Fachgruppe Sprachwissenschaft
Universität Konstanz
Arbeitspapier 71-1
Definiteness, Binding, Salience,
and Choice Functions
Urs Egli
Definiteness, Binding, Salience,
and Choice Functions
Urs Egli
University of Konstanz1
[email protected]
Contents
1. The Intuitions
2. Russell´s Iota Operator
3. Four Theories
3.1 Evans´ Theory of E-Type-Pronouns
3.2 Theory of Salience by Lewis
3.3 The Epsilon Operator and Ordinal Numbers
3.4 Thematization of the Rhema and Rhematization of the Thema
4. A New Formal Reconstruction: Modified Epsilon Operators
5. Prospects
1. The Intuitions
The definite article in combination with a descriptive phrase yields a definite noun
phrase. Examples are as follows:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
the island
the island which lies in the sun
the university
the university whose windows are lighted up
the one university
the other university
this university
the first island
1 I thank Barbara Partee, Christoph Schwarze, Arnim von Stechow, Klaus von Heusinger, Carla
Umbach, Anne Malchow, Uta Schwertel, and Antonio Quaranta for stimulating discussions on the
content of this paper. I thank Miriam Butt for providing English translations of German manuscripts
of parts of this work. A preliminary version of this paper was also presented at a colloquium of the
group of Professor Mahr at the Technische Universität Berlin. The work reported on in this paper
was supported by the Deutsche Forschungsgemeinschaft.
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U. EGLI
Noun phrases of this kind are also called descriptions because they can be used for
the description of an individual in a real, an experienced, or an imagined and
discussed situation. They identify an individual through the characterization
expressed by the predicative phrase. In a given situation, it is usually the case that
several individuals fit a given characterization. Descriptions can nevertheless be
used in these cases. The number of individuals can either be limited by providing
further characteristics of the individual, or by imposing a linear order on the
individuals contained in the situation. From the islands which may be contained
within a given situation, an island can be singled out as "this island" or as "the first
island" or as "the one island", and can thus be differentiated from the other islands.
The other islands are instead contrastingly referred to as "that island", "the other
island", or "the second island". A characterization becomes increasingly successful
as the number of items described by it decreases. "Give me the mammal" is
infelicitous when a dog is intended and there are both dogs and cats present. On the
other hand, it is felicitous to refer to "the mammal" in the Department of
Paleontology, where a single mammal is exhibited among a host of reptiles,
especially since the genus of the mammal may not be well-known.
When no other characteristic exists which would allow a more detailed
differentiation, I can say "take a coin", by which the first coin, the coin actually
chosen, is meant. "Take another coin" then refers to the second coin, etc. I can
identify the first and the second coin through properties which may only be known
to me: the first coin is Greek, the second Roman. In this situation the first coin is
more elaborately identified through the description "the Roman coin". In a
completely different situation, where the first coin is Parthian , the description is
identical with "the Parthian coin". The expression predicated of a described
individual can be inserted into the description as an adjective (in certain cases at
least), or as a relative clause. If the characterization is rendered unique through this,
the ordinal number "the first" is omitted because it is now no longer needed. The
process of inserting the predicate into the description can also be referred to as the
thematization of the rheme. This is because a predicate can be treated as the rheme,
and a subject or complement noun phrase as the theme.
Four important characteristics of descriptions have thus been arrived at so far:
1. The interpretation of descriptions is situation dependent.
2. It is possible to identify several descriptions with respect to a situation.
3. Based on the situation, or on the perspective taken of the situation, it is possible to
impose an artificial order on the set of individuals identified by a description
4. The thematization of the rheme.
DEFINITENESS, BINDING, SALIENCE AND CHOICE FUNCTIONS
3
The intuitions for definite noun phrases presented here are almost entirely in
agreement with the notions Schwarze (1986, 41ff) assumes for referential
expressions. However, in the position taken here, not quite as absolute a distinction
is made between the three kinds of referential expressions: n a m e s, deictic
expressions, and descriptions. Although a treatment of names is not included here, I
assume that they are much more context sensitive than has been generally assumed
("Hans" in one situation is not equivalent to "Hans" in another situation).
Within sentence semantics, the interpretation of the definite article "the" generally
relies on the methodology first proposed by Bertrand Russell. His method of
interpretation has one great disadvantage: it presupposes the uniqueness of
individuals which fall under a given characterization. According to Russell, if I
speak of "the king of France" then I presuppose that only exactly one king of France
exists. Note that Gottlob Frege has already discussed some of the issues raised by
this presupposition. It does not play much of a role whether I assume that a
sentence like "The king of France is wise" is false, when the precondition that there
be a king of France is not satisfied, as assumed by Russell, or whether I assume that
in this case the sentence is inapplicable and cannot be used, as proposed by Peter
Strawson. In the one case we can continue to rely on two-valued predicate logic, in
the other we have to assume some kind of three-valued predicate logic or partial
logic in which declarative sentences or statements are allowed to carry a third value
besides true and false (neither true nor false). This yields a "dreiwertige Logik der
Sprache [three-valued logic of language]" (the title of a book by Ulrich Blau).
If a true rendering of natural language is intended, then the uniqueness condition
should not be allowed to appear in the semantics of the definite article.
Furthermore, the dependence on situation and time must be accounted for.
Does a condition of uniqueness indeed need to be assumed? The discussion so far
appears to indicate that a meaningful description has to build on a characterization
which denotes at least one individual. What happens in the case where no
individual is described by a given characterization? It is relatively natural to express
that a round square can be neither round nor square, since round squares do not
exist. It can therefore be concluded from the non-existence of round squares that a
round square cannot be simultaneously round and square. This is a fifth property of
the definite article and descriptions: When no individual is selected by the
characterization B in a description, then it is quite natural in many situations to
express this as the fact that even the B cannot be a B.
Is there a logical constant which could replace the Russellian "the" and for which the
facts would fall out to be as described in this section? I will argue that a
modification of Hilbert's Epsilon Operator yields such a logical constant. The
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U. EGLI
operator is semantically equivalent to a selection function which assigns to every set
an individual selected from within that set, and to every empty set a random
individual from the domain of individuals assumed. It should already be
emphasized at this point that the intuitions behind this operator are the same as
those for "the" examined in this section. They interact with the intuitions for "a" so
that "a" is used instead of "the" when the discourse referent must be presupposed to
be unknown. I further propose that a lexeme for "the" exists which reconstructs the
intuitive meaning. It is also true of this lexeme that many of the meanings assumed
for sentences with "the" are not the direct meanings of such sentences, but can only
be arrived at through indirect derivations. This leads to the consequence that the
compositional semantics of sentences with "the" become relatively simple.
However, a few usages of "the" and "a" are not reconstructed. In particular, the
reference to genus or species as in the sentence "The lion is found in Africa" cannot
be accomplished by the semanteme as proposed because it reconstructs descriptions
referring to individuals. The theory argued for here would also not be crucially
affected if it became apparent that the uniqueness condition must be upheld after
all. It is possible to make a simple modification of the just discussed logical
mechanism with partial selection functions, which are not defined for empty sets.
The mechanism is applicable to the reconstruction of such intuitions.
2. Russell´s Iota Operator
Russell's (1905) "On Denoting" analyses natural language sentences of the subjectpredicate type. Some examples are given in (9)-(11).
(9) Every logician is a pipe smoker. (J. M. Bochenski)
(10) Every linguist today is also a logician.
(11) The king of France is wise. (B. Russell)
The interpretation of the sentences in (9) and (10) is performed in the same manner
as already employed by Frege in his work on ideography [Begriffsschrift] in 1879.
New and interesting is Russell's proposal for the definition in context, which in our
example is (11), of the phrase "the king of France". Many of the philosophers of that
time, for example Moore, later admitted that they did not understand the theory at
first. The theory perhaps only becomes fully understandable when its application is
not taken to be within the domain of natural language, but instead is applied to an
artificial language within the domain of logic. The paraphrase of (11) is thus (11a).
(11a) There is exactly one king of France and every king of France is wise.
DEFINITENESS, BINDING, SALIENCE AND CHOICE FUNCTIONS
5
It is immediately apparent that this sentence, which in natural language sounds
somewhat artificial, but is completely natural if rendered in quantification logic, is
false when there is no king of France. This is Russell´s solution of the so-called
problem of presupposition which Strawson, and before him Frege, pointed out. The
second problem is the dependence of such expressions on context and time, which
Strawson discussed as well.
The use of "the" under this definition is also problematic within the domain of logic.
Its technical description is that of a iota operator because it is represented by a
reversed "iota". The modification of the definition proposed by Frege and Carnap
which consists in attributing a value to the definite description if the uniqueness
condition fails, removes the technical difficulties, but it is so artificial that it cannot
be applied to natural language semantics.
A further development was made possible by Russell's formulation of type theory in
1908. The noun phrases in (9) to (11) can now once more receive a phrase meaning
thereby recasting Russell´s definition in context as an explicit definition. This goes
against the spirit of Russell in a certain sense, and was therefore proposed for the
first time only by Montague. The noun phrases now can be assigned the following
meanings.
(9b) "every logician" denotes the entire set of characteristics which every logician
has. If every logician smokes a pipe, this means that being a pipe smoker is one
of the characteristics every logician has.
(11b) "the king" denotes the set of characteristics for which it is true that every king
possesses them, whereby exactly one king exists.
The truth conditions for these constructions are logically equivalent to Russell's
formulation. By means of standard logic tools like lambda abstraction one can then
also determine type-logical entities which correspond to the words "every" and
"the". These have to be a function which assign every meaning of a noun phrase
meaning. One simplification, which is often employed, serves to reduce this
function to a simple relation between two general noun meanings. This implies a
view of sentences as being composed of a general noun as subject (of a different
type from what we have looked at so far), a copula, which corresponds to the
relation, and a second general noun. The current, widely accepted theory of
quantifiers examines such possible Aristotelian copulae, whereby the theory could
be formulated more elegantly through the notion of variable binding than it was
possible for Aristotle. Sometimes the first noun is then also referred to as the
restrictor of the quantifier. There are four examples of Aristotelian copulae. But the
expressions "most" or "more than two", for example, cannot be taken to be true
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U. EGLI
Aristotelian copulae anymore. There are many more quantifiers than Aristotle´s four
and Russell´s two or Frege´s one. Aristotle did not recognize the general nature of
the concept. This shows once more, even with due respect to tradition, that many
details did have to be developed further significantly.
Montague's third step, which is actually not immediately relevant for an analysis of
quantifiers, consists in intensionalising these concepts, i.e., their combination with
the notion of possible worlds.
The theory of generalized quantifiers and the theory of anaphora, to which the
examination of the definite noun phrase belongs, is also influenced by the classic
theory of Montague, who characterized quantifiers semantically as functions of sets
of individuals into sets of sets of individuals (the semantic counterparts of noun
phrases). Following Barwise and Cooper (1981), I here simplify the intensional
theory to an extensional version, which already contains everything of significance.
More recent research, like Keenan (1987), has attempted to distinguish between the
objects of the right type and those that really have the nature of a quantifier. In
effect, Keenan relies on a modified version of the classic cardinal number and the
structural properties in the tradition of Frege, Russell and Carnap (e.g., Carnap 1968,
137-143). A new development is also expressed in the works of van Benthem, who,
among other things, characterizes quantifiers by the conclusions that are valid for
them.
Let me digress shortly to give some information on the historical development of
the semantics of the quantifier "every" developed by Montague. It should be noted
that David Lewis already briefly pointed out in 1972 that Montague´s construction
of the universal generic character for the semantics of a noun phrase is closely
related to Lotze's theory of abstraction. In particular, it corresponds to his notion of
"Allgemeines Thier [general animal]". This animal has all the properties every
animal has, but not those properties which individual animals have in addition. If
every animal is mortal this means according to David Lewis that to be mortal is one
of the properties which constitute the general animal. David Lewis thus has applied
the theory of abstraction which was meant to provide an analysis of the general
concepts by giving them some sort of intension of comprehension to the problem of
giving a semantics of the universal noun phrase. Note that he also solved the
problem how to disentangle the two notions in the tradition, viz. the problem
whether the general animal has every property some animal has, i.e. conflicting
properties, or only the properties every animal has. These are two different notions
called the particular and the universal generic animal corresponding to some animal
and every animal, respectively. If the unrestricted quantifier "every" is considered
instead of the restricted quantifier "every animal", one arrives at something like "the
general (entity)" which has all the properties that every individual has, but not other
DEFINITENESS, BINDING, SALIENCE AND CHOICE FUNCTIONS
7
specific properties of the individuals. I believe that the origin of Frege's notion of
"generality" as a second-level notion can be traced back to this interpretation. Lotze
explicitly speaks of the notion of Allgemeinheit [generality]. Perhaps Frege has
followed the same line of argumentation for unrestricted quantifiers as David Lewis
has pursued for restricted quantifiers. If I am not mistaken, this is a further
contribution to the debate over the dependence of Frege on his teacher Lotze, and
one which could be added to Gottfried Gabriel's elegant discussion in his edition of
the first book on Lotze's logic in the Philosophical Library with Meiner, which
appeared in 1989. My work in 1979 already contains a discussion of the history of
the concept of the universal animal pointing to Stoic antecedents preserved in the
criticism of the Stoics by Sextus Empiricus.
3. Four Theories
3.1 Evans' Theory of E-Type Pronouns
A theory which is also regarded as promising today is the theory proposed by
Evans (1980, first published in 1977). It is called the theory of E-type pronouns and
uses definite description in order to reconstruct certain anaphoric pronouns. The
iota-operator is characterized by the following rule:
F ιx Gx is true exactly when there is a unique G and all G are F.
The condition that there be a unique G is called the uniqueness condition. In this
paper, the Evans-type pronouns are recast as epsilon-type pronouns. A good
comparison of DRT with the theory of E-type pronouns is found in Heim (1990, in
circulation since 1987 as a manuscript). Note, however, that we propose a twofold
modification of the theory of E-type pronouns in that we replace the ι- operator by
the ε-operator and in that we give different representations for E-type pronouns and
overtly anaphoric definite noun phrases, thereby being able to prove the
equivalence of formulations of donkey sentences with anaphoric pronouns and
donkey sentences with anaphoric definite noun phrases without stipulating this
equivalence.
3.2 Lewis' Theory of Salience
Important is also the theory formulated by Lewis in (1979). The present approach
tries to integrate this theory into the formulation of definiteness by the epsilon
operator. It proposes that individuals which fall under a given description are
ordered linearly according to a particular situation. This allows for the selection of
the most salient individual characterized by a given description, and also for the
selection of the next most salient, etc.
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U. EGLI
The proposal put forward here essentially involves a reformulation of the E-type
pronouns by substituting a modification of Hilbert's epsilon operator for Russell's
iota operator as a semanteme for definite and indefinite articles. I argue that some of
the problems surfacing recently in the literature can be solved through a
development of the theory of E-type pronouns which makes use of Hilbert's epsilon
operator.
Very important for the interpretation of definite noun phrases are anaphoric
relations, also those of non-pronomina, e.g. the island, the first, another, etc. In
relation to this, some questions need to be answered as a preliminary to deeper
issues:
a) old : new = the : a
b) mentioned first : mentioned second = that : this = the first : the second
c) only one mentioned : only two mentioned : more than two mentioned - the :
the one + the other : the first + the second + the third.
Identities which fall out from the distinctions listed above:
a) a bishop = the bishop = the other bishop = the first bishop
b) another bishop = the other bishop = the second bishop.
Contextual salience relations are formally determined by a selection function.
Intuitively, they correspond to the facts given by the text we interpret. Hilbert's
epsilon-operator is made use of in order to be able express these salience relations .
3.3 The Epsilon Operator and Ordinal numbers
Hilbert's epsilon operator is a variable binding term operator which semantically
expresses a selection function. This function assigns every non-empty set a selected
individual which is an element of the set. The model-theoretic definition can be
given as follows:
x/a
[[ εxα ]]g = f({a : [[ a ]]g
=1}), whereby f is a selection function determined
by the model.
Note that there is no analog to the uniqueness condition of the iota operator. Either
a weaker existence condition must be assumed, whereby the selection function
deviates from Hilbert's formulation in that it is not defined for the empty set, or one
can follow Hilbert and say that when the existence condition is not satisfied, the
epsilon operator is still meaningfully defined: it can express generality. This is
further motivated below. Formally, the following laws govern the epsilon operator.
DEFINITENESS, BINDING, SALIENCE AND CHOICE FUNCTIONS
9
The statements made previously can be derived from them.
F εx F x iff there exists an x such that F x.
F εx not F x iff for all x Fx.
I call these rules the first and second Hilbert rules. Note that the second Hilbert rule
is a consequence of the first.
Colloquially I refer to the epsilon expressions as "the selected x so that" or simply,
"the x such that", as the epsilon operator is used to reconstruct the intuitions for the
colloquial "the".
Each selection operator induces a salience hierarchy within each set. This means that
the intuitive order of the individuals which fall under a given description can be
reconstructed logically through a recursive definition by means of the epsilon
operator. This salience hierarchy can sometimes be interpreted according to Lewis
(1979). Formally, the n+1st island can be defined as the most salient island which is
not identical with the mth most salient island for every m less than n. A new theory
of ordinal numbers in the grammatical sense of a Latin grammar (i.e, a theory of the
colloquial use of "the first", "the second", etc.) can now be formulated. This theory is
an important step towards the solution of some of the currently most pressing
problems, and it utilizes a completely different method of reconstruction than the
one advocated within set theory. This method differs from the set theoretic
reconstruction of ordinal numbers just as much as the theory of generalized
quantifiers differs from the theory of cardinal numbers in general set theory. In
particular, the role of the ordinal numbers in language should be explicated within
anaphoric usage. Examples are noun phrases like (1). It is about the most salient,
most prominent, most conspicuous island being talked about. The second most
salient island can be described through the phrase "the second island", as opposed
to (8).
The formal definition of ordinal numbers in the sense discussed above can be
achieved through an simple definition, where n stands for the cardinal numbers for
which the Peano axioms are valid:
the 1st F = εx: Fx
the n+1st F = εx(Fx and for every m less than n+1: x is not identical to the mth F)
To sum up, we have arrived at the following intuitive justification for the use of the
choice functions for the representation of definiteness, indefiniteness and
universality:
Justification of the use as a translation of the definite article:
The definite noun phrase does not presuppose uniqueness in the way Russell
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thought it did. We can presuppose just existence and assume a salience hierarchy in
the sense of David Lewis. The definite noun phrase denotes the most salient
individual of the kind described.
Justification for the use of the operator as a translation of indefiniteness (cp. Tiles
1991):
The indefinite is an ideal element in the sense of Hilbert which is represented by a
concrete referent given (in the sense of Frege) by the epsilon term.2
Justification for the use of the operator as a translation of universal sentences:
There is no need of a existence presupposition. Intuitively, if even the (ideal) critic of
John does not criticize John, nobody criticizes John.
3.4 Thematization of the Rhema and Rhematization of the Thema
The notion of a salience hierarchy must be related to the thematization of the
rheme3, and to the rhematization of the theme. (2) is a description of the same
object as (1) if it is true that the island lies in the sun. The general rule expressing
this fact is as follows (Egli 1991, v. Heusinger 1992):
There exists a situation i in which a statement about an epsilon expression is
integrated into the expression. Schematically:
If F εjx Gx, then εjx Gx = εix (Fx und Gx).
This is the thematization of the rheme. The inverse rhematization of the theme
corresponds to the first rule of Hilbert:
If there exists a G, then G εix Gx.
In our framework we can identify the (relevant aspects of the) situation with a
specific choice function.
2 We do not discuss the situation already noted by Wittgenstein in the Tractatus of the phrase "one
number" and "another number" in the context of sentences like "If a number is identical to another,
then the second is identical to the first." I think that the problem of the identity of numbers which by
our explanation should be different, can only be solved in the context of a epistemic modal language.
The identification of pegs in one world which are different in another is possible in such a frame
work. The same applies to two occurrences of phrases like "a man" which mean the same thing
according to our reconstruction. The solution of this puzzle is that we add the rhemata to the themata
or differentiate by ordinal numbers or identify in some worlds but not in others.
3 The notion has been borrowed from Professor Danes, who has written about it since 1979.
DEFINITENESS, BINDING, SALIENCE AND CHOICE FUNCTIONS
11
4. A New Formal Reconstruction: Modified Epsilon Operators
Definition 1
Ausw(εi) ↔ ∀F(∃xFx ↔ FεiF)
Note on Type Assignment:
τ is a variable type. For every Type τ there are choice functions εi. In order to be
totally correct one must provide every εi with an index τ.
Fτ→ο(a)
εi
Type
ο, as a is of type τ
((τ→ο)→τ), where τ is ι in most applications.
Ausw
((τ→ο)→τ)→ο)
Status of εi and conventions of notation:
Let
ελxFx = εF = εxFx
εixFx = εiλxFx
λx(Fx ∧ Gx) = F ∧ G
εi is no operator, but a constant belonging to type ((τ→ο)→τ). ε is not a variable
binding operator. To represent the corresponding variable binding operator one
would have to simulate it by a sequence of epsilon and lambda, thereby applying
the constant εi to a λ-term.
We add the following axiom to those of Church´s proof theoretical version of type
theory which can be interpreted by the methods of Henkin semantics of Henkin
type theory where the constant ε0 represented by ι (in the non Russellian reading)
by Church and Henkin (Church 1940, Henkin 1950):
Axiom 14
Ausw(ε0τ)
Axiom 1 is equivalent to the formulation by Church. This is the only assumption we
have to make for our type theoretical calculus with polymorphic (typically
ambiguous) ε.
The intuition behind this axiom is the following: In the first approach we
presupposed many different epsilon operators depending on situations. They were
4 Church has the following version of the axiom: Fx → FexFx. He uses the sign ι instead of ε.
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U. EGLI
interpreted as different choice functions of the same type. The indices may also be
thought of as numbers which mark the different choice functions. According to this
view we need a logic not only for one choice function, but for a whole set of them.
The idea essential for the framework presented here is that we construct the
different constants εi from one and only one ε 0 . Furthermore, we try to integrate all
the contextual information of the text to be interpreted into only one choice function.
In order to do that we take the modification of an arbitrary choice function εn as a
fundamental construction. We will use the notation εn[F/a] for this concept.
Intuitively εn[F/a] attributes to every set different from F or not containing a as an
element the same individual as εn and the individual a to the set F.5 Explicit
definition of this concept in type theory is possible by the following definition by
cases.
Definition 2 (informal)
εn[F/a] =Def.
the function εn+1 such that
either
G = F and
[either
¬∃xFx ∧ εn+1(G) = a
or
or
∃xFx ∧ Fa ∧ εn+1(G) = a
or
∃xFx ∧ ¬Fa ∧ εn+1(G) = εn(G)]
G ≠ F and εn+1(G) = εn(G)
Or with an explicit definition using lambda:
εn[F/a] =Def.
= λGε0λb 6
(either
G = F and
[either
¬∃xFx ∧ a = b
or
∃xFx ∧ Fa ∧ a = b
or
∃xFx ∧ ¬Fa ∧ b = εn(G)]
or
G ≠ F and b = εn(G))
That formula is justified by the following intuition:
Suppose we already have determined which element is attributed by the old
function to the (arbitrary) predicate G. Then we have to choose between the
5 This extends the concept of a modified assignment of values to a variable. It is a different
extension from the dynamic binding of dynamic predicate logic.
6 The function ε is to be applied to arbitrary G.
DEFINITENESS, BINDING, SALIENCE AND CHOICE FUNCTIONS
13
following two alternatives: Either the predicate F is identical with G or not. In the
latter case the attribution of an element of G to G is not changed:
G ≠ F and εn+1(G) = εn(G)
If, however, F is identical to G, we have to distinguish between the following cases:
(i) there is no F and we assign a to the empty set. Therewith a becomes the
conventional element or the element expressing universality for the new
choice function. Then G is empty and is assigned this conventional element a.
(ii) There is an F and a is an F. In this case the assignment of an element to G is
changed to a. This is the interesting case which describes a change within the
discourse referents.
∃xFx ∧ Fa ∧ εn+1(G) = a = b
(iii) There is an F , but a is not an F
Then the attribution is not changed. This is something like the vacuous
binding.
∃xFx ∧ ¬Fa ∧ εn+1(G) = εn(G)
εn[F/a]
εn[F/a](G)
The modification ε n [F/a] of a selection function is also expressible canonically
within type theory by the following operator umw(ε i, F, a)) which modifies an
arbitrary ε-function such that F is assigned just a , whereas all other properties get
assigned what is assigned to G by εn.
If Ausw(εi), then Ausw(εi[F/a])
Theorem 1
Proof:
For all four cases of the definition of ε[F/a] we show: ∃xGx → (εn[F/a]).
Derivation of the rules of Thematisation and Rhematisation
∃x(Fx ∧ Gx) → εoF= εo[λx(Fx ∧ Gx)/εoF](F∧ G)
= εo[λx(Fx ∧ Gx)/εoF](F)
We can prove:
∃x(Fx ∧ Gx) →
∃ εi Ausw(εi) & εiF
= εi[λx(Fx ∧ Gx)/εiF](F)
= εi[λx(Fx ∧ Gx)/εiF](λx(Fx ∧ Gx))
The existence claims in Egli 1991 can be derived from pure type theory in this way.
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Derivation of Chrysippus sentences
(12) Jemand kommt herein und er setzt sich.
(13) Wenn jemand in Athen ist, ist er nicht in Rhodos.
(14) Wenn jemand spaziert, bewegt er sich.
We want to show:
Scope and information concerning binding can be expressed by the modified ε.
Definition of ε1:
ε1 = ε0[λx(x = x)/ε0F] [λx(Gx ∧ Fx)/ε0F]
Information about the antecedent: [λx(x = x)/ε0F]
Information about scope: [λx(Gx ∧ Fx)/ε0F]
Let V be the universal class (within the type considered) defined as V = λx(x = x).
ε1V denotes "chosen entity", it stands for he or she.7
We have the theorem by considerations very similar to those above concerning the
rule of thematisation
∃x(Fx ∧ Gx) → ε0F = ε1(F ∧ G) = ε1V
There follows
∃x(Fx ∧ Gx)
Fε1(F ∧ G) ∧ Gε1(F ∧ G)
Fε1F ∧ Gε1V
[i]
[ii]
[iii]
By conditionalisation we get from this derivation the direction from [i] to [iii]:
∃x(Fx ∧ Gx) → Fε1F ∧ Gε1F
Somehow more difficult is the proof of the direction from [iii] to [i]. The problem
lies in the fact that ε1F = ε1V = ε1F ∧ G ε1F must be proved and cannot be assumed.
We have
7 The varieties of pronouns of different gender cannot be represented in the present system.
Neither do we say anything about the difference between singular and plural.
DEFINITENESS, BINDING, SALIENCE AND CHOICE FUNCTIONS
15
Fε1F ∧ Gε1V,
where ε1 is defined as ε0[λx(x = x)/ε0F] [λx(Gx ∧ Fx)/ε0F]
We have to determine what ε1V is. For doing that we must use the definition of ε1.
There follows from the definition that ε 1 V = ε 0 F. ε 1 assigns – according to the
definition of the modification ε 1 – ε 0 F to V , as there is a V by logic. The only
applicable condition in the definition of the modified choice function thus says that
ε1V is identical with ε0F. From Fε0F one can derive logically ∃xFx , and from that one
can derive Fε1F. Furthermore we can derive logically the identity of ε0F with ε1F
from the supposition ∃x Fx. There follows
Fε1F ∧ Gε1F
By existential generalization we have
∃x(Fx ∧ G) q.e.d.
Thus we have proved both directions of the following formula:
∃x(Fx ∧ Gx) ↔ Fε1(F) ∧ Gε1(V)
↔ ∃xFx ∧ Gε1V
This is an analog of the so called theorem of existential sentences in dynamic
predicate logic. It is also a dual formulation of the Chrysippus sentences which uses
conjunction instead of the conditional. Chrysippus sentences are obtained by
negating both sides of the equivalence and the second conjunct and transformation
of the sentences according to well known rules of the propositional logic and the
logic of quantification.
16
U. EGLI
5. Prospects
The ideas developed here may be extended in the following directions, which I shall
state without entering into details:
1.
2.
3.
4.
5.
6.
7.
We may apply the analysis to donkey anaphora. Note, however, that not only
an equivalent of the standard reading is available, but also a reading which is
equivalent to one, in which one quantifier of the standard representation is
existential.
We may extend the analysis to intensional logic by using Ty2 .
We may give readings to sentences which have no plausible standard
formalization, like Bach-Peters sentences and Hob-Nob sentences.
We can apply the solution not only to particular and definite sentences, but
also to universal sentences. This gives a generalized treatment of both Ccommand and non-C-command anaphora thereby vindicating the unity of the
phenomenon of anaphora. The explanation does not make use of raising and
explains the binding phenomena of noun phrases left in situ.
In general, the solution allows a representation of a lot of readings which
sometimes must be declared impossible according to the different solutions,
but which exist nonetheless.
The solution can be reformulated in the metalanguage in various ways. We can
use the modified Henkin frame which says that a Henkin model consists of (i)
a system of nonempty domains corresponding to every type, (ii) which
contains the set theoretically definable constants of equality restricted to a
domain, of S and K and the connectives in the proper domains, and of Epsilon
in the proper domains, (iii) which contains a domain of truth values with but
two entities true and false, (iv) and which is closed under application. This
framework is equivalent to the usual one, but uses only set theoretical concepts
and no axioms of the object language for its definition. The valuation function
gets extremely simple, as we can dispense with modified valuations. We can
express the concepts defined in the semantic representation language directly
in the set theoretical model with minor changes.
It is also possible to have a metalogical formulation with models consisting of
a valuation function, an assignment function and a choice function and
ordinary Henkin models. This would be a way of proceeding much more in
accordance with the usual way of proceeding. It necessitates a prior indexing
of the sentences and requires indexing the terms with scope information. As an
alternative, we may try to recast the procedure in terms of a dynamics of
context change potential, which explains sentences as a function from one
choice function to another8.
8 This was suggested to my collaborator Dr. Klaus von Heusinger by Dr. J. Peregrin. Klaus von
Heusinger formulates a modification of this idea in this volume which we shall pursue further in the
project.
DEFINITENESS, BINDING, SALIENCE AND CHOICE FUNCTIONS
17
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