Theory of Graded Consequence and Fuzzy Logics

1. Theory of Graded Consequence and Fuzzy Logics Mihir K. Chakraborty Department of Pure Mathematics University of Calcutta India mihirc99@vsnl.com Soma Dutta Department of Pure Mathematics University of Calcutta India

2. Logic A logic consists of a language and a deductive apparatus i.e. a notion of consequence.

3. Notion of consequence: Due to Tarski • Notion of consequence due to Tarski (1930) is a function from the power set of all formulae to the power set of all formulae, mapping each set of formulae X to its consequence set C(X). Conditions imposed on the operator C are, X C(X) If X  Y then C(X) C(Y) C( C (X)) = C(X)

4. Notion of consequence: Due to Gentzen • Gentzen in (1934-1935) presented notion of consequence as a relation from the power set of all formulae to a single formula. i.e. a set of formulae X is related to a formula α by the consequence relation|―, denoted by X|― α, means α is a consequence of X. |― is postulated by • If α  X then X|― α (Overlap) • If X  Y then X|― α implies Y|― α (Monotonicity) • If for all β  Z, X |― β then XZ |― α implies X |― α (Cut)

5. Notion of Consequence in the context of fuzzy logics • There are two existing approaches (1) In fuzzy context Pavelka’s notion of consequence is a function assigning a fuzzy set of consequences to its corresponding fuzzy set of premises (2) On the other hand, in fuzzy set up Chakraborty’s notion of consequence is a fuzzy relation between a set of formulae and a single formula.

6. Pavelka’s notion of consequence • Pavelka in 1978, generalizing Tarskian tradition for consequence, proposed the notion of consequence as a function C from the set of all fuzzy subsets over formulae (F) to itself satisfying, X C(X) If X  Y then C(X) C(Y) C( C (X)) = C(X)

7. Graded consequence relation Chakraborty in 1987, propsed graded consequence relation following Gentzenian tradition in many-valued set up. Graded consequence relation |~ is a fuzzy relation from P(F) to F, satisfying, 1. If α  X then gr( X|~ α) = 1 (Overlap) 2. If X  Y then gr( X|~ α) ≤ gr( Y|~ α) (Monotonicity) 3. infβ Z gr( X|~β) * gr( XZ|~ α) ≤ gr( X|~ α) (Cut) Where ‘inf’ and ‘*’ of a complete Residuated lattice, are used to compute meta linguistic ‘for all’ and ‘and’ respectively. So a complete pseudo Boolean algebra <L, →, , ν, 0,1> also can be well considered as meta level algebraic structure.

8. Complete pseudo Boolean algebra • A complete pseudo Boolean algebra <L, →, , ν, 0,1> with 0 and 1 as the least and greatest elements respectively is a relatively pseudo complemented complete lattice with the least element. A relatively pseudo complemented lattice is a lattice such that for each a, b of the lattice there exists c in the lattice (which is known as the pseudo complement of a relative to b) such that c is the greatest of all such x satisfying a x ≤ b i.e. for each a, b in L, a →b exists such that a c≤ b iff c≤ a →b

9. Motivation behind graded consequence relation “If vagueness is present at the object-level language and hence multivalence is accepted in providing a semantics for object-level sentences, multivalence cannot be generally denied at the level of meta-concepts like consequence, consistency, tautologihood etc.”

10. Towards the semantic notion of graded consequence • Let us concentrate on the semantic notion of consequence from classical perspective. For a set X of formulae and a formula α, α is a semantic consequence of X if for all states of affairs T, if X is contained in T then α is a member of T, where T has been identified with a set of formulae which are true (1) under the valuation function T. A formal expression of the above statement would be, T (XT → α  T), abbreviated by X |= α, where all the notions are standard set theoretic and ‘→’ stands for the meta level connective ‘if-then’

11. Generalization due to Shoesmith and Smiley • This initial idea of semantic consequence has been generalized by D.J. Shoesmith & T.J.Smiley in1978.In accordance with their work T need not to range over all states of affairs. That is given an arbitrary collection of states of affairs i.e. a subset of P(F), α is a semantic consequence of X iff for all states of affairs T in that collection, if X is contained in T then α is a member of T.

12. Graded counterpart of the notion of semantic consequence • In many-valued set up let {Ti}iI be a collection of fuzzy subsets over formulae. Each Ti has been identified with a truth valuation under which a formula α gets the truth value Ti(α), the belongingness degree of α to the fuzzy subset Ti which may get the value other than 0 and 1. Then the graded notion of semantic consequence would say a formula α is a semantic consequence of a set of formulae X to the degree infi [ gr(XTi) → Ti(α)], which turns out to beinfi{infβX(X(β ) → Ti(β) ) →Ti(α) } i.e. infi{infβXTi(β) →Ti(α)}

13. Representation theorems • Given a graded consequence relation |~, there exists a collection {Ti}iI of fuzzy subsets of formulae for which gr( X|~ α)= gr( X|≈α), where gr(X|≈α) is the degree to which α is a semantic consequence of X. • For any collection {Ti}iI of fuzzy subsets of formulae |≈ (defined in the previous slide) is a graded consequence relation.

14. Existing fuzzy logics vs. the theory of graded consequence • To clarify the point of view of the existing fuzzy logics let us quote a relevant remark of Carlos Pelta (2004) “Until now the construction of superficial many-valued logics, i.e logics with arbitrary number ( bigger than two ) of truth values but always incorporating a binary consequence relation has prevailed in investigations on logical many-valuedness. That is many-valuedness has been excluded from the consequence operation and the meta logic of a logical system, although its object language may be many-valued.” • This is exactly the point where the theory of graded consequence differs.

15. Profile of the next slides • Firstly, to analyze existing literature of fuzzy logics. • Secondly, we will propose a methodology distinguishing various levels of propositions (sentences) in the study of a logical system to show how genuine is the theory of graded consequence is, in addressing meta-level multivalence.

16. From the existing literature

17. Fuzzy rule of inference: Goguen’s position • To Goguen a fuzzy rule of inference should be so designed that one can address “If you know P is true at least to the degree a and P Q at least to the degree b then Q istrue at least to the degree a.b.” • An immediate formal representation of the above rule is, ( P, a ) (P Q , b ) ( Q, a.b ) • This reveals, the rule is simply a subset of P(F×L) × F×L i.e a crisp relation.

18. Fuzzy Rule of Inference: Pavelka’s Position • Being motivated by Goguen’s fuzzy rule of inference, Pavelka proposed a many-valued rule r by a pair of components < r’, r’’ > of which r’ operates on formulas and r’’ operates on truth values and says how the truth value of the conclusion is to be computed from the truth values of the premises. • Let us see how Pavelka had put Modus Ponens as a many-valued rule----- P a P Q b Q a.b • This is quite similar to Goguen’s structure for many-valued Modus Ponens.

19. Fuzzy Rule of Inference: Hajek’s Position • For Peter Hajek the word ‘fuzzy logic’ suits more to the idea ‘partially true conclusion can be derived from partially true premises’ He proposed a fuzzy logical system, known as Rational Pavelka Logic (RPL) which results from the Łukasiewicz logic Ł by adding truth constants rfor each rational r in the unit interval [0,1] with two axioms determining the value of the truth constants denoted by r s and ~r • Now along with this extended set of axioms and the rule M.P one can obtain a  P b  (P Q) c Q i.e. ( P, a ) (P Q , b ) ( Q, a.b ) (identifying a  P by ( P, a )) as a derived rule in RPL.

20. Now with the usual understanding of validity of a rule i.e. whenever premises are true (1) conclusion is also true (1), the rule ( P, a ) (P Q , b ) ( Q, a.b ) offers the reading ‘if P is true at least to the degree a and P Q is true at least to the degree b then Q is true at least to the degree a.b’. That is Peter Hajek’s position also goes back to the idea, initiated by Goguen.

21. So, starting from Goguen to Hajek, no one intended to mean a fuzzy rule of inference, a special case of derivation as a fuzzy relation, rather they intended to view a fuzzy rule as a crisp relation between a set of premises, showing their truth value side by side and a conclusion, tagged with a truth value.

22. Fuzzy rule of inference in the theory of graded consequence • Graded Consequence differs here, i.e. in the theory of graded consequence a fuzzy rule of inference is indeed a fuzzy relation between a sets of formulae and a single formula. • So in this context the many-valued rule Modus Ponens will be a fuzzy relation RM.P ( {P, P Q }, Q ) with the grade inf P,Q gr({P, PQ }|≈ Q), where |≈ represents the semantic counterpart of graded consequence relation.

23. Example • Let us consider the formulas α, β ,α  β and β  α and a collection of fuzzy subsets consisting of T1 and T2. Assume  to be computed by Łukasiewicz implication. • Let αβα  ββ  α T1 .7 .3 .4 1 T2 .9 .7 .8 1 T3 .7 .9 1 .8 Then gr({α , α  β } |≈ β ) = .3 and gr({β, β  α } |≈ α) = .7 Hence as grade of RM.P ( {P, P Q }, Q ) i.e. inf P,Q gr({P, PQ }|≈ Q) considers all such gr({P, PQ }|≈ Q) it must be less equal to .3  .7 i.e. .3

24. Through distinction of levels of logic activities

25. Methodology for the process of distinguishing various levels of a logical system • First of all put all the initial logical entities at a layer and call it as level-0 language. Let A, B, C, --------- be the constituents of the level-0 language. • To depict the activities of level-0 entities we need another language, constituted by--- • The name of each level-0 entity, i.e ‘A’, ‘B’, ‘C’, ------- (These would be the constant symbols of the level-1 language.) • A set of required variables, predicates, connectives, quantifiers etc and hence level-1 wffs to talk about the level-0 entities

26. ‘Use’ and ‘Mention’ of a symbol • Let A be a symbol, used at level-0. • When one needs to talk about that very symbol, it has to be referred i.e. mentioned at the next level, higher to level-0 and this distinction has been made here by putting ‘’ mark over the symbol. • No symbol can be used and mentioned at the same level.

27. Three levels of a logic • Level-0 Let α, β, γ ---- be the formulae of the object language and X be a set consisting of some formulae. These belong to the language of level-0. • Level-1 Level-1 deals with the formula like ‘α’ is a semantic consequence of ‘X’, ‘α’ is a syntactic consequence of ‘X’, ‘X’ is inconsistent, ‘α’ is a semantic consequence of ‘X’ if and only if ‘α’ is a syntactic consequence of ‘X’ etc. • Level-2 Level-2 deals with the formula like, a “consequence relation” is sound, a “consequence relation” is complete etc. The value of these level-2 sentences would be determined by the value referred by “if ‘α’ is a syntactic consequence of ‘X’ then ‘α’ is a semantic consequence of ‘X’ ” and “if ‘α’ is a semantic consequence of ‘X’ then ‘α’ is a syntactic consequence of ‘X’ ” respectively.

28. Theory of graded consequence maintaining the level distinction mechanism Level-0 consists of • all formulae α,β,γ,---------- • all ordinary sets of formulae X,Y,Z,-------- • a particular set of fuzzy sets of formulae T1, T2, T3, -------- • all finite sequence of formulae < α1, α2, --------, αn>, ------- • Some particular sets of sequences of formulae, S12 = {< α1, α1 α2, α2>, < α3, α3 α4, α4>,--------} . . Smn =

29. At level-1 • ‘α’, ‘β’, ‘γ’,------, ‘X’, ‘Y’, ‘Z’,------, ‘T1’, ‘T2’, ‘T3’, -------- • ‘< α1, α2, --------, αn>’, ------- (These are the constant symbols of level-1) • Variables: x, T , < >SV Predicate symbols: , RM.P • Connectives: →, & Quantifiers: ,  Term: Constant symbols and variables are terms Wff : ‘α’ ‘X’, ‘α’ T, x  ‘X’, x  T ‘X’ ‘Y’ = x (x  ‘X’→ x  ‘Y’) ‘X’|= ‘α’ = T( ‘X’ T →‘α’ T)

30. Interpretation of level-1 language would be, • x ranges over all formulae of level-0 • T ranges over the collection T1, T2, T3, -------- • < >SV ranges over the set < α1, α2, --------, αn>, ------- • ‘α’ ‘X’ would get the value 1 if Interpretation of ‘α’ belongs to the set referred by ‘X’ and 0 otherwise. • Value of ‘α’ T would be the membership degree of α to the fuzzy set referred by T. • Now assuming a complete pseudo Boolean algebra as the structure for level-1 language, we can easily see how the level-1 sentences get many-valued truth value.

31. Analyzing Pavelka’s literature, it is quite clear that Pavelka’s fuzzy rule of inference is actually a crisp relation. But to make our claim strong, we propose to explore, what would happen if one likes to see Pavelka’s proposed fuzzy rule of inference indeed as a fuzzy relation. i.e instead of seeing the fuzzy rule Modus Ponens as r({(p,a), (p q, b)}, (q, a*b) ), let us see r as a fuzzy relation r({(p,a), (p q, b)}, q ) with the relatedness grade a*b

32. Pavelka’s Notion of Proof • For convenience, before rewriting Pavelka’s work maintaining level distinction let us have a glance at Pavelka’s notion of proof. • For a proof ω = < ω 1, ω 2, --------, ωn> from X, ώ(X), interpreted as value of the proof ω from X, is defined by, (i) if length of ω is 1 then ώ1(X) = X(Γω 1) or ώ1(X) = A(Γω 1), where Γω 1 denotes the target formula at step ω 1 and A has been interpreted as a fixed fuzzy set of axioms. (ii) if length of ω is i, (1< i ≤ n ) and ωi is a conclusion of a rule of inference, applied on the formulas, occurred at some steps i1, i2, -----in , preceding i, then ώi(X) = r’’(ώi1(X),-----, ώin(X) )

33. Pavelka’s logic in level distinction mechanism • Level-0 : • All formulae α,β,γ,---------- • All fuzzy sets of formulae X,Y,Z,-------- • All finite sequence of formulae < α1, α2, --------, αn>, ------- • All finite sequence over F×L, i.e < (α1,a1 ), (α2, a2),---, (αn, an) >, ---- • A particular set r’ containing all sequences of formulae of the form < α1, α1 α2, α2>, i.e r’ = {< α1, α1 α2, α2>, < α3, α3 α4, α4>,-----} • At level-1 we need name of each level-0 entity i.e • ‘α’, ‘β’, ‘γ’,------, ‘X’, ‘Y’, ‘Z’,------ • ‘< α1, α2, --------, αn>’, ------- • ‘< (α1,a1 ), (α2, a2),--------, (αn, an) >’, ------- (These are all constant symbols of level-1)

34. Variables : x ( ranges over all formulae of level-0) T ( ranges over a particular collection of fuzzy sets of formulae of level-0) < >SV (ranges over the set < α1, α2, -----, αn>, -------) <( ), ( )> ( ranges over all 2-length sequence over F×L • Predicate symbols: , r • Connectives: →1, →2, &,  • Quantifiers: ,  • Interpretation of &,  would be lattice meet and join respectively and that of →1,and →2 would be given by, a ═›1b = 1 if a ≤ b = 0 otherwise and a ═›2b = sup {z / a  z ≤ b } respectively.

35. Term: Constant symbols and variables are terms • Wff : ‘α’ ‘X’, ‘α’ T, x  ‘X’, x  T ‘X’ ‘Y’ = x (x  ‘X’→1 x  ‘Y’) ‘X’|= ‘α’ = T( ‘X’ T →2 ‘α’ T) r(‘< (α1,a1 ), (α2, a2) >’, ‘a3’) , r(<( ), ( )>, ‘α’) ‘< α1, α2, -----, αi>’D(‘X’, ‘αi’) = ‘αi’‘X’  ‘αi’A  Resr(‘αi’) Where Resr(‘αi’) would represent the wff <( ),( )>[ <( ), ( )>  ‘< (α1,a1 ),(α2, a2),- - -,(αi-1, ai-1)>’ & r(<( ), ( )>, ‘αi’)]

36. Example • Let ω = < ω 1, ω 2, --------, ωn> be a proof of Γωn from X and Γωn= r’ (Γωi , Γωk), i,k< n • Now in accordance with our proposal of seeing the many-valued rule modus ponens as a fuzzy relation, (in Pavelka’s framework) it would be read as {(Γωi , ώi(X)),(Γωk , ώk(X))} is related to Γωn with the relatedness grade r’’(ώi(X), ώk(X)) • But as ώi(X), ώk(X) are the values of the sub-proofs of < ω 1,ω 2, -----, ωn>, cannot be used within the structure of a rule.

37. Γω1ώ1(X) Γω2ώ2(X) - - - - Γωiώi(X) - - Γωkώk(X) - - Γωn = r’ (Γωi , Γωk) i.e. as proposed, here the application of the fuzzy rule modus ponens can be illustrated as Γωi ώi(X) Γωk-----------------------ώk(X) Γωn and the grade of the fuzzy relation is r’’(ώi(X), ώk(X))

38. Merits of graded consequence over other existing fuzzy logics • Classically notion of consequence and notion of inconsistency (consistency) are equivalent in the sense that taking one as primitive another can be obtained. • In Pavelka, notion of consistency has been introduced as,‘a fuzzy set X of formulae is said to be consistent if C(X) ≠ 1 i.e. C(X) is not the whole set of formulae’. Thus, according to Pavelka’s definition, consistency is a crisp notion. So, the question arises, how could Pavelka make the notion of consistency, a two-valued notion and the notion of consequence, if grade is being attached to it, commensurate to each other in a sense similar to that in classical logic?

39. Peter Hajek in his fuzzy logical system has shifted from classical position to address ‘partially true conclusion can be derived from partially true premises’ i.e. notion of consequence has been modified in many-valued set up by accommodating a rule like ( P, a ) (P Q , b ) ( Q, a.b ) But no such similar treatment has been found for the notion of inconsistency. According to Hajek a set X would be inconsistent if ō i.e. (ō, 1) can be derived from X. The question arises whether he would not like to give any status to X if (ō, a) follows from X, where a is non-unit.

40. In this respect theory of graded consequence seems more stable. Theory of graded consequence has offered the notion of graded inconsistency which has been established to be equivalent to the notion of graded consequence.

41. Theory of graded consequence in presence of negation (~), a object language connective GC1. If αX then gr(X|~ α) = 1 GC2. If X Y then gr(X|~ α)≤ gr(Y|~ α) GC3. infβ Z gr( X|~β) * gr( XZ|~ α) ≤ gr(X|~ α) GC4. There is some k>0 such that for any α, inf β gr({α, ~α} |~β)≥k GC5. gr(X{α}|~β) * gr( X{~α}|~β) ≤ gr( X|~β)

42. Graded Inconsistency Axioms • Let INCONS be a fuzzy subset over the set of all sets of formulae. For each set of formulae X, INCONS(X), the degree to which X is inconsistent is postulated by, • If X Y then INCONS(X) ≤ INCONS(Y) • INCONS(X{~ y}) * INCONS(XY) ≤INCONS(X), for each y in Y • There is some k>0 such that for any α, INCONS({α, ~α})≥k

43. As in classical case, graded notion of consequence and inconsistency both are equivalent i.e. • Given a graded consequence relation |~, graded notion of inconsistency can be obtained by defining INCONS by, INCONS(X)=infαgr(X|~ α) • Given INCONS satisfying all graded inconsistency axioms, a graded consequence relation can be defined by gr(X|~ α)= 1 , if αX = INCONS(X{~α}) , otherwise

44. Crisp consequences generated from a graded consequence relation and their properties Let |~ be a graded consequence relation characterized by GC1 to GC3. Define a class of crisp consequences Ci for each i L by Ci(X) ={ β/ gr(X |~ β) ≥ i }. Then Ci turns out to be a Tarskian consequence operator in the sense that Ci satisfies (i) X Ci(X) (ii) If X  Y then Ci(X) Ci(Y) (iii) Ci( Ci (X)) = Ci(X) Additionally let us assume |~ satisfies GC4 i.e. there is a k > 0 such that for any α, infgr({α , ~ α }) k . Theorem: For each i  L, Ci({α, α }) = F for any α,if i  k and there is some α, for which Ci({α, α }) F if i > k or i is non-comparable with k.

45. Proof: We shall prove the theorems in two stages. Stage-1: Let us take an arbitrary α1 such that infgr({α1 , ~ α1 }) = a k Now for any i  L, either i a or i > a or i is non-comparable with a. Case-I Let i a As infgr({α1 , ~ α1 }) = a , for all , gr({α1 , ~ α1 }) a i  Ci({α1,  α1 }) = F Case-II Let i > a Ci({α1,  α1 }) = { / gr({α1,  α1 })  i } { / gr({α1,  α1 }) a } = Ca({α1,  α1 }) = F

46. Claim is that Ci({α1,  α1}) is a proper subset of Ca({α1,  α1 }) i.e. there is some  which does not belong to Ci({α1,  α1 }). If not, then for all , gr({α1,  α1 })  i  infgr({α1 , ~ α1 }) = i > a This is a contradiction to the assumption infgr({α1 , ~ α1 }) = a  Ci({α1,  α1 })  F Case-III Let a and i be non-comparable and sup { a , i}= j As a and i are non-comparable and gr({α1 , ~ α1 }) a, for all β, for no γ, gr({α1 , ~ α1 } γ) = i  Ci({α1,  α1 }) = { / gr({α1,  α1 }) > i } Claim is that Ci({α1,  α1 }) = Cj({α1,  α1}) i.e. { / gr({α1,  α1 }) > i }= { / gr({α1,  α1 })  j } i.e.there is no such β such that i < gr({α1,  α1 }) < j or i < gr({α1,  α1 }) but gr({α1,  α1 }) is non-comparable with j. -----------------(a)

47. For the first case, if possible let for some β, i < gr({α1,  α1}) = l < j  l a [ Since for all γ, gr({α1 , ~ α1 } γ) a ] But i < l < j together with l a contradict the fact that sup { a , i}= j That is there is no such β such that i < gr({α1,  α1 }) < j For the second case, let us assume there is some β for which i < gr({α1,  α1 }) = l but l and j are non-comparable. Then again as a l , l is an upper bound of a and i. Then as j = sup { a , i}, j can not be non-comparable with l. Hence (a) is proved.  Ci({α1,  α1 }) = Cj({α1,  α1 })  F [By case-II, since j > a]

48. Case-II Let i > k Then three subcases arise. Subcase (i)a < i Subcase (ii)a and i are non comparable Subcase (iii) k < i < a For (i) and (ii) as we already have in stage-1 infgr({α1 , ~ α1 }) = a, by case-I and case-II of stage-1 we can conclude Ci({α1,  α1 })  F Subcase (iii) Let k < i < a Since k < i , it is not that for any α, infgr({α , ~ α }) i i.e. there is some α2 such that infgr({α2 , ~ α2 }) = j where either j < I or j is non-comparable with i. Then in either cases Ci({α2,  α2 })  F [By already proved results in stage-1] Combining all these three cases, we can conclude that there is some α such that Ci({α,  α })  F, for i > k

49. Case-III Let i be non-comparable with k Then again two subcases arise. Subcase (i) a and i are non-comparable Subcase (ii) i a [ As k and i are non-comparable and k a, the case for a > i would not arise.] For subcase (i) again as infgr({α1 , ~ α1 }) = a, by previous result Ci({α1,  α1 })  F Subcase (ii) Let i a Since i is non-comparable with k, there is no such γ such that infgr({γ , ~ γ }) = i Also it is not that for any α, infgr({α , ~ α }) > i

50. There is some α3 such that infgr({α3 , ~ α3 }) = j, where j < i or j is non-comparable with i. But j < i can not be the case. Because if j < i then as j  k, we have k < i. This contradicts the assumption that i is non-comparable with k. Now if j is non-comparable with i, Ci({α3,  α3 })  F [By previous result as infgr({α3 , ~ α3 }) = j] Combining all the above subcases, we can conclude that there is some α such that Ci({α,  α })  F, where i is non-comparable with k.