Semantics: Handout 13
(A Revised Version of) the Grammar of PTQ
Yılmaz Kılıçaslan
December 24, 2009
1 Introduction
Having seen Montague’s Intensional Logic (IL), we can now look at a revised version of the
grammar he offered for English in the PTQ paper. That is, we are now ready to see his
syntactic treatment of English with a maximum understanding of the semantic motivation
behind each part of the analysis. Though restricted for expository purposes, the grammar of
PTQ illustrates the treatment of a variety of problems involving:
-
quantifier words
the interaction of quantifier words with pronouns and conjunctions
the “intensional constructions” discussed in the previous lectures and
others besides.
The grammar of PTQ has two components:
1. a syntactic component, where a “fragment” of English is given a syntactic analysis with a
semantic motivation behind it, and
2. a component of translation, where each English expression generated by the syntactic
component is translated into an expression of IL.
2 Syntax
The syntax of the English fragment is formulated like the syntax of the languages we have
seen in the previous lectures in that complex expressions are built up “from the inside out”,
i.e. by means of recursive definitions specifying how complex phrases are to be formed out of
simpler ones. As with the language L2E (cf. Handout 6), the syntactic rules will not only
concatenate input expressions, possibly with other symbols added as well, but may also
perform transformation-like manipulations on the inputs.
2.1
The Syntactic Categories of English in the PTQ Grammar
The specification of the English fragment begins with the recursive definition of an infinite
set of syntactic categories:
1
(1)
SYNTACTIC CATEGORIES OF ENGLISH1
1. t is a syntactic category.
2. CN is a syntactic category.
3. IV is a syntactic category.
4. If A and B are any syntactic categories, then A/B and A//B are syntactic categories.
This system is based on the categorial grammars developed by Ajdukiewicz (1935). In a
categorial grammar an expression of category A/B or of A//B2 typically combines with an
expression of category B to give an expression of category A. The notation is conciously
chosen to suggest a kind of algebraic cancellation operation (cf. x/y.y = x), and the result is
that the specification of what categories combine with what others is implicit in the category
labels themselves. As an example, in the PTQ grammar expressions of category t(t/e) (socalled term-phrases such as John and Mary) combine with expressions of category t/e
(intransitive verb phrases such as walk and seek a unicorn) to yield expressions of category t
(sentences).
It should be noted that the complex English category symbols are, as it were, “read right to
left”, whereas the complex types of IL, to which they bear an obvious resemblance, are “read
left to right”. That is, a logical expression of type <a, b> combines with an expression of type a to
give an expression of type b, whereas an expression category A/B combines with an expression of
category B to give an expression of category A.
Since the slash notation becomes cumbersome for complex categories, Montague introduced
mnemonic abbreviations for them. A revised version of his list of category names is
summarized in Table 1:
TABLE 1
Category Categorial Nearest Transformational
Name
Definition Equivalent
of Name
Basic
Expressions
t
t
Sentence
None
CN
CN
Common Noun
man, woman, park, fish,
pen, unicorn, price
IV
IV
Verb Phrase and Intransitive Verb (IV is run, walk, talk, rise,
mnemonic for ‘IntransitiveVerb Phrase’) change
T
t/IV
Noun Phrase and Proper Name (T is John, Mary, Bill, ninety,
mnemonic for ‘Term Phrase’)
he0, he1, he2, ...
IAV
IV/IV
Verb Phrase Adverb (IAV is mnemonic rapidly, slowly,
for ‘Intransitive Adverb’)
voluntarily, allegedly
1
In the original version of the PTQ grammar, among basic syntactic categories was e and CN and IV were
derived categories. Bennett (1974) observed that a great deal of complexity in the system can be avoided if these
two categories are treated as basic categories.
2
We will return to the difference between these two categories in a moment.
2
TV
IV/T
Transitive Verb
(=IV/(t/IV))
find, lose, eat, love, date,
be, seek, conceive
T/CN
(t/IV)/CN
Determiner
every, the, a(n)
t/t
t/t
Sentence Adverb
necessarily
IV/t
IV/t
Sentence-complement Verb
belive, assert
IV//IV
IV//IV
Infinitive-complement Verb
try, wish
IAV/T
(IV/IV)/T
Preposition (one that forms a VP- in, about
modifying prepositional phrase)
2.2
The Syntactic Rules of English in the PTQ Grammar
S1.
The set of basic expressions of category A, denoted BA, is included in the set of phrases
of category A, denoted PA.
S2.
If δ  PT/CN and ζ  PCN, then F2(δ, ζ)  PT, where F2(δ, ζ) = δ′ζ, and δ′ is δ except in
the case where δ is a and the first word in ζ begins with a vowel; here, δ′ is an.
S3.
If ζ  PCN and   Pt, then F3,n(ζ, )  PCN, where F3,n(ζ, ) = ζ such that ′, and ′
comes from  by replacing each occurence of hen or himn by {he / she / it} or {him /
her / it} respectively, according as the first BCN in ζ is of {masc. / fem. / neuter}
gender.
S4.
If α  PT and δ  PIV, then F4(α , δ)  Pt, where F4(α , δ) = αδ′, and δ′ is the result of
replacing the first verb (i.e. member of BIV, BTV, BIV/t, or BIV//IV) in δ by its third person
singular present form.
S5.
If δ  PTV and β  PT, then F5(δ, β)  PIV, where F5(δ, β) = δβ if β does not have the
form hen, and F5(δ, hen) = δ himn.
S6.
If δ  PIAV/T, α  PT, then F5(δ, α)  PIAV.
S7.
If α  PIV/t,   Pt, then F11(α , )  PIV, where F11(α , ) = α that .
S8.
If δ  PIV//IV and β  PIV, then F17(δ, β)  PIV, where F17(δ, β) = δ to β.
S9.
If δ  Pt/t,   Pt, then F6(δ, )  Pt, where F6(δ, ) = δ.
S10.
If δ  PIV/IV and β  PIV, then F7(δ, β)  PIV, where F7(δ, β) = βδ.
S11a. If , ψ  Pt, then F8(, ψ)  Pt, where F8(, ψ) =  and ψ.
S11b. If , ψ  Pt, then F9(, ψ)  Pt, where F9(, ψ) =  or ψ.
S12a. If δ, γ  PIV, then F8(δ, γ)  PIV.
S12b. If δ, γ  PIV, then F9(δ, γ)  PIV.
S13.
If α, β  PT, then F9(α, β)  PT.
S14.
If α  PT and   Pt, then F10,n(α , )  Pt, where either
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(i) α does not have the form hek, and F10,n(α, ) comes from  by replacing the first
occurence of hen, or himn by α and all other occurences of hen, or himn by {he /
she / it} or {him / her / it} respectively, according as the first BCN or BT in α is
{masc. / fem. / neuter}or
(ii) α = hek, and F10,n(α, ) comes from  by replacing all occurences of hen or himn by
hek or himk, respectively.
S16.
If α  PT and δ  PIV, then F10,n(α, δ)  PIV.
S17.
If α  PT and δ  PIV, then F11(α, δ), F12(α, δ), F13(α, δ), F14(α, δ), F15(α, δ)  Pt,
where:
F11(α, δ) = αδ′ and δ′ is the result of replacing the first verb in δ by its negative third
person singular present;
F12(α, δ) = αδ′′ and δ′′ is the result of replacing the first verb in δ by its third person
singular future;
F13(α, δ) = αδ′′′ and δ′′′ is the result of replacing the first verb in δ by its negative third
person singular future;
F14(α, δ) = αδ′′′′ and δ′′′′ is the result of replacing the first verb in δ by its third person
singular present perfect;
F15(α, δ) = αδ′′′′′ and δ′′′′′ is the result of replacing the first verb in δ by its negative
third person singular present perfect;
3 Translation into IL
In contrast to his earlier treatment in ‘English as a Formal Language, Montague in the PTQ
grammar did not give a direct model-theoretic interpretation of English parallel to the modeltheoretic interpretations of the languages we have seen in the previous lectures. Instead, he
proceeded indirectly, first translating each English expression into an expression of IL. The
model-theoretic interpretation of IL (cf. Section 3.2, Handout 12) thus serves indirectly as the
interpretation of the English fragment.
Translating from English into IL follows a rigorously formalized procedure that satisfies the
following requirements:
1. Each basic expression of English is translated into one and only one expression of IL.
2. There is a uniform correspondence between the categories of English and the types of IL.
3. For each syntactic rule of English there is a corresponding translation rule which specifies
the translation of the output of the syntactic rule in terms of the translations of the input(s)
to it.
These requirements insure that the translation procedure obeys a kind of compositionality
principle of its own: the translation of the whole English expression is determined by the
translation of its parts and the syntactic rules used in forming it. And since the expressions of
IL are temselves given a compositional model-theoretic interpretation, it turns out that the
indirect model-theoretic interpretation that results from the translation procedure is likewise
compositional.
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3.1
The Correspondence between Categories of English and Types of IL
It should be noted that there is a confusing divergence in notation between the complex
English category symbols and the complex types of IL. The former are, as it were, “read right
to left”, whereas the latter are “read left to right”. A logical expression of type <a, b>
combines with an expression of type a to give an expression of type b, whereas an expression
of category A/B combines with an expression of category B to give an expression of category
A.
The parallelism between categories of English and the types of IL makes the correspondence
between the two systems rather easy to state. There is one complicating factor, however.
Montague observed that there would be certain cases where syntactically distinct categories of
English would correspond to the same logical type, and to make this distinction he introduced
the notational device of the “double slashes”. Thus, A/B and A//B are distinct categories but
are to correspond to the same logical type. For example, verb phrase adverbs (e.g. rapidly,
allegedly etc.) are given the category label IV/IV, while infinitive-complement verbs (e.g. try,
wish etc.) are classified as IV//IV, and both are translated into the same type of IL (cf. Table
2).
The correspondence between the categories of English and the types of intensional logic can
be stated in the form of a recursive definition of a function f which maps English categories
into types:
(2)
A FUNCTION MAPPING SYNTACTIC CATEGORIES OF ENGLISH INTO TYPES OF IL
1. f(t) = t
2. f(CN) = f(IV) = <e, t>
3. For all categories A and B, f(A/B) = f(A//B) = <<s, f(B)>, f(A)>
Note that clause 3 states that expressions of category A/B or of category A//B, whatever the A and B in
question, are to denote functions applying to the intensions of expressions of category B, rather than to
their extensions. This will allow for a proper treatment of oblique-context creating elements (cf.
Handouts 11 and 12). It is, of course, true that not all functors are intensional. However, as intension is
a “stronger” notion than extension, in that the extension of an expression at a certain index is
determined by the intension, it is adequate to unify extensional and intensional functors by
assimilating the class of the former to that of the latter.
Table 2 shows the types corresponding to our syntactic categories along with the names of
denoted objects:
TABLE 2
Category Corresponding Type
Name
Name of Semantical
Object Denoted by this Type
t
t
truth value
CN
<e, t>
set of individuals
5
IV
<e, t>
set of individuals
T
<<s, <e, t>>, t>
set of properties of individuals
IAV
<<s, <e, t>>, <e, t>>
function from properties of
individuals to sets of individuals
TV
<<s, <<s, <e, t>>, t>>, <e, t>>
function from properties of properties
of individuals to sets of individuals
T/CN
<<s, <e, t>>, <<s, <e, t>>, t>>
function from properties of properties
of individuals to sets of properties of
individuals
t/t
<<s, t>, t>
set of propositions
IV/t
<<s, t>, <e, t>>
function from propositions to sets of
individuals
IV//IV
<<s, <e, t>>, <e, t>>
function from properties individuals
to sets of individuals
IAV/T
<<s,<<s,<e,t>>,t>>,<<s,<e, t>>,<e,t>>
function from properties of properties
of individuals to functions from
properties of individuals to sets of
individuals
Before going on to discuss the translation rules of the PTQ fragment, we must say a word
about the translations of basic expressions of English. A function g (not to be confused with
the function g which assigns values to variables IL) assigns a translation to each of the
(finitely many) basic expressions of English. Thus, g plays much the same role in the
translation procedure as the function F does in a model in assigning interpretations to the
basic expressions. In many cases, g simply translates a basic expression of English into a nonlogical constant of IL (of the appropriate type prescribed by f). For example, the basic
expression man of category CN is translated into some constant of IL of type <e, t>.
Montague chose to designate each constant of IL that translates a word of English by a primed
variant of the English word, e.g. g(man) = man’.3 A few basic expressions of English, such
as names and logical words like every and necessarily, are given special translations which
are not simply constants of IL, though of course they must be expressions of the appropriate
type. Below are the translation of the quantifier words every, the and a:
g(every) = P( Qx( P{x}  Q{x} )
g(the) = P( Qy(x( P{x}  x = y)  Q{y} ))
g(a) = P( Qx( P{x}  Q{x} )
3.2
Translation Rules
3
Strictly speaking, this is a meta-language convention. The actual constant of IL corresponding to man might be
c273, <e, t>, or some such constant; man’ is merely our meta-language name for it.
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T2.
If δ  PT/CN and ζ  PCN, then F2(δ, ζ) translates into δ′(ˆζ′).
T3.
If ζ  PCN and   Pt, and ζ and  translate ζ′, ′ respectively, then F3,n(ζ, ) translates
into λxn(ζ′ (xn)  ′).
T4.
If α  PT and δ  PIV, and α, δ translate into α′, δ′ respectively, then F4(α , δ) translates
into α′(ˆδ′).
T5.
If δ  PTV and β  PT, and δ, β translate into δ′, β′ respectively, then F5(δ, β) translates
into δ′(ˆ β′).
T6.
If δ  PIAV/T, α  PT, α, δ translate into α′, δ′ respectively, then F5(α , δ) translates into
δ′(ˆα′).
T7.
If α  PIV/t,   Pt, and α,  translate into α′, ′ respectively, then F11(α , ) translates
into α′(ˆ′).
T8.
If δ  PIV//IV and β  PIV, and δ, β translate into δ′, β′ respectively, then F17(δ, β)
translates into δ′(ˆ β′).
T9.
If δ  Pt/t,   Pt, and δ,  translate into δ′, ′ respectively, then F6(δ, ) translates into
δ′(ˆ′).
T10. If δ  PIV/IV and β  PIV, and δ, β translate into δ′, β′ respectively, then F7(δ, β)
translates into δ′(ˆβ′) .
T11a. If , ψ  Pt, and , ψ translate into ′, ψ′ respectively, then F8(, ψ) translates into
(′  ψ′).
T11b. If , ψ  Pt, and , ψ translate into ′, ψ′ respectively, then F9(, ψ) translates into
(′  ψ′).
T12a. If δ, γ  PIV and δ, γ translate into δ′, γ′ respectively, then F8(δ, γ) (δ and γ) translates
into λx(δ′(x)  γ′(x)).
T12b. If δ, γ  PIV and δ, γ translate into δ′, γ′ respectively, then F9(δ, γ) (δ or γ) translates
into λx(δ′(x)  γ′(x)).
T13. If α, β  PT and α, β translate into α′, β′ respectively, then F9(α, β) translates into
λP(α ′(P)  β ′(P)).
T14. If α  PT and   Pt, and α,  translate into α′, ′ respectively, then F10,n(α , )
translates into α′(ˆλxn′).
T16. If α  PT and α, δ translate into α′, δ′ respectively, then F10,n(α, δ) translates into
λyα′(ˆ λxn(δ′(y))).
S17.
If α  PT and δ  PIV and α, δ translate into α′, δ′ respectively, then:
F11(α, δ) translates into α′(ˆδ′);
F12(α, δ) translates into Fα′(ˆδ′);
F13(α, δ) translates into Fα′(ˆδ′);
F14(α, δ) translates into Pα′(ˆδ′);
F15(α, δ) translates into Pα′(ˆδ′).
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4 Anaphoric Pronouns as Bound Variables; Scope Ambiguities and
Relative Clauses
4.1 Subscripted pronouns
Among the basic expressions of category Term Phrase (T) are the subscripted pronouns he0,
he1, he2, … These pronouns serve roughly as free variables in the English syntax. However,
they do not translate directly into the individual variables v0,e, v1,e, etc. of IL but, like all
other term phrases, translate into expressions denoting sets of properties of individuals. The
schema for their translations is as follows:
hen translates into P( P{xn} )
(1)
The subscripted pronouns in the PTQ play a role in the production of:
-
anaphoric pronouns,
de dicto / de re ambiguities, and
scope ambiguities in sentences with more than one term phrase,
relative clauses.
In the following subsections, we will have a brief look at each of these constructs in turn.
4.2 Anaphoric pronouns
In the PTQ fragment the subscripted pronouns behave much as the variables v1, v2, etc. did in
the language L2E (cf Handout 6). They are “bound” by the operation of replacing a variable
with a term phrase, thus making the operation of variable being a kind of invisible one with
respect to the surface form of the sentence (cf. rule S14). However, the semantic effect of the
operation is exactly like that of the familiar variable binding operation in formal languages. If
there is more than one occurrence of a certain free variable in a sentence, then only the first of
these is replaced by the “quantifying” term phrase, and the remaining instances of this
variable becoming bound anaphoric pronouns.
4.2 de dicto / de re ambiguities
The de dicto / de re ambiguity is captured by allowing for two different derivations for the
same sentence. While the de dicto reading of a term phrase is produced by integrating it to the
analysis tree directly (e.g. via rule S4 or rule S5), the de re reading is generated by inserting
the term phrase into the analysis tree via a quantification rule (e.g. rule S14). More precisely,
what matters here is whether the term phrase is introduced into the analysis tree before or
after an oblique-context-creating element (such as believe).
4.3 Scope ambiguities in sentences with more than one term
In addition to the de dicto / de re ambiguities, the PTQ grammar also gives rise to simple
scope ambiguities with two quantifiers. Regardless of whether they happen to be introduced
into the analysis directly or by a quantification rule (e.g. rule S14), whether a term phrase will
8
be interpreted as having narrower or wider scope than another phrase depends upon whether it
is introduced into the tree before or after that other phrase.
4.4 Relative clauses
A syntactic operation involving the “binding” of subscripted pronouns is also used in the rule
for forming relative clauses. For reasons of expository simplicity, Montague did not choose to
include normal relative clauses in English in the PTQ grammar (e.g. the man whom Mary
loves) but instead included relative clauses formed with such that (e.g. the man such that
Mary loves him) which are characteristic of mathematical jargon, but hardly a part of
colloquial English. These relative clauses are simpler than the normal ones in that they
involve neither WH-pronouns nor movement of embedded noun phrases.
The syntactic rule responsible for relative clauses in the PTQ grammar is S3. This rule
combines a common noun and a clause to give another common noun; it does not combine a
term phrase with a clause to give another term phrase. The reason for this choice is semantic:
the requirement of producing the correct meaning of a relative clause structure (i.e.
determiner, noun and clause) from the meanings of its parts in a way consistent with the
analysis determiner-noun constructions elsewhere requires an analysis of this sort. As Partee
(1975) notes, this can probably best be appreciated by comparing the meaning of the fish with
that of the fish which walks. The meaning of the in the fish is analyzed so as to require that
there is one only and only one fish. But in the fish which walks it is clearly not intended that
there be one and only one fish (or a unique fish identifiable to the hearer in the context of the
utterance), but merely that there be a unique individual that is both a fish and is something
that walks. But if fish which walks is a constituent interpreted as denoting the set of things
which are both fish and are things which walk, then the meaning of the in the full phrase the
fish which walks is exactly the same as that of the in all other constructions. Thus we will get
correct semantic results for relative clauses in such cases if the combination of a noun ζ and a
sentence  (containing a free variable x) is interpreted as denoting the intersection denoted by
ζ with the set of all x such that . Montague’s translation rule for S3 (i.e. T3) has this effect.
PS: The content of this handout is adopted from Dowty et al (1981).
References
Bennett, M. (1974) Some Extensions of a Montague Fragment. UCLA PhD. Dissertation;
reproduced by the Indiana University Linguistics Club, 1975.
Dowty D.R., Wall R. E. and Peters S. (1981) Introduction to Montague Semantics. Dordrecht,
The Netherlands: Kluwer Academic Publishers.
Montague, R. (1970) English as a formal language. Linguaggi nella e nella Tecnica, pp. 189224, ed. by B. Visenti, et al., Milan: Edizioni di Comunitá; reprinted in Formal
Philosophy: Selected Papers of Richard Montague, pp. 108-221, ed. By R. H. Thomason,
New Haven: Yale University Press, 1974.
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