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Review of  Foundations of Intensional Semantics

Reviewer: Igor Yanovich
Book Title: Foundations of Intensional Semantics
Book Author: Chris Fox Shalom Lappin
Publisher: Wiley-Blackwell
Linguistic Field(s): Computational Linguistics
Book Announcement: 16.3631

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Date: Mon, 19 Dec 2005 15:48:50 +0300
From: Igor Yanovich <>
Subject: Foundations of Intensional Semantics

AUTHORS: Fox, Chris; Lappin, Shalom
TITLE: Foundations of Intensional Semantics
PUBLISHER: Blackwell Publishing
YEAR: 2005

Igor Yanovich, Moscow State University

The main aim of the book is the development of a viable intensional
semantic theory that is able to solve the hyperintensionality problem
without resorting to the introduction of impossible worlds. The key is
treating intensions as primitives, not as functions from possible worlds
to something else. Of course, the overall architecture of the semantic
theory must be changed in order to implement this move, but the
theory proposed by the authors is shown to be able to deal with a
considerable range of natural language semantic phenomena.

The book is a must reading for any semanticist who had ever asked
herself what intensions actually are. Some parts of it may happen to
be too technical (and mathematical) for some linguists, but the joy of
understanding the results will definitely pay back one's effort to
understand the unfamiliar technicalities twice.


The title ''Foundations of Intensional Semantics'' may seem too wide
for a 200-page book, but the suspicious reader should not worry
about that, for the problem that Fox & Lappin (henceforth, F&L) are
hoping to solve is very important (though usually neglected) for any
semantic theory looking deeper than just into extensions. The problem
is: how can we distinguish between intensions of such propositions
as ''2+2=4'' and ''2*2=4''?

Obviously, the truth-conditions of these two statements are the same
in our world, that is, their extensions in our world are equal. It is not
very strange, since there are a lot of propositions that have the same
extension in our world, like ''All elephants have one head''
or ''Mushrooms are not turtles''. What is much more troublesome is
that the truth-conditions of ''2+2=4'' and ''2*2=4'' are equal in just any
possible world, since there is no possible world that does not respect
mathematical laws. Since the intension of a proposition is traditionally
taken to be a division of the set of all possible worlds into the worlds
where this proposition is true and those where it is false (or, to put it in
different worlds, a function from worlds to truth values), it follows that
the intensions of the propositions in question are equal too: in both
cases, the intension is just the set of all possible worlds.

Why is that a problem? Consider (1) and (2):

(1) John believes that 2+2=4.
(2) John believes that 2*2=4.

It is easy to imagine a situation where (1) is true while (2) is false:
believing that ''2+2=4'' is true is not the same thing as believing
that ''2*2=4'' is true. For our semantic theory to acknowledge this fact,
it should assign these two mathematical statements different
denotations. But that is exactly what the classical Montagovian
intensional semantics fails to do, since the finest distinction it can
possibly provide is a distinction between intensions as sets of worlds.
Thus two propositions are not intensionally equal iff there is a world in
which one of them is false while the other is true. Since there is no
such possible world where either 2+2=4 or 2*2=4 does not hold, these
two propositions cannot be distinguished. That is exactly why our
problem is called the hyperintensionality problem: in order to assign
right meanings to sentences like (1) and (2) we need more fine-
grained means to distinguish proposition meanings then Montagovian
intensions--functions from possible worlds to something else--are.

If we accept that standard analysis of intentions, two obvious options
arise with respect to this problem: it is either to give up our hope to
distinguish intensions of ''2+2=4'' and ''2*2=4'', hoping that some day
someone will solve the problem; or to introduce ''impossible worlds''--
that is, such worlds where some but not all of tautological (in all
possible worlds) sentences are true. Since the first answer is by no
means satisfying, let us see what happens if we try to develop an
impossible world theory.

The strategy to implement our goal is simple: if we want to distinguish
between two propositions that are both tautologies or contradictions,
we invent an impossible world where one of these two propositions
preserves its meaning, while the other does not. In the case
of ''2+2=4'' and ''2*2=4'', we may introduce a world in which numbers, +
and = have their usual meanings, but x*y is interpreted as (x*y+1). We
treat every other two tautologies or contradictions this way, each time
introducing an impossible world in which some meaning postulate
does not hold or some constant's interpretation is different from what it
is in the possible worlds. Thus every two propositions will have distinct
intensions, unless they are composed of exactly the same elements--if
it is not the case, we may always introduce a world where some
element that is different for the two propositions receives a non-
intended interpretation.

At first sight, impossible worlds seem to work well, since accepting
them allows us to distinguish between (1) and (2). However, the price
is high. We will not be able to have more than one proposition that is
true for all worlds: if we would have two such propositions, we should
have introduced an impossible world where one of them is false. So
there will be no true meaning postulates holding for all worlds. The
problem is not dramatic, since we may shrink the domain of worlds we
are interested in and formulate our meaning postulates for this domain

Another problem is the interpretation of constants--it cannot be
constant for all worlds any more, since there must be impossible
worlds where the constants are interpreted in a non-intended way.
Again, we may choose a set consisting of possible worlds only where
all ''constants'', that are not, technically speaking, constants, but which
we want to treat as such for the domain we are interested in, receive
their usual interpretations. Similar problems will arise in different
cases, too--for instance, if we want to compute inferences, we also
need to stay inside a domain containing possible worlds only.

In other words, choosing the impossible worlds strategy leads us to
introduction of a discriminated type of worlds, that is needed only to
distinguish between intensions of propositions, but is never used for
any reason other than that.

What the book under review contributes is a different answer to the
hyperintensionality problem--an answer that can solve it, but still does
not force us to have any impossible worlds.

This alternative solution is as simple as elegant: since classical
intensions, functions from possible worlds, cannot distinguish certain
fine-grained meaning distinctions, we should introduce a new kind of
intensions. These new intensions of F&L are primitives of the theory,
not functions. They are the same objects as individuals or possible
worlds. Besides, the domain of propositional intensions under F&L has
less rich structure imposed on it than it has under the standard
intensional semantics. Usually it is assumed that the propositional
domain is a lattice in which there is a partial order relation--that is,
entailment. F&L reject this common wisdom, proposing that the
domain of propositions is a prelattice, and that entailment is a
preorder relation in this lattice, rather than a partial order. After we
have replaced the ordering relation, it is possible to have two different
equality relations, one of which is intensional identity, and the other is
extensional identity. Weakening the ordering relation allows us to
have several propositions that mutually entail each other, but still are
not in the intensional identity relation.

Of course, if we adopt that, the overall architecture of the semantic
theory must be changed. Is it worth doing such a thing ''just'' to solve
the hyperintensionality problem? Though answers to such questions
usually depend more on taste than on reason, F&L show that such
theory is at the very least worth investigating.


Chapter 1 states the main goal of the monograph: to re-evaluate the
basics of current intensional semantic theories-descendants to
Montague's intensional semantics. In order to make the comparison
with the new proposals easier, the chapter contains a brief overview
of the standard Montague system.

In Chapter 2, the authors introduce mathematical apparatus that will
be needed further, and prove that if the entailment between
propositions is a partial order in a lattice, then mutual entailment
equals to identity. More informally, if the entailment relation is partial
order, there cannot be two distinct propositions that have the same
truth-conditions in all worlds.

This leaves just two logical possible solutions for the
hyperintensionality problem. The first is the impossible worlds strategy
sketched above. The second, adopted by F&L in the rest of the book,
is modeling the entailment relation as a preorder in a bounded
distributive prelattice rather than a partial order in a lattice--this will
allow us to have two propositions with the same Montagovian
intensions, but still distinct. In other words, it allows us to find a more
fine-grained intensions than the standard intensions are.

The rest of the chapter discusses previous work--on the one hand,
the impossible world approaches or approaches that were claimed not
to use impossible worlds but actually need them, as the authors show
in this chapter; and on the other hand, Bealer's (1982) intensional
logic and Turner's (1992) Property Theory, which are the
predecessors of the authors' proposal.

Chapter 3 gives the first idea of how the authors' solution may be
implemented. The authors formulate a very simple intensional theory
that is clearly not enough to model natural language semantics, but
still shows that constructing such theory that does not need to
collapse mutually entailed propositions is possible.

Chapters 4 and 5 respectively present two non-toy theories treating
intensions as primitives. The first one is Fine-Grained Intensional
Logic (FIL). FIL preserves the original Montague's system as much as
possible, while implementing the main idea of the book. FIL is a higher-
order theory, just of the familiar type for natural language

The other theory is called Property Theory with Curry Typing (PTCT).
Unlike FIL, PTCT is essentially first-order. However, being first-order
does not prevent it from accounting for various phenomena that are
often cited as arguments in favor of higher-order theories, like
generalized quantifiers. Though the form of the theory is not as
familiar for linguists as that of FIL, and its formal power is provably
smaller, it is shown that PTCT should not be considered inferior to FIL
with respect to treatment of natural language phenomena before
actual testing.

Both theories have two different identity predicates, and it is shown
that in both of them intensional identity entails extensional identity, but
not vice versa. Thus, the authors' proposal is successfully
implemented. From this point of the book, what becomes important is
not the implementation of a semantics allowing for more fine-grained
distinctions, but the degree of viability of this theory and its good and
bad points in comparison to other semantic theories.

Chapter 6 discusses the introduction of arithmetic into FIL and PTCT
and the treatment of proportional quantifiers such as MOST. Though
Peano arithmetic can be added both to FIL and to PTCT, such
addition makes PTCT incomplete, and thus one of its main
advantages over higher-order theories such as FIL is lost. However, if
we add to PTCT not Peano arithmetic, which has both addition and
multiplication, but Presburger arithmetic, which has only addition, the
theory will not lost completeness. Such a theory, of course, will be
weaker in expressive power than a theory with Peano arithmetic, but it
will still suffice to give proper meanings to proportional quantifiers.

Chapter 7 shows that PTCT can easily handle all known cases of
anaphora (including bound-variable anaphora, coreference anaphora,
and donkey anaphora) and ellipsis (including VP ellipsis, gapping, and
ACD), as well as the combination of the two, such as binding into the
elided constituent where both strict and sloppy readings must be

The proposed mechanism for anaphora resolution is the resolution of
a type parameter. Under this treatment, pronouns are typed free
variables, where a type contains a free variable in it too. In the case of
a bound variable, when such a variable is in the scope of an
abstraction operator over a different variable, it can be bound by this
operator and substituted for the variable abstracted over. If the
pronoun is free, then it must find some contextual predicate to fill in its
type. Usually, such predicate may be obtained from previous

Ellipsis treatment of F&L is very similar to that. The meaning of a
clause with VP ellipsis is the statement that the argument (or the list of
arguments, if there is more than one) belongs to some unidentifyed
type. The simplest way to obtain the value for this type from the
context is to get a needed type via applying abstraction to the
antecedent clause. This accounts for VP ellipsis, gapping and
pseudogapping in a similar manner.

Chapter 8 proposes a mechanism for generating representations with
underspecified scope relations. The idea is that such a representation
consists of a list of all possible permutations of scope-bearing
elements and of the core relation of a sentence to which any of the
permutations may be applied. The proposed treatment has several
advantages: underspecified representations are normal terms, and
not some meta-expressions, as in other current theories for
underspecification; this fact is not only pleasant from the general
economy considerations, but also has a direct welcome consequence--
we may use F&L's underspecified representations as premises in
inference processes without computing their meaning (that is,
choosing only one scope reading from the list). There is also a natural
way to account for scope constraints in natural language: such
constraints may be formulated as filters on lists of permutations. For
instance, we may formulate a constraint saying that a given quantifier
may never have the widest scope in the sentence.

Chapter 9 once again, at more length, discusses the problems of the
balance between formal strength and expressive power of a logic.
From a practical point of view, if we want to have a very expressive
logic, we will not be able to build a theorem prover for it; if we want to
have a logic that can be implemented, we need to give up on some
expressive power. However, if it is the case that we do not really need
that much power modeling natural language semantics, then we will
have no reason to choose a higher-order undecidable logic. That is
exactly what the authors are arguing for: a first-order PTCT-based
logic is enough to treat most, if not all, natural language phenomena.

Chapter 10 concludes the book, summing up the main results
achieved and discussing possible directions for future work.


Just as it is not possible to cover all questions that can be raised
concerning such cardinal changes in the architecture of the semantic
theory in a 200-page book, it is not possible even to mention all these
questions in a review of any reasonable length. I will confine myself
with a very limited number of issues: first, I will briefly discuss some of
the problems for the ''empirical'' side of the authors' analyses of
anaphora and scope presented in Chapters 7 and 8, and then I will
turn to more general architectural issues, considering the
consequences of switching from the standard theory to F&L's system
on a single example--the treatment of de se/de re readings.

1. Anaphora (Chapter 7). As it is stated, F&L's proposal for anaphora
does not account for the ''binding principles'' effects. However, since
the analysis in the book just provides the general mechanism for
resolving pronouns, it is reasonable not to demand that much, in the
case we can set up some additional constraints that will account for
the anaphora facts of real human languages. The question is whether
it is possible to provide such additional constraints within F&L's
system, or not.

If we are forced to accept a very rich language for description of
syntactic relations (and we may be actually forced to, see the
discussion of Chapter 8 below), it seems to me that it will not be
problematic to account for these effects.

However, note that F&L's system shares the problem of the classical
Chomsky-Reinhart binding theory. F&L have distinct representations
for bound-variable and coreferent readings of pronouns: in the former
case, they are just bare variables, and in the latter, they are variables
bound by a universal quantifier (that ensures maximality of
interpretation) and restricted by a type judgement. (See Jacobson
(1999), who provides the following argument against such a view: If
the meanings of bound-variable and coreferent pronouns are
different, then why are there no languages that have different words
for these two types of pronouns? See also Kratzer (2005) for a recent
analysis explaining this peculiar fact under the classical binding

Also, it would be interesting to see how F&L's system may account for
paycheck readings of pronouns. If we just try to apply the standard
procedure for resolving the coreference anaphora for the
paycheck ''it'' in ''x who put her paycheck to the Bank A was wiser than
y who put it to the bank B'', we would get an interpretation like this: ''z
belongs to type A, and type A is the type of objects that were put to
the Bank A by x'', which is indeed just the simple coreference reading
for this sentence, not the paycheck reading. So something special
must be done here.

2. Scope (Chapter 8). Just as in the case of anaphora, the questions
for the mechanism for generating underspecified scope
representations are, first, whether it is sufficiently powerful to express
natural language scope constraints, and whether it is restrictive
enough to the extent it will not overgenerate, after the needed scope
constraints are defined.

As the authors show, it is easy to define a filter to the effect that some
quantifier (''a certain'' in the authors' example) will be assigned the
widest scope. Informally, this filter says ''There is no scope-bearing
expression that has scope wider than 'a certain'''. What if there are
two ''a certain''-s in a sentence? In order not to arrive at a derivation
failure, we just need to improve our filter a bit, restating it as following
(again, informally): ''There is no scope-bearing expression _other
than ''a certain''_ that takes scope wider than 'a certain'''.

The constraint requiring some quantifier to take non-widest scope is
easy to implement too: ''The quantifier A is not allowed to be the first in
the scope sequence'' (where the first in the sequence receives the
widest scope). Farkas (1997) describes a constraint on the Hungarian
determiner ''egy-egy'', that is similar to English indefinite article, but
must always take non-widest scope. The peculiarity of this Hungarian
determiner is that it is OK in the scope of quantifiers over individuals
and situations, but not over worlds. The constraint for GQs formed with this
determiner will be '''Egy-egy' is not allowed to be the first in the scope
sequence, and there must be a quantifier over individuals or over
situations in the sequence that is prior to 'egy-egy'''.

The weak point of F&L's scope system is that they have to introduce
constraints that have references to syntactic relations. For instance,
their (233) is a constraint preventing a quantifier inside a relative
clause to take scope over the quantifier that is the head of the clause.
Informally, it says ''there cannot be that A scopes over B and B is in
the relcl_embed relation to A''. Of course, this relation relcl_embed
should be defined based on syntax, and implicitly allowing such
semantic correlates of syntactic relations is not the good thing to do
without a very serious reason: it means that the semantics can use
any part of the syntax to determine in the meaning constraints, that is
surely not the most restrictive view of the grammar. Moreover, to be
able to state such relations in the syntax will also require a relatively
complex view of what syntactic relations may be used by the grammar.
(Note we will need to have semantic correlates of rather complex
syntactic relations. For instance, (233) is not sufficient to account for
the relative clause scope island, since it does not rule out cases when
the quantifier that is inside the island scopes over some other
quantifier outside the island that is not the head of the relative clause.
To make (233) account for those cases as well, we need to replace
the relcl_embed relation with the relation ''A is outside the relative
clause inside of which B is''. This relation needs even richer syntactic
language to express than relcl_embed needs.)

3. The overall structure of the semantic theory.

The main idea of the book under review is that we should replace
intensional meanings that are functions from possible worlds to
something else with intensions as primitives. The case of proposition
meanings is relatively simple: now the truth of all text-level
propositions will be checked in some fixed, ''real'', world, and functions
taking propositions as arguments will just have a slot for expressions
in a new primitive proposition type, not in the familiar <s,t> type.

But what will become of other expressions? Consider the well-known
problem of de re / de se readings.

(3) Ann wants to marry a doctor.
a. Ann wants to marry any person, if this person is a doctor.
b. Ann wants to marry a specific person, and she wants that this
person were a doctor.
c. Ann wants to marry a specific person, and she does not know (or
care) if he is a doctor or not, but actually he is.

Under the standard intensional semantics, we would say that these
three meanings are generated like this:

(4) a. For every w' from the worlds compatible with Ann's desires,
there is x: doctor(x)(w') & marry(x, Ann)(w').
b. There is x: for every w' from the worlds compatible with Ann's
desires, doctor(x)(w') & marry(x, Ann)(w').
c. There is x: for every w' from the worlds compatible with Ann's
desires, doctor(x)(w) & marry(x, Ann)(w').

So there are two parameters that distinguish the three readings: first,
the existential quantifier introduced by the DP ''a doctor'' may take
scope either higher or lower than ''want'', and second, the person Ann
wants to marry may be a doctor either in the real worlds w, or in the
worlds of Ann's desires w'.

But first let us consider a simpler problem--the scope of intensional
verbs. The most straightforward way to treat intensional verbs
like ''want'' under F&L will not involve any quantification over
worlds: ''want'' will just take a proposition argument, and its meaning
will ensure that the semantics is like usual. However, to allow for
scope ambiguities, we should treat ''want'' as other scope-bearing
elements (L stands for lambda, ''there is'' for an existential quantifier,
and {} is the array of scope-taking elements subject to permutations):

''want''(marry(''a doctor'')(a))(a) =>
Lx.''want''(marry(x)(a) & doctor(x))(a); {there is x} =>
Lr.Lx.r(marry(x)(a) & doctor(x))(a); {there is x; want}.

Two permutations are generated given the lambda-term and the array
above: 1) Ann wants that there is a doctor and she marries him; 2)
There is a doctor such that Ann wants to marry him. In other words,
the scope ambiguity is as easy to get as under the standard analysis.

The real problem is how to handle the evaluation of the ''doctor''
predicate--we should be able to interpret it both in w and in w'. A
single predicate of the form will not do: the interpretation
of such predicate will be constant for all worlds, and there is no way
we can account for the difference between the ''doctor in w''
and ''doctor in w''' readings. Moreover, what would such a predicate
mean? Would be true for all individuals who are doctors at least in one
world? Or only for those who are doctors in all worlds where they
exist? Or, maybe, for those who are doctors in more than 50% of
worlds where they exist?

So we are left with two options to distinguish the w and w' readings:
either to use the good old world argument, or to accept that the
instances of ''a doctor'' DP in (3) may be interpreted as two primitive
predicates ''doctor1(x)'' and ''doctor2(x)''. In both cases we need to
find some way to assign right interpretations: under the world
argument option, we have to explain where the value of the world
argument comes from; under the different predicates option, we must
find some reasonable rules to govern the interpretation and to ensure
that they are indeed the right predicates. The two ways seem more or
less equivalent, so below for expository purposes I will use only the
first option.

Here is the first problem: under the standard account, the proposition
must take as its argument some possible world w and return a truth-
value. Thus the world argument is a part of the tree, and it can bind
the variable that is a world argument of some predicate below
according to the usual binding rules: in principle, any expression may
bind a variable in its c-command domain, if the types and binding
constraints do not rule out such a construal. But when we switch to
F&L's system, matters get complicated: in the case of (4c), there will
be no binder for the variable, since there will be no explicit world
argument of the matrix proposition. Of course, we may implement
some rule that will allow all variables over worlds to be assigned the
value of the fixed world in which we evaluate our sentence, that is, to
allow binding by an implicit world argument, to the same effect that
had binding by an explicit world argument in the standard story.

But what to do with the case when the predicate should be evaluated
in w', not in w? Under the standard account, this is accomplished via
binding the world argument by a universal quantifier over possible
worlds introduced by ''want''. But under F&L, there is no such
quantifier. Moreover, propositions are primitives, and not sets of
worlds, so there is no place in the interpretation of (3) that could
possibly supply the world argument.

All that is left is to try to bind the world argument by a quantifier, much
like free variables are existentially closed under File Change
Semantics. What we will get under this kind of approach will be like the
interpretations in (5) (it contains only cases when the existential
quantifier introduced by ''a doctor'' scopes over ''want''):

(5) a. There is w': there is x: Ann wants that marry(x)(a) & doctor(x)
b. For all w': there is x: Ann wants that there is x: marry(x)(a) & doctor
c. There is x: for all w': Ann wants that there is x: marry(x)(a) & doctor
d. There is x: Ann wants that there is w': marry(x)(a) & doctor(x)(w').
e. There is x: Ann wants that for all w': marry(x)(a) & doctor(x)(w').

(5a) says that in there is a world where there is a person Ann wants to
marry, and this person is a doctor in this world. This is too weak, since
it does not even require that Ann wants to marry anyone in the
evaluation world. (5b) says that in all worlds there is such person who
is a doctor in this world and who Ann wants to marry. It is too strict: the
truth of (5b) depends on whether each world to have at least one
doctor in it. (5c) says that there is a person that Ann wants to marry,
and this person is a doctor in all possible worlds. This is also too strict:
intuitively, if (3) is true, it does not mean that Ann wants to marry a
person that cannot be a non-doctor or not to exist in any world. (5d)
says that Ann wants that she marries some person x and that x is a
doctor at least in one possible world. Now, it is too weak: Ann wants
this person to be a doctor in the world where she marries him, not in
just any world.

Finally, (5e) says that Ann wants this person to be a doctor in all
possible worlds, even in those in which she does not marry him. This
is the best one of all the interpretations in (5), but is it good enough? It
is not: suppose that Ann wants to marry Phil. He is studying medicine
at the time, and Ann knows that. But she wants to marry Phil only after
he has become a doctor. In this situation, (3) is true (on the reading
(3b)), but (5e) is false, since Phil is definitely not a doctor in the real
world where we evaluate the sentence. Hence, none of the
interpretations in (5) is a correct interpretation for (3).

So even if we solve the first problem, that is, obtaining the doctor(w)
reading, the second problem, obtaining the doctor(w') reading, seems
to be unsolvable. And if it really is unsolvable, than it undermines the
whole F&L's proposal and forces us to use some kind of the
impossible worlds approach.

But in the end, even if it turns out that F&L's proposal cannot be
implemented, at least we will have some serious evidence to return to
the standard view. So the main goal of the book--to bring the
fundamental problems of intensional semantics back to the light--will
be achieved.


Bealer, G. (1982). ''Quality and Concept'', Clarendon Press, Oxford.

Farkas, D. (1997). ''Dependent Indefinites'', in F. Corblin, D. Godard
and J.-M. Marandin (eds.), Empirical Issues in Formal Syntax and
Semantics, Peter Lang Publishers, pp. 243-268.

Jacobson, P. (1999). ''Binding without pronouns and pronouns without
binding'', in Oehrle, R. and Kruiff, G-J. (eds.), ''Binding and Resource
Sensitivity'', Kluwer Academic Press.

Kratzer, A. (2005). ''Minimal Pronouns'', paper presented at CSSP

Turner, R. (1992). ''Properties, propositions and semantic theory'', in
Rosner, M. and Johnson, R. (eds.) ''Computational Linguistics and
Formal Semantics'', Studies in Natural Language Processing,
Cambridge University Press, Cambridge, pp. 156-180.

Igor Yanovich is a graduate student at Moscow State University. He
has done work on indefinite pronouns, variable-free binding, and
some aspects of negation in Russian on the formal semantics side,
and on relative clause attachment, errors in subject-verb agreement,
and acquisition of binding on the side of psycholinguistics. He is also
one of the organizers of the Moscow Formal Semantics Reading
Group, as well as the annual Formal Semantics in Moscow workshop.

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