A Note on Intensionalization

Journal of Logic, Language and Information, Apr 2013

Building on Ben-Avi and Winter’s (2007) work, this paper provides a general “intensionalization” procedure that turns an extensional semantics for a language into an intensionalized one that is capable of accommodating “truly intensional” lexical items without changing the compositional semantic rules. We prove some formal properties of this procedure and clarify its relation to the procedure implicit in Montague’s (1973) PTQ.

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A Note on Intensionalization

Philippe de Groote 0 Makoto Kanazawa 0 0 P. de Groote INRIA Nancy, Grand-Est, 615, rue du Jardin Botanique, 54600 Villers-ls-Nancy, France Building on Ben-Avi and Winter's (2007) work, this paper provides a general intensionalization procedure that turns an extensional semantics for a language into an intensionalized one that is capable of accommodating truly intensional lexical items without changing the compositional semantic rules. We prove some formal properties of this procedure and clarify its relation to the procedure implicit in Montague's (1973) PTQ. Ben-Avi and Winter (2007) proposed a procedure for intensionalization, a method for mapping an object of an extensional type (i.e., type based on atomic types e and t ) into an object of a corresponding intensional type (based on e, t , and s). They used this mapping to convert an extensional semantics for a fragment of natural language, where all lexical items have extensional denotations, into an intensional semantics which assigns intensionalized denotations to the same lexical items and - which is capable of accommodating new lexical items with truly intensional denotations without any change in the grammar. This is supposed to allow a modular development of natural language semantics, where the purely extensional fragment is first presented in the simplest possible terms, without the machinery of possible worlds. Intensionalization can also be discerned, in a particularly simple form, in the semantics of various intensional logics, which are usually built on top of the standard, extensional language of propositional or first-order logic. The higher-order character of natural language semantics makes it a less trivial task to find a suitable definition of intensionalization. This paper presents an alternative intensionalization procedure that is more general than Ben-Avi and Winters (2007) in two respects. First, Ben-Avi and Winters procedure is only applicable to objects of types that are either quasi-relational or e-basedin other words, types that contain no subtype of the form (1 n t ) 1 m e. This limitation stems from the fact that their type change scheme replaces t by s t , but leaves e unchanged. In contrast, the present scheme uniformly replaces each atomic type a in an extensional type by s a to produce an intensional type ; this uniformity allows the intensionalization procedure to be defined at arbitrary extensional types. Second, unlike Ben-Avi and Winters (2007) method, where the input is a single extensional object, the method defined here takes a set of extensional objects of type parameterized by objects of type s, and returns an object of type . This allows a construction of an intensional model from a class of extensional models viewed as possible worlds; as a result of this, there is no need to stipulate a sharp distinction between lexical items that are logical constants and others whose denotations are unconstrained, as in Ben-Avi and Winters construction. We give the formal definition of intensionalization in Sect. 2. Naturally, the present intensionalization procedure generalizes the way an intensional language is usually built on top of an extensional one in formal logic. We take the extensional language whose sentences are simply typed -terms (of type t ) containing constants, and interpret it in an intensional model constructed from a class of extensional models, assigning each constant the denotation obtained from its denotations in the extensional models by intensionalization. For the intended application to natural language semantics, constants are to stand for extensional lexical items, and a closed -term of type t is to represent a possible sentence meaning, expressing how the denotations of words may be combined to give a truth value. After intensionalization, a constant of type will now denote an object of type ; a closed -term of type t will now express a recipe for combining intensionalized denotations of constants to yield an object of type t = s t , or a set of possible worlds. To give a concrete example, in giving an extensional semantics for a certain fragment of English, we may use the -terms every man (find (a unicorn)) to represent the subject wide scope and object wide scope readings of the sentence every man finds a unicorn. Here, each word whose (extensional) denotation is of type is represented by the corresponding constant (in small capitals) of type . We assume that determiners have type (e t ) (e t ) t , commoun nouns have type e t , and transitive verbs have type ((e t ) t ) e t . (Note that a transitive verb can directly combine with its object by functional application in the case of subject wide scope reading.) Given an extensional model M assigning each constant c of type its denotation M (c), the truth value of the sentence under the two readings is given by the denotation of the above -terms in M . Now, suppose that we systematically intensionalize the semantic types for all syntactic categories, so that determiners, common nouns, and transitive verbs will now denote objects of type (e t ) (e t ) t , e t , and ((e t ) t ) e t , respectively. Given an intensional denotation for each constant, the same -terms (1) and (2) will still be meaningful, provided that the variables ve and uet are now understood to range over objects of type e and e t , respectively; the denotations of (1) and (2) will then be objects of type t = s t . The resulting compositional semantics will be able to accommodate intensional transitive verbs like seeks, treating them as belonging to the same syntactic categoryand hence having the same semantic typeas extensional transitive verbs like finds. Whereas the denotation of finds will be an object that is constructed from a collection of objects of type ((e t ) t ) e t , the denotation of seeks will not reduce in the same way to objects of type ((e t ) t ) e t . The compositional semantics, however, will be agnostic to the distinction, and meaning recipes of the exact same form as (1) and (2) will account for two readings of every man seeks a unicorn. An important desideratum for intensionalization is that the resulting intensional semantics is conservative over the original extensional semantics. This desideratum is somewhat misstated in Ben-Avi and Winters (2007) paper for a reason related to their treatment of non-logical lexical items. They demand that validity and consequence be preserved in moving from extensional models to intensional models. Since their treatment of non-logical lexical items in effect amounts to treating them as free variables, this is an unreasonably strong requirement, one that easily fails in the intensionalization of as simple a language as the language of propositional logic augmented with a non-logical constant of type t t . In order to satisfy this desideratum, Ben-Avi and Winter (2007) had to severely restrict the admissible types of non-logical lexical itemsspecifically, they were limited to those types that contain at most one instance of t , and only at the tail position. In the present construction of intensional models from classes of extensional models, the desideratum is, simply put, that the truth conditions of sentences (i.e., closed -terms of type t ) in the extensional language be preserved. More precisely, this means that the truth value of a sentence in an extensional model M is to coincide with the truth value of in any intensional model constructed form extensional models in which M is the actual world. The preservation of validity and consequence for sentences follows as a corollary. These results are proved in Sect. 3. Even though the validity of an open formula in a class of extensional models should not be expected to be preserved in intensional models built from them, it may be of some technical interest to see to what limited extent this expectation may be satisfied. For an open formula in the extensional language to be valid in a class of intensional models, all instances of , including formulas in an expanded language containing some truly intensional items, must be valid as well. Particularly simple examples of such formulas are provided by propositional modal logic, where a propositional variable in a tautology may be replaced by any modal formula to produce a valid modal formula. This is a rather trivial case, but there are other more interesting cases. In Sect. 4, we define a class of safe open formulas, and show that for those formulas, validity in extensional models guarantees validity in intensional models based on them. The present intensionalization procedure based on the mapping is simple and natural, but not so familiar. In Sect. 5, we compare it to the procedure implicit in Montague (1973), which is closer to the practice of linguists. The two procedures can easily be seen to be equivalent. The chief difference is that in Montagues approach, not only the denotations of the lexical items but also the meaning recipes associated with the analysis trees (or LFs) of natural language sentences must be modified. (Sect. 5 does not depend on Sect. 4 and can be read immediately after Sect. 3.) It is not entirely clear to us whether a fully general intensionalization procedure such as the one given in this paper is called for in natural language semantics. It seems to us that the usual practice in linguistics is to introduce as few instances of s as is required for proper linguistic analysis, rather than adding as many instances of s as there are instances of e and t .1 Assigning fully intensionalized types to extensional lexical items may be seen as a manifestation of Montagues strategy of generalizing to the worst case. Instead, it may be more congenial to the current practice of linguists to adopt a type-shifting mechanism to intensionalize (and extensionalize) denotations of phrases and apply it only when it is needed, and only to the extent that it is needed.2 Nevertheless, the very fact that a completely general intensionalization procedure exists, with a definition so natural as to look almost inevitable, and its equivalence to a mechanism behind Montagues PTQ, may interest some researchers concerned with foundational aspects of semantics. 2 Intensionalization We begin with some necessary definitions. The set of types over a set A of atomic types is defined inductively as follows3: Every atomic type a A is a type over A. If 1 and 2 are types over A, then 1 2 is a type over A. The type constructor is assumed to be right-associative, so that stands for ( ). We let n abbreviate , with repeated n times. Given a family (Da )aA of base domains, the domain D of objects of type is defined for each type over A by recursion: 1 Cf. von Fintel and Heim (2011, section 1.3.1). 2 The paper by Partee and Rooth (1983) contains two type-lifting principles to this effect. 3 In Montague semantics, is usually written , , but we prefer the notation standard in -calculus. In concrete examples, we often assume A = {e, t } and Dt = {1, 0}. The elements of De are individuals and 1 and 0 are truth values. Let s be a new atomic type not in A. Elements of Ds are called indices or possible worlds. The intensionalization of type , written , is defined by For example, if = (e t ) (e t ) t , then = ((s e) s t ) ((s e) s t ) s t . We use simply typed -calculus ( la Church) as our metalanguage to denote objects in D, for types over A {s}. The type of a variable is indicated by a superscript at its first occurrence. If is a -term of type and is a -term of type , then the application is a -term of type and denotes the object in D that is the value of on argument . If is a -term of type , then the -abstract x . is a -term of type and denotes the function from D to D which maps x to . The application of -terms is assumed to be left-associative, so that stands for ( ) . Application binds stronger than -abstraction, so that x . means x .( ), rather than (x . ) . A sequence of s is collapsed into one, so that x11 . . . xnn . abbreviates x11 . . . xnn . . For each type over A, we define the intensionalization and extensionalization combinators at type by mutual recursion as follows: int = x s y.int (i s .xi (ext y i )), ext = y j s x .ext (y(int(ks .x ))) j. int = x s y11 . . . ynn i s .xi (ext1 y1 i ) . . . (extn yn i ), ext = y j s x11 . . . xnn .y(int1 (ks .x1)) . . . (intn (ks .xn)) j, For example, if q Ds(et)(et)t , then = y1(se)st y(se)st i s .qi (ze.y1(ks .z)i )(ze.y2(ks .z)i ). 2 If qi is the quantifier some for all i , i.e., q = ks x et x et .e(ze.(x1z)(x2z)), 1 2 where e D(et)t is the first-order existential quantifier over individuals and Dttt is conjunction, then int(et)(et)t q equals y1(se)st y(se)st i s .e(ze.(y1(ks .z)i )(y2(ks .z)i )). 2 Lemma 2 For any type over A, x s .ext (int x ) = x s .x . = x s j s z .ext (int x (int (ks .z))) j = x s j s z .ext (int (i s .x i (ext (int (ks .z))i ))) j = x s j s z .(i s .x i ((ks .z)i )) j by induction hypothesis does not hold in general. We call an object y D quasi-extensional if y = int (ext y). By Lemma 2, it is clear that y D is quasi-extensional if and only if y = int x for some x Ds . We call y D truly intensional if it is not quasi-extensional. Note that if x Dtn t (i.e., x is a truth function), then inttn t (ks .x ) coincides with its usual Boolean generalization to type (s t )n s t . For instance, the intensionalization of conjunction Dttt is intttt (ks .) = x st yst i s .(x i )(yi ), that is, the intersection operation on subsets of Ds .4 In general, the intensionalization int (ks .x ) corresponding to a logical constant x D does not necessarily agree with other existing ways of lifting x to an object in D . For instance, the first-order universal quantifier over individuals e D(et)t can be naturally lifted to using the universal quantifier se D((se)t)t over individual concepts (i.e., functions from possible worlds to individuals), but this differs from its intensionalization, int(et)t (ks .e) = y(se)st i s .e(extet y i ) 4 As is customary, we often identify sets with their characteristic functions. The point here is that the intensionalization of ks .x is -definable in terms of x , but the universal quantifier over objects of a higher type is not -definable in terms of the universal quantifier over objects of a lower type.5 Another example is generalized conjunction, defined for each type of the form 1 n t by = y1 y2 z11 . . . znn .(y1z1 . . . zn)(y2z1 . . . zn). = y1(se)st y(se)st zsei s .et (extet y1 i )(extet y2 i )(extez i ) 2 = y1(se)st y(se)st zsei s .et (x e. y1(ks .x )i )(x e. y2(ks. x )i )(zi ) 2 = y1(se)st y(se)st zsei s .(y1(ks. zi )i )(y2(ks. zi )i ), 2 which does not equal et = y1(se)st y(se)st zsei s .(y1zi )(y2zi ). 2 3 Preservation of Extensional Semantics Under Intensionalization The object language of our study is that of typed -terms built up from basic expressions consisting of constants, each of some type over A, and countably many variables for each type over A. We use boldface variables v1, v2, . . . and boldface in the object language to avoid confusion with the metalanguage. Because of the presence of constants, whose interpretation we can pick at will, this choice of the object language is general enough to encompass most extensional languages of formal logic, and is also adequate as a language for representing meanings of expressions in extensional fragments of natural language in the style of Montague semantics. The intensionalization of the semantics of these languages serves as a foundation on which to build richer languages including intensional constructs within the usual framework of possible world semantics. An extensional model M of our object language consists of base domains (Da )aA and an assignment of a denotation M (c) D to each object language constant c of type . An intensional model consists of base domains (Da )aA{s} together with an assignment of a denotation M (c) D to each object language constant c of type . 5 In fact, in the particular case of the universal quantifier, -definability goes the other way around: ks . is -definable in terms of by (The same can be said of equality at different types.). We are interested in those intensional models that are built from extensional models by means of intensionalization. An object language expression has the denotation [[]]M,g in an (extensional or intensional) model M relative to an assignment g of values to variables. In an extensional model, g(vl) D, whereas in an intensional model, g(vl) D. We let g[x /vl] denote the assignment that is like g except that it assigns x to vl. Note that if is an object language expression of type , the denotation of in an intensional model belongs to D. If is a closed object language expression, [[]]M,g does not depend on g, so we let [[]]M = [[]]M,g for an arbitrarily chosen g. Given an indexed collection I = (Mi )iI of extensional models with the same base domains (Da )aA, we create an intensional model MI based on I, with Ds = I . For an object language constant c of type , we let be its denotation in MI . Let us consider a very simple example to illustrate the above definition. Let A = {t } and let the vocabulary of the object language include constants c of type t and q of type t t . Fix Dt = {1, 0}. There are eight possible extensional models for this language: M (c) is either 0 or 1, and M (q) must be one of x t .0, x t .1, x t .x , , and x t .x . Let C = {M1, . . . , M8} be the set of these eight models and let I be a subset of {1, . . . , 8}. We can construct an intensional model MI out of I = (Mi )iI by the above definition. The denotation MI (c) of c is (the characteristic function of) a subset of I , and MI (q) is a function from the power set of I to the power set of I . We have [[qc]]MI = MI (q)MI (c) = (ysei s .Mi (q)(yi ))(i s .Mi (c)) Having defined MI , one can then expand it to a model M for a larger vocabulary including additional constants, such as the necessity operator (of type t t , with denotation in Dtt ), whose denotation is truly intensional. We shall show that equalities exemplified by (3) are completely general and hold for all sentences in the extensional object language. If g is an intensional assignment (suitable for MI ), define an extensional assignment gi by Lemma 3 Let be an object language expression of type , and let g be an intensional assignment suitable for MI that is quasi-extensional. 1. ext[[]]MI,g i = [[]]Mi ,gi . 2. If the -normal form of is not a -abstract, [[]]MI,g = int(i s .[[]]Mi ,gi ). Proof We prove both 1 and 2 simultaneously by induction on , assuming that is in -normal form. Note that if is not a -abstract, 1 follows from 2 by Lemma 2. If is a constant c, then 2 holds by the definition of MI (c). If is a variable vl, 2 holds since g is quasi-extensional. Suppose = , where is of type and is of type . Since is in -normal form, is not a -abstract. Hence by induction hypothesis, and the condition in 2 holds of . It remains to consider the case where is a -abstract u . , where = and is of type . By induction hypothesis, ext [[ ]]MI,h i = [[ ]]Mi ,hi for all assignments h that are quasi-extensional. ext [[u . ]]MI,g i = x .ext ([[u . ]]MI ,g(int (ks .x ))) i by induction hypothesis Note that the condition in part 2 of Lemma 3 does not hold of -abstracts. A simple counterexample is the I combinator uab.uab: = y1(sa)sb ysai s .(za .y1(inta (k.z))i )(exta y2 i ) 2 = y1(sa)sb ysai s .(za .y1(k.z)i )(y2i ) 2 (The inequality assumes |Da | 2 and |Ds | 2.) Remark 4 A special case of Lemma 3 is when g = h for some extensional assignment h, where h is defined by In this case, we have gi = h for all i I . This special case itself can be proved directly by induction. Remark 5 The content of Lemma 3 can be stated entirely within simply typed -calculus, as follows. If is a -term of type , with free variables z11 , . . . , znn , let be the -term of type obtained from by replacing each occurrence of a A by s a in the type annotation of . Then we have = i s .[(x1s1 i )/z11 , . . . , (xnsn i )/znn ], Proof Immediate from Lemma 3. Now assume t A and fix Dt = {1, 0}. We call an object language expression of type t a formula, and a closed formula a sentence. A pointed possible world model is a pair of the form (MI , i ) where i I . The extensional model Mi is the actual world of a pointed possible world model (MI , i ). We say that a sentence is true in a pointed possible world model (MI , i ) if [[]]MI i = 1. Corollary 7 For every sentence in the object language and every extensional model M , the following are equivalent: 1. is true in M . 2. is true in any pointed possible world model whose actual world is M . Let C be a class of extensional models. Call a sentence extensionally valid in fCorifal[[li]n]Mdex=ed 1coflolercatilolnMs I=C,( Manid)i inItecnosniosinsatilnlyg voaflimdoidneCls iifn [[C]t]hMaIt s=hareitsh.1e same base domains. Similarly, is an extensional consequence of 1, . . . , n in C if [[1]]M = = [[n]]M = 1 implies [[ ]]M = 1 for all M C, and is an intensional consequence of 1, . . . , n in C if [[1]]MI [[n]]MI [[ ]]MI for all indexed collections I consisting of models in C with the same base domains.6 1. If is extensionally valid in C, then is intensionally valid in C. 2. If is an extensional consequence of 1, . . . , n in C, then is an intensional consequence of 1, . . . , n in C. In the presence of conjunction () and implication () in the object language, the consequence relation between 1, . . . , n and can be defined as the validity of 1 . . . n , both in the extensional and in the intensional sense. (Recall that truth-functional connectives behave as desired in intensional models.) This allows us to concentrate on validity. 4 Intensionally Valid Schemata Corollary 8 does not quite give what Ben-Avi and Winter (2007) were aiming for, because in their method of intensionalization, the denotation of a non-logical constant 6 The intensional consequence relation as defined here corresponds to local consequence in modal logic (Blackburn et al. 2001). of type is not restricted to quasi-extensional objects in D . They start from a class C of extensional models that is closed under arbitrary change in the denotations of nonlogical constants and obtain by intensionalization a class C of intensional models that is again closed under arbitrary change in the denotations of non-logical constants. (In their method, intensionalization is only used to determine the denotations of logical constants in intensional models.) Replacing non-logical constants with free variables, we can say in our setting that what they were aiming for was preservation of validity of (and the consequence relation among) open formulas or schemata. This is clearly an unreasonably high demand and is impossible to achieve in any general terms7; be that as it may, it will be instructive to see the limited extent to which the present method of intensionalization preserves validity of open formulas. The generalization of the notion of validity to open formulas is the standard one. Let C be a class of extensional models. For an object language expression of type t , we say that is extensionally valid in C if [[]]M,g = 1 for all M C and all extensional assignments g suitable for M ; we say that is intensionally valid in C if [[]]MI ,g = i s .1 for all indexed collections I consisting of models in C built on the same base domains and all intensional assignments g suitable for MI . Lemma 3 does not imply that the validity of an open formula is preserved when one moves from extensional models to intensional models created out of them, because not all intensional assignments are quasi-extensional. For example, let = utt utt or, in a more readable style, utt utt Let , , , have the usual interpretation in Mi for all i I . Then is extensionally valid in I = { Mi | i I }, but it is easy to see that there are intensional assignments g such that The reason that an extensionally valid formula with FV() = intensionally valid is related to the failure of the equality [[ ]]MI = int (i s .[[ ]]Mi ) when is a (closed) -abstract. Let FV() = {u11 , . . . , unn }. We have [[]]Mi ,g = 1 for all extensional assignments g if and only if [[u11 . . . unn .]]Mi = x11 . . . xnn .1. Also, [[]]MI ,g = i s .1 for all intensional assignments g if and only if [[u11 . . . unn ]]MI = y11 . . . ynn i s .1. Now suppose [[]]Mi ,g = 1 for all i I and all extensional assignments g. Then [[u11 . . . unn .]]Mi = x11 . . . xnn .1, and this clearly implies need not be int1n t (i s .[[u11 . . . unn .]]Mi ) = y11 . . . ynn i s .1. 7 As mentioned in the introduction, Ben-Avi and Winter (2007) opted to restrict the types of non-logical constants to those with a very special form. But since [[u11 . . . unn .]]MI = int1n t (i s .[[u11 . . . unn .]]Mi ) need not hold, we cannot infer [[u11 . . . unn .]]MI = y11 . . . ynn i s .1. The open formula (4) should be clearly distinguished from its closure utt ) utt vt )) or the open formulas in one free variable utt ) utt vt ), utt ) utt vt ). The three formulas (5), (6), (7), unlike (4), remain valid in intensional models. As mentioned above, the intensionalization of the universal quantifier still only quantifies over objects in D , so the intensional validity of (5), (6), (7) does not imply the intensional validity of (4). The intensional validity of (6) and (7) illustrates the fact that extensionally valid open formulas may remain intensionally valid in certain restricted cases. In what follows, we give one sufficient condition for an extensionally valid to be intensionally valid. Fix an indexed collection I = (Mi )iI of extensional models. We call an object language constant c rigid (in I) if Mi (c) = M j (c) for all i, j I . Let V be a set of object language variables. Let be an object language expression. We define two predicates V -safe and V -protected by simultaneous induction as follows: is V -safe if and only if one of the following conditions holds: 1. is a constant or a variable. 2. = and either is V -protected and is V -safe, or is V -safe, all constants that occur in are rigid, FV( ) V = , and is -protected. 3. = v. and is V -safe. is V -protected if and only if one of the following conditions holds: 1. is a constant or a variable not in V . 2. = and is V -protected and is V -safe. 3. = v. and is V {v}-protected. More informally, if u11 . . . unn . occurs in a V -safe formula as an argument of a variable in V , then cannot contain any non-rigid constants or variables in V , and must be a {u11 , . . . , unn }-safe formula that does not start with one of u11 , . . . , unn . Lemma 9 Let V be a set of object language variables, and let be an object language expression of type . Suppose that g is an intensional assignment such that for all variables u FV() V , we have g(u ) = int (ks .x ) for some x D . The following hold: 1. If is V -protected, [[]]MI,g = int(i s .[[]]Mi ,gi ). 2. If is V -safe, ext[[]]MI,g i = [[]]Mi ,gi . Proof We prove 1 and 2 by simultaneous induction on . Note that the equality in 1 implies the equality in 2, so when is V -protected, it suffices to prove the former. Induction basis. Case 1. is a constant c. In this case, is V -protected. We have [[c]]MI,g = MI (c) = int(i s .Mi (c)) = int(i s .[[c]]Mi ,gi ) by the definition of MI (c). Case 2. is a variable v. In this case, is V -protected if and only if v V . We have ext[[v]]MI,g i = ext g(v) i = gi (v) = [[v]]Mi ,gi by the definition of gi , so the equality in 2 holds. If v V , then by the assumption on g, we have [[v]]MI,g = g(v) = int(i s .gi (v)) = int(i s .[[v]]Mi ,gi ), so the equality in 1 holds. Induction step. Case 1. = , where is of type and is of type . Case 1a. is V -protected and is V -safe. In this case, is V -protected (as well as V -safe). By induction hypothesis, [[ ]]MI,g = int(i s .[[ ]]Mi ,gi ) and ext [[ ]]MI,g i = [[ ]]Mi ,gi . Hence and the condition in 1 is satisfied. Case 1b. is V -safe, all constants that occur in are rigid, FV( ) V = , and is -protected. In this case, is V -safe. Note that FV( ) V = implies that g(u) = int(ks .x ) for some x D for all u FV( ). Hence, the induction hypothesis applies to both and and we get ext[[ ]]MI,g = [[ ]]Mi ,gi and f[[or]]aMllIu,g = int (i s .[[ ]]Mi ,gi ). The fact that g(u) = int(ks .x ) for some x D FV( ) also implies that gi and g j agree on FV( ) for all i, j I . Since Mi (c) = M j (c) for all i, j I for all constants c in , we see that [[ ]]Mi ,gi = [[ ]]M j ,g j for all i, j I . Thus, and the condition in 2 holds. Case 2. = v . , where is of type and = . Case 2a. is V {v }-protected. In this case, is V -protected. By induction hypothesis, [[ ]]MI ,h = int(i s .[[ ]]Mi ,hi ) holds of all h satisfying the following condition: for all u FV( ) (V {v }), there is an x D such that h(u ) = int (ks .x ). (8) for all u FV( ) V , there is an x D such that h(u ) = int (ks .x ). ext [[v . ]]MI ,g i = x .ext ([[v . ]]MI ,g (int (ks .x ))) i (Note that FV( ) FV() {v } and h = g[int (ks .x )/v ] satisfies (9).) Thus, the condition in 2 is satisfied. This completes the induction step. Remark 10 Remark 5 applies, mutatis mutandis, to Lemma 9 as well. Theorem 11 Let be a formula of the object language that is FV()-safe. If is extensionally valid in a class C of extensional models, then is intensionally valid in C. Proof Let I = (Mi )iI be an indexed collection of extensional models in C that share the same base domains. Let g be an arbitrary intensional assignment suitable for MI . Then [[]]MI,g = extt [[]]MI,g = i s .[[]]Mi ,gi by Lemma 9. Thus, if [[]]Mi ,h = 1 for all i I and all extensional assignments h suitable for Mi , then [[]]MI,g = i s .1. Here are some examples illustrating the scope of applicability of Theorem 11. First, all tautologies of propositional logic are intensionally valid. It is easy to see that all formulas built from propositional variables in V are V -safe, because propositional variables are V -safe and truth-functional connectives are V -protected. In fact, we need not invoke Theorem 11 in this case, because all objects in Dst are quasi-extensional. Of course, the fact that propositional tautologies are intensionally valid just means that the power set of Ds is a Boolean algebra. A less trivial example is Aristotelian syllogisms, which are of the form Q1ue1t ue2t Q2ue3t ue4t Q3ue5t ue6t , where uet , . . . , ue6t are not necessarily distinct variables and Q1, Q2, Q3 are not 1 necessarily distinct constants of type (e t ) (e t ) t . Formulas of this form are {u1, . . . , u6}-safe, so if they are extensionally valid, one can instantiate u1, . . . , u6 by expressions denoting truly intensional properties (functions from individual concepts to sets of possible worlds). What about first-order logic? Of the usual Hilbert-style axioms, e(ve.ue1t ve ue2t ve) (e(ve.ue1t ve) e(ve.ue2t ve)), where (of type t t t ) is written as an infix operator. This object language uet , ue2t }-safe and is hence intensionally valid, assuming the usual formula is { 1 interpretation of e and . In contrast, x (x ) (t ) s = t ((s) (t )), which is not {uet , te}-safe. Indeed, it is not intensionally valid, because not all individual concepts are constant functions. Another axiom that is not intensionally valid, this time from first-order logic with equality, is which is rendered as where = stands for the equality between individuals in De. This formula is not {se, te, vet }-safe and is not intensionally valid. This is just the well-known failure of substitutivity in intensional contexts. Here are a couple of more artificial examples. Let I be an object language constant that denotes the identity function on Det in all models in C. Then is extensionally valid in C, but not intensionally so. This is because the intensional (utt utt ) utt (vt ), where , , , , have the usual interpretation. This formula is extensionally valid, but not intensionally so. Observe that it is not {utt , vt }-safe because FV(vt ) {utt , vt } = . Note that FV()-safety is by no means a necessary condition for to have the property in Theorem 11. For one thing, may be an instance of an FV( )-safe formula while not itself FV()-safe. 5 Montagues Typing in PTQ The mapping which replaces each occurrence of e and t by s e and s t , and the associated intensionalization and extensionalization combinators (int and ext) are not familiar to linguists. In linguistics, a common practice nowadays is to use the fewest instances of s that are necessary for adequate semantic analysis, rather than systematically replacing each occurrence of an atomic type by its intensional counterpart. In Montagues original work, however, there was a systematic placement of s in the semantic types associated with syntactic categories. In PTQ (Montague 1973), syntactic categories are built from basic categories e and t by means of two connectives / and //. The semantic type f ( A) associated with a syntactic category A was defined by the following recursion: f (e) = e, f (t ) = t, h(e) = e, h(t ) = t, f ( A/B) = f ( A//B) = (s f (B)) f ( A). This gives rise to the following association between an extensional semantic type and its intensional counterpart h(): This mapping h() looks quite different from the above mapping . For example, if = (e t ) t , then we have Note that the number of occurrences of s in the two types is different: it is three for and two for h(). Nevertheless, there is a systematic correspondence between the two approaches. First, note that f ( A) is the type of the extension of an expression of syntactic category A. The type of the intension of an expression of syntactic category A is s f ( A). Thus, what we should really be comparing to is not h(), but s h(). It is easy to see that the number of occurrences of s in and in s h() is the same for all . Indeed, we can go from one type to the other by repeatedly applying the operation of changing the order of arguments: a = s h(a), The domains of the two types in (10) are of course related by the combinator and its inverse, C,,, which shows that the two types are isomorphic (di Cosmo 2005). It easily isomorphic; indeed, this is witnessed by the pair of combinators P and Q defined as follows: P = x i s ysh(). P (x(Q y))i, Q = ysh()x.Q (i s. yi (P x)). x .Q( P x ) = x .x , ysh(). P(Q y) = ysh().y. This allows us to define the PTQ version intPTQ and extPTQ of intensionalization and extensionalization combinators in terms of int and ext: intPTQ = x s i s ysh().intPTQ(j s .x j (extPTQ y j ))i, extPTQ = ys(sh())h() j s x .extPTQ(i s .yi (intPTQ(ks .x ))) j. intPTQ = x si s y1sh(1) . . . ynsh(n).xi (extPT1Q y1 i ) . . . (extPTnQ yn i ), extPTQ = ysh() j s x11 . . . xnn .y j (intPT1Q(ks .x1)) . . . (intPTnQ(ks .xn)). For example, if j De, then int(PeTQt)t (ks uet .uj) = i s ys(se)t .yi (ks .j). This is the simply typed -calculus expression corresponding to PTQs translation of John. A PTQ model M of our object language consists of base domains (Da )sA{s} together with an assignment of an intension M (c) Dsh() to each object language constant c of type . An object language expression has the intension [[]]PMT,Qg in a PTQ model M relative to an assignment g of values to variables, where g(vl) Dsh() for every variable vl of type : [[c]]PMT,Qg = M (c), PTQ PTQ PTQ [[vl.]]M,g = i s x sh().[[]]M,g[x/vl]i Note that these are recursive clauses for the intensions of object language expressions. In the case of the PTQ fragment, our object language expressions roughly correspond to meaning recipes associated with analysis trees of English expressions.8 The last two clauses of the above recursive definition can be recast in terms of extensions (not to be confused with extensionalization), i.e., values of intensions at particular indices, as follows. Writing [[]]PMT,Qg,i for [[]]PMT,Qg i , we have PTQ PTQ [[vl.]]M,g,i = x sh().[[]]M,g[x/vl],i . The former says that the extension of is the extension of applied to the intension of . This semantic recipe was called intensional functional application by Heim and Kratzer (1998), and it appears in PTQ in the form of the Intensional Logic (IL) expression ( ), where and translate and , respectively. The two intensional interpretations of are related by the following equations. For any object language expression of type , we have [[]]PMT,Qg = P ([[]]QM,Qg) for PTQ model M and PTQ assignment g, PTQ [[]]M,g = Q ([[]]PM,Pg) for intensional model M and assignment g. Here, P M is the PTQ model such that ( P M )(c) = P(M (c)) for each constant c of type , and P g is the PTQ assignment such that ( P g)(vl) = P(g(vl)) for each variable vl. The definitions of Q M and Q g are similar. The above equations can be proved by straightforward induction on . In particular, when is a sentence (i.e., closed object language expression of type t ), we have [[]]PMTQ = [[]]QM for every PTQ model M . As before, given an indexed collection I = (Mi )iI of extensional models with the same base domains (Da )aA, we can create a PTQ model M PTQ based on I, with I 8 The correspondence is not exact, however, for reasons we choose not to go into here. Ds = I , by letting for each object language constant c of type . Then Q M PTQ = MI and we can easily prove analogues of Lemma 3 and Theorem 6. In particIular, for every sentence , we have intIL[x s ] = y1sh(1) . . . ynsh(n ). x (extIL1 [y1]) . . . (extILn [yn ]), extIL[ysh()] = x11 . . . xnn . y((intIL1 [ x1])) . . . ((intILn [ xn ])), where the right-hand sides of the equations are IL expressions and equality is syntactic equality. Thus, intIL[x s ] is an IL expression of type h() whose only free variable is x , and extIL[ysh()] is an IL expression of type whose only free variable is y. (We write intIL[], where is an IL expression of type s , for the result of replacing x s by in intIL[x s ].) When these IL expressions are translated into simply typed -terms (of type s h() and s , respectively), they come out as equivalent to the following, obtained from (11): intPTQ x s = i s y1sh(1) . . . ynsh(n ).x i (extPT1Q y1 i ) . . . (extPTnQ yn i ), extPTQ ysh() = j s x11 . . . xnn .y j (intPT1Q(ks .x1)) . . . (intPTnQ(ks .xn )). The translation in question is which is a straightforward rendering of the semantics of IL given in Montague (1973).9 Note that this translation gives a simply typed -term that represents the intension of 9 This translation of IL expressions (without constants) into simply typed -calculus must not be confused with (12), which gives the PTQ-style intensional compositional semantics to expressions of our object language, which are simply typed -terms, not IL expressions. We are indebted to Reinhard Muskens and Yoad Winter for helpful discussions. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


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Philippe de Groote, Makoto Kanazawa. A Note on Intensionalization, Journal of Logic, Language and Information, 2013, 173-194, DOI: 10.1007/s10849-013-9173-9