Unicorns and Meaning Postulates

Montague's most influential article was ‘The Proper Treatment of Quantification in Ordinary English’ (Montague 1973). It presented a fragment of English that covered several phenomena which were in those days discussed extensively. One of the examples gave rise to the trademark of Montague grammar: the unicorn (several publications on Montague grammar are illustrated with unicorns).
Consider the two sentences John finds a unicorn and John seeks a unicorn. These are syntactically alike (subject-verb-object), but are semantically very different. From the first sentence follows that there exists at least one unicorn, whereas the second sentence is ambiguous between the so called de dicto reading which does not imply the existence of unicorns, and the de re reading from which existence of unicorns follows.
The two sentences are examples of a traditional problem called ‘quantification into intensional contexts’. Traditionally the second sentence as a whole was seen as an intensional context, and the novelty of Montague's solution was that he considered the object position of seek as the source of the phenomenon. He formalized seek not as a relation between two individuals, but as a relation between an individual and a more abstract entity, see Section 2.2. Under this analysis the existence of a unicorn does not follow. The de re reading is obtained in a different way, see Section 2.5.
It was Montague's strategy to apply to all expressions of a category the most general approach, and narrow this down, when required, by meaning postulates. So initially, find is also considered to be a relation between an individual and such an abstract entity, but some meaning postulate restricts the class of models in which we interpret the fragment to only those models in which the relation for find is the (classical) relation between individuals.
As a consequence of this strategy, Montague's paper has many meaning postulates. Nowadays semanticists often prefer to express the semantic properties of individual lexical items directly in their lexical meaning, and then find is directly interpreted as a relation between individuals. Nowadays meaning postulates are mainly used to express structural properties of the models (for instance the structure of time axis), and to express relations between the meanings of words. For a discussion of the role of meaning postulates, see Zimmermann 1999.

2.2 Noun Phrases and Generalized Quantifiers

Noun phrases like a pig, every pig, and Babe, behave in many respects syntactically alike: they can occur in the same positions, can be conjoined, etc. But a uniform semantics seems problematic. There were proposals which said that every pig denotes the universally generic pig, and a pig an arbitrary pig. Such proposals were rejected by Lewis (1970), who raised, for instance, the question which would be the color of the universal pig, all colors, or would it be colorless?
Montague proposed the denotation of a descriptive phrase to be a set of properties. For instance, the denotation of John is the set consisting of properties which hold for him, and of every manthe set of properties which hold for every man. Thus they are semantically uniform, and then conjunction and/or disjunction of arbitrary quantifier phrases (including e.g. most but not all) can be dealt with in a uniform way.
This abstract approach has led to generalized quantifier theory, see Barwise & Cooper 1981 and Peters & Westerståhl 2006. By using generalized quantifier theory a remarkable result has been achieved. It concerns ‘negative polarity items’: words like yet and ever. Their occurrence can be licensed by negation: The 6:05 has arrived yet is out, whereas The 6:05 hasn't arrived yet is OK. But there are more contexts in which negative polarity items may occur, and syntacticians did not succeed in characterizing them. Ladusaw (1980) did so by using a characterization from generalized quantifier theory. This was a great success for formal semantics! His proposal roughly was as follows. Downward entailing expressions are expressions that license inferences from supersets to subsets. No is downward entailing because from No man walks it follows thatNo father walks. A negative polarity item is acceptable only if it is interpreted in the scope of a downward entailing expression, e.g. No man ever walks. Further research showed that the analysis needed refining, and that a hierarchy of negative polarity items should be used (Ladusaw 1996).

2.3 Logic and Translating

An expression may directly be associated with some element from the model. For instance, walkwith some set of individuals. Then also the operations on meanings have to be specified directly, and that leads to formulations such as:

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G3 is that function f ∈ ((2I)A×A)A[size=9]ω such that, for all x ∈ Aω, all u,t ∈ A and all i ∈ I : f(x)(t,u)(i) = 1 if and only if t = u. (Montague 1970a, 194)[/size]

Such descriptions are not easy to understand, nor convenient to work with. Montague (1973, 256) said, ‘it is probably more perspicuous to proceed indirectly.’ For this purpose he introduced a language, called ‘intensional logic’. The operation described above is then represented by ∧λtλu[t= u]. The λt says that it is a function that takes t as argument, likewise for λu. So λtλu[t = u] is a function which takes two arguments, and yields true if the arguments are equal, and otherwise false. The preceding ∧ says that we consider a function from possible worlds and moments of time to the thus defined function.
Two features of the Montague's ‘intensional logic’ attracted attention.
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[*]It is a higher order logic. In those days, linguists, philosophers and mathematicians were only familiar with first order logic (the logic in which there are only variables for basic entities). Since in Montague semantics the parts of expressions must have meaning too, a higher order logic was needed (we have already seen that every man denotes a set of properties).

[*]The logic has lambda abstraction, which in Montague's days was not a standard ingredient of logic. The lambda operator makes it possible to express with higher order functions, and the operator made it possible to cope differences between syntax and semantics. For instance, in John walks and he talks there is only one occurrence of John, whereas in the logic John should occur with the predicate walk and with the predicate talk. The use of lambda-operators enables us to plug in the meaning of John at several positions. The importance of lambdas is expressed by Partee at a talk on ‘The first decade of Montague Grammar’: ‘Lambdas changed my life’ (Partee 1996, 24). Nowadays lambdas are a standard tool in all papers in semantics. In section 4.1 an example will be given that illustrates the power of lambdas.

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This motivation for using translations (a tool for obtaining perspicuous representations of meanings) has certain consequences.
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[*]Translation is a tool to obtain formulas which represent meanings. Different, but equivalent formulas are equally acceptable. In the introduction of this article it was said that Montague grammar provided a mechanical procedure for obtaining the logical translation. As a matter of fact, the result of Montague's translation of Every man runs is not identical with the traditional translation, although equivalent with it, see the example in Section 4.1.

[*]The translation into logic should be dispensable. So in Montague semantics there is nothing like ‘logical form’(which plays such an important role in the tradition of Chomsky).

[*]For each syntactic rule which combines one or more expressions there is a corresponding semantic rule that combines the corresponding representations of the meanings. This connection is baptized the rule-to-rule hypothesis (Bach 1976). Maybe it is useful to emphasize that (in case the syntactic operation is meaning preserving) the corresponding semantic rule can be the identity mapping .

[*]Operations depending on specific features of formulas are not allowed. Janssen (1997) criticized several proposals on this aspect. He showed that proposals that are deficient in this respect are either incorrect (make wrong predictions for closely related sentences), or can be corrected and generalized, and thus improved.

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The method of using a logic for representing meanings has a long history. One might point to philosophers such as Dalgarno and Leibniz who developed formal languages in order to express philosophy clearly. In the 19th century there were several proposals for artificial languages in order to make mathematical argumentation more transparent, for instance by Frege and by Peano. Frege's ‘Begriffsschrift’ (Frege 1879) can be seen as the birth of predicate logic: he introduced quantifiers. His motivation came from mathematical needs; he did not use his Begriffsschrift in his papers on natural language. Russell (1905) used logic to represent the meanings of natural language. A classical example in his paper is the analysis of The king of France is bald. Syntactically it has the form subject-predicate, but if it would be constructed logically as a subject-predicate, then the king of France, which denotes nothing, cannot be the subject. So there is a difference between the syntactic form, and the logical form: natural language obscures the view of the real meaning. This became known as the ‘misleading form thesis’. Therefore philosophers of language saw, in those days, the role of logic as a tool to improve natural language. An interesting overview of the history of translating is given in Stokhof 2007.
Note, however, that Montague semantics has nothing to do with the aim of improving natural language or providing its logical form.

2.4 Intensionality and Tautologies

Montague defined the denotation of a sentence as a function from possible worlds and moments of time to truth values. Such a function is called an ‘intension’. As he said (Montague 1970a, 218), this made it possible to deal with the semantics of common phenomena such as modifiers, e.g. in Necessarily the father of Cain is Adam. Its denotation cannot be obtained from the truth value of The father of Cain is Adam : one has to know the truth value for other possible worlds and moments of time. The intensional approach also made it possible to deal with several classical puzzles. Two examples from Montague 1973 are: The temperature is rising, which should not be analyzed as stating that some number is rising. And John wishes to catch a fish and eat it should not be analyzed as stating that John has a particular fish in mind, but that he wants to eat the fish he will catch.
Intensional semantics has been criticized for the fact that all tautologies get the same meaning (are synonymous). Indeed, a tautology as John is ill or he is not ill gets as intension the function that constantly yields true, and the same holds for other tautologies. If one is interested in discriminating semantically between tautologies, then a refinement of the notions ‘meaning’ and ‘equivalence’ is needed: ‘meaning’ should see distinctions between tautologies, and ‘equivalence’ should be sensitive for the thus refined notion of meaning. The oldest proposals to account for this problem is by Lewis (1970): propositions are structured by including in their meanings also the meanings of their parts. Then indeed Green grass is green and White snow is white have different meanings. However, lexical synonyms still pose a problem. Since woodchuck andgroundhog are names for the same species, John believes that Phil is a groundhog is, under this view, equivalent with John believes that Phil is a woodchuck. One could consider belief contexts a separate problem, but most authors see it as part of the problem of equivalence of all tautologies.
Later several proposals for dealing with this are given. Bäuerle and Cresswell (2003) give an overview of the older proposals, and Fox and Lappin (2005) review more recent ones. The latter authors explain that there are two strategies: the first is to introduce impossible worlds in whichwoodchuck and groundhog are not equivalent, and the second is to introduce an entailment relation with the property that identity does not follow from reciprocal entailment. Fox and Lappin follow the second strategy.