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April 28, 1979 12:56 PM[IVY]<krl>document>rep-quant-old

previous versions of quantification

Label and Quantifier scoping

I have adopted the convention of labelling a descriptor (i.e. creating an Instance slot of the unit, whose contents is that descriptor) using a flag like the footnote flag, but with a name followed by a colon. Universal quantification is indicated by footnoting the outer scope with labels indicating the variables, e.g.

In Maine, every dog has bitten some part of every postman who is unfriendly.

# Maine
self/Instance: The↑1 location from1: WhichIs Universal {My dog, My badguy}
a Biting with
biter = [↑dog: A Dog; Location(Maine)]
target = WhichIs PartOf
[↑badguy: A Postman
Location(Maine)
WhichIs Unfriendly])

This says that for every combination of things matching the labelled descriptions, there exists a Biting with the appropariate properties. Note that if you ignore all the footnotes, it becomes the corresponding existential.

John called each of his friends who owns a book which contains every letter in some alphabet.

# John
self/Instance:
The↑1 caller from a Call with
callee = [↑friend: WhichIs FriendOf John
Which Owns A Book
Which↑2 Contains
[↑letter: A Letter
WhichIs In An↑3 Alphabet]]

1: WhichIs Universal {My friend}
2: WhichIs Universal {My letter}
3: WhichIs Dependent {My friend}

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Units having to do with sets, sequences, and quantifiers

# SequenceDescriptor↑1
1: Comment ("One of the standard descriptor types, used to explicitly set out a finite sequence
by describing the elements individually")
self:A Description
Which Describes \A Sequence with
length = the length of a Sequence
thatIs My elements
elements: SequenceOf (\A Description)
The rangeSequence from
a SequenceMap with
base = the referent of a Description thatIs My self
mapDescription = \Which Describes This element

# Set↑11: HasFunctional (self, SetOf, itemDescription)
HasFunctional (MemberOf(self), MemberOf, self)
HasFunctional (self, HasElement, MemberOf(self))
itemDescription: A↑2 Description
2: Comment ("each element of the set matches this description")
cardinality: Or(An Integer, An Infinity)

# Sequence↑11: HasFunctional (self, SequenceOf, itemDescription)
itemDescription: A↑2 Description
2: Comment ("each element of the sequence matches this description")
length: Or(An Integer, An Infinity)

# SetSelection↑11: HasFunctional (rangeSet, SetOfAll, selector)
HasFunctional (rangeSet, Subset, base)
HasFunctional (base, Superset, rangeSet)
base: A Set
The↑2 entities from a DescriptionSpace thatIs *M*2: A Default
selector: A Description
rangeSet: A Set
SetOf(/MemberOf(My base))
SetOf(/DescribedBy(My selector))
element:↑3 MemberOf(My base)
3: Comment ("This is a funny slot, in specifing the selector, the descriptor
’This element’ refers to the specific element being checked for
selection, not one element defined for the whole selection
process. We need to decide just what this is a special case
of")

# SetMap↑11: Comment ("Represents a mapping where the mapDescription
describes the range element in terms of the base element
which is referred to as ’This element’ the range set can be
smaller than the base set because two base elements can
map into the same range element")
base: A Set
mapDescription: A Description
rangeSet: SetOf(\DescribedBy(My mapDescription))
element: MemberOf(My base)

# SequenceSelection↑11:Comment ("Preserves ordering")
HasFunctional (rangeSet, Subsequence, base)
HasFunctional (base, Supersequence, rangeSet)
base: A Sequence
selector: A Description
rangeSequence:A Sequence
SequenceOf(/MemberOf(My base))
SequenceOf(/DescribedBy(My selector))
element: MemberOf(My base)

# SequenceMap↑11: Comment ("Represents a 1-1 mapping where the mapDescription
describes the range element in terms of the base element
which is referred to as ’This element’. It preserves
ordering")
base: A Sequence
mapDescription: A Description
rangeSequence:
A Sequence with length = the length of a Sequence thatIs My base
SequenceOf(\DescribedBy(My mapDescription))
element: MemberOf(My base)

# UniqueElement↑11: HasFunctional (self, TheOne, description)
self: WhichIs MemberOf
A Set with cardinality = 1
The rangeSet from a SetSelection with
base = The entities from a DescriptionSpace
thatIs My universe
selector = My description
universe: A DescriptionSpace; *M*↑22: A Default
description: A Description

# UniversallyQuantifiedDescription↑1
1: HasFunctional (description, Universal, universalVariables)
description: A Description
universalVariables: SetOf(\[A Description; WhichIs NestedIn My description])

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