Syntactic Categories and Semantic Types

According to Montague the purpose of syntax is to produce the input for the semantics:

اقتباس :: I fail to see any interest in syntax except as a preliminary to semantics. (Montague 1970c, 223)

Although the syntax was in his eyes subordinate, he was fully explicit in his rules in which he used some ad hoc syntactic tools.
In Montague 1970a, the relation between syntactic categories and semantic types is given only by a list. Montague (1973) defines a systematic relation which amounts to the same relation as one would have in categorial grammar. However, Montague's syntax is not a categorial syntax because the rules are not always category driven and because some of the rules are not concatenation rules.
For each of these two aspects, proposals have been put forward to change the situation. One direction was to stay closer to the ideals of categorial grammar, with only type driven rules, sometimes allowing for a restricted extension of the power of concatenation rules. See, for example, Morrill 1994 and Carpenter 1998. The other approach was to incorporate in Montague grammar as much as possible the insights from syntactic theories, especially originating from the tradition of Chomsky. A first step was made by Partee (1973), who let the grammar produce structures (labelled bracketings). A syntactically sophisticated grammar (with Chomskyan movement rules) was used in the Rosetta translation project (Rosetta 1994).
Montague introduced the and in John walks and Mary sings not from a given lexical entry, but as effect of a rule. This is known as treating and syncategorematically. He did this for all determiners, and also for negation. For John walks and sings a different rule is needed than forJohn walks and Mary sings because syntactically the first one is a conjunction of verb phrases and the second one of sentences. However, the two meanings of and are closely related and a generalisation is missed. As general solution it was proposed to use rules (or alternatively general principles) that change the category of an expression to another category; a change that corresponds with a semantic rule that ‘lifts’ the meaning. For instance, the meaning of and as a connective between verb phrases is obtained by lifting the meaning of sentence connective ∧ to λPλQλx[P(x) ∧ Q(x)]. Classical papers about the approach with lifting rules are Partee and Rooth 1983, Partee 1987 and Hendriks 2001. In a monograph (Winter 2001) the whole complex of conjoined phrases is considered.
Nowadays the syntactic side usually plays no important role in publications on Montague semantics. Montague's method to present fragments with a fully explicit syntax is largely abandoned. One rather focuses on a semantically interesting phenomenon, suggesting rules which are only explicit concerning the semantic side. Whether and how the phenomenon fits together with the treatment of other phenomena is not considered. But Partee in Janssen 1997 and Jacobson 2014 give arguments against this tendency. Jacobson 2014 indeed provides a fragment.

3.4 Pragmatics

The meaning of sentences is sometimes determined by factors from the context of use; e.g. whether I am happy is true, depends on who the speaker is. Other examples are here and this. Montague writes about these factors in his paper ‘Pragmatics’ (Montague 1968) and in Montague 1970b. He indicates how this could be done by introducing additional parameters (besides the time and the possible world). His papers focus on the formal apparatus, and he works it out only for the pronoun I.
Several authors followed Montague's approach, and extended, when needed, the list of parameters. A classical example is Kaplan 1989, which deals with demonstratives and indexicals. He uses ‘context’ as a parameter, which consists at least in agent, moment of time, location, and possible world. The content of a sentence, with respect to a context, is a proposition, and the linguistic meaning, or character, of an expression is a function from contexts to contents. This difference between content and meaning is exploited to develop his (influential) theory of demonstratives (she, her, that) and indexicals (I, today).
Cresswell (1973, 111) has another opinion. He argues that the approach with parameters requires that a finite list of contextual features is given in an advance. He considers that to be impossible and provides an alternative. His proposal is not followed by other authors.
Presuppositions and implicatures are often considered as belonging to pragmatics. The aim of a recursive approach to presupposition was always in the air, for the practical reason that it seems the only way to deal with presuppositions for infinitely many sentences. An example of a compositional treatment is Peters 1979. But the phenomena are complex, and later treatments are not always completely compositional; several correcting factors have to be taken into consideration (Beaver 1997).
Finally, there is pragmatics in the sense of using a language in practical situations. Declarative sentences can be used to ask questions, and to give orders, and sometimes sentences are not used literally, but metaphorically. On this aspect of pragmatics not much has been written, but Cresswell (1973) explains that formal semantics has all the ingredients to cope with it.

3.5 Ontology

Montague's ‘intensional logic’ is a higher order logic. This aspect provoked a very critical attack by Hintikka:

اقتباس :: It seems to me that this is the strategy employed by Montague Grammarians, who are in fact strongly committed to compositionality. […]. There is a price to be paid however. The higher order entities evoked in this “type theoretical ascent” are much less realistic philosophically and psycholinguistically than our original individuals. Hence the ascent is bound to detract from the psycholinguistic and methodological realism of one's theory. (Hintikka 1983, 20)

Hintikka's criticism has not found many supporters. Ironically, Hintikka's alternative (game theoretical semantics), is encapsulated in the traditional Tarskian approach (see Hodges 1997 or Caicedo et al. 2009); they define the meaning of a formula as a collection of sets of assignments.
In Montague's approach possible worlds are basic objects without internal or external structure. Phenomena having to do with belief, require external structure, such as an accessibility relation for belief-alternatives. Counterfactuals require a distance notion to characterize worlds which differ minimally from each other. Structures on possible worlds are used frequently.
Sometimes an internal structure for possible worlds is proposed. A possible world determines a set of propositions (those propositions which are true with respect to that world), and in Fox and Lappin 2005, the reverse order is followed. They have propositions as primitive notions, and define possible worlds on the basis of them. Also Cresswell (1973) provides a method for obtaining possible worlds with internal structure: he describes how to build possible worlds from basic facts. None of these proposals for internal structure have been applied by other authors than the proposers.
The philosophical status of certain entities is not so clear, such as pains, tasks, obligations and events. These are needed when evaluating sentences like e.g. Jones had a pain similar to the one he had yesterday. In ‘On the nature of certain philosophical entities’ (Montague 1969), Montague describes how these notions can be described using his intentional logic; they are properties of moments of time in a possible world. Of these notions, only events occur in papers by other authors, albeit not in the way Montague suggested. They are seen as basic, but provided with an algebraic structure allowing, e.g., subevents (Link 1998, ch. 10–12; Bach 1986a).
The set E may include whatever one would like to consider as basic entities: numbers, possible objects, and possible individuals. Whether an individual is considered to be really living or existing at a certain moment of time or in a certain possible world is not given directly by the model; one has to introduce a predicate expressing this. Normally the set E has no internal structure, but for mass nouns (which have the characteristic property that any part of water is water), a structure is needed, see Pelletier & Schubert 2003. Also plurals might evoke a structure on the set E, e.g. when sum-individuals are used (see Link 1983, 1998 (ch. 1–4), and Bach 1986a). Also when properties (loving John) are considered as entities for which predicates may hold (Mary likes loving John) structure is needed: property theory gives the tools to incorporate them (see Turner 1983).

3.6 Psychology

When Montague grammar emerged, the leading theory about syntax was Chomskyan grammar. That approach claimed that it revealed processes that went on in the brain, and that linguistics was a branch of biology. In those days it was experimentally shown that the passive transformation was a real process in the brain. Chomskyan grammar still is a leading theory, and although most of the theory has changed considerably (there is no passive transformation anymore), it still considers itself to be revealing psychologically real processes. Montague had no psychological claim for his theory; on the contrary, he considered linguistics as a branch of mathematics and not of psychology (Thomason (ed.) 1974, 2).
However, the field remained interested in psychological aspects. Partee (1977) explained to a meeting of psychologists that the theory cannot be applied directly to psychology because of the huge numbers of entities in the models (infinite numbers of functions from functions to functions). Partee (1979) argues that there is a deep gap between the mathematical view and the psychological view, especially concerning propositional attitude verbs and the behavior of proper names in such contexts, and she says that this gap has, somehow, to be bridged.
An argument often put forward in defense of compositionality concerns its psychological motivation. The principle explains how a person can understand sentences he has never heard before; Frege (1923, 55) already mentioned this argument This motivation for compositionality is attacked by Schiffer (1987). On the one hand he argues that compositionality is not needed in an explanation of that power, and on the other hand that a compositional approach does not work. His argumentation is illustrated by Tanya believes that Gustav is a dog. Schiffer considers several compositional theories and argues that none of these theories offers a plausible account for the proposition that is supposed to be the content of Tanya's belief. So there is nothing from which the meaning of the sentence can be formed compositionally. Hence compositionality cannot hold. Partee (1988) discusses Schiffer's arguments against compositionality, and explains that Schiffer does not make a sufficient distinction between semantic facts and psychological facts. Partee points out the analogy between these problems with belief and those with the semantics of proper names (how can one correctly use proper names without being acquainted with the referent). The latter is discussed and explained by Kripke (1972). Partee proposes to solve the problems of belief along the same lines. Schiffer (1988) replies to this paper, but he does not react to her analogy, nor to the main point: that a semantic theory is to be distinguished from a psychological theory.
An extensive discussion of the relation between Montague semantics and psychology is given in the last chapter in Dowty 1979. He starts his chapter with a description of the situation. ‘Contemporary linguists, unlike many philosophers of language, almost invariably profess to be concerned with the “psychological reality” of the theoretical concepts they postulate in semantics analysis’ (Dowty 1979). He works out this point and then describes his own position. ‘To get the point right away, let me confess that I believe that the model theoretic intension of a word has in principle nothing whatsoever to do with what goes on in a person's head when he uses a word.’ Nevertheless, he tries to show that the notion of intension is a fundamental and indispensable concept from the point of view of ‘psychological semantics’. He gives three reasons. The first is that semantics provides a theory that explains entailment (and synonymity, validity contradiction etc.]), all notions that must somehow be part of a theory of language understanding. Secondly, the theory of truth and reference must be a bottom line in any general account of ‘meaning’ in natural language. And thirdly, when certain ways of compositionally deriving the meanings from their parts can be shown to be necessary in a theory of truth and reference, then it may be concluded that the same compositional analysis is necessary in a theory of language understanding.
These examples illustrate the general opinion that psychological reality can only very indirectly be associated with what is going on in Montague semantics; only a few articles discuss the connection.

4. Concluding Remarks

4.1 Legacy

Montague revolutionized the field of semantic theory. He introduced methods and tools from mathematical logic, and set standards for explicitness in semantics. Now all semanticists know that logic has more to offer than first order logic only. Finally, recall that Barbara Partee said: ‘lambdas really changed my life’; in fact lambdas changed the lives of all semanticists.

4.2 Further Reading

A recent introduction is Jacobson 2014. It is a gentle introduction to the field, especially for linguists and philosophers. It presents several successes obtained by the approach. Older introductions are Dowty et al 1981 and Gamut 1991, which are more technical and prepare for Montague's original paper. An overview of the history of the field is given by Partee and Hendriks (1997). Collections of important papers are Portner and Partee (eds.) 2002 and Partee 2004. The ‘Handbook of compositionality’(Werning et al 2011) discusses many aspects of the approach. The most important journal in the field are Linguistics and Philosophy, Natural Language Semantics, and Semantics and Pragmatics.

4.3 Example

A small example is presented below, it consists of the two sentences John is singing, and Every man is singing. The example is not presented in Montague's original way, but modernized: there is a lifting rule, the determiner is a basic expression, and intensional aspects are not considered.
The grammar has four basic expressions:
1. John is an expression of the category Proper Name. Its denotation is an individual represented in logic by John.
2. The Intransitive Verb sing denotes a set (the set of singers), and is represented by the predicate symbol sing.
3. The Common Noun man, which denotes a set, represented by man.
4. The Determiner every. Its denotation is λPλQ∀x[P(x) → Q(x)]; an explanation of this formula will be given below.
The grammar has three rules.
1. A rule which takes as input a Proper Name, and produces a Noun Phrase. The input word is not changed: it is lifted to a ‘higher’ grammatical category. Semantically its meaning is lifted to a more abstract, a ‘higher’ meaning: the representation of the denotation of John as Noun Phrase is λP[P(John)]. An explanation of the formula is as follows. P is a variable over properties: if we have chosen an interpretation for P, we may say whether P holds for John or not, i.e. whetherP(John) is true. The λP abstracts from the possible interpretations of P: the expression λP[P(John)] denotes a function that takes as input properties and yields true if the property holds for John, and false otherwise. So the denotation of John is the characteristic function of the set of properties he has.
2. A rule that takes as input a Noun Phrase and an Intransitive Verb, and yields as output a Sentence: from John and sing it produces John is singing. The corresponding semantic rule requires the denotation of the Noun Phrase to be applied to the denotation of the Intransitive Verb. This is represented as λP[P(John)](sing). When applied to the argument sing, the function represented by λP[P(John)] yields true if the predicate sing holds for John, so precisely in casesing(John) is true. So λP[P(John)](sing) and sing(John) are equivalent. The latter formula can be obtained by removing the λP and substituting sing for P. This is called ‘lambda-conversion’.
3. A rule that takes as inputs a Determiner and a Common Noun, and yields a Noun Phrase: fromevery and man it produces every man. Semantically the denotation of the Determiner has to be applied to the denotation of the Common Noun, hence λPλQ∀x[P(x) → Q(x)](man). By lambda conversion (just explained) this is simplified to λQ∀x[man(x) → Q(x)]. This result denotes a function that, when applied to property A, yields true just in case all man have property A.
The example given with the last rule helps us to understand the formula for every : that denotes a relation between properties A and B which holds in case every A has property B.
The next step is now easy. Apply the rule for combining a Noun Phrase and an Intransitive Verb to the last result, producing Every man is singing. The output of the semantic rule is λQ∀x[man(x) → Q(x)](sing). By lambda conversion we obtain ∀x[man(x) →sing(x)], which is the traditional logical representation of Every man is singing.
Note the role of lambda-operators:
1. John and every man are interpreted in a similar way: sets of properties. These sets can be represented due to lambda-operators.
2. Every man and sing are syntactically on the same level, but semantically sing has a subordinated role: it occurs embedded in the formula. This switch of level is possible due to lambda-operators.

Syntactic Categories and Semantic Types

3.4 Pragmatics

3.5 Ontology

3.6 Psychology

4. Concluding Remarks

4.1 Legacy

4.2 Further Reading

4.3 Example

Syntactic Categories and Semantic Types :: تعاليق

Syntactic Categories and Semantic Types