NOTES ON INTENSIONAL THEORIES
Joe Sneed
Colorado School of Mines
Golden, CO 80401
05-16-93
Please do not cite or quote without permission of the author.
ABSTRACT
The question of wether intensional languages are more
expressive than non-intensional languages is raised within the
framework of a semantic view of theories. From perspective, the
question is this. Are there model classes that can be
characterized by theories using intensional concepts that can not
be characterized by theories that do not use intensional
concepts? A precise formulation of this question is suggested,
but no answer is given.
To approach this question, model theory of first order
theories is summarized (Sec. II) and the semantic approach to
theories using non-intensional, theoretical augmentations of
first order theories is reviewed (Sec. III).
Intensional augmentations of first order theories are
sketched (Sec. IV) but not rigorously defined. This intensional
language provides the apparatus for attributing language use and
intensional attitudes to individuals whose behavior is the object
investigation. It also provides apparatus for talking about
translation from the attributed language to the investigator's
language.
The initial question then becomes wether there are model
classes that can be characterized by intensional augmentations of
first order logic that can not be captured by non-intensional
theoretical augmentations (Sec. V).
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NOTE ON NOTATION
ASCII precludes the use of sub- and super-scripts. Thus,
notation that would normally appear as sub-scirpts appears in
`[]'`s while superscripts appear in `{}'`s. This convention
iterates in the obvious way. This is not beautiful, but it is
(hoepfully) consistent and intelligible.
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TABLE OF CONTENTS
I. INTRODUCTION
I.1 PURPOSE
I.2 APPROACH
II. DESCRIPTIVE LANGUAGE -- L[d]
II.1 FIRST ORDER SYNTAX (FOS)
II.2 SEMANTICS
II.2.1 INTERPRETATION
II.2.2 TRUTH AND MODELS
II.3 DESCRIPTIVE THEORIES
III. THEORETICAL AUGMENTATIONS OF THE DESCRIPTIVE LANGUAGE --
L[t]
III.1 SYNTAX
III.2 SEMANTICS
III.2.1 INTERPRETATION
III.2.1 TRUTH AND MODELS
III.3 THEORETICAL THEORIES
IV. INTENSIONAL AUGMENTATIONS OF THE DESCRIPTIVE LANGUAGE --
L[i]
IV.1 INTENSIONAL FIRST ORDER SYNTAX (IFOS)
IV.1.1 FIRST ORDER SYNTAX PREDICATES
IV.1.2 QUOTE NAMES
VI.1.3 TRANSLATION PREDICATE
VI.1.4 TOKEN PREDICATE
VI.1.5 TOKEN CONCATENATION PREDICATE
VI.1.6 INTENSIONAL ABSTRACTION OPERATOR
VI.1.7 INTENSIONAL ATTITUDE PREDICATES
IV.1.8 FORMATION RULES FOR L[i]
IV.2 SEMANTICS
IV.2.1. INTERPRETATION
IV.2.1.1 DESCRIPTIVE PREDICATES (P[d])
IV.2.1.2 FIRST ORDER SYNTAX PREDICATES
IV.2.1.2.1 ATTRIBUTED LANGUAGE PREDICATES
IV.2.1.2.2 DESCRIPTIVE LANGUAGE PREDICATES
IV.2.1.3 QUOTE NAMES
IV.2.1.4 SEMANTIC INTERPRETATION OF L[a]
IV.2.1.5 TRANSLATION PREDICATE
IV.2.1.6 TOKEN PREDICATE
IV.2.1.7 TOKEN CONCATENATION PREDICATE
IV.2.1.8 INTENSIONAL ABSTRACTION PREDICATE
IV.2.1.9 INTENSIONAL ATTITUDE PREDICATES
IV.2.1.9.1 INTERPRETATION
IV.2.1.9.2 INTENSIONAL ATTITUDES AND
INTENSIONAL ABSTRACTION
IV.2.2 TRUTH AND MODELS
IV.2.2.1 TRUTH DEFINITION
IV.2.2.2 OPACITY OF INTENSIONAL CONTEXTS
IV.2.2.3 THEORY OF MEANING?
IV.3 INTENSIONAL THEORIES
IV.3.1 ATTRIBUTED LANGUAGE THEORY
IV.3.2 PURELY INTENSIONAL THEORY
IV.3.3 PURELY DESCRIPTIVE THEORY
IV.3.4 FULL THEORY
IV.3.5 LINGUISTIC ACTIONS
IV.3.6 PSYCO-PHYSICAL LAWS
IV.3.7 HOLISTIC THEORIES
IV.3.8 MODELS FOR INTENSIONAL THEORIES
IV.3.8.1 THEORETICAL MODELS
IV.3.8.2 NON-THEORETICAL MODELS
IV.3.9 INDETERMINACY OF INTENSIONAL CONCEPTS
IV.3.10 TRANSLATION MANUALS
V. COMPARISON OF THEORIES
I. INTRODUCTION
I.1 PURPOSE
What is an intensional theory of behavior? Are intensional
theories of behavior more powerful than non-intensional theories?
Is there a kind of behavior that can only be "explained" by
intensional theories? Here I try to address these questions
using a semantic conception of theory.
Consider a situation in which an "external observer" is
trying to understand the behavior of a some number of individuals
-- human beings, animals, black boxes and possibly other kinds of
things as well. Intuitively, the observer sees only overt
behavior of these individuals -- how they move about in space
relative to each other, change color, produce sounds, etc.. There
is no a priori reason to believe that the individuals
communicate with each other (or the external observer), use
language, have beliefs, desires, etc.. Such "intensional
attributes" may be imputed to individuals in an effort to explain
their behavior, but they are not a part of the behavior to be
explained. 'Understanding' or 'explaining' the behavior of these
individuals is taken in a minimal sense of distinguishing, in a
general way, kinds of behavior that may occur from that which may
not. This kind of understanding may lead to an ability to
predict and/or control behavior, but it need not.
A bit more precisely, the observer uses some fixed
vocabulary to describe the behavior of the individuals. Her task
is to distinguish in some general way the kinds of behavior she
observes (or countenances as possible) of these individuals form
kinds of behavior he does not observe (and countenances as
impossible) . That is she wants to have a theory about the
behavior of these individuals. Are there kinds of behavior that
could ONLY be characterized by attributing intensional states to
some of the individuals?
These questions can be given precise formulation by:
A) viewing the enterprise of producing a theory as
characterizing a class of non-theoretical models using
theoretical models;
B) distinguishing intensional theoretical concepts from
non-intensional theoretical concepts.
On my account of the matter, intensional concepts are
essentially related to language. So we need to talk both about
the language used by the observer to construct his theory as well
as whatever language she might attribute to individuals about
which she is theorizing.
For present purposes, languages are essentially formal
devices (of a specific kind) that characterize classes of models
(in a specific way). Since we know a good bit about the model
theory of first order logic, it is expedient to begin thinking
about our question in terms of languages conceived as syntax for
first order logic (FOL).
Restricting our attention to FOL de facto excludes form
consideration one important feature of the syntactic view of
theories -- "laws" that operate across different models of the
theory -- so-called "constraints". I say 'de facto' because I
don't know how to formulate constraints in FOL. But, there may
be a way. Should the formulation of the question presented here
lead us to conclude that intensional theories are no more
expressive than non-intensional, theoretical theories, the
question of wether consideration of constraints would make a
difference should be considered.
I.2 APPROACH
I will begin by considering a descriptive language L[d] -
some specific instance of a first order syntax --
L[d] =
its interpretations with finite domains drawn from an "ur-domain"
H -- I[H,L[d]], and the set of sub-sets of I[H,L[d]], |M[d], that
can be characterized by finite sets of sentences L[d] -- L[d]
theories. Here P[d] is a finite set of predicate types and S[d]
is the set of sentence types of L[d].
We will then consider theoretical, but non-intensional
augmentations of L[d] of the form
L[t] = <,V,Q,K,concat[t],F[t],S[t]>
where P[t] are theoretical predicates. We will consider
interpretations of L[t] with domains drawn from a domain K -- H
emmended with theoretical individuals -- I[K,L[t]] and |M[t] --
the class of sub-sets of I[K,L[t]] that can be characterized by
finite sets of sentences of L[t]. Each member t of I[K,L[t]] has
a descriptive fragment Ram(t) which is a member of I[H,L[d]].
Thus sub-sets of I[K,L[t]] determined by sentences of L[t]
correspond, via Ram, to subsets of I[H,L[d]]. In some cases, the
Ram images of sub-sets of I[K,L[t]] can not be characterized by
any finite set of L[d] sentences. In these cases, L[t] is
stronger than L[d].
Finally we will consider an intensionlal, theoretical
augmentation of L[d] -- L[i]. The intensional language L[i] will
contain sufficient syntactical apparatus to permit the
attribution of language use and intensional attitudes to some
individuals. The objects of intensional attitudes are taken to
be sets of non-theoretical models -- sub-sets of I[H,L[d]].
Intensional theoretical augmentations are distinguished from
non-intensional theoretical augmentations of L[d] essentially in
that the former have singular terms that denote sets of
non-theoretical models. The formal apparatus I use to do this is
somewhat baroque. Many, I suspect, would deny that my L[i] is an
intuitively adequate, and even internally coherent, rendition of
an intensional language. Some effort is devoted to anticipating
these objections.
The predicates of a first order intensional syntax (FOIS)
L[i] will be:
P[i] =
where P[d] are the predicates of the "underlying" descriptive
language. The remainder are theoretical predicates analogous to
the theoretical predicates P[t] in the non-intensional
theoretical augmentation.
Some idea of the intended interpretations of these
predicates is required to understand why L[i] might plausibly be
viewed as an intensional language.
P[L[a]] and P[L[d]] are the predicates required to
characterize the syntax of the attributed language L[a] and
the descriptive language L[d].
Consonant with the semantic conception of theory, these syntaxes
are viewed as kinds of set-theoretic structures and specific
languages are viewed as instances (models) of these structures.
The individuals in these structures are taken to be symbol types.
They are regarded as theoretical individuals.
There must be singular terms in L[i] that can be interpreted as
referring to at least some symbol types in L[a] and/or L[d].
These are needed for L[i]-sentences which both USE and MENTION
L[d] sentences. For example,
Whatever Bill wants he gets.
|\/(X)(Y) [ Prefer(b,that(`Y'),that(`-Y'))) --> Y ]
If Bill believes that pizza tastes good then pizza tastes
good.
Believe(b,that(`P')) --> P.
Apparently, we can get by with quote-names for sentences only.
But including quote-names for other symbol types appears to be
cost free. Thus,
P[qn] is a set of L[i]-singular term types which will be
interpreted as quote-names of L[d] and L[a] symbol-types,
including sentences.
` and ' are quote mark symbol types used in constructing
(via concat[i]) quote-names for L[d] and L[a] symbol types.
They are L[i]-logical symbols analogous to quantifiers.
trans(`s[a]',`s[d]') means that L[a]-sentence s[a] is a
translation of L[d]-sentence s[d].
I will describe below an L[i]-interpretation relative sense of
translation.
token(x,`s[a]') means that non-theoretical individual x is a
token for the attributed sentence type s[a].
concat[token] is required to formulate "laws" requiring that
the concatenation structure of the non-theoretical
individuals identified as tokens for linguistic symbols have
a set-theoretic structure isomorphic to a finite fragment of
the structure of the abstract (theoretical) symbol types
providing the interpretation for P[L[a]] and P[L[d]].
Strictly speaking contat[token] is a non-intensional
theoretical predicate.
'that' is a unary operation interpreted so that
'that(`s[a]')' denotes the class of models determined by the
L[a]-sentence s[a].
To interpret the that-operation in this way, we include the set
of all sub-sets of I[H,L[a]] in the domains of all
interpretations of L[i]. These we regard as theoretical
individuals. Other theoretical individuals are required to
provide interpretations for sets of symbol types.
P[a] is a set of intensional attitude predicate types.
Members of P[a] will be interpreted with sets of tuples whose
first member is a non-theoretical individual and whose other
members are sub-sets of I[H,L[a]]. Thus the objects of
intensional attitudes are taken to be sets of non-theoretical
models.
This intuitive sketch of interpretations of L[i] will be
filled out to characterize a set of interpretations I[K[i],L[i]].
Finite sets of L[i]-sentences may -- intensional theories -- will
be regarded as characterizing sub-sets of I[K[i],L[i]].
Examples, of sentences that might be in intensional theories
are:
assert(x,that(`s[a]') iff |/\(y) [token(y,`s[a]') & d(x,y) & ...
where d(x,y) is some purely descriptive relation and '...'
indicates more conditions, either descriptive or intensional;
prefer(x,that(`s[a]'),that(`-s[a]')) & d(x,....) &
trans(`s[a]',`s[d]') --> s[d].
The second is an example of a putative "psyco-physical law". It
purports to provide descriptive conditions under which
"preference leads to action".
Providing an empirically acceptable intensional theory of
some body of behavior is no part of the present enterprise. No
claim is made that examples considered would be a part of such a
theory. It is, however, claimed that a plausible account of the
ontology and logical structure of intensional theories has been
provided.
L[i]-models have purely descriptive fragments just as do
L[t]-models. Thus it is possible, to regard the model classes
determined by intensional theories -- sub-sets of I[K[i],L[i]] --
as determining purely descriptive model classes -- sub-sets of
I[H,L[d]], via a Ram-functor, in just the same way as it is
possible to regard L[t]-theories as determining purely
descriptive model classes.
Are intensional theories essential to characterizing some
kinds of behavior? Using the apparatus sketched above, the
question is roughly this.
Is there some descriptive language L[d] such that there are
purely descriptive model classes determined by a theory in
some intensional theoretical augmentation of L[d] that can
not be determined by a theory any non-intensional
theoretical augmentation L[d]?
It should be noted that the question is not wether the content of
intensional theories can be reproduced by purely descriptive
theories. Rather, it is wether the content of intensional
theories can be reproduced by non-intensional, but still
non-purely-descriptive theories.
At this point, I don't have an answer to this question.
But, I think the apparatus sketched sketched above and described
more fully below formulates the question with sufficient
precision to admit of a rigorous answer. That is, there is a
theorem to be proved -- but I can't prove it.
II. DESCRIPTIVE LANGUAGE -- L[d]
II.1 SYNTAX
The non-theoretical, or descriptive language L[d] is some
specific instance of the syntax of first order logic (FOS) with a
finite number of individual constants and a finite number of
predicates of order's less than some fixed n.
For our purposes, it is convenient to think of instances of
FOS as set-theoretic structures with the usual formation rules
being part of the definition of a set-theoretic predicate that
characterizes these structures. The sets appearing in these
structures are to be interpreted sets of SYMBOL TYPES. Symbol
types are abstract entities whose "instances" are SYMBOL TOKENS.
We will interpret symbol tokens to be individual physical
objects. How symbol types are related to their tokens will be
explained below (VI.2.1.4).
More precisely, we may think of FOS's as set-theoretic
structures of the following form:
L =
where:
P =
is an m+1-tuple consisting of n-tuples
P[i] =
whose elements are predicate types of order i; 0 =< i =< m
(Constants are 0-order predicate types.); V is a p-tuple of
variable types; Q a 2-tuple of quantifier types; K is q-tuple of
sentential connective types (all the usual ones). concat is a
tertiary relation (binary operation) on the set of all symbol
types appearing in P, V, Q, and K that characterizes the way
symbol types in these sets are concatenated to form members of F
-- the set of formula types. S, a sub-set of F -- is the set of
sentence types. Different formula types in F (and sentence types
in S) are distinguished by the way they are constructed by
iteration of concatenations. Sentence types are distinguished
from other formula types in the usual way via the concept of
"bound variable".
Note that no delimiting symbols like '(' are used here. I
assume these can be avoided by use of Polish notation and
formation rules that attend to the arity of predicates and
connectives. On could, as well, introduce set of delimiting
symbols into the tuple L.
On this view, the class of set-theoretic structures that are
FOS's is determined by defining a set-theoretic predicate 'is an
FOS'. The usual "formation rules" for formulas and sentences in
FOS appear as clauses in this definition.
Thus we may think of L[d] as some specific set-theoretic
structure
L[d] =
in which P[d] is a tuple consisting of some small number of
predicate types, F[d] and S[d] are sets consisting respectively
the formula types and sentence types constructed from the members
of P[d] using concat[d].
The motivation for this is that we will need to provide a
formulation of the first order syntax (FOS) within a first order
syntax (FOS) -- provide a first order syntactical theory of first
order syntax -- and to speak about different "models" for this
syntax in which different physical objects are taken to be the
symbol tokens corresponding to the FOS-symbol-types.
II.2 SEMANTICS
II.2.1 INTERPRETATION
Interpretations for L[d] all have finite domains -- h -- of
individuals drawn from some (possibly) infinite set of
"ur-individuals" -- H. Intuitively, H is just the set of all
individuals whose behavior interests "the observer" -- say,
plovers and their predators. Specific H-interpretations are
specific instances of this behavior in which a few individuals
participate -- say, plovers and their predators in front of my
beach house on July 4, 1992.
We may consider the infinite set I[H,L[d]] of all
interpretations of L[d] constructed in this way. Intuitively,
this is the set of all possible data the observer might have
about the behavior of the kinds of individuals that interest her.
Note that the set I[H,L[d]] will generally be infinite, though
each member of it is a set-theoretic structure over a finite
domain.
In the usual formulations of FOL predicate types are
assigned set of tuples from h. So it is here too. An
H-interpretation is just an 2-tuple
i = ; f{P} =
f{j} : {set of j-order predicate types} --> Pot(h{j}).
('Pot(X)' denotes the power set of X; X{0} = X. 'h{j}' denotes
the set of all j-tuples formed from members of h.)
Note that a semantic interpretation of L does NOT
characterize a model for the set-theoretic structure L. Rather,
it assumes that such a model is at hand and characterizes a
semantic interpretation for this model.
II.2.2 TRUTH AND MODELS
Members of S[d] -- sentence types -- are assigned
interpretation relative truth (i-truth) in the usual way. Thus,
sentences of L[d] characterize sub-sets of I[H,L[d]] -- the
sub-sets of H[I,L[d]] in which they are true. Members or these
sub-sets are the models for sentences. Intuitively, we may think
of sentences of L[d] as "denoting" their model classes.
II.2.3 DESCRIPTIVE THEORIES
Sets of sentences T[d] of L[d] are linguistic expressions of
descriptive theories. From a semantic point of view, the
"theories" ARE the intersection of the classes of models
characterized by (denoted by) the sentences. The model class of
T[d] is M[T[d]].
Let |M[d] be the set of all sub-sets of I[H,L[d]] that the
observer can characterized using L[d]. Intuitively, |M[d] is the
set of all possible, purely descriptive theories. These theories
make use of no conceptual apparatus beyond the non-theoretical,
descriptive vocabulary.
III. THEORETICAL AUGMENTATIONS OF THE DESCRIPTIVE LANGUAGE --
L[t]
III.1 SYNTAX
The simplest way to augment L[d] is simply to add additional
predicate types to those already appearing in L[d] to produce
L[t].
L[t]= <,V,Q,K,concat[t],F[t],S[t]>.
Note that the "logical symbol types" remain unchanged. The
relation concat[t] must be different from concat[d] simply
because it has a bigger domain.
III.2 SEMANTICS
III.2.1 INTERPRETATION
Intuitively, we want to allow for the possibility that some
new kinds of individuals will be needed to satisfy some of our
new, theoretical predicates. So consider a set of individuals K
so that H _< K. Then K-interpretations of L[t] -- theoretical
interpretations -- will look like:
i[t] = .
We allow that k may be infinite, but require that k intersect H
be non-empty and finite. Intuitively, we may employ an infinite
number of theoretical individuals, but always in connection with
some finite number of non-theoretical individuals.
III.2.2 TRUTH AND MODELS
This is no different from L[d] except that model for
sentences in L[t] have the set-theoretic structure of i[t].
III.3 THEORETICAL THEORIES
The additional theoretical apparatus may be used to
construct sentences T[t] that characterize classes of theoretical
models M[T[t]] for the theoretical language -- sub-sets of
I[K,L[t]]. From M[T[t]] we may obtain a sub-set of I[H,L[d]] by
doing two things:
1) from the members of M[T[t]] delete all the pairs
containing P[t]{k[l]}'s;
2) from the sets of tuples of individuals paired with
P[d]{k[l]}'s delete all tuples containing members of
K-H.
Intuitively, 1) eliminates all interpretations of theoretical
predicates; 2) eliminates theoretical individuals from
interpretations of descriptive predicates. Call the sub-set of
I[H,L[d]] obtained in the way 'Ram(M[T[t]])'. As above,
Ram(M[T[t]]) is a theory (in the semantic sense) about H-P
behavior -- behavior of individuals in H described with
predicates P. 'Ram' is technically a "forgetful functor"
sometimes called the Ramsey functor to suggest the historical
origin of its use in explaining the logical form of empirical
theories.
IV. INTENSIONAL AUGMENTATIONS OF THE DESCRIPTIVE LANGUAGE --
L[i]
Intentional theories, on the account offered here,
essentially involve the attribution of "language use" to some
individuals and the attribution of "sentence tokenhood" to some
other individuals. They also involve attribution of "intensional
attitudes" to the same individuals to which language use is
attributed. Intensional theories MAY (but do not essentially)
involve the attribution of intensional attitudes to observed
individuals that are "shared" by the external observer and
linguistic communication between observer and observed. For the
moment, I ignore this latter aspect of intensional theories. A
somewhat different formulation of the view that intensional
theories have this holistic character may (I think.) be
attributed to Davidson.
IV.1 INTENSIONAL FIRST ORDER SYNTAX (IFOS)
Intensional augmentations of the descriptive language add
intensional attitude predicates together with the requisite
linguistic apparatus to make them work. The linguistic apparatus
permits the observer to talk about the syntactic structure of the
attributed language and identify some observed individuals as
symbol tokens in this language. In addition it provides a means
for describing translation between the observer's descriptive
language and the language whose use she attributes to some
individuals. The key feature of this linguistic apparatus is a
FOS characterization of FOS -- an FOS theory whose models are
FOS's. Both the attributed language and the descriptive language
L[d] are required to be models for this theory. We consider
first the syntax needed for this theory.
Viewed as a set-theoretic structure
L[i]= .
In addition to the logical symbols V, Q and K, L[i] contains
{`,'} which will be used to construct quote names.
The predicates of a first order intensional syntax (FOIS)
L[i] will be:
P[i] = >
where P[d] are the predicates of the "underlying" descriptive
language. The remainder are theoretical predicates analogous to
the theoretical predicates P[t] in the non-intensional
theoretical augmentation. These are discussed in more detail
below.
IV.1.1 FIRST ORDER SYNTAX PREDICATES
The essential feature of intensional theories is a
"language" (call it 'L[a]') whose use is attributed to
individuals. It may be FOS or some other formal structure like
FOS. This language must have two essential features:
1) it must consist of an infinite set of "sentences"
recursively definable over a a finite "alphabet".
2) the sentences must provide a way of characterizing
(denoting) some sub-sets of I[H,L[d]] -- the SAME set of
interpretations that the observer works with.
The attributed language L[a] may be some specific instance
of FOS -- the observer's, or some other. Essentially,
L[a]-sentences function as L[i]-names for possible states of
affairs the observer can describe in L[d] only via USE of
L[d]-sentences. That sentences of L[a] have this property will
be a formal requirement on the interpretation of an intensional,
theoretical augmentation of L[d].
Clearly, we can attribute languages to individuals that are
both stronger and weaker than the observer's L[d] -- in terms of
the model classes they can characterize. For the purpose of
considering the "theoretical power" of intensional languages it
seems natural to require that the the language attributed to
individuals be no stronger than the observer's language.
My discussion will be restricted to attributed languages
that are instances of FOS, though there may well be other formal
structures that satisfy the two conditions above.
To use L[i] to attribute use of some FOS -- L[a] -- we must
first provide L[i] with predicates suitable for describing the
set-theoretic structure of FOS. Ultimately we will use these
predicates to produce an FOS-theory whose models are these
set-theoretic structures. To do this we consider those FOS's in
which only predicates needed for our immediate purpose appear.
Thus, we suppose that there are one-place predicates for all
the symbol types appear in the tuple that is an FOS together with
a 3-place concatenation relation. That is, we have
P[FOS] =
where:
P^ =
is an m+1-tuple consisting of n-tuples
P^{i} =
and P^{i[j]} is simply a one-place predicate; V^ is a p-tuple of
one-place predicates; Q^ a q-tuple of one place predicates;
concat^ a 3-place predicate; F^ and S^ are one-place predicates.
Interpretations of these P[FOS] are the sorts of things that
could be FOS's in the set-theoretic sense -- provided they are
models for FOS-sentences T[FOS] that provide a theory for FOS
structures. They are "potential models" for an FOS theory of
FOS.
So that we can talk about translation between the
descriptive and attributed languages, we need to equip L[i] with
two instances of P[FOS] -- one for the attributed language L[a]
and one for the observer's language L[d]. Call these,
respectively,
PL[a] and PL[d].
Intuitively, these predicates will be true of sentence types and
other symbol types in these languages. Together, they will be
require (by any intensional theory) to be models for T[FOS],
VI.1.2 QUOTE NAMES
L[i] contains apparatus for forming quote-names of symbol
types in L[a] and L[d]. This is needed to talk about attributed
language use and translation between L[a] and L[d]. Quote-names,
rather than simple constants, are needed because we must be able
to read read the intended referent of the name from the syntactic
form of the name. Why this is so will become evident when we
consider intended interpretations for L[i].
The apparatus we employ consists of a predicate P[qn]
interpreted as a set of L[i]-singular term types the form `x'
together with two symbol types ` and ' The meta-linguistic
formation rules for L[i] will assure that concat[i](`,x,')
appears in P[qn] iff x is a symbol type of L[a] of L[d]. See
below IV.1.7.
VI.1.3 TRANSLATION PREDICATE
The syntax of L[i] will contain a predicate 'trans'.
Intuitively, 'trans(`s[a]',`s[d]')' means sentence type s[a] in
L[a] is a translation of sentence type s[d] in L[d]. Just how we
construe 'translation' will be explained below (IV.2.2.3).
VI.1.4 TOKEN PREDICATE
The syntax of L[i] will also contain a predicate 'token'.
Intuitively, 'token(i,`s[a]')' means that individual i is a token
for sentence type s[a] in the attributed language.
VI.1.5 TOKEN CONCATENATION PREDICATE
In attributing language use, an intensional theory will
identify some non-theoretical individuals as tokens for symbol
types in the attributed language. In any model for the theory,
there will be at most a finite number of symbol tokens. In
contrast, there will be an infinite number for formula and
sentence types in F[a] and S[a]. "Laws" of the intensional
theory will require that these symbol tokens have the same
set-theoretic structure as some finite fragment of L[a]. More
precisely, token will be required to be a homomorphism between
the interpretation of concat[token] and the interpretation of
concat[a].
Intuitively, this is a part of the way an intensional theory
"connects" abstract linguistic structures with infinite numbers
of symbol types to observable behavior of a finite number of
individuals. The rest of the way involves saying how observable
behavior involving putative symbol tokens is related to
intensional attitudes -- i. e. characterizing linguistic behavior
in intensional terms.
VI.1.6 INTENSIONAL ABSTRACTION OPERATOR
The syntax of L[i] will contain a unary operation symbol
denoted by 'that'. Intuitively, when s[a] is a sentence of the
attributed language 'that('s[a]')' denotes the class of models
for S[a] into the power set of H-interpretations of L[a] --
Pot(I[H,L[a]]).
VI.1.7 INTENSIONAL ATTITUDE PREDICATES
In addition to predicates intended to describe the syntax of
L[a], L[d], their semantic relations and physical
representations, an intensional language augments L[d] with
predicates P[a] intended to attribute intensional attitudes to
some individuals. Intuitively, these predicates describe
relations between some individuals to whom language use is
attributed and other abstract (theoretical) individuals which are
classes of H-interpretations -- sub-sets of I[H,L[a]] -- denoted
by sentences of the attributed language L[a].
Thus,
a(x,that(`s[a]'),that(`-s[a]'))
might be intuitively interpreted as
'x prefers that(s) to that(not-s).'
To this end, we add to L[d], P[a] an m-1-tuple of predicate
types of orders between 2 and m. Intuitively, we intend the
first place in these predicates to be occupied by a
non-linguistic individual and the remainder of the places to be
occupied by quote-names of sentence tokens of L[a]. We allow for
multiple intensional objects, but only one bearer of these
objects -- no group minds.
IV.1.8 FORMATION RULES FOR L[i]
The formation rules for L[i] work to characterize it in much the
same way that they work in any FOS to obtain:
The major exception is the formation of quote-names for L[d] and
L[a] symbol types. To do this, we need, for each predicate P in
P[L[d]] and PL[a] (except the concat predicate), a clause of the
form
For all X, if P(X) then P[qn](concat[i](`,X,')).
This rather liberal attitude to what is to count as a sentence in
L[i] means that any restrictions on what is "meaningful" will be
made in the semantics for L[i].
Since sentence types in L[a] are effectively treated as
singular terms in L[i], it may appear that L[i]-quantification
into intensional contexts is ruled out. However, this need not be
the case. Consider;
A) There is someone whom Bill believes to have killed Cockrobin.
which one might render in L[i] as:
A') |/\(x) [ Person(x) & Believes(b,that(`Kill(x,c)') ]
The syntax of L[i] apparently can be chosen to admit such a
rendition. If there is a problem, it comes with specifying
interpretation relative truth conditions for sentences like A'.
IV.2 SEMANTICS
IV.2.1 INTERPRETATION
Interpretation, is analogous to that provided for
theoretical augmentations for L[d] above (III.2.1). All
L[i]-predicates except P[d] will be treated as theoretical
predicates. A domain of "ur-individuals" K (H _< K) provides for
theoretical individuals. There are two kinds of theoretical
individuals. First, there are those to provide interpretations
for symbol-type predicates in P[L[a]] and P[L[d]]. Second, the
interpretation of intensional abstraction and intensional
attitude predicates ( See Sec. IV.2.6 and .7 below) requires
enlarging the domain k of EVERY interpretation L[i] to include
Pot(I[H,L[d]]). We regard regard these as THEORETICAL individuals
-- members of K-H. For those who might have ontological scruples
about this enlargement, I suggest restricting the discussion to
finite H's. Intuitively, it would not be too interesting to
discover that the need for an intensional vocabulary hinged on
wanting to talk about infinite sets.
Unlike interpretations for simple theoretical augmentations,
the interpretations of some predicates will have restrictions on
them that go beyond those of set-theoretic type. In most cases,
these restrictions PARTIALLY, but not completely, specify the
meaning of these predicates. One might avoid these restrictions
by including sentences in intensional theories whose models are
restricted in these ways. However, it is not immediately evident
that all restrictions we impose can be replicated syntactically
in this way.
VI.2.1.1 DESCRIPTIVE PREDICATES (P[d])
Interpretations may be provided for the descriptive
fragments of IFOS's, the usual way. These interpretations will
be restricted to H intersect k and simply have the form:
i[d] = .
Intuitively, descriptive predicates are required to be
interpreted with sets of tuples of non-theoretical individuals.
IV.2.1.2 FIRST ORDER SYNTAX PREDICATES
IV.2.1.2.1 ATTRIBUTED LANGUAGE PREDICATES
An interpretation i[a] of P[L[a]] consists of functions
assigning members of members of Pot(k-(h U Pot(I[H,L[d]])) to the
one-place predicates in P[L[a]] and some sub-set of Pot((k-(h U
Pot(I[H,L[d])){3}) to concat[a]. Thus interpretations of these
predicates are restricted to theoretical (abstract) individuals
which are not sets of H-interpretations for L[d]. These
theoretical individuals are introduced just for the purpose of
providing interpretations for symbol types. The only interesting
about them the set-theoretic structure that will be imposed on
them by the "laws" of TL[a]. Depending on our T[L[a]], there may
or may not be non-isomorphic interpretations of P[L[a]].
IV.2.1.2.2 DESCRIPTIVE LANGUAGE PREDICATES
The interpretation i[L[d]] of of L[i]-predicates intended to
describe the syntax of L[d] is structurally the same as i[L[a]].
Intuitively, however, this interpretation should be
considered as "fixed". This means the observer considers only one
syntactic representation of his language even though her theory
of FOS syntax might allow for multiple models. The observer
countenances possibly multiple interpretations of attributed
language because he has no preconceived idea about which of the
possibly multiple models for T[L[a]] observed individuals might
be using. But it's simply hard to see what intuitive sense could
be made of letting in multiple interpretations of the observer's
syntax.
Formally, this means that as we consider model classes
determined by L[i]-sentences we require the interpretation of
P[L[d]] to be the same in all these. These considerations become
otiose when T[FOS] is categorical.
IV.2.1.3 QUOTE NAMES
The functions f{P[qn]}, f{{`,'}} assign disjoint sub-sets of
k - (h U Pot(I[H,L[d]])) to P[qn], {`,'} respectively. The
interpretations ` and ' are required to be distinct for
interpretations of anything else. The interpretation of P[qn]
will depend on the interpretation already given for the
attributed language predicates P[L[a]]. The formation rules for
L[i] assure that, for each predicate P in P[L[d]] and P[L[a]]
For all X, if P(X) then P[qn](concat[i](`,X,')).
(See Sec. IV.2.4 above) Thus, we need only to stipulate further
that
f{P[qn]}('X') = X.
VI.2.1.4 SEMANTIC INTERPRETATION OF L[a]
In addition to interpretation for the descriptive and
linguistic predicates of L[i], an K-interpretation for L[i] must
also provide a SEMANTIC interpretation for the attributed
language L[a]. Interpretation of the attributed linguistic
predicates P[L[a]] provides a syntactic interpretation.
Intuitively, it attributes the use of an FOS to some individuals
(at least in M[T[L[a]]] -- models for T[L[a]] the FOS theory of
L[a]) ). But attribution of full language use requires as well
the the attribution of "meaning" to this syntax.
This suggests that K-interpretations for intensional
languages L[i] have as component parts H-interpretations of the
attributed language L[a]. Formally, this just amounts to
functions that map the linguistic predicates P[a] into the
appropriate types of sets of the domain h < H of the
interpretation i[L[a]] of the descriptive predicates. Thus
i[*] = .
Note that the f[*{P[L[a]]}]'s that appear in in the semantic
interpretation i[*] of the attributed language L[a] are different
from the f{P[L[a]]}'s that appear in i[L[a]]. The latter simply
assign sub-sets of k-(h U Pot(I[H,L[d]])) to all the predicates
regardless of arity. The intended interpretation is sets of
symbol types. The former assign sub-sets of h{n} depending on the
arity n. The intended interpretation is the "meaning" of the
symbol types assigned to the latter.
In L[i]-models for T[L[a]] where the interpretations for
P[L[a]] are FOS's, the semantic interpretation i[*] will provide
"denotations" for members of S[a] -- the sentence types of L[a]
-- via the usual definition of interpretation relative truth.
They may be viewed as denoting their model classes M[s[a]] <
I[H,L[a]]. Outside M[T[L[a]]] we may still assign semantic
interpretations to P[L[a]], but lacking the structure of FOS, the
definition of truth will generally lead to nonsense. More
precisely, recursive definitions will not be able to move away
from their basic cases for lack of structures that satisfy their
conditions.
Note that including i[*] in i[i] is a departure from the
usual way of interpreting FOS. At this point, and only at this
point, we depart from the usual practice of simply assigning
set-theoretic objects to PREDICATES. However, i[*] is described
in the meta-language for L[i] in just the same way as the rest of
i[i] so that semantic paradox is apparently avoided.
VI.2.1.5 TRANSLATION PREDICATE
Now that we have agreed that an interpretation of L[i] must
include an interpretation of the descriptive predicates -- i[d]
-- as well as an interpretation of the attributed language --
i[*] -- we can explain how to interpret 'trans'.
Note first that the arguments of an atomic L[i]-sentence
'trans(a,b)' will be L[i]-singular-terms. They will not BE L[a]-
and L[d]-sentences. Our intention is to interpret 'trans' so
that 'trans(a,b)' will be true only if the singular terms a an b
refer to sentence types. To this end, we begin by interpreting
'trans' as a sub-set of:
f{P[L[a]]}(S[a]) X f{P[L[d]]}(S[d]).
That is, it is interpreted as a set of ordered pairs of
L[a]-L[d]-sentence types. This interpretation of 'trans' assumes
we have already assigned the interpretations to the sentence-type
predicates in P[L[a]] and P[L[d]].
Intuitively, we want to impose further conditions on the
interpretation of 'trans' that capture the idea of INTERPRETATION
RELATIVE sameness of meaning. Having already assigned
interpretations to L[d] and L[a] what (if any) L[d] and
L[a]-sentence types have the same meaning? For example, we could
interpret 'trans(a,b)' to mean something like 'a has the same
syntactic structure as b and the same interpretation of all
predicates'. More precisely,
trans(`s[a]',s`[d]') is true in iff
there is a one-one mapping from symbols
in s[a] to symbols in s[d]
which preserves
syntactic structure and corresponding
predicate symbols are assigned the same
interpretation by both i[*] and i[d].
According to this interpretation, 'trans' entails material
equivalence -- i.e. 'trans(a,b)" is i-true only if 'a' and 'b'
are both i-true or both i-false. But, it is stronger than
material equivalence. One can think of weaker requirements for
the truth of trans that would still be plausible and still entail
material equivalence. One might call this interpretation of
'trans' 'literal translation'.
On any plausible weaker interpretation interpretation, the
truth of trans-sentences depends on the syntactic structure of
its arguments and on specific pairs of interpretations.
Note that a more expressive L[i] could be obtained by
replacing 'trans' with a a one-way translation predicate
'include' and defining 'trans(a,b)' as 'include(a,b) &
include(b,a)'.
IV.2.1.6 TOKEN PREDICATE
Intuitively, domains of non-theoretical individuals consist
of things that can BE symbol tokens -- including sentence tokens
in L[a] and L[d] -- as well as things that can have intensional
attitudes to model classes denoted by sentence types and other
things as well that have descriptive properties. The
interpretation of 'token' is thus simply a sub set of h X (k-h).
Typically, we expect it to be a many-one mapping INTO k-h. That
is, many physical objects may count as tokens of the same symbol.
And some symbol types will not have corresponding tokens. For
example, only some small number of the infinite number of
sentence tokens will be "represented" by tokens in any given
interpretation.
IV.2.1.7 TOKEN CONCATENATION PREDICATE
The predicate concat[token] is simply to be interpreted with
a set of 3-tuples from H intersect k -- that is with 3-tuples of
non-theoretical individuals. Thus it is a non-intensional,
theoretical predicate. Intuitively, it is theoretical because
"tokenhood" and what counts as concatenated tokens is something
that is IMPUTED by the theory -- not something that is a part of
the behavioral data for the theory. One can imagine there being
several ways of imputing tokenhood an concatenation among tokens
that would be compatible with the same behavioral data.
IV.2.1.8 INTENSIONAL ABSTRACTION OPERATOR
The unary operator 'that' is interpreted so that, in the
case that 's[a]' is interpreted in i[i] as denoting an L[a]
sentence type, 'that(s[a])' denotes M[s[a]] -- the H-model class
of s[a]. In all other cases, we simply let 'that(s[a])' denote
the null-set.
IV.2.1.9 INTENSIONAL ATTITUDE PREDICATES
IV.2.1.9.1 INTERPRETATION
Interpretations for the intensional attitude predicates
A =
are more subtle. Intuitively, we take the OBJECTS of individual
i's intensional attitudes to be sets of H-interpretations of the
purely descriptive, non-linguistic and non-intensional, part of
the intensional language. Thus, a K-interpretation with
descriptive domain h assigns to predicates in A{n+1} some sub-set
of
h X (PotI[H,L[a]]){n}.
Each n+1-tuple in this set consists of an individual member of h
plus an n-tuple of SETS of H-interpretations for the attributed
language L[a].
More formally, the interpretation of L[i] will contain
f{A} =
so that
f{n+1} : A{n+1} --> h X (Pot(I[H,L[a]])){n}.
IV.2.1.9.2 INTENSIONAL ATTITUDES AND INTENSIONAL INTENSIONAL
ABSTRACTION
Consider the intensional attitude L[i]-sentence
a(c,t)
where a is in the set of intensional attitude predicates A{2} c
in h, t is a singular term (either a member of P[0] of the form
that(`s[a]')). Intuitively, the idea is that intensional
attitude L[i]-sentences like this one are i-true only if the
singular term denotes s (in interpretation i) the model class of
some L[a]-sentence type.
So far, we have effectively stipulated that the singular
term t denotes a model class of an L[a]-sentence if it is of the
form 'that(`s[a]')'. It still remains open that other
L[i]-constants might be i-interpreted as denoting model classes
for L[a]-sentences or, indeed, other sub-sets of I[H,L[d]]. For
our purposes, it seems clear that this should be ruled out. That
is, the only singular terms in L[i] that denote sub-sets of
I[H,L[d]] are of the form 'that(s[a])'. Intuitively, this means
that the only apparatus in L[i] that can "directly" refer to
these model classes is that provided by L[a]. It is just here
that the potential for increased strength in determining models
classes in I[H,L[d]] could arise.
It should also be noted here that the "observer" the user of
L[i] does not USE sentence types in L[a], she only MENTIONS them
via singular terms of L[i] that denote them. The observer can
also talk about the model classes characterized by these L[a]
sentences both in attributions of intensional attitudes and -- so
far as the preceding discussion has taken us -- in other contexts
as well. We have said nothing yet that rules out attributing
descriptive (L[d]) predicates to model classes. Thus, we might
say something like
heavier_than(george,that(it's raining)).
It is not completely obvious that we want to rule this out.
In general, we do not want to preclude attributing descriptive
properties to theoretical individuals (See III.3 above). For
example, we attribute (descriptive) spatial properties to
(arguably, theoretical) electrons. However, for present
purposes, it appears natural to regard a predicate's
attributability to model classes as a sufficient condition for
taking it to be intensional. Thus, we require descriptive
predicates to be interpreted with sets tuples of non-intensional
individuals -- either non-theoretical or theoretical, but
non-intensional. An intensional individual is just a member of
Pot(I[H,L[d]]).
IV.2.2 TRUTH AND MODELS
An interpretation of L[i] will have the form:
i[i] =
Each member of the tuple i[i] will be described below.
IV.2.2.1 TRUTH DEFINITION
A definition of 'true in interpretation i[i]' for
L[i]-sentences can be provided in the something like the usual
way. The interpretation of predicates and singular terms leads
in the obvious way to i[i]-truth definitions for atomic
sentences. Once this is done, sentential connectives and
quantifiers work as they usually do.
IV.2.2.2 OPACITY OF INTENSIONAL CONTEXTS
It should be noted that referential opacity of intensional
contexts will fall out of this truth definition in a natural way.
Suppose that
a = b
and
a(x,that(`P(a)')
are both i[i]-true. It will not then generally be the case that
a(x,that(`P(b)'))
is also i[i]-true. For,
that(`P(a)') /= that(`P(b)').
that(`P(a)') denotes the set of interpretations in which the
denotation of 'a' is in the sub-set of h denoted by 'P', while
that(`P(b)') denotes this set of interpretation in which the
denotation of 'b' is in the sub-set of h denoted by 'P'. These
two sets of interpretations are isomorphic under permutation of
tuples and in the functions that comprise the
interpretations. But, they are not identical.
This may be clarified by making explicit one feature of our
concept of interpretation. Our interpretations are tuples of
functions
f{i}: {P{i} --> Pot(k{i}).
These functions are sets of ordered pairs of the form
where p is a linguistic symbol type (predicate of constant) and M
is a set of tuples from k.
In semantic formulations of theories (for example, those
given by informal definition of a set-theoretic predicate), it is
usually just the VALUES of the f{i}'s that appear in the theory's
"models" -- the M's. These values appear a members of ordered
tuples. Their position in these tuples serves to identify and
distinguish them -- e. g. to say which set of tuples is the
"heavier-than" relation and which is the "longer-than". Here, it
is the arguments of the f{i}'s that identify these sets of
tuples.
This entails that there is some ambiguity involved in
talking about L[d]-theory "determining a model classes". On one
hand, it is perfectly clear to say that L[d]-sentences determine
sub-sets of I[H,L[d]]. Members of these subs sets are
INTERPRETATIONS of L[d] in the sense just described. But these
sets are not EXACTLY the same as sets of models for a
"corresponding" theory provided by an informal definition of a
set-theoretic predicate. In fact, there will generally be a
many-one correspondence between the set of interpretations of
L[d] determined by an L[d]-theory and the set of models
determined by a "corresponding" informal definition of a
set-theoretic predicate. L[d]-interpretations that differ
"trivially" in that different constants and/or predicates are
assigned to the same tuples will all correspond to the same model
for the set-theoretic predicate.
Intuitively, our concept of interpretation makes explicit
EXACTLY HOW linguistic symbol-types are (literally) mapped onto
(small parts of) the world. It is just this explicitness that
makes intensional contexts opaque. The singular term
'that(`P(a)')' denotes a different set of interpretations than
the singular term `that(`P(b)')' just because the constants 'a'
and 'b' are mapped onto the world in different ways.
Note, as well, that essential to referential opacity of
intensional contexts is the fact that intensional objects --
that(`s[a]') -- denote, in interpretation i[i], set-theoretic
objects constructed from individuals outside the domain h of
non-theoretical individuals. Aside from the L[d] symbol types,
members of H-h appear as well. On one view of intensional
objects, these individuals H-h are "possible individuals", while
those in h are "actual individuals". On the present view,
members of H-h are no less real than members of h. On this view,
theories -- wether expressed in formal or informal languages --
generally have multiple models. All these models consist of
real, actual individuals. The purpose of theorizing is not to
characterize the world as a whole, but rather a number of small,
possibly overlapping, fragments of the world.
IV.2.2.3 THEORY OF MEANING?
One might expect, having provided a semantics for L[i], to
be able to say something about the logical consequence relation
among intensional members of S[i]. Are there interesting,
general things to note about when s is true in all
interpretations in which s' is true? For example, can we define
"logical consequence" in such a way that
Bill believes someone killed Cockrobin.
turns out to be a logical consequence of
Bill believes Socks killed Cockrobin.
Clearly, we can not. The reason is evident. We have placed no
limitations at all on how the sets of H-interpretations assigned
to intensional attitude predicates are to be related. This is
rather like failing to place conditions on truth value
assignments that make them "normal" with respect to the truth
functional connectives. "Somehow" configurations of
interpretations of intensional attitude predicates must be
constrained by properties of the objects of these attitudes. To
show "just how" is to provide a "theory of meaning" for
intensional sentences.
There are basically two ways to proceed. One way is to
enrich the concept of L[i]-interpretation in such a way that
limitations on how sub-sets of I[H,L[d]] are assigned to
intensional attitude predicates are built-in to the concept of
interpretation. The other way is to leave the concept of
L[i]-interpretation relatively weak and allow the meaning of
intensional concepts to be further constrained by the laws of
intensional theories.
Following the second line, one might (at most) regard the
concept of interpretation provided here as a preliminary step
toward an interesting theory of meaning. Returning to our
initial question, is there anything interesting that can be said
about the expressive power of intensional languages without
saying more about a theory of meaning for such languages?
IV.3 INTENSIONAL THEORIES
Intensional theories are simply sets of L[i]-sentences in
T[i] with some distinguished sub-sets.
IV.3.1 ATTRIBUTED LANGUAGE THEORY
T[L[a]] are FOL sentences characterizing the syntax of L[a].
In the case that L[a] is an FOL it is apparent what these
sentences must like. Their models must have the set-theoretic
structure characteristic of the syntax of FOL. T[L[a]] will also
require that the token relation be a homeomorphism between
concat[token] and a fragment of concat[a].
IV.3.2 PURELY INTENSIONAL THEORY
T[a] is the "purely intensional" part of T[i]. It
characterizes the structure required of the entities that are
models for intensional attitude predicates.
An example of a plausible T[a] is the purely qualitative
fragment of Jeffrey decision theory (the Jeffrey-Bolker axioms)
-- more precisely an FOL axiomitization of this theory slightly
modified to accommodate the present model theoretic conception of
the objects of the intensional attitudes. This theory deals with
the intensional attitudes
x believes a is at least as likely as b = more_likely(x,a,b)
x weakly prefers a to b = pref(x,a,b)
Recalling that intensional objects are SETS of interpretations,
two requirement of this theory can be rendered as:
A) If b =< a then more_likely(x,a,b).
B) If more_likely(x,a,b) & more_likely(x,b,c)
then
more_likely(x,a,c)
To see how such a theory might reproduce plausible
inferences about beliefs, note that belief simpliciter is
rendered in this theory as
C) believe(x,a) iff more_likely(x,a,that(`P v -P'))
Now note that
that(`P(a)') =< that(`|/\xP(x)').
Thus, using A), B) and C), from
believe(x,that(`P(a)'))
we may infer
believe(x,that(`|/\P(x)'))
Transitivity is also required of the pref relation:
If pref(x,a,b) & pref(x,b,c) then pref(x,a,c)
but no plausible conditions connecting set-theoretic properties
of a and b with pref (analogous to A) above) are readily
apparent. This suggest that interesting inferences involving
pref alone are not likely to be found.
IV.3.3 PURELY DESCRIPTIVE THEORY
T[d] is the "purely descriptive" part of T[i]. It is what
the observer believes about the situations in question that can
be expressed in the purely descriptive vocabulary.
IV.3.4 FULL THEORY
Clearly, no plausible T[i] can be just the union
(conjunction) of T[L[a]], T[a] and T[d]. Nor can it be any purely
set-theoretic (truth functional) combination of them. There has
to be some quantificational link among the components.
T[i] must be require some kind of connection between (some
of) the sentence tokens whose intensional abstractions fill the
intensional object places in T[a] and the predicates appearing in
T[d]. Crudely, the attribution of intensional attitudes must have
some "descriptive import".
How should this work?
IV.3.5 LINGUISTIC ACTIONS
First, consider "linguistic actions" like asserting,
questioning, commanding, etc.. Within the present framework, it
seems natural to regard these as MANIFISTED BY "descriptive" or
"observable" relations between non-linguistic individuals and
sentence tokens. But these descriptive relations are connected
by T[i] to attributions of intensional attitudes. That is, Hans
shouting 'Tur schliessen!' (a relation between non-theoretical
individual Hans and a disturbance in the ambient atmosphere -- a
token for the sentence type "Tur schliessen!" and also a
non-theoretical individual) counts as Hans commanding that the
door be shut only in models where certain intensional relations
are also attributed to Hans, 'Tur schliessen' and perhaps other
sentence tokens as well.
Thus, in T[i] one might expect to find sentences like:
command(x,that(`s[a]') iff |/\(y) [token(y,`s[a]') & d(x,y) & ...
where d(x,y) is some purely descriptive relation and '...'
indicates more conditions, either descriptive or intensional;
IV.3.6 PSYCO-PHYSICAL LAWS
What about other, non-linguistic, actions? Intuitively, the
obvious tack here is to suppose that T[i] contains
"psyco-physical laws". That is, T[i] requires that some
configurations of intensional attitude attributions entail the
"truth" of some of the sentence tokens appearing as objects in
these attitudes. For example, if T[a] is Jeffrey decision theory
then for SOME sentences s among the sentences of L[a] it is
plausible to suppose T[i] contains something roughly like:
pref(x,that(`s[a]'),that(`-s[a]')) & d(x,...) &
trans(`s[a]',`s[d]') --> s[d].
That is, under certain descriptive conditions described by
'd(x,...)' x's preferring that(`s[a]') to that(`-s[a]') entails
s[d] when `s[d]' is a translation of `s[a]' -- e.g. when s[d]
describes something that x can do in circumstances d(x,...).
Similarly, one might expect that for some perception
predicates like "sees" laws of the following form might appear:
s[d] & l(`s[d]',...) & d(x,...) & trans(`s[a]',`s[d]') -->
sees(x,that(`s[a]')).
That is, under certain descriptive conditions described by
'd(x,...)` and for certain kinds of sentences described by
l(`s[d]',...), whenever s[d] (is true) x sees that s[d]. A
"causal" theory of perception might be formulated in this way.
IV.3.7 HOLISTIC THEORIES
Having considered the possibility that T[i] might contain
psyco-physical (and psysi-psychological) laws to clarify our
conception of the apparatus permitted in L[i], we may consider
not the possibility that T[i] can have "descriptive import"
without sentences of these forms. That is, T[i] might contain no
sentences that had purely descriptive or purely intensional
sentences connected to others as consequents in in universally
quantified implications. Roughly, there are no sentences in T[i]
that might count (even as conditional, partial) definitions of
intensional predicates in terms of descriptive (or conversely).
This kind of holism is commonplace in theories from physical
science. In such theories various theoretical concepts are so
tightly interwoven with each other and with non-theoretical
concepts that only in a few, very special models of the theory
can one make inferences from fully non-theoretical sentences to
fully theoretical (and conversely). Nevertheless, such theories
do have descriptive, non-theoretical import. That is, they serve
to characterize an non-trivial class of non-theoretical models.
IV.3.8 MODELS FOR INTENSIONAL THEORIES
IV.3.8.1 THEORETICAL MODELS
Interpretations of L[i] have the form:
i[i] =
The set of all such interpretations -- relative to a fixed
ur-domains H, and K of non-theoretical and theoretical
individuals (H _< K) is I[H,K,L[i]]. Each set of L[i]-sentences
T[i] -- an L[i]-theory -- determines of sub-set of I[H,K,L[i]],
M[T[i]]. The set of all sub-set of I[H,K,L[i]] that can be
determined in this way is |M[L[i]].
IV.3.8.2 NON-THEORETICAL MODELS
Each i[i] in I[H,K,L[i]] corresponds to exactly one member
of I[H,L[d]] via a functor Ram such that
Ram(i[i]) =
where h = k intersect H and k and i[d] are respectively the first
an second members of i[i]. Intuitively, Ram just wipes out
everything but the first and second members of i[i].
Extending Ram to operate on sets, we note that each
L[i]-theory T[i] determines a sub-set of I[H,L[d]], Ram(M(T[i])).
This we call 'the descriptive content' of T[i]. The "empirical
claim" of T[i] is that all L[d]-descriptions of observed behavior
-- L[i]-descriptions of behavorial data -- are to be found in
Ram(M(T[i])).
IV.3.8 INDETERMINACY OF INTENSIONAL CONCEPTS
The question of wether some specific description of of
putative behavorial data -- some specific member of I[H,L[d]],
i[d] -- is in the content of L[t] is essentially this. Is there
SOME theoretical augmentation if i[d], i[i] that is in M[T[i]]?
Except in very special cases, when the answer to this question is
affirmative, there will be multiple theoretical augmentations of
i[d] to models for T[i]. That is, the intensional theoretical
concepts required to demonstrate that i[d] is in the content of
T[i] will not be uniquely determined. More intuitively, there
may be a variety of ways to impute linguistic behavior and
intensional attitudes to members of h that satisfy the laws of
the intensional theory T[i]. This may be so even though the
theory T[i] is non-trivial -- in the sense that Ram(M[T[i]]) is a
proper sub-set of I[H,Li[d]].
This kind of indeterminacy of theoretical concepts is common
in theories from the physical sciences. Indeed, it remains even
when these theories are strengthened by conditions -- so-called
'constraints' -- that operate across different models for the
theory. Thus, there is every reason to expect that intensional
theories will exhibit the same lind of indeterminacy.
IV.3.9 TRANSLATION MANUALS AND THEIR INDETERMINACY
Indeterminacy of attributions of "meaning" to attributed
language is one aspect of the indeterminacy of intensional
concepts has received considerable attention within the framework
of somewhat different formulations of the issues at hand.
Intuitively, we may regard the triple
as a meta-linguistic "translation manual" (in the Quinean sense)
between the observer's descriptive language L[d] and the language
L[a] which he attributes to some of the individuals he observes.
At least we may make this intuitive identification in M[T[L[a]]]
-- those models for the descriptive-linguistic part of the
language L[i] in which interpretation of the symbol-type
predicates of the attributed language have the formal properties
of an FOS.
In the absence of further restriction on the models of
interest, it is clear that there will be a multiplicity of
possible translation manuals. Further restriction on the models
is provided by intensional theories T[i] can not be counted on to
completely eliminate this. There will still be a multiplicity of
translation manual triples intensional augmentations of an i[d]
that are in M[T[i]]. That is, there will generally be a
multiplicity of translation manuals compatible with the
behavioral data.
Some features of this multiplicity are worth noting. First,
there may be multiple possibilities for the i[*] associated with
some fixed i[d]. Clearly, our intensional theory of
L[d]-described, H- behavior will allow for different instances of
this behavior, i. e. different i[d]'s. It could be the case that
each of these i[d]'s had associated with it (in M[T[i]]) exactly
one corresponding i[*]. In this case we would say that the
theory T[i] uniquely determined the interpretation of the
attributed language. In the case that there were multiple i[*]'s
associated with the same i[d] we would say that T[i] countenanced
an "indeterminacy of translation". This appears to be the kind
of SEMANTIC translation indeterminacy discussed by Quine.
There is, however, a further possibility for SYNTACTIC
translation indeterminacy that becomes explicit in this
formulation. For a fixed there may be multiple
possibilities for interpreting 'trans'. Wether there are depends
(in part) on how strong a notion of "syntactic translation" we
build into the interpretation of 'trans'. Intuitively, this
indeterminacy appears to be identifiable as that commonly
encountered (even by true bilinguals) in rendering text of one
language into another.
It should also be noted that some semantic translation pairs
might be compatible only with a null-set interpretation of
'trans'. That is, on some acceptable semantic "translations"
there might be no way to identify sentences as having the same
meaning.
V. COMPARISON OF THEORIES
Our question is roughly this. Are there any descriptive
model classes that can be characterized by an intensional theory
that can not be characterized by a non-intensional theory? More
precisely, are there any descriptive model classes that can be
characterized by an intensional theory than can not be
characterized by a theory using only non-intensional theoretical
concepts?
There are at least two interesting ways to make this
question precise. One way is to take the descriptive language
L[d] and ur-domain H to be fixed; the other is to consider all
possible descriptive languages and ur-domains.
First, consider the case of a fixed L[d] and H. Here the
question is:
Is it the case that:
Given L[d] and H, for all intensional theoretical
augmentations of L[d], L[i] and all L[t]-theories T[i]
there is some non-intensional theoretical augmentation
of L[d] L[t] and L[t]-theory T[t] such that
Ram(M[T[i]]) = Ram(M[T[t]])?
Intuitively, we have settled on the kind of behavior to be
theorized about by fixing L[d] and H. We simply want to know
wether there is anything we can say about this kind of behavior
using intensional concepts that we could not say using
non-intensional, theoretical concepts.
Next, consider the more sweeping question: is there any
kind of behavior that demands intensional concepts for its
characterization?
Is it the case that:
For all L[d] and H, for all intensional theoretical
augmentations of L[d], L[i] and all L[t]-theories T[i]
there is some non-intensional theoretical augmentation
of L[d] L[t] and L[t]-theory T[t] such that
Ram(M[T[i]]) = Ram(M[T[t]])?
I confess that, at this point, I have no idea how to answer
either of these questions. Supposing you conjecture that the
answer to the second question is negative, the natural strategy
is to try to produce an counterexample. But, even at the
intuitive level, it's not clear to me what kind of use of an
intensional language might provide counterexamples. Some kinds
of potential counter examples would clearly be unconvincing -- e.
g. those that depended on things like the cardinality of domains
and arity of predicates. Should one be able to produce them,
reformulating the question to rule them out would appear to be in
order.