A Lattice Theoretic Look: A Negated Approach to Adjectival (Intersective, Neutrosophic and Private) Phrases

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Abstract—The aim of this paper is to provide a contribution to Natural Logic and Neutrosophic Theory. This paper considers lattice structures built on noun phrases. Firstly, we present some new negations of intersective adjectival phrases and their settheoretic semantics such as non-red non-cars and red non-cars. Secondly, a lattice structure is built on positive and negative nouns and their positive and negative intersective adjectival phrases. Thirdly, a richer lattice is obtained from previous one by adding neutrosophic prefixes neut and anti to intersective adjectival phrases. Finally, the richest lattice is constructed via extending the previous lattice structures by private adjectives (fake, counterfeit). We call these lattice classes Neutrosophic Linguistic Lattices (NLL).
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  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 set-theoretic semantics such as  non-red non-cars  and  red non-cars .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;pre-orders, 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 rela-tions 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, phi-losophy 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 ver-sions 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 model-theoretic 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-: non-Albanian (cf. non-student) ”in their book [7]. Inthis sense,  non-red cars  would interpret the intersection of theof   non-red things  and the set of   cars . Negating intersectiveadjectives without nouns (red things) would be complementsof the set of   red things , in other words,  non-red things . Wemean by  non-red things  are which the things are which are not red  . Remark that  non-red 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,  non-cars . We mean by non-carsthat 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,  non-red cars  is the intersectionthe set of things that are not red with cars. In other words, 408 978-1-5090-5795-5/17/$31.00 ©2017 IEEE  Fig. 1: An example of   cars  and  red  in a discourse universe non-red cars  are the cars but not red (the region  c  in Figure1). Another candidate for the negated case,  non-red non-cars refers to intersect the set of non-red things (things that are notred) with non-cars (the region  d  in Figure 1). The last one,  red non-cars  has meaning that is the set of intersection of the setof red things and the set of non-cars (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  non-red cars ,  red non-cars  and  non-red non-cars  from red cars  we already had.The intersective theory and conjunctives suits well intoboolean semantics [7], [8] which proposes very close relation-ship 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 set-theoretic no-tions of intersection and union [9], [10]. On the other hand,correlative conjunctions might help to interpret negated in-tersective 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 non-A and non-B  can be used interchangeablywhere  A  is an intersective adjective and  B  is a noun. Therefore,we say “ neither red (things) nor pencils  ”and “ both non-red (things) and non-pencils  ”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 set-theoretic view-point. 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 prop-erty 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 (pre-order)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 type-theoretical 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 non-carsin 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 “non-fake non-cars” 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 . Continu-ing 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 lat-tice. The structures will be hold neither join nor meetsemi-lattice 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 sub-lattice 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 struc-ture  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 EFERENCES[1] L. S. Moss,  Natural logic and semantics . In Logic, Language andMeaning (pp. 84-93), Springer Berlin Heidelberg, 2010[2] J. F. van Benthem, A brief history of natural logic, College Publications,2008.[3] F. Smarandache,  A Unifying Field in Logics: Neutrosophic Logic. Neu-trosophy, Neutrosophic Set, Neutrosophic Probability: Neutrsophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability. Infinite Study ,2005.[4] F. Smarandache,  Matter, antimatter, and unmatter  . CDS-CERN (pp. 173-177). EXT-2005-142, 2004.[5] F. Smarandache,  Neutrosophic Actions, Prevalence Order, Refinement of  Neutrosophic Entities, and Neutrosophic Literal Logical Operators , APublication of Society for Mathematics of Uncertainty, 11, Volum 10,pp. 102-107, 2015.[6] F. Smarandache,  Neutrosophy: Neutrosophic Probability, Set, and Logic: Analytic Synthesis & Synthetic Analysis , 1998.[7] E. L. Keenan and L. M. Faltz,  Boolean semantics for natural language ,Vol. 23, Springer Science & Business Media, 2012.[8] Y. Winter and J. Zwarts,  On the event semantics of nominals and adjec-tives: The one argument hypothesis , Proccedings fo Sinn and Bedeutung,16, 2012.[9] F. Roelofsen,  Algebraic foundations for the semantic treatment of inquis-itive content  , Synthese, 190(1), 79-102, 2013.[10] L. Champollion,  Ten men and women got married today: Noun coordi-nation and the intersective theory of conjunction , Journal of Semantics,ffv008, 2015.[11] G. M. Hardegree,  Symbolic logic: A first course , McGraw-Hill, 1994.[12] B. A. Davey and H. A. Priestley,  Introduction to lattices and order  ,Cambridge University Press, 2002.[13] H. Uchida and N. L. Cassimatis,  Quantifiers as Terms and Lattice-Based Semantics , 2014.[14] S. Chatzikyriakidis and Z. Luo,  Adjectives in a modern type-theoreticalsetting , In Formal Grammar, Springer Berlin Heidelberg, 159-174, 2013.[15] B. Partee,  Compositionality and coercion in semantics: The dynamicsof adjective meaning , Cognitive foundations of interpretation, 145-161,2007.[16] P. C. Hoffher and O. Matushansky,  Adjectives: formal analyses in syntaxand semantics , Vol. 153, John Benjamins Publishing, 2010. 411
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