Theory of Graded Consequence and Fuzzy Logics

Mihir K. ChakrabortyDepartment of Pure MathematicsUniversity of CalcuttaIndiamihirc99@vsnl.com

Soma Dutta Department of Pure Mathematics University of Calcutta India

Logic

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

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)

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 XZ |― α implies

X |― α (Cut)

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.

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)

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( XZ|~ 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.

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

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.”

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 (XT → α T), abbreviated by X |= α, where all the notions are standard set theoretic and ‘→’ stands for the meta level connective ‘if-then’

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.

Graded counterpart of the notion of semantic consequence In many-valued set up let {Ti}iI 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(XTi) → Ti(α)], which turns out to be infi{infβX(X(β ) → Ti(β) ) →Ti(α) } i.e. infi{infβXTi(β) →Ti(α)}

Representation theorems

Given a graded consequence relation |~, there exists a collection {Ti}iI 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}iI of fuzzy subsets of formulae |≈ (defined in the previous slide) is a graded consequence relation.

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.

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.

From the existing literature

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 is true 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.

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.

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 r for 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.

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.

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.

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, PQ }|≈ Q), where |≈ represents the semantic counterpart of graded consequence relation.

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 α β α β β 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 ) = .7Hence as grade of RM.P ( {P, P Q }, Q ) i.e. inf P,Q gr({P, PQ }|≈ Q) considers all such gr({P, PQ }|≈ Q) it must be less

equal to .3 .7 i.e. .3

Through distinction of levels of logic activities

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

‘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.

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.

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 =

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 termsWff : ‘α’ ‘X’, ‘α’ T, x ‘X’, x T ‘X’ ‘Y’ = x (x ‘X’→ x ‘Y’) ‘X’|= ‘α’ = T( ‘X’ T →‘α’ T)

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.

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

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) )

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)

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 otherwiseand a ═›2b = sup {z / a z ≤ b } respectively.

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’)]

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.

Γω1 ώ1(X) Γω2 ώ2(X) - - - -

Γωi ώi(X) - -

Γωk ώk(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))

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?

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.

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.

Theory of graded consequence in presence of

negation (~ ), a object language connective

GC1. If αX then gr(X|~ α) = 1

GC2. If X Y then gr(X|~ α)≤ gr(Y|~ α)

GC3. infβ Z gr( X|~β) * gr( XZ|~ 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)

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,

1. If X Y then INCONS(X) ≤ INCONS(Y)

2. INCONS(X{~ y}) * INCONS(XY) ≤INCONS(X), for each y in Y

3. There is some k>0 such that for any α, INCONS({α, ~ α})≥k

As in classical case, graded notion of consequence and inconsistency both are equivalent i.e.

1. Given a graded consequence relation |~, graded notion of inconsistency can be obtained by defining INCONS by, INCONS(X)=infαgr(X|~ α)

2. Given INCONS satisfying all graded inconsistency axioms, a graded consequence relation can be defined by

gr(X|~ α)= 1 , if αX

= INCONS(X{~ α}) , otherwise

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 ( C i (X)) = Ci (X)

Additionally let us assume |~ satisfies GC4 i.e. there is a k > 0 such

that for any α, infgr({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.

Proof: We shall prove the theorems in two stages. Stage-1:Let us take an arbitrary α1 such that

infgr({α1 , ~ α1 }) = a kNow for any i L, either i a or i > a or i is non-comparable with a. Case-I Let i a

As infgr({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

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 infgr({α1 , ~ α1 }) = i > aThis is a contradiction to the assumption infgr({α1 , ~ α1 }) = a Ci({α1, α1 }) F

Case-III Let a and i be non-comparable and sup { a , i}= jAs 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)

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 }) < jFor 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]

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 infgr({α1 , ~ α1 }) = a, by case-I and case-II of stage-1 we can

conclude Ci({α1, α1 }) FSubcase (iii) Let k < i < a Since k < i , it is not that for any α, infgr({α , ~ α }) i i.e. there is some α2 such that infgr({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

Case-III Let i be non-comparable with k Then again two subcases arise.Subcase (i) a and i are non-comparableSubcase (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 infgr({α1 , ~ α1 }) = a, by previous

result Ci({α1, α1 }) FSubcase (ii) Let i aSince i is non-comparable with k, there is no such γ such that

infgr({γ , ~ γ }) = i Also it is not that for any α, infgr({α , ~ α }) > i

There is some α3 such that infgr({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 infgr({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.

Graded consequence relation taking object language into consideration

We now take up some examples to illustrate the situation. In these examples the two levels are explicitly shown . Some more insights would also emerge through these exercises.

A specific case

Object language Connectives : , ~ , & Algebraic structure for object language

An MV algebra <L, , ν, *, →o,¬o , 0, 1> i.e. algebra of Łukasiewicz logic where *, →o and ¬o are the operators to compute &, and ~ respectively.

Algebraic structure for meta language

A complete pseudo Boolean algebra <L, →m , , ν, 0,1>

Some logical rules

Given any collection {Ti}iI of fuzzy subsets over formulae, the semantic notion of graded consequence relation allows these following rules to hold.

1. gr(X{α}|~β) ≤ gr(X|~ α β)Corollaries: gr(|~ α α)= 1 gr(|~β) ≤ gr(|~ α β) gr(|~ β (α β)) = 12. gr(X{α}|~γ) ≤ gr(X{α &β }|~γ) 3. gr(X|~ α &β ) ≤ gr(X|~ α)Corollaries:1. gr({α &β }|~ α) = 12. gr(|~ α &β α)= 13. gr({α, β }|~ γ) ≤ gr({α&β }|~γ)4. gr({α & ~ α } |~ β ) = 1 but not necessarily gr({α , ~ α } |~ β ) = 1 5. gr({α & (α β) } |~ β ) = 1 but not necessarily gr({α , (α β) } |~

β ) = 1

<[0, 1], , ν, *, →o,¬o , 0, 1> is considered to be a standard algebra of Łukasiewicz logic with respect to the truth functions,

a*b = max(0, a+b-1) a →o b = 1 ,if a ≤ b = 1-a+b ,otherwise ¬oa = 1-a [0, 1] with respect to the adjoint pair (→m , ) forms a complete

pseudo Boolean algebra, known as Gödel algebra. The operator →m turns out to be

a →m b = 1 , if a ≤ b = b , otherwise So <[0, 1], , ν, *, →o,¬o , 0, 1> and <[0, 1], →m , , ν, , 0, 1> are

the respective standard algebras for object language and meta language of a graded consequence relation, characterized in the previous slide.

So there are several characterization for a graded consequence relation depending upon the structures associated to object language and meta language. In this regard our proposal is as follows:

Level-0 Lukasiewicz Gödel Product Kleene ---------

Level-1 Gödel Lukasiewicz Lukasiewicz Lukasiewicz --------

As discussed earlier a residuated lattice structure

<L, *, →, 0, 1> suffices for the meta theory of a graded consequence relation. Let us see what necessary and sufficient conditions GC4 suggests

Theorem1: If inf β gr({α, ~ α} |~β)>0 then for each

a L such that 0<a<1 there is a z L such that 0<z<1 and

a*z =0.

Theorem2: In a finite lattice if

(i) for each a L such that 0<a<1 there is a z L such that 0<z<1 and a*z =0 and

(ii) for each non-zero, non-unit a, z in L a*z < a z then

inf β gr({α, ~ α} |~β)>0

Model for a graded consequence relation satisfying GC4 : Example 1

Let us consider the following lattice structure L

1 and the composition

a b table for *

c * 0 a b c d 1

0 0 0 0 0 0 0

d a 0 0 d d 0 a

b 0 d 0 d 0 b

0 c 0 d d 0 0 c

d 0 0 0 0 0 d

1 0 a b c d 1

The residum corresponding to the * is given by

→ 0 a b c d 1 Assume <L, *, →, 0,1> to be

0 1 1 1 1 1 1 the algebra for the meta logic of graded

a a 1 1 1 1 1 consequence. Now let us define the truth

b b 1 1 1 1 1 function corresponding to the object

c c 1 1 1 1 1 language connective ~ by the operator ¬,

d 0 a b c d 1 0 a b c d 1

1 0 a b c d 1 ¬ 1 c c a 1 1

Note: (I) ¬ satisfies x ≤ y implies ¬ y ≤ ¬ x

Let T1, T2, T3 be a finite collection of fuzzy subsets over formulae such that, α ~ α β

T1 a c 0

T2 b c d

T3 d 1 a

Then gr({α, ~ α} |~β) = d

Hence 0 < inf β gr({α, ~ α} |~β) ≤ d

Model for a graded consequence relation satisfying GC4 : in a known algebraic structure

Example 2: Meta Level structure: <L, *, →, 0, 1> where L = {0, 1/5, 2/5, 3/5, 4/5, 1} and the table for * and →are considered to be Lukasiewicz t-norm and its corresponding

residum respectively. i.e.

* 0 1/5 2/5 3/5 4/5 1 and → 0 1/5 2/5 3/5 4/5 1 0 0 0 0 0 0 0 0 1 1 1 1 1 11/5 0 0 0 0 0 1/5 1/5 4/5 1 1 1 1 12/5 0 0 0 0 1/5 2/5 2/5 3/5 4/5 1 1 1 13/5 0 0 0 1/5 2/5 3/5 3/5 2/5 3/5 4/5 1 1 14/5 0 0 1/5 2/5 3/5 4/5 4/5 1/5 2/5 3/5 4/5 1 11 0 1/5 2/5 3/5 4/5 1 1 0 1/5 2/5 3/5 4/5 1

Let us compute the object level connective by the largest R-implication operator jLR, defined by jLR(a, b) = b if a =1

= 1 otherwise i.e. jLR 0 1/5 2/5 3/5 4/5 1 0 1 1 1 1 1 11/5 1 1 1 1 1 1 2/5 1 1 1 1 1 1 3/5 1 1 1 1 1 1 4/5 1 1 1 1 1 1 1 0 1/5 2/5 3/5 4/5 1

Note: A R-implication is a fuzzy implication defined by j(a, b) = sup{ x [0,1] / i(a, x) ≤ b} where i is a fuzzy intersection (t-norm).

¬a is computed by jLR(a, 0) i.e. a* ¬a a ¬a

0 0 1 Here ¬ satisfies

1/5 1/5 1 (1) x ≤ y implies ¬ y ≤ ¬ x and

2/5 2/5 1 (2) ¬ ¬ x ≤ x

3/5 3/5 1

4/5 4/5 1

1 0 1

Let T1, T2, T3 be a finite collection of fuzzy subsets over formulae such that, α β ~ α

T1 3/5 2/5 1

T2 4/5 1/5 1

T3 3/5 1/5 1 gr({α , ~ α } |~ β ) = infi{(Ti(α) * Ti(~ α))→Ti(β)} = 3/5

Hence 0 < inf β gr({α , ~ α } |~ β ) ≤ 3/5

Example 3

Let us stick to the same algebraic structure as the previous for meta language and define a truth function for the object level connective ~ by the following table

a* ¬a a ¬a 0 0 1 Here ¬ satisfies 0 1/5 4/5 (1) x ≤ y implies ¬ y ≤ ¬ x and 1/5 2/5 4/5 (2) ¬ ¬ x = x 1/5 3/5 3/5 α ~ α β

1/5 4/5 2/5 T1 3/5 3/5 2/5 With this definition of

0 1 0 T2 4/5 2/5 0 T1, T2, T3,

T3 3/5 3/5 1/5 gr({α , ~ α } |~ β )= 4/5

i.e. 0 < inf β gr({α , ~ α } |~ β ) ≤ 4/5

Important observations

1. If the object level connective ~ satisfies x ≤ y implies ¬ y ≤ ¬ x then it is not necessary that

gr(X { α} |~ β) ≤ gr(X {~ β} |~ ~ α). According to the last example gr({ α} |~ β) = 1 whereas gr({~ β} |~ ~ α) = 2/5

In case of Pavelka the above assumption leads to the result

C(X { α})(β) ≤ C(X {~ β})(~ α) In designing graded consequence relation we would not like to

have gr(X { α} |~ β) ≤ gr(X {~ β} |~ ~ α) and

gr(X |~ ~ ~ α) ≤ gr(X |~ α) together. Because presence of the

above two along with GC1 and GC2 yield infβ gr({α , ~ α } |~ β ) always to be 1.

Interrelation between Meta level and Object level implication determines some logical rules

Let →o and →m be the respective implication functions for object level and meta level of a graded consequence relation.

1. Let →m ≤ →o

(i) gr(X{α}|~β) ≤ gr(X|~ α β)Corollaries: gr(|~ α α)= 1 gr(|~β) ≤ gr(|~ α β) gr(|~ β (α β)) = 1

2. Let →o ≤ →m

(i) gr(X|~ α β) ≤ gr(X{α}|~β) (ii) gr(X|~ α) * gr(Y|~ α β) ≤ gr(XY|~β) (iii) gr(X|~ α β) * gr(Y|~ β γ) ≤ gr(XY {α} |~ γ) but not

necessarily gr(X|~ α β) * gr(Y|~ β γ) ≤ gr(XY |~ α γ)

Features of graded consequence relation: a summary It is local In the sense that the logic emerges out of a context

given by a set of valuations {Ti}iI to the wffs. The set {Ti}iI may be interpreted variedly

(1) as truth values (2) as opinions of experts (3) as beliefs of agents (4) as situations or worlds.

Syntactic part is quite well-defined and it depends upon a context {Ti}iI in the sense that the rules of inferences are weighted ; the weight arises out of the context {Ti}iI. and may be interpreted as the strength of the rule.

The degree of a rule is the extent to which the conclusion of the rule follows from the premises of the rule taking the context {Ti}iI. into consideration.

Hence syntactic derivations are graded; the grade may be interpreted as the strength of derivability.

So semantics i.e. context plays one of the main roles in proposing a logic with a graded notion of consequence.

Soundness completeness issues are well formulated.

Completeness would be a matter of degree.

Interrelation between the respective algebraic structures of object level and meta level plays a considerable role in proposing a logic with graded notion of consequence.

The (truth) value set is not fixed but has some minimum

algebraic structure.

Usually the structures for the object language and meta language are distinct.

The user of the language has freedom of selection of both object level and meta level structures appropriate for his/her purpose.

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