Applied Logics - A Review and Some New Results

Applied Logics - A Review and Some New Results ICLA 2009 Esko Turunen Tampere University of Technology Finland January 10, 2009

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Some historical remarks Modern mathematical logic was born some 150 years ago due to the need of clarifying the basis of mathematical reasoning: to answer questions like what is a well founded theory, what is a mathematical proof, etc. Disregarding maybe electrical circuits, mathematical logic was not much applied outside mathematics. Even today applying logic means for most of us solving mathematical problems by tools constructed by logicians - model theory is a good example. However, the last few decades have dramatically changed the situation. Indeed, there are traffic signal systems and natural reservoir regulation that are based on logical formalism, there are data mining soft wares that use first order logic, etc just to mention examples from my home country. However, the logics we are talking about are not Boolean logic but its generalizations and modifications.

Some historical remarks In the early twentieth century L. E. J. Brouwer represented his intuitionist logic, a more rigid logic than that of North Whitehead s and Bertrand Russell s formalism, today known as classical mathematical logic. A contemporary Jan Lukasiewicz was a pioneer investigator of multi-valued logics; his three-valued propositional calculus was introduced in 1917. However, this logic and his more general infinite valued logic was for decades far from the main stream in logic research. It was only in 1935 when Morchaj Wajsberg showed that infinite valued sentential logic was complete with respect to the axioms conjectured by Lukasiewicz. In 1957 C.C. Chang gave an algebraic proof for the same fact. Lukasiewicz predicate logic does not enjoy the completeness property. However, proved by Scarpellini in 1962, the predicate version of Lukasiewicz infinite valued logic becomes complete after adding an infinitary rule of inference. In 1965 Zadeh introduced Fuzzy Sets and this relieved a research boom in many valued logics.

Pavelka logic MV algebra Semantics Syntax In Lukasiewicz infinite valued propositional logic there are four axioms and Modus Ponens as a rule of inference. Formulae are valuated on the real unit interval [0, 1]. Unlike in classical logic, = α α&α. In 1979 Jan Pavelka extended Lukasiewicz logic by adding truth constants: they generalize the symbols and of classical logic. For each real in [0, 1] there is a truth constant in the formal language F. Unfortunately the language is no more countable (this problem was solved by Hájek who showed that it is enough to have a truth constant for each rational in [0, 1]). Pavelka also introduced a formal fuzzy theory and the concepts partial tautology and partial proof: He also proved that they coincide. Most remarkable is that everything that can be done in Boolean logic can be done in Lukasiewicz-Pavelka graded logic, too.

Pavelka logic MV algebra Semantics Syntax To understand Lukasiewicz Pavelka logic we need some algebra. An MV-algebra L = L,,, 0 is a structure such that L,, 0 is a commutative monoid, i.e., x y = y x, (1) x (y z) = (x y) z, (2) holds for all elements x, y, z L and, moreover, x 0 = x (3) x = x, (4) x 0 = 0, (5) (x y) y = (y x) x. (6)

Pavelka logic MV algebra Semantics Syntax Denote x y = (x y ) and 1 = 0. Then L,, 1 is another commutative monoid and hence x y = y x, (7) x (y z) = (x y) z, (8) x 1 = x (9) holds for all elements x, y, z L. It is obvious that x y = (x y ), thus the triple,, satisfies De Morgan laws. A partial order on the set L is introduced by By setting x y iff x y = 1 iff x y = 0. (10) x y = (x y) y, (11) x y = (x y ) [= (x y) y] (12) for all x, y, z L the structure L,, is a lattice.

Pavelka logic MV algebra Semantics Syntax Moreover, x y = (x y ) holds and therefore the triple,,, too, satisfies De Morgan laws. However, the unary operation called complementation is not a lattice complementation. By stipulating x y = x y (13) the structure L,,,,, 0, 1 is a residuated lattice with the bottom and top elements 0, 1, respectively [Ono s FL ew algebras!]. In particular, a Galois connection x y z iff x y z (14) holds for all x, y, z L. The couple, is an adjoint couple. Lattice operations on L can now be expressed via x y = (x y) y, (15) x y = x (x y). (16)

Pavelka logic MV algebra Semantics Syntax A standard example of an MV algebra is Lukasiewicz structure L: the underlying set is the real unit interval [0, 1] equipped with the usual order and, for each x, y [0, 1], Moreover, x y = min{x + y, 1}, (17) x = 1 x. (18) x y = max{0, x + y 1}, (19) x y = max{x, y}, (20) x y = min{x, y}, (21) x y = min{1, 1 x + y}, (22) x y = max{x y, 0}. (23) Boolean algebras are MV algebras where = and =.

Pavelka logic MV algebra Semantics Syntax Proved by Turunen in 1995, Pavelka s program is realizable in any injective MV algebra. Thus, assume a language F of sentential logic with truth constants is given. Any mapping v : F a L such that v(a) = a for all truth constants a extends into F by v(α imp β) = v(α) v(β) and v(α and β) = v(α) v(β). Such mappings v are called valuations. The degree of tautology is C sem (α) = {v(α) v is a valuation }. Fix a fuzzy set T F of wffs and consider valuations v such that T (β) v(β) for all wffs β. If such a valuation v exists, the T is called satisfiable. We say that T is a fuzzy theory and formulae α such that T (α) 0 are the non logical axioms of the fuzzy theory T. Then we consider values C sem (T )(α) = {v(α) v is a valuation, v satisfies T }.

Pavelka logic MV algebra Semantics Syntax There are eleven logical axioms denoted by a set A. A fuzzy rule of inference is a scheme α 1,, α n, a 1,, a n r syn (α 1,, α n ) r sem (α 1,, α n ), where the wffs α 1,, α n are premises and the wff r syn (α 1,, α n ) is the conclusion. The values a 1,, a n and r sem (α 1,, α n ) L are the corresponding truth values. The mappings r sem : L n L are semi continuous, i.e. r sem (α 1,, j Γ a k j,, α n ) = j Γ r sem (α 1,, a kj,, α n ) holds for all 1 k n. Moreover, fuzzy rules are required to be sound in a sense that r sem (v(α 1 ),, v(α n )) v(r syn (α 1,, α n )) holds for all valuations v.

Pavelka logic MV algebra Semantics Syntax The following are examples of fuzzy rules of inference, denoted by a set R: Generalized Modus Ponens : a Lifting rules : α, α imp β, a, b β a b α, b a imp α a b where a is an inner truth constant. Rule of Bold Conjunction: α, β, a, b α and β a b Proved by Turunen in 1997, any classical rule of inference has a sound counterpart in Pavelka logic!

Pavelka logic MV algebra Semantics Syntax A meta proof w of a wff α in a fuzzy theory T is a finite sequence α 1, a 1.. α m, a m where (i) α m = α, (ii) for each i, 1 i m, α i is a logical axiom, or is a non logical axiom, or there is a fuzzy rule of inference in R and wff formulae α i1,, α in with i 1,, i n < i such that α i = r syn (α i1,, α in ), (iii) for each i, 1 i m, the value a i L is given by a if α i is the axiom a 1 if α a i = i is in A T (α i ) if α i is a non logical axiom r sem (a i1,, a in ) if α i = r syn (α i1,, α in ) The value a m is called the degree of the meta proof w.

Pavelka logic MV algebra Semantics Syntax Since a wff α may have various meta proofs with different degrees, we define the degree of deduction of a formula α to be the supremum of all such values, i.e., C syn (T )(α) = {a m w is a meta proof for α in T }. A fuzzy theory T is consistent if C sem (T )(a) = a for all inner truth values a. Any satisfiable fuzzy theory is consistent. Theorem (Completeness of Pavelka style sentential logic) In consistent fuzzy theories T, C sem (T )(α) = C syn (T )(α), α F. Thus, in Pavelka style fuzzy sentential logic we may talk about tautologies of a degree a and theorems of a degree a for all truth values a L, and these concepts coincide.

Basic ideas A continuous valued extension In 1977 Belnap introduced four possible values associated with a formula α in first order logic. They are true, false, contradictory and unknown: 1. if there is evidence for α and no evidence against α, then α obtains the value true 2. if there is no evidence for α and evidence against α, then α obtains the value false 3. a value contradictory corresponds to a situation where there is simultaneously evidence for α and against α and, finally, 4. α is labeled by value unknown if there is no evidence for α nor evidence against α. More formally, the values are associated with ordered couples 1, 0, 0, 1, 1, 1 and 0, 0, respectively.

Basic ideas A continuous valued extension In 1998, 2007, Perny, Tsoukias and Özturk imposed - being unaware of MV algebras a continuous valued extension of. Given an ordered couple B(α), B( α), graded values are to be computed via t(α) = min{b(α), 1 B( α)}, (24) k(α) = max{b(α) + B( α) 1, 0}, (25) u(α) = max{1 B(α) B( α), 0}, (26) f (α) = min{1 B(α), B( α)}. (27) The intuitive meaning of B(α) and B( α) is the degree of evidence for α and against α, respectively. Moreover, the set of 2 2 matrices of a form [ ] f (α) k(α) u(α) t(α) is denoted by M. However, assuming a Boolean structure in M leads to anomalies.

New results Example We show how Belnap s ideas can be extended to a Pavelka style fuzzy sentential logic. We star by presenting some new algebraic results. Let L = L,,, 0 be an MV algebra. The product set L L can be equipped with an MV structure by setting a, b c, d = a c, b d, (28) a, b = a, b, (29) 0 = 0, 1 (30) for each ordered couple a, b, c, d L L. The order on L L is defined via a, b c, d if and only if a c, d b, (31)

New results Example The lattice operations are defined by and an adjoin couple, by a, b c, d = a c, b d, (32) a, b c, d = a c, b d, (33) a, b c, d = a c, b d, (34) a, b c, d = a c, (d b). (35) Notice that a c = a c and (d b) = (d b) = d b = b d.

New results Example Definition Given an MV-algebra L, denote the structure described via (28) - (35) by L EC and call it the MV algebra of evidence couples induced by L. Moreover, denote {[ a M = ] } b a b a b a b a, b L L and call it the set of evidence matrices induced by evidence couples. Then we have Theorem There is a one to one correspondence between L L and M: if A, B M are two evidence matrices induced by evidence couples a, b and x, y, respectively, then A = B if and only if a = x and b = y.

New results Example Next we observe that the MV structure descends from L EC to M in a natural way: if A, B M are two evidence matrices induced by evidence couples a, b and x, y, respectively, then the evidence couple a x, b y induces an evidence matrix C = [ (a x) (b y) (a x) (b y) (a x) (b y) (a x) (b y) Thus, we may define a binary operation on M by [ a ] [ b a b x ] y x y a b a b x y x y = C. ].

New results Example Similarly, if A M is an evidence matrix induced by an evidence couple a, b, then the evidence couple a, b induces an evidence matrix [ a b A a = b ] a b a. b In particular, the evidence couple 0, 1 induces the following evidence matrix [ 0 F = ] [ ] 1 0 1 1 0 0 1 0 1 =. 0 0 Theorem Let L be an MV algebra. The structure M = M,,, F as defined above is an MV-algebra (called the MV algebra of evidence matrices).

New results Example Our main algebraic result is the following Theorem L is an injective MV algebra if, and only if the corresponding MV algebra of evidence matrices M is an injective MV algebra. A immediate consequence is that, starting from an injective MV algebra L, the corresponding M valued sentential logic is a sound and complete logic in Pavelka sense. Applications of this logic are now intensively studied in decision making theory.

New results Example To illustrate the use of this logic, assume we have the following four non logical axioms and evidence couples [0, 1] [0, 1]: Statement formally evidence (1) If wages rise or prices rise there will be inflation (p or q) imp r 1, 0 (2) In case of inflation, Government will stop it or people will suffer r imp (s or t) 0.9, 0.1 (3) If people will suffer, Government will lose popularity t imp w 0.8, 0.1 (4) Government will not stop inflation and will not lose popularity non s and non w 1, 0

New results Example 1 T is satisfiable and therefore consistent. Indeed, the evidence matrix induced by the following evidence couples satisfies T. Atomic formula Evidence couple p 0.3, 0.8 q 0, 1 r 0.3, 0.8 s 0, 1 t 0.2, 0.9 w 0, 1

New results Example 2 What can be said on logical cause about the claim wages will not rise, formally expressed by non p? The above consideration on evidence couples associates with (non p) an evidence couple 0.3, 0.8 = 0.7, 0.2 and the corresponding valuation v is given by the evidence matrix v(non p) = [ 0.7 0.2 0.7 0.2 0.7 0.2 0.7 0.2 ] = [ 0.2 0 0.1 0.7 and the degree of tautology of (non p) is less than or equal to v(non p). 3 We prove that the degree of tautology of the wff (non p) cannot be less that v(non p), thus it is equal to v(non p). ],

New results Example To this end consider the following meta proof: (1) (p or q) imp r 1, 0 non logical axiom (2) r imp (s or t) 0.9, 0.1 non logical axiom (3) t imp w 0.8, 0.1 non logical axiom (4) non s and non w 1, 0 non logical axiom (5) non w 1, 0 (4), GS2 (6) non s 1, 0 (4), GS1 (7) non t 0.8, 0.1 (5), (3), GMTT (8) non s and non t 0.8, 0.1 (6), (7), RBC (9) non (s or t) 0.8, 0.1 (8), GDeM1 (10) non r 0.7, 0.2 (9), (2), GMTT (11) non (p or q) 0.7, 0.2 (10), (1) GMTT (12) non p and non q 0.7, 0.2 (11), GDeM2 (13) non p 0.7, 0.2 (12), GS1

New results Example 4 By completeness of T we conclude C sem (T )(non p) = C syn (T )(non p) = [ 0.2 0 0.1 0.7 We interpret this result by saying that, from a logical point of view, the claim wages will not rise is (much) more true than false, is not contradictory but lacks some information. Two marginal notes 1 The meta logic of Pavelka logic is Boolean logic. 2 In a philosophical sense, evidence matrices are not really truth-values. ].

Introduction New results Example