A Lattice Theoretic Look: A Negated Approach toAdjectival (Intersective, Neutrosophic and Private)Phrases
Selc¸uk Topal
Department of MathematicsFaculty of Arts and SciencesBitlis Eren UniversityBitlis, TurkeyEmail: selcuk.topal@yandex.com
Florentin Smarandache
Mathematics & Science DepartmentUniversity of New Mexico705 Gurley Ave.,Gallup, NM 87301, USAEmail: smarand@unm.edu
Abstract
—The aim of this paper is to provide a contributionto Natural Logic and Neutrosophic Theory. This paper considerslattice structures built on noun phrases. Firstly, we present somenew negations of intersective adjectival phrases and their settheoretic semantics such as
nonred noncars
and
red noncars
.Secondly, a lattice structure is built on positive and negativenouns and their positive and negative intersective adjectivalphrases. Thirdly, a richer lattice is obtained from previous oneby adding neutrosophic preﬁxes
neut
and
anti
to intersectiveadjectival phrases. Finally, the richest lattice is constructed viaextending the previous lattice structures by private adjectives(fake, counterfeit). We call these lattice classes
Neutrosophic Linguistic Lattices
(
NLL
).
Keywords: Logic of natural languages; neutrosophy;preorders, orders and lattices; adjectives; noun phrases;negation
I. I
NTRODUCTION
One of the basic subﬁelds of the foundations of mathematicsand mathematical logic, lattice theory, is a powerful toolof many areas such as linguistics, chemistry, physics, andinformation science. Especially, with a set theoretical view,lattice applications of mathematical models in linguistics area common occurrence.Fundamentally, Natural Logic [1], [2] is a human reasoningdiscipline that explores inference patterns and logics in naturallanguage. Those patterns and logics are constructed on relations between syntax and semantics of sentences and phrases.In order to explore and identify the entailment relations amongsentences by mathematical structures, it is ﬁrst necessary todetermine the relations between words and clauses themselves.We would like to ﬁnd new connections between natural logicand neutrosophic by discovering the phrases and neutrosophicclauses. In this sense, we will associate phrases and negatedphrases to neutrosophic concepts.Recently, a theory called Neutrosophy, introduced bySmarandache [4], [6], [5] has widespread mathematics, philosophy and applied sciences coverage. Mathematically, itoffers a system which is an extension of intuitionistic fuzzysystem. Neutrosophy considers an entity, “
A
” in relation to itsopposite, “
anti
−
A
” and that which is not
A
, “
non
−
A
”,and that which is neither “
A
” nor “
anti
−
A
”, denoted by“
neut
−
A
”.Up to section 3.3, we will obtain various negated versions of phrases (intersective adjectival) because Neutrosophyconsiders opposite property of concepts and we would liketo associate the phrases and Neutrosophic phrases. We willpresent the ﬁrst
NLL
in section 3.3. Notice that all models andinterpretations of phrases will be ﬁnite throughout the paper.II. N
EGATING
I
NTERSECTIVE
A
DJECTIVAL
P
HRASES
Phrases such as “red cars” can be interpreted the intersectionof the set of
red things
with the set of
cars
and get theset of “red cars”. In the sense of modeltheoretic semantics,the interpretation of a phrase such as
red cars
would be theintersection of the interpretation of
cars
with a set of
red individuals
(the region
b
in Figure 1). Such adjectives arecalled intersective adjectives or intersecting adjectives. As tonegational interpretation, Keenan and Faltz told that “similarly,intersective adjectives, like common nouns, are negatable bynon: nonAlbanian (cf. nonstudent) ”in their book [7]. Inthis sense,
nonred cars
would interpret the intersection of theof
nonred things
and the set of
cars
. Negating intersectiveadjectives without nouns (red things) would be complementsof the set of
red things
, in other words,
nonred things
. Wemean by
nonred things
are which the things are which are
not red
. Remark that
nonred things
does not guarantee that
those individuals
have to have a colour property or somethingelse. It is changeable under incorporating situations but wewill might say something about it in another paper. On theother hand, negating nouns (cars) would be complements of the set of
cars
, in other words,
noncars
. We mean by noncarsthat the things are which are not cars. Adhering to the spirit of intersective adjectivity, we can explore new meanings and theirinterpretations from negated intersective adjectival phrases byintersecting negated (or not) adjectives with negated (or not)nouns. As was in the book,
nonred cars
is the intersectionthe set of things that are not red with cars. In other words,
408
9781509057955/17/$31.00 ©2017 IEEE
Fig. 1: An example of
cars
and
red
in a discourse universe
nonred cars
are the cars but not red (the region
c
in Figure1). Another candidate for the negated case,
nonred noncars
refers to intersect the set of nonred things (things that are notred) with noncars (the region
d
in Figure 1). The last one,
red noncars
has meaning that is the set of intersection of the setof red things and the set of noncars (the region
a
in Figure 1).
red
−
x
is called
noun level partially semantic complement
.
−
redx
is called
adjective level partially semantic complement
.
−
red
−
x
is called
full phrasal semantic complement
. In summary, weobtain
nonred cars
,
red noncars
and
nonred noncars
from
red cars
we already had.The intersective theory and conjunctives suits well intoboolean semantics [7], [8] which proposes very close relationship between
and
and
or
in natural language, as conjunctionand disjunction in propositional and predicate logics thathave been applied to natural language semantics. In theselogics, the relationship between conjunction and disjunctioncorresponds to the relationship between the settheoretic notions of intersection and union [9], [10]. On the other hand,correlative conjunctions might help to interpret negated intersective adjectival phrases within boolean semantics becausethe conjunctions are paired conjunctions (neither/nor, either/or,both/and,) that link words, phrases, and clauses. We mightreassessment those negated intersective adjectival phrases inperspective of correlative conjunctions. “
neither A nor B
” and “
both nonA and nonB
can be used interchangeablywhere
A
is an intersective adjective and
B
is a noun. Therefore,we say “
neither red (things) nor pencils
”and “
both nonred (things) and nonpencils
”equivalent sentences. An evidencefor the interchangeability comes from equivalent statements inpropositional logic, that is,
¬
(
R
∨
C
)
is logically equivalentto
¬
R
∧¬
C
[11]. Other negated statements would be
¬
R
∧
C
and
R
∧¬
C
. Semantically,
¬
R
∧¬
C
is
full phrasal semanticcomplement
of
R
∨
C
, and also
¬
R
∧
C
and
R
∧ ¬
C
arepartially semantic complements of
R
∨
C
.We will explore full and partially semantic complementsof several adjectival phrases. We will generally negate thephrases and nouns by adding preﬁx “
non
”, “
anti
” and “
neut
”.We will use interpretation function
[[ ]]
from set of phrases(
Ph
) to power set of universe (
P
(
M
)
) (set of individuals) toexpress phrases with understanding of a settheoretic viewpoint. Hence,
[[
p
]]
⊆
M
for every
p
∈
Ph
. For an adjective
a
(negated or not ) and a plural noun
n
(negated or not ) ,
a n
will be interpreted as
[[
a
]]
∩
[[
n
]]
. If
n
is a positive plural noun,
non
−
n
will be interpreted as
[[
non
−
n
]] = [[
−
n
]] =
M
\
[[
n
]]
.Similarly, if
a
is a positive adjective,
non
−
a
will beinterpreted as
[[
non
−
a
]] = [[
−
a
]] =
M
\
[[
a
]]
. While we will add
non
to both nouns and adjectives as preﬁx, “
anti
” and “
neut
”will be added in front of only adjectives. Some adjectivesthemselves have negational meaning such as
fake
. Semanticsof phrases with
anti
,
neut
and
fake
will be mentioned in nextsections.III. L
ATTICE
T
HEORETIC
L
OOK
We will give some fundamental deﬁnitions before we startto construct lattice structures from those adjectival phrases.A lattice is an algebraic structure that consists of a partiallyordered set in which every two elements have a uniquesupremum (a least upper bound or join) and a unique inﬁmum(a greatest lower bound or meet) [12]. The most classicalexample is on sets by interpreting set intersection as meet andunion as join. For any set
A
, the power set of
A
can be orderedvia subset inclusion to obtain a lattice bounded by
A
and theempty set. We will give two new deﬁnitions in subsection 3.2to start constructing lattice structures.
Remark 3.1:
We will use the letter
a
and
red
for intersectiveadjectives, and the letter
x
,
n
and
cars
for common plural nounsin the name of abbreviation and space saving throughout thepaper.
A. Individuals
Each element of
[[
ax
]]
and
[[
−
ax
]]
is a distinct individual andbelongs to
[[
x
]]
. It is already known that
[[
a x
]]
∩
[[
−
a x
]] =
∅
and
[[
ax
]]
[[
−
ax
]] = [[
x
]]
. It means that no common elementsexist in
[[
ax
]]
and
[[
−
a x
]]
. Hence, every element of those setscan be considered as individual objects such as Larry, John,Meg,.. etc. Uchida and Cassimatis [13] already gave a latticestructure on power set of all of individuals (a domain or auniverse).
B. Lattice
L
IA
Intersective adjectives (red) provide some properties fornouns (cars). Excluding (complementing) a property froman intersective adjective phrase also provide another property for nouns. In this direction, “
red
” is an property fora noun, “
non
−
red
” is another property for the noun aswell.
red
and
non
−
red
have discrete meaning and sets ascan be seen in Figure 1. Naturally, every set of restrictedobjects with a property (red cars) is a subset of those ob jects without the properties (cars).
[[
red x
]]
and
[[
−
red x
]]
are always subsets of
[[
x
]]
. Neither
[[
red x
]]
≤
[[
−
red x
]]
nor
[[
−
red x
]]
≤
[[
red x
]]
since
[[
red x
]]
∩
[[
−
red x
]]
byassuming
[[
−
red x
]]
=
∅
and
[[
red x
]]
=
∅
. Without loss of generality, for negative (complement) of the noun
x
and theintersective adjective
red
(positive and negative) are
−
x,red
−
x
409
Fig. 2: Lattice on
cars
and
red
and
−
red
−
x
.
[[
red
−
x
]]
and
[[
−
red
−
x
]]
are always subsets of
[[
−
x
]]
.Neither
[[
red
−
x
]]
≤
[[
−
red
−
x
]]
nor
[[
−
red
−
x
]
≤
[[
red
−
x
]]
since
[[
red
−
x
]]
∩
[[
−
red
−
x
]]
by assuming
[[
−
red
−
x
]]
=
∅
and
[[
red
−
x
]]
=
∅
. On the other hand,
[[
x
]]
∩
[[
−
x
]] =
∅
and
[[
x
]]
[[
−
x
]] =
M
(
M
is the universe of discourse) and also
[[
red x
]]
,
[[
−
red x
]]
,
[[
red
−
x
]]
and
[[
−
red
−
x
]]
are by two discrete.We do not allow
[[
red x
]]
[[
−
red x
]]
and
[[
red x
]]
[[
−
red
−
x
]]
and
[[
−
red x
]]
[[
red
−
x
]]
and
[[
−
red x
]]
[[
−
red
−
x
]]
to take placesin the lattice in Figure 2 because we try to build the latticefrom phrases only in our language. To do this, we deﬁne a setoperation
and an order relation
≤
as the follows:
Deﬁnition 3.2:
We deﬁne a binary set operator
for ourlanguages as the follow: Let
S
be a set of sets and
A, B
∈
S
.
A
B
=
C
:
⇔
C
is the smallest set which includes both
A
and
B
, and also
C
∈
S
.
Deﬁnition 3.3:
We deﬁne a partial order
≤
on sets as thefollow:
A
≤
B
if
B
=
A
BA
≤
B
if
A
=
A
B
Example 3.4:
Let
A
=
{
1
,
2
}
, B
=
{
2
,
3
}
, C
=
{
1
,
2
,
4
}
, D
=
{
1
,
2
,
3
,
4
}
and
S
=
{
A,B,C,D
}
.
A
A
=
A
,
A
C
=
C
,
A
B
=
D
,
B
C
=
D
,
C
D
=
D
.
C
≤
C, A
≤
C, A
≤
D, B
≤
D, C
≤
D
Notice that
≤
is a reﬂexive, transitive relation (preorder)and
is a reﬂexive, symmetric relation.Figure 3 illustrates a diagram on
cars
and
red
. The diagramdoes not contain sets
{
b,d
}
,
{
a,b
}
,
{
a,c
}
and
{
c,d
}
becausethe sets do not represent linguistically any phrases in thelanguage. Because of this reason,
{
a
}
{
c
}
and
{
a
}
{
b
}
and
{
d
}
{
c
}
and
{
d
}
{
b
}
are
{
a,b,c,d
}
=
M
. This structurebuilds a lattice up by
and
that is the classical setintersection operation.
L
IA
= (
L,
∅
,
,
)
is a lattice where
L
=
Fig. 3: Hasse Diagram of lattice of
L
IA
= (
L,
∅
,
,
)
Fig. 4: The Lattice
L
N IA
{
M, x,
−
x, red x, red
−
x,
−
red x,
−
red
−
x
}
. Remark that
L
IA
= (
L,
∅
,
,
) = (
L,
∅
,
≤
)
. We call this latticebrieﬂy
L
IA
.
C. Lattice
L
N IA
In this section, we present ﬁrst
NLL
. Let
A
be the color
white
. Then,
non
−
A
=
{
black,red,yellow,blue,...
}
,
anti
−
A
points at
black
, and
neut
−
A
=
{
red,yellow,blue,...
}
. Inour interpretation base,
anti
−
black cars
(
a
black cars
) is aspeciﬁc set of
cars
which is a subset of set
non
−
black cars
(
−
black cars
).
neut
−
black cars
(
n
black cars
) is a subset of
−
black cars
which is obtained by excluding sets
black cars
and
a
black cars
from
−
black cars
. Similarly,
anti
−
blackcars
(
a
black
−
cars
) is a speciﬁc set of
−
cars
which is a subset of set
non
−
black non
−
cars
(
−
black
−
cars
).
neut
−
black
−
cars
(
n
black
−
cars
) is a subset of
−
black
−
cars
which is obtained byexcluding sets of
black
−
cars
and
a
black
−
cars
from
−
black
−
cars
.The new structure represents an extended lattice equipped with
≤
as can be seen in Figure 4. We call this lattice
L
N IA
.
D. Lattice
L
N IA
(
F
)
Another
NLL
is an extended version of
L
N IA
by private ad jectives. Those adjectives have negative effects on nouns such
fake
and
counterfeit
. The adjectives are representative elementsof, called
private
, a special class of adjectives [14], [15], [16].
410
Fig. 5: The lattice
L
N IA
(
F
)
Chatzikyriakidis and Luo treated transition from the adjectivalphrase to noun as
Private Adj
(
N
)
⇒ ¬
N
in inferentialbase. Furthermore, they gave an equivalence “
real gun
(
g
)
iff
¬
fake gun
(
g
)
” where
[

g is a real gun

] =
real gun
(
g
)
and
[

f is not a real gun

] =
¬
real gun
(
f
)
in order toconstitute a modern typetheoretical setting. In light of thesefacts,
fake car is not a car (real)
and plural form:
fake cars arenot cars
. Hence, set of fake cars is a subset of set of noncarsin our treatment.On the one hand, compositions with private adjectives andintersective adjectival phrases do not effect the intersectiveadjectives negatively but nouns as usual. Then, interpretationof “
fake red cars
” would be intersection of set of
red things
and set of
non
−
cars
.Applying “
non
” to private adjectival phrases,
non
−
fake cars
are cars (real),
[[
non
−
fake cars
]] = [[
cars
]]
whereas
[[
fake cars
]]
⊆
[[
non
−
cars
]]
.
non
−
fake cars
willbe not given a place in the lattice. Remark that phrase “nonfake noncars” is ambiguous since
fake
is not a intersectiveadjective. We will not consider this phrase in our lattice.
f
x
is incomparable both
−
black x
and
−
black
−
x
except
−
x
ascan be seen in Figure 5. So, we can not determine that set of
fake cars
is a subset or superset of a set of any adjectivalphrases. But we know that
[[
fake cars
]]
⊆
[[
non
−
cars
]]
.Then, we can see easily
[[
fake black cars
]]]
⊆
[[
blacks non
−
cars
]]
by using
[[
fake cars
]]
[[
black things
]]
⊆
[[
−
cars
]]
[[
black things
]]
.Without loss of generality, set of
fake black cars
is asubset of set
black non
−
cars
and also set of
fake non
−
black cars
is a subset of set
non
−
black non
−
cars
. Continuing with
neut
and
anti
, set of
fake neut black cars
is a subsetof set of
neut black non
−
cars
and also
fake anti black cars
is a subset of set of
anti black non
−
cars
. Those phrasesbuild the lattice
L
N IA
(
F
)
in Figure 5.Notice that when
M
and empty set are removed fromlattices will construct, the structures lose property of lattice. The structures will be hold neither join nor meetsemilattice property as well. On the other hand, set of
{
n
black
f
x,
−
black
f
x,
n
black
−
x,
−
black
−
x
}
equipped with
≤
is theonly one sublattice of
L
N IA
(
F
)
without using
M
and emptyset.IV. C
ONCLUSION AND
F
UTURE
W
ORK
In this paper, we have proposed some new negated versionsof set and model theoretical semantics of intersective adjectivalphrases (plural). After we ﬁrst have obtained the lattice structure
L
IA
, two lattices
L
N IA
and
L
N IA
(
F
)
have been built fromthe proposed phrases by adding ‘neut’, ‘anti’ and ‘fake’ stepby step.It might be interesting that lattices in this paper can beextended with incorporating coordinates such as
light red cars
and
red cars
. One might work on algebraic properties as ﬁltersand ideals of the lattices considering the languages. Somedecidable logics might be investigated by extending syllogisticlogics with the phrases. Another possible work in future, thisidea can be extended to complex neutrosophic set, bipolarneutrosophic set, interval neutrosophic set [17], [18], [19],[20].We hope that linguists, computer scientists and logiciansmight be interested in results in this paper and the results willhelp with other results in several areas.V. T
HANKS
We would like to thank the reviewers for their valuablecomments, corrections and contributing to the quality of thepaper.R
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