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C00002 00002	.cb GRADE SCHOOL GRAMMAR AS AN AI LANGUAGE
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.cb GRADE SCHOOL GRAMMAR AS AN AI LANGUAGE


	In grade school I was told that an adjective was a modifier
of a noun and an adverb modified adjectives or verbs.  After long
thought, I have come to the conclusion that my teachers were essentially
right, perhaps contrary to modern linguistics.  Of course, 1930s
grade school linguistics was a confused amalgam of syntax and
semantics, but it turns out that this is what we need for AI.

	In this paper I shall present a language for computers
to use in expressing and communicating among themselves certain kinds of facts
about the world.  Although this language corresponds in many ways to
grade school sentence diagrams, correspondence with English is
not a goal in itself, and I will deviate from it whenever it
suits the AI purpose.

	Our intention is that programs should be written using the
formalism.  Such programs will be capable of expressing internally
some part of what can be expressed in English.  Should a deficiency
in the formalism be found before programs are written, we won't
necessarily abandon the formalism.  We may decide that it is better
to write programs using it even if their capabilities are limited
than to pursue perfection at the cost of elaboration at the
present time.

	We proceed by giving examples:

.item←0
	#. An adjective modifies nouns in the sense that an adjective
will be represented as operating on nouns like a function.  Thus
we will write ⊗brown[dog] corresponding to "brown dog", and treat
⊗brown somewhat as a function taking the ⊗noun ⊗concept ⊗dog into
a noun phrase ⊗brown[dog].

	#. The word "somewhat" in the previous item refers to the
fact that we shall not suppose any extensionality principle for
adjectives or other modifiers.  Thus we might have ⊗brown[x] and
⊗brown1[x] always yielding the same concept as a result but
not necessarily having %2brown_=_brown1%1.  That's why we
write ⊗brown[x] instead of ⊗brown(x).  We could write
⊗apply[brown,x], and then there would be no excuse at all for
confusion.

	In my opinion, prematurely defining identity conditions
for classes of entities is a vice.  It is one of the main faults
of Montague's approach to English grammar.  This remark is not
intended to convince, but merely to announce a position.  The
reader may insert at any point, any identity conditions he
likes compatible with what has been so far asserted, but he
risks having to take them back later.

	#. As in grade school, ⊗the will be treated as a modifier,
so ⊗the[brown[dog]] represents "the brown dog".

	#. Adverbs may be used to modify adjectives, so we
have ⊗the[dark[brown][dog]] for "the dark brown dog".

	#. Now we begin on semantics.  We write

	%2denotes(the[brown[dog]], fido, xi)%1

to assert that in the context xi, ⊗the[brown[dog]] denotes ⊗fido.
⊗denotes, ⊗fido and xi are not part of the "inner language" whose
semantics we are giving.  However, we intend to use first order
logic to express this semantics.  Note that since ⊗denotes is
an ordinary predicate, its arguments are surrounded by ordinary
parentheses, and ⊗denotes cannot serve as a argument of
functions and predicates.

	#. At this point we can venture an axiom - or rather
an instance of an axiom, namely

	%2denotes(the[dog],fido,xi) ∧ isbrown(fido) 
⊃ denotes(the[brown[dog]],fido,xi)%1.

This involves properties of ⊗the and of ⊗brown as an adjective.  A more
detailed definite description denotes the same object as a less detailed
description, provided the less detailed description denotes, and the
more detailed description applies to the object.  There would not be
a corresponding axiom for the adjective ⊗alleged; indeed there wouldn't
be a predicate ⊗isalleged corresponding to ⊗alleged in the same way that
⊗isbrown corresponds to ⊗brown. 

	#. ⊗the[of[the[brown[dog]],collar]] represents "the brown dog's collar".

Note the two occurrences of ⊗the. 
I haven't decided whether ⊗of shall be used exclusively to
represent the posessive or to include other relations - most
likely the latter.  If so, then the meaning will depend on
the context xi.

	#. Collars as posessions will have the following "extensionality"
property.

	%2denotes[the[brown[dog]],x,xi) ∧ denotes[the[of[the[brown[dog]],
collar]],y,xi] ∧ denotes[Fido,x,xi] ⊃ denotes[the[of[Fido,collar]],y,xi]%1.