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.

INTRODUCTION: THE HYPERINTENSIONALITY PROBLEM

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 only.

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.

BOOK OVERVIEW

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 semanticists.

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 generated.

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 discourse.

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.

DISCUSSION

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 theory.)

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

REFERENCES

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 2005.

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.

ABOUT THE REVIEWER:
ABOUT THE REVIEWER

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.