G1: BN = {Harry, Pat, London, Kim,} (BasicNames)Exercise: Rewrite this grammar using the notations from Notes 1 for set membership and set inclusion.
G2: BIV = {runs, jumps, sleeps} (Basic Intransitive Verbs)
G3: BTV = {sees, loves, admires, detests} (Basic Transitive Verbs)
Convention: for all categories A, BA is included in PA.
For the time being think of the B (Basic) sets as Lexical Items and the P (Phrasal) sets as potential extensions of the sets by adding derived or complex elements. G4: If α is a member of PN and β is a member of PIV then αβis a member of PS.
G5: If α is a member of PTV and β is a member of PN then αβ is a member of PIV.
G6: If α is a member of PS and β is a member of PS then α and β is a member of PS
S0: the members of PS denote truth-values, as spelled out in what follows.
S1: the members of PN and BN denote members of A, in particular: "Harry" denotes Harry, "Pat" denotes Pat, "London" denotes London, "Kim" denotes Kim.
By convention we will represent the denotation of elements in our grammars by writing them with a following ': so Pat denotes Pat' etc.
S2: the members of PIV and BIV denote functions from individuals to truth-values.
S3: the members of PTV and BTV denote functions from individuals to functions from individuals to truth-values.
S4. If α is a member of PN and denotes α' and β is a member of PIV and denotes β' then αβ denotes β'(α') = 1 iff α' is a member of β' E.g.:The denotation of "Kim runs" = 1 if Kim runs, otherwise 0. S5. If α is a member of PTV and denotes α' and β is a member of PN and denotes β' then αβ denotes α'(β') which is that function h from individuals to truth-values such that for any individual x h(x) = α'(&beta')(x); = 1 iff x is a member of α'(β')
E.g.: the denotation of "Pat loves London" = 1 iff Pat is member of the set of London-lovers.
S6. If φ and ψ are members of PS of with denotations &phi'; and ψ' respectively then φ and ψ = 1 iff φ' = 1 and ψ' = 1.
Comments on the grammar so far: like the semantic interpretation, the grammar
just given follows Montague's lead in being firmly rooted in set theory. That
is, it defines sets of expressions. You need to be clear about this if you
compare this sort of grammar with the phrase-structure grammars and their
descendents familiar from the generative tradition. What the latter generate
are representations of structures: phrase-structure diagrams, phrase-markers
and so on. There is a natural correspondence between the derivation of a
member of a complex category here and the phrase-structure trees of TG and
other theories, but it is deceptive. The closest correspondent in early TG
was the notion of a T-marker (as opposed to a P[hrase]-marker).
A T-marker was a collection of P-markers and a record of their
transformational history. [Illustrate on board.]
(using j and m to stand for John and Mary)
John loves Mary,S : [love'(m)](j)
/\
/ \
/ \
/ \
John,N : j loves Mary,IV : love'(m)
/ \
/ \
/ \
/ \
/ \
loves, TV : love' Mary, N : m
This object is an analysis tree, of which more below, annotated with
the syntactic category of the string so far, and a representation of the
denotation of the expression. We will continue to use such representations. We
will also take up the question of the theoretical status of the representation
of the semantic value of the exressions.
Montague's general theory was set forth in all its gory formal details in his
"Universal Grammar" (1970b). There a clear distinction was made between
rules and operations. Each rule specifies a sequence of k
categories (the arguments), another category (the result), and a
k-place operation.
What is an operation? A function from some set to that same set. What
is the set in this case: the set of sequences of strings of symbols in the
language of the grammar. So in general, the operations of the grammar are
functions from sequences of strings of symbols to (one member) sequences of
strings of symbols. This set -- call it Σ -- is a superset of the set of
well formed written strings of symbols of English. So the grammar is a way of
picking out the subsets of Σ that are members of various syntactic
categories of the language.
The interpretation of the result of a syntactic rule is a function of the interpretation of the expressions combined and the manner of their combination.
Warning! There are several notations for categorial grammars kicking around. In this one (following Steedman 2000 and others) the argument category is always on the right. In another popular notation the argument is always "under" the slash, so where we have here a/b the other notation would have b\a.Now we can straightforwardly specify the relation between the syntax and the semantics. In an interpretation that ignores intensionality we would have the function m relating categories and types like this:
Now rather than stating individual rules we state once and for all rules for functional application to the right (FA>) and to the left FA<. In line with the rule-to-rule constraint on interpretations, we simply invoke the mappings defined above and write the interpretation rule as part of the schema, separated the syntax from the semantics with a colon:
- {Harry, Pat, London, Kim} ⊆ BNN ⊆ PN
We will use a highly abbreviated form of such statements:
{Harry, Pat, London, Kim} ⊆ N
- {runs, jumps, sleeps} ⊆ S\N
- {sees, loves, admires, detests} ⊆ (S\N)/N
If α ∊ Pa/b and β ∊ Pb, then αβ ∊ Pa and D(αβ) = D(α)(D(β)). Abbreviated:
FA>: a/b b ==> a : D(a/b)(D(b)) If β ∊ Pb and α ∊ Pa\b, then βα ∊ Pa and D(βα) = D(α)(D(β)). Abbreviated:
FA<: b a\b ==> a : D(a\b)(D(b))
History: Categorial grammars were invented by the Polish logician and
philosopher Kasimierz Ajdukiewicz (1935), although the basic idea has roots in
work by Łukasiewicz, Frege, and ultimately Aristotle. Ajdukiewicz's
formulation used bidirectional functors, built from the basic categories
n and s (think: names and sentences). Functors
are written as fractions (for convenience here we may also use |), so we have:
FA: a a
___ b = b ___ ==> a
b b
Or:
a|b b = b a|b ==> a
(Again: argument on the right!)
Later, several linguists and logicians took up Ajdukiewicz's ideas, most
notably Yehoshua Bar-Hillel and Joachim Lambek, who used the slashes / and \
to indicate right and left concatenation respectively. At the conference at
Stanford that went into Davidson and Harman (1972), Peter Geach made creative
use of Ajdukiewicz's ideas. A sign of the times: Geach's paper was called "A
programme for syntax" and at the same conference Jim McCawley gave a paper
called "A program for logic." The two papers went right past each other.
Joachim Lambek published several papers on categorial systems around 1960
(Lambek 1958, 1961), as did Haskell Curry (1961). Sir John Lyons (1966)
considered categorial grammars as a base for transformational grammar, as did
David Lewis (1972). Lyons' work was an important step toward X-Bar theory
(Jackendoff 1977) and in turn drew on questions and ideas of Zellig Harris
(1951)). Lyons asked whether the occurrence of "N" on both sides of rules
like "NP --> Det N" was more than a pun.
Categorial grammars of the directional type were early shown to be weakly
equivalent to context-free (CF) grammars. At the time many people bought the
supposed demonstration by N. Chomsky of the inadequacy of context-free
grammars for natural language and hence directional categorial grammars, so
interest died out (with the notable exceptions mentioned in the last
paragraph). Another honourable exception: Gilbert Harman 196? -- FILL IN! --
his paper may well be considered a precursor to Gazdar etc. to be mentioned
below for realizing the importance and possibilities of feature systems.
Chomsky's dismissive and polemical response may be found in Chomsky (FILL IN).
To be is to be the value of a variable.
Willard Van Orman Quine "On what there is"