of a modal operator a rigid designator names always the same
object whereas the semantic value of a non-rigid designator depends on the
possible world at stake. According to Kripke all
proper names are rigid designators but most definite descriptions are non-rigid
designators. (There are definite descriptions like „the sum of two plus two“
that are rigid because in every possible world always the same object
meets the description. Definite descriptions of that kind are called de
facto rigid by Kripke).
2. Modal logic and scope
Obviously, the standard formal tool to deal with
questions of rigid and non-rigid designation is modal logic. As a matter of
fact the standard outlook of modal logic nowadays reflects a certain historical
development. In the beginning, the modal operators had been designed in such
a way that all predications inside their scope had to be evaluated always
with respect to the possible world at stake as determined by the modal operator.
Thus, standing inside the scope of a modal operator was thought to be a necessary
and sufficient condition for every predicative expression in order to get
affected by that modal operator. (With nested modal operators things get a
little bit more complicated: Then a predicative expression might stand inside
the scope of several modal operators but is only affected by one of them).
Later it was found that this strategy is unsatisfactory because there
are simple examples that can’t be analysed by mere scope distinction (Hazen
1976):
(1)
Under certain counterfactual circumstances,
everyone who has flown to the moon would not have flown to the moon.
The problem arises because on the one hand the
expression „everyone who has flown to the moon“ must stand inside the scope
of the modal operator in order to allow for an adequate analysis but on the
other hand this expression refers to the actual world and is thus independent
of the modal operator.
To deal with these problems and thus to increase the expressive power
of the formal modal language the so-called actuality
operator had been introduced (Crossley/Humberstone
1977): Expressions standing inside the scope of an actuality operator always refer to the actual world, even inside the
scope of a modal operator. An actuality
operator protects against the influence of modal operators, so to say. Now,
the example can be rendered as follows:
(2)
ą "x (A (Fx) ® Ų Fx)
Thus, in modal logic with actuality operator it is no longer the case that it is a necessary
and sufficient condition to stand inside modal scope for an expression to
be evaluated with respect to the possible world at stake because expressions
in the scope of an actuality operator
are always evaluated with respect to the actual world.
But once we have a language in which not all expressions inside modal
scope depend on the respective modal operators one
can ask whether there is no better way to indicate dependencies and independencies
from modal operators than by scope distinctions and supplementary operators
as the actuality operator.
3. Wehmeier’s
modal logic with subjunctive marker
(Wehmeier
2002) presents an alternative to standard modal logic, his modal logic with
subjunctive marker, S5* (in (Rückert 2002) Wehmeier’s modal logic
is used to solve Fitch’s paradox of knowability).
An expression inside modal scope depending on the respective modal operator
needs to be indicated with a subjunctive marker „*“ (this
corresponds to the subjunctive mood in natural language). Otherwise it refers
to the actual world (corresponding to the indicative mood in natural language).
In Wehmeier’s logic the above mentioned example
looks as follows:
(2*)
ą "x (Fx
® Ų F*x)
The subjunctive marker „*“ functions
in analogy to the well-known individual variables, it is nothing else but
a variable over possible worlds that needs to be bound up by a modal operator.
Thus, expressions that contain subjunctive markers that are not bound by modal
operators and thus appear freely are incomplete.
4. Definite descriptions
Now, we apply Wehmeier’s
modal logic to the problem of rigid designation, starting with an analysis
of definite descriptions. An expression of the kind „the teacher of Alexander“ is ambigous. Its two readings are:
(a) the man
who taught Alexander
(b) the man
who would have taught Alexander
According to Kripke
this difference has to be reflected by scope distinctions: The subjunctive
mood in (b) indicates that the definite description has to stand inside modal
scope whereas the indicative mood in (a) indicates that the definite description
has to stand outside modal scope (or, as an ‘actualised’ definite description
inside modal scope, respectively). But according to Kripke
we are dealing with one and the same definite description in both cases.
Things look different in the framework of Wehmeier’s
S5*. The two readings (a) and (b) get the
following translations:
(a*) (ix) (Tx)
(b*) (ix) (T*x)
(a*) is the
formal representation of a rigid designator: Even inside modal scope, it always
designates the one who taught Alexander in our world, namely Aristotle. On
the other hand (b*) is not even the formal representation of a designator
at all. It contains a freely occurring subjunctive marker and is thus incomplete:
One needs to add information about the circumstances in order to determine
a designated object. Thus, if we apply Kripke’s
rigidity criterion to definite descriptions within the framework of Wehmeier’s logic we get the following result: All definite
descriptions are rigid designators.
Of course, we can define another concept of rigidity to account for
some of Kripke’s intuitions. „(ix) (Tx)“ is a rigid2 designator iff
the following holds:
(3)
š ((ix) (Tx) =* (ix) (T*x))
In this sense Kripke’s
de facto rigid definite descriptions
are rigid2 and all other definite descriptions are non-rigid. The
idea behind the concept of rigidity2 is the following: One and
the same language can be spoken in different possible worlds. When an inhabitant
of another possible world uses an expression in indicative mood like „the
man who taught Alexander“ he refers to his own world.
Now, a designator is rigid2 if it names the same object no matter
in whatever possible world it is used.
5. Proper names
In an expression like „the teacher of Alexander“ we detected an ambiguity earlier on that resulted from a
difference between the indicative and the subjunctive mood. The expression
has two readings because as a matter of fact it does not contain a verb in
the indicative or the subjunctive mood. The ambiguity can be resolved by using
other expressions that contain a verb or by using a formal language that makes
the difference explicit by help of a subjunctive marker.
Now to proper names. An expression like „Aristotle“ does not contain a verb either. But is there also a indicative/subjunctive-ambiguity? In our natural language
we have no means to build subjunctive versions of proper names. But that’s
a contingent fact. Even if in most cases we use an expression like „Aristotle“
to refer to the man that was named „Aristotle“ in our world, one can imagine
contexts in which we are confronted with something like subjunctive versions
of proper names: „Imagine a world in which there was a young man that was
given the name ‘Aristotle’. Aristotle was a very good wrestler and he...“.
I think, most of us understand this story in a way that the word „Aristotle“ in the second sentence does not refer to (our) Aristotle
but to the young wrestler whoever he is.
Anyway, in a formal language we are free to introduce subjunctive counterparts
of proper names reflecting the fact that in other possible worlds the same
names might have been given to other objects than in our world. In the possible
world of our example, people using the name „Aristotle“
name the young wrestler and not (our) Aristotle. Nevertheless, this
is no sufficient reason to say that say speak another language. And thus an
expression like „Aristotle“ (like „the teacher of
Alexander“) has two readings, too:
(c*) a
(d*) a*
(c*) is certainly
a rigid designator, and (d*) isn’t a designator at all because it contains
a freely occurring subjunctive marker. In this respect there’s no difference
between proper names and definite descriptions. Also, most proper names are
non-rigid2 too, because
(4)
š (a =* a*)
is false for most proper names „a“. (One
might argue that there are rigid2 proper names in our language
as for example „2“. A language in which „2“ does
not designate the successor of the number 1 couldn’t be the same language
as ours.) Again no essential difference between proper names
and definite descriptions.
6. Conclusion
I tried to show that accepting Wehmeier’s modal logic S5*
as a more appropriate formal tool than standard modal logic leads to new insights
concerning the concepts of rigid and non-rigid designation. Against Kripke it is not true that there is a difference in principle
between proper names and definite descriptions, the first being rigid designators,
the latter non-rigid designators. According to Kripke’s
conception of rigidity both, proper names and definite descriptions,
are (in general) rigid. And according to another conception of rigidity, rigidity2,
both are (in general) non-rigid2.
For these reasons, also Kripke’s modal argument
against the description theory of proper names is not valid. Nevertheless
his thesis is correct: Proper names and definite descriptions are not synonymous
because (in general) there is for a proper name „a“ no
definite description „(ix) (Gx)“
such that
(5)
š (a* =* (ix) (G*x))
holds. (Maybe a definite description like
„the object named ‘a’“ does the job but this does
not yield any substantial form of description theory because the proper name
itself appears in the definite description.
7. Bibliographie
Crossley, J. / Humberstone,
L. (1977): ‘The Logic of ‘Actually’’, Reports
on Mathematical Logic, 8, S. 11-29
Hazen, A. (1976): ‘Expressive Completeness
in Modal Language’, Journal of Philosophical
Logic, 5, S. 25-46
Kripke, S. (1980): Naming and Necessity,
Rückert, H. (2002): ‘A Solution to Fitch’s
Paradox of Knowability’, to appear in Gabbay, D. / Rahman, S. / Symons,
J. / Van Bendegem, J. (ed.): Logic, Epistemology and the Unity of Science, Kluwer
Academic Publishers, in preparation
Wehmeier, K. (2002): ‘Descriptions in the Mood’,
to appear in Kahle, R. (ed.): Intensionality - an Interdisciplinary Discussion, AK
Peters (Lecture Notes in Logic), in preparation