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.
Multi-valued logics were initially introduced by Łukasiewicz [
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 [
There are more probabilistic ontology reasoners than fuzzy or possibilistic ones
and most of them reason with probabilist ontologies [
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
[
Several generic reasoning logics are discussed in the literature, like probabilistic,
non-monotonic or non-axiomatic reasoning [
Syllogistic reasoning reduced to the proportional inference rules deduction,
induction and abduction are employed in the Non-Axiomatic Reasoning System
(NARS) [
There is one implementation mentioned in the literature that is close to the concept
of syllogistic cases: Syllogistic Epistemic REAsoner (SEREA) implements
polysyllogisms 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 [
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
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.
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)
where F={1, 2, 3, 4} identifies the four possible combinations of Φ1 with Φ2, namely syllogistic figures. Every figure produces 43=64 moods and the whole syllogistic system has 4x64=256 moods.
Syllogistic cases are an elementary concept of the fuzzy-syllogistic systems n ,
for calculating truth ratios [
For three sets, 7 distinct spaces δi, i=[
We extend the ancient binary truth classification of moods, to a fuzzy
classification with truth values in [
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] 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:
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 [
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 Φ1
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
* A full list of all syllogistic cases Δj, j=[
Fig 1. 9 syllogistic cases Δj of the mood 2/1IA1I4 of the fuzzy-syllogistic systems 2 . true: Δ76=δ1δ2δ4δ6δ7 ...
n/1I * 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 [
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 fuzzymoods, 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
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.
Although existing learning approaches generate ontological concepts and their
relationships through probabilistic analysis of the data [
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 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.
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 |Λt| 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:
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
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 [
y 6/1I=few few
6/1 I=
6/5 I=m an
y 6/1I=few few 6/1 I= 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.
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 Φ3 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) Φ1: Some bicycles are good for children Φ2: All bicycles are good for sports Φ3: Some sports are good for children 3 : (3/2IA1I3, 6/6=1.0) Φ1: Most bicycles are good for children Φ2: All bicycles are good for sports Φ3: Several sports are good for children 5
The FSSs , 2 , 6 were introduced as the fundamental logic of the fuzzysyllogistic reasoner (FSR) and its usage was exemplified on a sample fuzzysyllogistic 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
6/4 I=most
6/5 /16 fIe=w 6I/=1Im=anfyew Bicycle 6/1 I= few 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.
few 6/1 I= Bicycle 6/1 I= few 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. 6