There is in my opinion no important theoretical difference between natural languages and the artificial languages of logicians; indeed, I consider it possible to comprehend the syntax and semantics of both kinds of language within a single natural and mathematically precise theory. On this point I differ from a number of philosophers, but agree, I believe, with Chomsky and his associates. ("Universal Grammar" 1970)
Montague published what soon became known as Montague grammar[1] in three papers:
The meaning of a sentence obtained by the rule is obtained by
applying the function for NP to the function for VP.
The types of VP and NP might appear unintuitive because of the question as to the meaning of a noun phrase that is not simply a term. This is because meanings of many noun phrases, such as "the man who whistles", are not just terms in predicate logic, but also include a predicate for the activity, like "whistles", which cannot be represented in the term (consisting of constant and function symbols but not of predicates). So we need some term, for example x, and a formula whistles(x) to refer to the man who whistles. The meaning of verb phrases VP can be expressed with that term, for example stating that a particular x satisfies sleeps(x) snores(x) (expressed as a function from x to that formula). Now the function associated with NP takes that kind of function and combines it with the formulas needed to express the meaning of the noun phrase. This particular way of stating NP and VP is not the only possible one.
Key is the meaning of an expression is obtained as a function of its components, either by function application (indicated by boldface parentheses enclosing function and argument) or by constructing a new function from the functions associated with the component. This compositionality makes it possible to assign meanings reliably to arbitrarily complex sentence structures, with auxiliary clauses and many other complications.
The meanings of other categories of expressions are either similarly function applications, or higher-order functions. The following are the rules of the grammar, with
the first column indicating a non-terminal symbol, the second column one possible
way of producing that non-terminal from other non-terminals and terminals,
and the third column indicating the corresponding meaning.
meaning
S
NP VP
NP
name
NP
DET CN
NP
DET RCN
DET
"some"
DET
"a"
DET
"every"
DET
"no"
VP
intransverb
VP
TV NP
TV
transverb
RCN
CN "that" VP
RCN
CN "that" NP TV
CN
predicate
Here are example expressions and their associated meaning, according to the above grammar, showing that the meaning of a given sentence is formed from its constituent
expressions, either by forming a new higher-order function, or by applying
a higher-order function for one expression to the meaning of another.
expression
meaning
a
man
a man
sleeps
a man sleeps
man that dreams
a man that dreams
a man that dreams sleeps
The following are other examples of sentences translated into the predicate logic by the grammar.
InDavid Foster Wallace's novel Infinite Jest, the protagonist Hal Incandenza has written an essay entitled Montague Grammar and the Semantics of Physical Modality. Montague grammar is also referenced explicitly and implicitly several times throughout the book.
^"Universal grammar". Theoria 36 (1970), 373–398. (reprinted in Thomason, 1974)
^"English as a Formal Language". In: Bruno Visentini (ed.): Linguaggi nella società e nella tecnica. Mailand 1970, 189–223. (reprinted in Thomason, 1974)