Proceedings of the KI 2015 Workshop on Formal and Cognitive Reasoning




Approximate Reasoning with Fuzzy-Syllogistic
                 Systems
                                       Bora İ Kumova
    İzmir Institute of Technology; Department of Computer Engineering; 35430 Turkey
                              borakumova@iyte.edu.tr

         Abstract. The well known Aristotelian syllogistic system consists of 256
         moods. We have found earlier that 136 moods are distinct in terms of equal
         truth ratios that range in τ=[0,1]. The truth ratio of a particular mood is
         calculated by relating the number of true and false syllogistic cases the mood
         matches. A mood with truth ratio is a fuzzy-syllogistic mood. The introduction
         of (n-1) fuzzy existential quantifiers extends the system to fuzzy-syllogistic
         systems n�, 1<n, of which every fuzzy-syllogistic mood can be interpreted as a
         vague inference with a generic truth ratio that is determined by its syllogistic
         structure. We experimentally introduce the logic of a fuzzy-syllogistic ontology
         reasoner that is based on the fuzzy-syllogistic systems n�. We further introduce
         a new concept, the relative truth ratio rτ=[0,1] that is calculated based on the
         cardinalities of the syllogistic cases.

         Keywords. Syllogistic reasoning; fuzzy logic; approximate reasoning.


1     INTRODUCTION
    Multi-valued logics were initially introduced by Łukasiewicz [10], as an extension
to propositional logic. After Zadeh generalised multi-valued logics within fuzzy logic
[19], he discussed syllogistic reasoning with fuzzy quantifiers in the context of fuzzy
logic [20]. However, this initial fuzzification of syllogistic moods was experimentally
applied on only a few true moods and did not systematically cover all moods. The
first systematic application of multi-valued logics on syllogisms were intermediate
quantifiers and their reflection on the square of opposition [14]. However only set-
theoretic representation of moods as syllogistic cases allow 64analysing the fuzzy-
syllogistic systems n� mathematically exactly, such as by calculating truth ratios of
moods [6] and their algorithmic usage in fuzzy inferencing [7]. Here we present a
sample application of n� for fuzzy-syllogistic ontology reasoning.
    Learning from scratch can be modelled probabilistically, as objects and their
relationships need to be first synthesised from a statistically significant number of
perceived instances of similar objects. This leads to probabilistic ontologies [4], [15],
[11], in which attributes of objects may be synthesised also as objects.
    There are more probabilistic ontology reasoners than fuzzy or possibilistic ones
and most of them reason with probabilist ontologies [8]. Several ontology reasoners
employ possibilistic logic and reason with fuzzy ontologies. The most popular
reasoning logic being hyper-tableau, for instance in HermiT [12]. Other experimental
reasoning logics are also interesting to analyse, such as fuzzy rough sets and
Łukasiewicz logic [3] in FuzzyDL [1], Zadeh and Gödel fuzzy operators in DeLorean
[2], Mamdani inference in HyFOM [18] or possibilistic logic in KAON [15]. Fuzzy-

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    Proceedings of the KI 2015 Workshop on Formal and Cognitive Reasoning




syllogistic reasoning (FSR) can be seen as a generalisation of both, fuzzy-logical and
possibilistic reasoners.
    A fuzzy-syllogistic ontology (FSO) extends the concept of ontology with the
quantities that led to the ontological concepts. A FSO is usually generated
probabilistically, but does not preserve any probabilities like probabilistic ontologies
[11] or probabilistic logic networks [5] do. A FSO can be a fully connected and
bidirectional graph.
    Several generic reasoning logics are discussed in the literature, like probabilistic,
non-monotonic or non-axiomatic reasoning [17]. Fuzzy-syllogistic reasoning in its
basic form [21] is possibilistic, monotonic and axiomatic.
    Syllogistic reasoning reduced to the proportional inference rules deduction,
induction and abduction are employed in the Non-Axiomatic Reasoning System
(NARS) [16]. Whereas FSR uses the original syllogistic moods and their fuzzified
extensions [22].
    There is one implementation mentioned in the literature that is close to the concept
of syllogistic cases: Syllogistic Epistemic REAsoner (SEREA) implements poly-
syllogisms and generalised quantifiers that are associated with combinations of
distinct spaces, which are mapped onto some interval arithmetic. Reasoning is then
performed with concrete quantities, determined with the interval arithmetic [13].
    First the fuzzy-syllogistic systems n� are discussed, thereafter fuzzy-syllogistic
reasoning is introduced, followed by its sample application on a fuzzy-syllogistic
ontology and the introduction of relative truth ratios rτ.

2    FUZZY-SYLLOGISTIC SYSTEMS
   The fuzzy-syllogistic systems n�, with 1<n fuzzy quantifiers, extend the well
known Aristotelian syllogisms with fuzzy-logical concepts, like truth ratio for every
mood and fuzzy quantifiers or in general fuzzy sets. We discuss first the systems n�
and introduce them further below as the basic reasoning logic of FSR.

2.1 Aristotelian Syllogistic System �
    The Aristotelian syllogistic system � consists of inclusive existential quantifiers
ψ, ie I includes A and O includes E as one possible case:
          Universal affirmative: All S are P: ψ=A: {x| x∉P-S ∧ x∈P∩S}
          Universal negative: All S are not P: ψ=E: {x| x∈S-P ∧ x∉P-S}
Inclusive existential affirmative: Some S are P: ψ=I: A ∪ {x| (x∉S-P ∧ x∉P-S ∧
                           x∈P∩S) ∨ (x∉S-P ∧ x∈P∩S)}
 Inclusive existential negative: Some S are not P: ψ=O: E ∪ {x| (x∈S-P ∧ x∉P-S
                         ∧ x∉P∩S) ∨ (x∈S-P ∧ x∉P∩S)}
   A categorical syllogism ψ1ψ2ψ3F is an inference schema that concludes a
quantified proposition Φ3=Sψ3P from the transitive relationship of two given
quantified proportions Φ1={Mψ1P, Pψ1M} and Φ2={Sψ2M, Mψ2S}:
           ψ1ψ2ψ3F = (Φ1={Mψ1P, Pψ1M}, Φ2={Sψ2M, Mψ2S}, Φ3=Sψ3P)



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               Table 1: Binary coding of the 7 possible distinct spaces for three sets.
                                             Syllogistic Case Δ95
    Binary code Δj=
                           Venn Diagram                              Space Diagram+
     δ1δ2δ3δ4δ5δ6δ7*

                            M            S     S­M­P P­S­M M­S­P M∩S­P M∩P­S S∩P­M M∩S∩P
     Δ96=1111110#
                                  P                δ1        δ2      δ3      δ4      δ5       δ6       δ7
*
 Binary coding of all possible distinct space combinations Δj, j=[1,96] that can be generated for three sets.
#
 δi=0: space i is empty; δi=1: space i is not empty; i=[1,7].
+
  Every circle of a space diagram represents exactly one distinct sub-set of M∪ P∪ S.

    where F={1, 2, 3, 4} identifies the four possible combinations of Φ1 with Φ2,
namely syllogistic figures. Every figure produces 4 3=64 moods and the whole
syllogistic system � has 4x64=256 moods.

2.2 Syllogistic­Cases
    Syllogistic cases are an elementary concept of the fuzzy-syllogistic systems n�,
for calculating truth ratios [6] of the moods algorithmically [7].
    For three sets, 7 distinct spaces δ i, i=[1,7] are possible, which can be easily
identified in a Venn diagram (Table 1). There are in total j=96 distinct combinations of
the spaces Δj=δ1δ2δ3δ4δ5δ6δ7, j=[1,96] [22], which constitute the universal set of
syllogistic moods. Out of this universe, we determine for every mood true and false
matching space combinations (Fig 1).

2.3 Fuzzy­Syllogistic Moods
    We extend the ancient binary truth classification of moods, to a fuzzy
classification with truth values in [0,1]. For this purpose, first the above set-theoretical
definitions of the quantifiers of a particular mood are compared against the set of all
syllogistic cases Δj, j=[1,96], in order to identify true and false matching cases:
                 True syllogistic cases: Λt = j=1∪96Δj∈(ΦΔ1∩ΦΔ2) →Δj∈ΦΔ3
                False syllogistic cases: Λf = j=1∪96Δj∈(ΦΔ1∩ΦΔ2) → Δj∉ΦΔ3
    where Λt and Λf is the set of all true and false matching cases of a particular mood,
respectively (Fig 1) and ΦΔ is a proposition in terms of syllogistic cases. For instance,
the two premisses Φ1 and Φ2 of the mood IAI4 of the syllogistic system �, match the
10 syllogistic cases Λt = {Δ4, Δ19, Δ67, Δ24, Δ43, Δ46, Δ68, Δ74, Δ48, Δ76}, which are all true
for the conclusion Φ3 as well. Thus the mood has no false cases Λf = Ø.
    The truth ratio of a mood is then calculated by relating the amounts of the two sets
Λt and Λf with each other. Consequently the truth ratio becomes either more true or
more false τ ∈ {τf, τt}:
                 More true: τt ∈ {|Λf|<|Λt| → 1-|Λf|/(|Λt|+|Λf|)} = [0.545,1]
                  More false: τf ∈ {|Λt|<|Λf| → |Λt|/(|Λt|+|Λf|)} = [0,0.454]



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      Proceedings of the KI 2015 Workshop on Formal and Cognitive Reasoning




    where |Λt| and |Λf| are the numbers of true and false syllogistic cases, respectively.
A fuzzy-syllogistic mood is then defined by assigning an Aristotelian mood ψ1ψ2ψ3F
the structurally fixed truth ratio τ:
                                  Fuzzy-syllogistic mood: (ψ1ψ2ψ3F, τ)
    The truth ratio identifies the degree of truth of a particular mood, which we will
associate further below in fuzzy-syllogistic reasoning with generic vagueness of
inferencing with that mood.
    The analysis of the Aristotelian syllogistic system � with these concepts reveals
several interesting properties, like � has 136 distinct moods, 25 true moods τ=1, of
which 11 are distinct, and 25 false moods τ=0, of which 11 are distinct, and that � is
almost point-symmetric on syllogistic cases and truth ratios of the moods [22], [9].

2.4 Fuzzy­Syllogistic System 2�
   In the fuzzy-syllogistic system (FSS) 2�, the universal cases A and E are excluded
from the existential quantifiers I and O, respectively:
     Exclusive existential affirmative: Some S butNotAll are P: ψ=I: {x| (x∉S-P ∧
                        x∉P-S ∧ x∈P∩S) ∨ (x∉S-P ∧ x∈P∩S)}
    Exclusive existential negative: Some S butNotAll are not P: ψ=O: {x| (x∈S-P ∧
                        x∉P-S ∧ x∉P∩S) ∨ (x∈S-P ∧ x∉P∩S)}
    For instance the mood IAI4 of �, becomes 2/1IA1I4 in 2�. Because of the exclusive
existential quantifier 2/1I, the case Δ46 is no more matched by of the first premiss Φ 1
and the conclusion Φ3 becomes false for the case Δ19 (Fig 1).




true: Δ4*=δ4δ6δ7                                        false: Δ19=δ2δ6δ7             true: Δ67=δ1δ2δ7




true: Δ24=δ2δ4δ6δ7                                       true: Δ43=δ1δ6δ7             true: Δ68=δ1δ2δ6δ7




true: Δ74=δ1δ2δ4δ7                                      true: Δ48=δ1δ4δ6δ7            true: Δ76=δ1δ2δ4δ6δ7
*
    A full list of all syllogistic cases Δj, j=[1,96], can be found elsewhere [22].
          Fig 1. 9 syllogistic cases Δj of the mood 2/1IA1I4 of the fuzzy-syllogistic systems 2�.

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     Proceedings of the KI 2015 Workshop on Formal and Cognitive Reasoning




       Table 2. Value ranges of affirmative quantifiers of various fuzzy-syllogistic systems n�
     Syllogistic System                                                           Quantifier ψ*
    Aristotelian       �        A=all                                             I=some (including A)
                       2
                           �    A=all                                            2/1
                                                                                       I=some (excluding A)
                       3
                           �    A=all                             3/2
                                                                        I=most                                           3/1
                                                                                                                               I=several
                       4
                           �    A=all                     4/3
                                                                I=most                      4/2
                                                                                                  I=half                         4/1
                                                                                                                                       I=several
        Fuzzy          5
                           �    A=all               5/4
                                                          I=many                   5/3
                                                                                         I=most          5/2
                                                                                                               I=several                  5/1
                                                                                                                                                I=few
                       6
                           �    A=all   6/5
                                              I=many                     6/4
                                                                               I=most       6/3
                                                                                                  I=half          6/2
                                                                                                                        I=several               6/1
                                                                                                                                                      I=few
                       n
                           �    A=all   n/n-1
                                                I                                                  ...                                                   n/1
                                                                                                                                                               I
*
    Column breadths are not drawn proportional to the overall value range and to the other quantifiers.

     The analysis of the FSS 2� shows that 2� has 70 distinct moods, 11 true moods
τ=1, of which 5 are distinct, and 40 false moods τ=0, of which 13 are distinct, and that
2
  � is not point-symmetric [22], [9].

2.5 Fuzzy­Syllogistic System n�
    By using (n-1) fuzzy-existential quantifiers, the total number of fuzzy-syllogistic
moods of the FSS n� increases to (2n)3. The sample mood IAI4 of � can now be
generalised to n/k1IAk2I4, 1<n, 0<k1,k2<n of n�. n/k1IAk2I4 consists of (n-1)2 fuzzy-
moods, all having the very same 9 syllogistic cases (Fig 1).
    Same linguistic terms used in different FSSs do not necessarily equal each other.
For instance, "most" may have different value ranges in the FSSs 3�, 4�, 5�, 6� and
therefore are in general not equal 3/2I≠4/3I≠5/3I≠6/4I, respectively. Likewise for "half" in
4
 � and 6� the quantifiers may not exactly equal 4/2I≠6/3I, respectively (Table 2).

3      FUZZY-SYLLOGISTIC ONTOLOGY
   A fuzzy-syllogistic ontology (FSO) consists of concepts, their relationships and
assertions on them, whereby all quantities are given with fuzzy-quantifications:
                               Fuzzy-syllogistic ontology: FSO=k(C, R, A)
    where C is the set of all concepts, R is the set of all directed relationships between
the concepts and A is the set of all assertions. A FSO may be specified top-down or
may be transformed from any existing ontology, provided that all quantities are
determined systematically, in compliance with one of the FSSs k�, 1<k≤n, (Table 2).
In a bottom-up approach, a FSO may be learned from given domain data.

3.1 Learning Fuzzy Quantifiers
    Although existing learning approaches generate ontological concepts and their
relationships through probabilistic analysis of the data [4], [15], [11], the quantities
that actually imply the concepts and relationships, are not preserved in the ontology
[8]. Therefore we sketch here briefly how to learn such quantities of a FSO.
    For any directly connected triple concept relationship of the FSO, seven distinct
relationships are possible (Table 1). The quantity of every such relationship has to be
stored with the FSO. Since the relationships may be bi-directional or a concept may

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be involved in multiple triple relationships (Fig 2), the quantities of all these cases
need to be stored too.
   The objective of learning a FSO=k(C, R, A) is, to update the FSO against changing
domain data and to determined the most appropriate FSS k� out of n�, 1<k≤n.

3.2 Relative truth ratio
    Relative truth ratios are calculated from the exact quantities of all syllogistic cases
of a particular mood, rather than from just the amount of the cases:
                                         Relative true: rτt = λf<λt → λf/(λf+λt)
                                         Relative false: rτf = λt<λf → λt/(λt+λf)
    where λt = j=1∑|Λt| |Δtj| and λf = j=1∑|Λf| |Δfj| is the total number of elements
accumulated over all true and false syllogistic cases, respectively. Where |Λ t| and |Λf|
is the number of true and false cases of the mood, respectively. Accordingly, we can
re-define a fuzzy-syllogistic mood with relative truth ratio rτ:
                   Fuzzy-syllogistic mood with relative truth ratio: (ψ 1ψ2ψ3F, rτ)
    The structural truth ratio τ of a particular mood represents the generic vagueness
of the mood and is constant, whereas the relative truth ratio rτ adjusts τ by weighting
every case of the mood with its actual quantity.

4     FUZZY-SYLLOGISTIC REASONING
    The fuzzy-syllogistic systems �, 2� and 6� are currently implemented
experimentally as the reasoning logic of the fuzzy-syllogistic reasoner (FSR), for
reasoning over FSOs [21]. Our objective is to generalise the logic of the reasoner to
n
 � and to use it as a cognitive primitive for modelling other cognitive concepts within
a cognitive architecture. We now sketch the algorithmic design of the FSR.



                         Race                                                              Race
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               fe




                                                                               fe
                                                                             I=
            I=




                                   6/




                                                                                                    6/
           1




                                                                            1




                                                                                         I=few
                        I=few

                                      5




                                                                                                       5
         6/




                                                                          6/
                                   I=




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                                           A=




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            I=




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                        6/1




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                                                 y




                                                                                                           y




                                   6/1
                                        I=few                                                       6/1
                                                                                                         I=few
            1




                                                                            1
         6/




                                                                         6/




    Child                                               Sport        Child                                               Sport
                   I=few
                  6/1
                                                                                  I=few
                                                                                   6/1
                                                  all                                                              all
                                 6/1




                                                                                                  6/1




            6/                                                               6/
              5                                                                5
               I=                                                             I=
                                I=few




                                                                                                 I=few




                                                A=                                                              A=
     6/                                                               6/
       4                                                                4
        I=        m                                                     I=       m
           m        an                                                    m       an
            os        y                                                    os       y
              t                                  ew                          t                                    ew
                                          6/1 I
                                               =f                                                          6/1 I
                                                                                                                =f
                        Bicycle                                                       Bicycle
          6
            �: (6/4I1I1I1, 40/47=0.851)                        6
                                                                 �: (6/5I1I1I2, 40/48=0.833)
Φ1: Most bicycles are good for children             Φ1: Many children have bicycles
Φ2: Few sports are good for bicycles                Φ2: Few sports are good for bicycles
Φ3: Few sports are good for children                Φ3: Few sports are good for children
 Fig 2. Sample fuzzy-syllogistic ontology with affirmative relationships and the best matching
                     fuzzy-syllogistic moods from the syllogistic figures 1 and 2.

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    Proceedings of the KI 2015 Workshop on Formal and Cognitive Reasoning




4.1 Reasoning Algorithm
    FSR is concerned with identifying for any given concept c∈C, all possible triple
concept relationships r∈R, r={M,P,S}, of the given FSO=k(C, R, A) and to reason
with the most appropriate fuzzy-syllogistic moods of its FSS k�. Whereby associated
assertions a∈A may be used for exemplifying a particular reasoning.
    For instance, for the concept c=Bicycle, multiple triple relationships r={Bicycle,
Child, Sports} exist in the sample FSO=6(C, R, A) (Fig 2). The reasoner iterates for
the FSSs k�, k=[2,n], and for their moods, in order to match the moods with the
closest fuzzy-syllogistic quantities of relationships r. The reasoner determines the FSS
k=6 and the mood 6/k1IAk2I4, 0<k1,k2<6 as best matches for this example.
    In the below example with � , I in Φ 3 may include A and therefore is less true.
Whereas in 3�, 3/1I in Φ3 is still too general. The best matching quantifiers are found in
6
 � (Fig 3).

�: (IAI3, 10/10=1.0)                                             �: (3/2IA1I3, 6/6=1.0)
                                                                 3

Φ1: Some bicycles are good for children                          Φ1: Most bicycles are good for children
Φ2: All bicycles are good for sports                             Φ2: All bicycles are good for sports
Φ3: Some sports are good for children                            Φ3: Several sports are good for children

5     CONCLUSION
    The FSSs �, 2�, 6� were introduced as the fundamental logic of the fuzzy-
syllogistic reasoner (FSR) and its usage was exemplified on a sample fuzzy-
syllogistic ontology (FSO). The relative truth ratio rτ of a mood was introduced,
which adapts the structural truth ratio τ of the mood to the amount of elements of its
syllogistic cases. FSR with FSOs is a generic possibilistic reasoning approach, since
the employed reasoning logic n� is generic.
    We are currently implementing a sample educational application that extends an
existing probabilist ontology learning tool and generates a FSO= k(C, R, A) for a given


                         Race                                                              Race
                                                                                  w
                 w
               fe




                                                                               fe
                                                                             I=
            I=




                                   6/




                                                                                                    6/
           1




                                                                            1




                                                                                         I=few
                        I=few

                                      5




                                                                                                       5
         6/




                                                                          6/
                                   I=




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                                           A=




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                 w




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            I=




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                        6/1




                                                                                         6/1
                                                 y




                                                                                                           y




                                   6/1
                                        I=few                                                       6/1
                                                                                                         I=few
            1




                                                                             1
         6/




                                                                          6/




    Child                                               Sport        Child                                               Sport
                   I=few
                  6/1
                                                                                  I=few
                                                                                   6/1
                                 6/1




                                                                                                  6/1




                                                  all                                                              all
            6/                                                               6/
              5                                                                5
               I=                                                             I=
                                I=few




                                                                                                 I=few




                                               A=                                                               A=
     6/                                                               6/
       4                                                                4
        I=        m                                                     I=       m
           m        an                                                    m       an
            os        y                                                    os       y
              t                                  ew                          t                                    ew
                                          6/1 I
                                               =f                                                          6/1 I
                                                                                                                =f
                        Bicycle                                                       Bicycle
          6
            �: (6/4IA1I3, 6/6=1.0)                            6
                                                                �: (6/5IA1I4, 8/9=0.888)
Φ1: Most bicycles are good for children            Φ1: Many children have bicycles
Φ2: All bicycles are good for sports               Φ2: All bicycles are good for sports
Φ3: Few sports are good for children               Φ3: Few sports are good for children
 Fig 3. Sample fuzzy-syllogistic ontology with affirmative relationships and the best matching
                    fuzzy-syllogistic moods from the syllogistic figures 3 and 4.

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    Proceedings of the KI 2015 Workshop on Formal and Cognitive Reasoning




domain. For a user-chosen concept C from the ontology FSO, FSR is then used to
reason with all associated quantities R and present the user all associated scenarios A.

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