Strong non-standard completeness for fuzzy logics Tommaso Flaminio Dipartimento di Matematica e Scienze Informatiche Pian dei Matellini 44, 53100 Siena, (Italy) e-mail:
[email protected]
Abstract In this paper we are going to introduce the notion of strong non-standard completeness (SNSC) for fuzzy logics. This notion naturally arises from the well known construction by ultraproduct. Roughly speaking, to say that a logic C is strong non-standard complete means that, for any countable theory Γ over C and any formula ϕ such that Γ 6`C ϕ, there exists an evaluation e of C-formulas into a C-algebra A such that the universe of A is a non-Archimedean extension [0, 1]? of the real unit interval [0, 1], e is a model for Γ, but e(ϕ) < 1. Then we will apply SNSC to prove that various modal fuzzy logics allowing to deal with simple and conditional probability of infinite-valued events are complete with respect to classes of models defined starting from non-standard measures, that is measures taking value in [0, 1]? . Keywords: Non-standard completeness, ultraproduct construction, conditional probability, fuzzy events.
1
Introduction
In H´ ajek approach to fuzzy logics (see [20]), when we fix a (left-)continuous t-norm ∗, we also fix a propositional calculus: ∗ is taken as the truth function of strong conjunction &, its residuum ⇒ becomes the truth function for the implication, the pseudo-complementation ¬x = x ⇒ 0, says us how the negation behaves, and the min (which is definable starting from ∗ iff ∗ is continuous) is taken as truth function for the weak conjunction ∧. Definition 1.1 Given a (left-)continuous t-norm, the propositional calculus P C(∗) has propositional variables p1 , p2 , . . ., connectives &, →, ∧ and the truth constant 0 for 0. Formulas are defined as usual: each propositional variable is a formula, 0 is a formula, and if ϕ, ψ are formulas, then so are ϕ&ψ, ϕ → ψ and ϕ ∧ ψ.
1
Further definable connectives are as follows: ϕ∨ψ ¬ϕ ϕ≡ψ
is is is
((ϕ → ψ) → ψ) ∧ ((ψ → ϕ) → ϕ), ϕ → 0, (ϕ → ψ)&(ψ → ϕ).
Moreover if ∗ is continuous, then ϕ ∧ ψ is ϕ&(ϕ → ψ). The Monoidal T-norm based Logic (MTL) was introduce by Esteva and Godo in [9] to characterize the class of tautologies common to all left-continuous t-norms. If C is either an MTL schematic extension or an MTL expansion (see Section 1.1 for a formal definition of schematic extension or expansion), then to say that C enjoys the standard completeness means that, for each C-formula ϕ, ϕ is theorem of C iff ϕ is a tautology for the class of those C-algebras whose lattice reduct is the real unit interval [0, 1] (or a finite subset of [0, 1]). MTL was proved to be standard complete by Jenei and Montagna in [26]. The standard completeness of MTL means that MTL actually is the logic of left-continuous t-norms, and therefore MTL is the ground fuzzy logic, in fact as Esteva and Godo showed in [9], the sufficient and necessary condition for a t-norm to define a residuum is left-continuity. All the main fuzzy logics which have been introduced in those last years are schematic extensions of MTL. A general way to state the standard completeness theorem is the following: Theorem 1.2 Let Γ ∪ {ϕ} be a C-theory such that Γ 6`C ϕ. Then there exists a standard C-algebra, say [0, 1]C and a [0, 1]-evaluation e such that e(γ) = 1 for each γ ∈ Γ, but e(ϕ) < 1. Depending on Γ we can now state the following different notions of standard completeness for C: • If Γ = ∅, then we say that C satisfies SC (that is C is standard complete). • If Γ 6= ∅, and Γ is finite, then we say that C satisfies FSSC (that is C is finite strong standard complete). • If Γ 6= ∅, and Γ is denumerable, then we say that C satisfies SSC (that is C is strong standard complete). Clearly SSC implies FSSC and FSSC implies SC. In those last years the research on fuzzy logic has lead to various results in the direction of making the relations between SC, FSSC and SSC clear. Especially in [10] the basic requirements as to guarantee a logic to be SC, FSSC and SSC have been introduced. In this paper we will introduce the notion of strong non-standard completeness (SNSC). Roughly speaking we will say that a fuzzy logic C enjoys SNSC if C is complete in a strong sense (i.e. deductions from infinite theories) with respect to algebras whose universe is a non-Archimedean extension [0, 1]? of the real unit interval [0, 1]. Remember that in this direction, a first approach to a non-standard interpretation for a fuzzy logics was provided by the famous Di Nola’s Theorem with respect to MV-algebras (i.e. the algebraic semantic for Lukasiewicz logic): 2
Theorem 1.3 (Di Nola, [8]) Up to isomorphism, every MV-algebra is an algebra of [0, 1]? -valued functions over some set, where [0, 1]? is an ultrapower of the real unit interval. This paper is organized as follows: (1) In the next subsection we will give a slightly more technical introduction for MTL, its algebras and its main schematic extensions. (2) In Section 2, we will introduce some known results about universal algebra. In particular we will introduce the ultraproduct construction, and we will also quote, as an example, how the hyperreal field is built by using this construction. (3) Then in the third section, we will define the notion of strong non-standard completeness and we will show that FSSC implies SNSC. Moreover we will also use the ultraproduct construction to provide a new proof for SSC of some finitevalued logics. (4) In Section 4, we will apply the results of the previous section to provide a partial solution for a problem left in [16, 20] about the (non-)standard completeness for some modal fuzzy logics allowing a treatment of probability of infinite-valued events. (5) Finally, in Section 5, we will end with some remarks and comments about this paper, and we will also discuss some future works.
1.1
A more technical introduction
The language of MTL consists of a denumerable class of propositional variables V = {p1 , p2 , . . .}, the truth constant 0 and the binary connectives &, → and ∧. Further definable connectives are: ϕ∨ψ ¬ϕ ϕ≡ψ
is is is
((ϕ → ψ) → ψ) ∧ ((ψ → ϕ) → ϕ), ϕ → 0, (ϕ → ψ)&(ψ → ϕ).
The following definition provides an Hilbert-style calculus for MTL. Definition 1.4 The following are the axioms of MTL: (A1) (ϕ → ψ) → ((ψ → χ) → (ϕ → χ)) (A2) (ϕ&ψ) → ϕ (A3) (ϕ&ψ) → (ψ&ϕ) (A4) (ϕ ∧ ψ) → ϕ (A6) (ϕ&(ϕ → ψ)) → (ϕ ∧ ψ) (A5) (ϕ ∧ ψ) → (ψ ∧ ϕ) 3
(A7a) (ϕ → (ψ → χ)) → ((ϕ&ψ) → χ) (A7b) ((ϕ&ψ) → χ) → (ϕ → (ψ → χ)) (A8) ((ϕ → ψ) → χ) → (((ψ → ϕ) → χ) → χ) (A9) 0 → ϕ The only inference rule is Modus Ponens: (MP):
ϕ ϕ→ψ ψ
For every set Γ of MTL-formulas and for every MTL-formula ϕ, we say that ϕ follows from Γ in MTL (and we write Γ `M T L ϕ) iff there is a finite sequence ψ1 , . . . , ψn of MTL-formulas such that ψn = ϕ and for i = 1, . . . , n, either ψi is an axiom of MTL, or ψi ∈ Γ, or ψi can be derived by a rule of MTL from a finite number of formulas ψj and j < i. Definition 1.5 (1) A commutative integral bounded residuated lattice is a structure A = hA, u, t, ∗, ⇒, 0, 1i such that: (i) hA, u, t, 0, 1i is a lattice with largest element 1 and least element 0 (w.r.t. the lattice ordering ≤), (ii) hA, ∗, 1i is a commutative monoid where 1 is the identity element, (iii) ∗ and ⇒ form an residuated pair, i.e. for all x, y, z ∈ A, z ≤ (x ⇒ y) iff x ∗ z ≤ y.
(1)
An MTL-algebra is a commutative integral bounded residuated lattice satisfying the pre-linearity property: i.e. for each x, y ∈ A, (iv) (x ⇒ y) t (y ⇒ x) = 1. An MTL-algebra is said linearly ordered iff the lattice order is total. (2) Let A be an MTL-algebra. Then an evaluation of MTL into A (A-evaluation for short) is a map e from MTL-formulas into A such that e(0) = 0 and for each formulas ϕ and ψ: e(ϕ&ψ) = e(ϕ) ∗ e(ψ), e(ϕ → ψ) = e(ϕ) ⇒ e(ψ), e(ϕ ∧ ψ) = e(ϕ) u e(ψ), e(ϕ ∨ ψ) = e(ϕ) t e(ψ). For every formula ϕ of MTL, for every MTL-algebra A and for every evaluation e of MTL into A (or A-evaluation), we say that (A, e) is a model of ϕ (and we write (A, e) |= ϕ) iff e(ϕ) = 1. As usual, if Γ is a set of MTL-formulas, Γ |=M T L ϕ means that every model of all γ ∈ Γ also is a model of ϕ.
4
As shown in [9], the class of MTL-algebras forms a variety. Indeed, (1) can be proved to be equivalent to a set of equations, as done by H´ajek in [20] with respect to the Basic Logic BL. Definition 1.6 (1) An axiom schemata given by a formula Φ(p1 , . . . , pn ) is the set of all formulas Φ(ϕ1 , . . . , ϕn ) resulting by the substitution, for each i = 1, . . . , n, of ϕi for pi in Φ(p1 , . . . , pn ). (2) A logical calculus C 0 is a schematic extension of a logical calculus C if it results form C by adding (finitely or infinitely many) axiom schemata to its axioms. (3) Let C 0 be a schematic extension of C and let A be an C-algebra. Then A is a C 0 -algebra if all axioms of C 0 are A-tautologies. The following are the main schematic extensions of MTL, main references are due to [9, 20, 30]. Definition 1.7 The logic WCMTL is obtained as schematic extension of MTL by (W C) ¬(ϕ&ψ) ∨ ((ψ → ϕ&ψ) → ϕ) The logic IMTL is obtained by adding to MTL the axiom: (IN V ) ¬¬ϕ → ϕ. The logic WNM is obtained by adding to MTL the axiom: (W N M ) ¬(ϕ&ψ) ∨ ((ϕ ∧ ψ) → (ϕ&ψ)). Adding the axiom (INV) to WNM, what we obtain is the logic NM. The logic SMTL is obtained by adding to MTL the following axiom: (ST R) ¬(ϕ ∧ ¬ϕ). The logic ΠMTL is obtained by adding to SMTL the following axiom: (Π1) ¬¬ψ → [((ϕ&ψ) → (χ&ψ)) → (ϕ → χ)]. The logic BL is obtained by adding to MTL the axiom: (DIV ) ϕ&(ϕ → ψ) ≡ ϕ ∧ ψ. If we add to BL the axiom (INV), (STR) we respectively obtain Lukasiewicz logic L and the logic SBL. For each k ∈ N, the k + 1-valued Lukasiewicz logic Lk is obtained by adding to L the two further axioms: (K1) (k − 1)ϕ ≡ kϕ, (K2) (tϕt−1 )k ≡ kϕt for each integer t = 2, . . . , k − 2 that does not divide k − 1
5
where kϕ is an abbreviation for ¬(¬ϕ& . . . &¬ϕ) (k-times) and ϕt is an abbreviation for ϕ& . . . &ϕ (t-times). Product logic Π is obtained by adding to SBL the axiom (Π1), while G¨ odel logic G is obtained by adding to SBL the following further axiom: (CON ) ϕ → ϕ ∧ ϕ. Let C stands for any of the above introduced systems. C-algebras are obviously defined by translating into equations the previous logical axioms. A C-algebra is called standard if its lattice-reduct is the real unit interval [0, 1], or a finite subset of [0, 1]. For instance the lattice reduct of any standard MTL-algebra is h[0, 1], min max, 0, 1i, while the standard Lk -algebra is finite (for each k ∈ N): 1 k−1 Sk = h{0, k1 , . . . , k−1 k , 1}, ⊕, ¬, 0i, where for a, b ∈ {0, k , . . . , k , 1}, a ⊕ b = min{1, a + b}, and ¬a = 1 − a, is the standard Lk -algebra. As shown in [33], MTL as well as its schematic extensions are algebraizable logics in the sense of Block and Pigozzi (cf [2]). This means that each schematic extension of MTL is complete with respect to its class of algebras. In [9], following a technique developed by H´ajek in [20], Esteva and Godo also proved that each schematic extension of MTL is complete with respect to the class of the related linearly ordered structures. As for standard completeness the following shows the relation between the MTL schematic extensions above presented and SC, FSSC, and SSC. Main contributions are due to [7, 9, 10, 20, 22, 23, 30, 26]. Theorem 1.8 (1) Let C stand for either MTL, IMTL, WNM, NM, or G, Lk . Then C enjoys SSC. (2) Let C stand for either ΠMTL, WCMTL, BL, SBL, Π, or L. Then C enjoys FSSC, but C does not enjoy SSC. Other fuzzy logics can be defined by expanding the language of an MTLextension by means of new connectives and/or propositional constants. Following the tradition, if a logic C is obtained by enlarging the language of an MTL-extension C 0 , then we will say that C is an expansion of C 0 . As regards to SC, FSSC and SSC it is clear that if a logic C 0 does not enjoy one of the previous three properties, than each C 0 -expansion C will not enjoy it. The P L0 , LΠ, and LΠ 12 logics are the Lukasiewicz expansions which will turn out to be useful in the following part of this paper. These are so defined. Definition 1.9 ([5, 12, 24]) (1) The language of P L0 is obtained by adding the binary operator &Π to Lukasiewicz language. Formulas are defined as usual. Axioms of P L0 are those of L and the following for &Π : (P 1) (γ&Π ϕ) (γ&Π ψ) ≡ γ&Π (ϕ ψ), (P 2) ϕ&Π (ψ&Π γ) ≡ (ϕ&Π ψ)&Π γ, (P 3) ϕ → ϕ&Π 1, 6
(P 4) ϕ&Π ψ → ψ&Π ϕ, (P 5) ϕ&Π ψ → ϕ. Rules are Modus ponens and (ZD): from ¬(ϕ&Π ϕ), deduce ¬ϕ. (2) The logic LΠ is obtained by joining together L and product logic. Formulas are consequently defined. Axioms are those of L and Π plus the following further axioms: (L∆) ∆(ϕ → ψ) → (ϕ →Π ψ), (Π∆) ∆(ϕ →Π ψ) → (ϕ → ψ), (Dist) ϕ&Π (γ ψ) ≡ (ϕ&Π γ) (ϕ&Π ψ), where ∆ϕ stands for ¬Π ¬ϕ and ϕ ψ stands for ¬(ϕ → ψ). The only deduction rule is modus ponens. (3) The logic LΠ 21 is obtained by adding to the language of LΠ the propositional constant 12 and the axiom (Half )
1 2
≡ ¬ 12 .
It is easy to write the (quasi-)equations which characterize the classes of algebras related to the above introduced logics. Just to quote the prototypical examples the relative standard algebras are so defined: (1) The standard PMV-algebra (also called P L0 -algebra in [24]) is the algebra [0, 1]P M V = h[0, 1], ⊕, ¬, Π , 0i where, for all a, b ∈ [0, 1], a ⊕ b = min{1, a + b}, ¬a = 1 − a, and a Π b = ab (the usual product on [0, 1]). (2) The standard LΠ-algebra is [0, 1]LΠ = h[0, 1], ⊕, ¬, Π , ⇒Π , 0i, where for arbitrary a, b ∈ [0, 1] a ⊕ b, ¬a, and a Π b behave as in [0, 1]P M V . a ⇒Π b = 1 if a ≤ b and a ⇒Π b = b/a otherwise. (3) The standard LΠ 21 -algebra is [0, 1]LΠ = h[0, 1], ⊕, ¬, Π , ⇒Π , 21 , 0i, such that its { 12 }-free reduct is [0, 1]LΠ . From what above stated, and given that L does not enjoy SSC, it follows that neither P L0 , LΠ nor LΠ 21 do enjoy SSC. On the other hand all these logics enjoy SC and FSSC (see [5, 12, 24]). Notation 1 Henceforth C will always denote one of the above introduced logical systems. In other words, when we will refer to a logic C, we will always mean that C is either an MTL schematic extension in the sense of Definition 1.7, or C is any among PL0 , LΠ, or LΠ 12 . 7
2
Background of universal algebra
In this section we are going to introduce some preliminary notions and basic results of universal algebra we will use in the following sections. For a further reading on universal algebra we refer to [3]. First of all remember that, given a set I a filter over I is an F ⊆ P(I) such that: • I ∈ F, • If A, B ∈ F , then A ∩ B ∈ F , • F is upward closed, that is, if A ∈ F and B ⊇ A, then B ∈ F . An ultrafilter over I is a maximal filter over I. Remember by [3] that a family F of sets has the finite intersection property (f.i.p. in symbols) if every finitely many elements of F have non-empty intersection. A simple application of Zorn’s Lemma prove the following: Lemma 2.1 Let X be a non-empty family of subset of a given set I. Then, if X satisfies f.i.p., then there exists an ultrafilter U over I extending X. The following definition will turn out to be useful in the following sections. Definition 2.2 ([3]) Let {Ai | i ∈ I} be a familyQof algebras of a given type, and Q let U be an ultrafilter over I. Define θU on i∈I Ai by: for each a, b ∈ i∈I Ai (a, b) ∈ θU iff {i ∈ I | a(i) = b(i)} ∈ U, where a(i) stands for the i-th projection of a (and b(i) analogously). We define the ultraproduct A? =
Q
i∈I
Ai /U
to be A? =
Q
i∈I
Ai /θU .
When Ai = Aj = A for all i, j ∈ I we will use to call it an ultrapower and we will denote it by A? = AI /U. Q The structure A? = i∈I Ai /U introduced in the previous theorem is obtained by using the, so called, ultraproduct construction. It is useful to quote, as an example, how the ultraproduct construction allows to introduce the ordered hyperreal field R? built up from the ordered real field R = hR, +, ·, ≤, 0, 1i. Example 2.3 (Construction of the hyperreal field) Let R = hR, +, ·, ≤, 0, 1i be the ordered real field, let N be the set of natural numbers and let RN be the set of all functions from N to R, namely 8
RN = {hri i | i ∈ N, ri ∈ R}. If hri i and hsi i are elements of RN , it is possible to define the operations of sum and product as follows: hri i + hsi i = hri + si i hri i · hsi i = hri · si i Let now U be a free ultrafilter over N (i.e. an ultrafilter containing all the co-finite subsets of N , see [4]) and let ≈ be the following relation over RN : for each hri i, hsi i ∈ RN , hri i ≈ hsi i iff {i ∈ N | ri = si } ∈ U. It is straightforward to notice that ≈ is an equivalence over RN . Hence let R? be the quotient RN / ≈. The elements hri iU = r of R? are called hyperreals numbers. Now we can define the operations of sum and product and the order relation over R? as follows: (i) The operations of sum and product are so defined (without danger of confusion we will still denote them by + and · respectively): – hriU + hsiU = hr + siU , – hriU · hsiU = hr · siU . (ii) hriU ≤ hsiU iff {i ∈ N | ri ≤ si } ∈ U. The structure R? = hR? , +, ·, ≤, 0, 1i is an ordered field called the hyperreal field. The elements of R? \ R are usually called non-standard numbers. Just to quote an example the following are non-standard numbers: (i) Ω = h1, 2, 3, . . .iU is a non-standard number greater (in the sense of R? ) than each natural numbers. For this reason it is called an infinite. (ii) ε = h1, 12 , 13 , . . .iU is a non-standard number strictly less (in the sense of R? ) than each n1 (for n ∈ N), but greater than 0. For this reason ε is called infinitesimal. Clearly, the presence of infinitesimal numbers makes R? a non-Archimedean field. In the next section we will use the following lemma which, roughly, says that if we start from a finite family of finite algebras then the ultraproduct construction does not lead to new elements like infinite ones or infinitesimals. Lemma 2.4 ([3]) Let {Ai | i ∈ I} be a finite set of finite algebras, say {B1 , . . . , Bn } (I Q can be infinite), and let U be an ultrafilter over I. Then the ultraproduct i∈I Ai /U is isomorphic to one of the algebras B1 , . . . , Bn . Namely to that Bj such that {i ∈ I | Ai = Bj } ∈ U. 9
3
Strong non-standard completeness
As we have recalled in Section 1.1 there are MTL schematic extensions and Lukasiewicz expansions enjoining FSSC, but not SSC. Therefore there are logics (in the sense of this paper) whose standard completeness does not preserve under logical deductions from infinite (denumerable) theories. As we will see in the next section, infinite theories arise quite often when we deal with modal fuzzy logic, and for this reason, in this section, we will introduce the notion of strong non-standard completeness to overcome this difficulty. In what follows we will say that a C-algebra (C being any logic) A is a non-standard C-algebra if its lattice reduct is a non-Archimedean extension [0, 1]? of the real unit interval [0, 1]. Each non-standard C-algebra is denoted by [0, 1]?C . Definition 3.1 Let C be a logic. Then we say that C enjoys the strong nonstandard completeness (SNSC in symbols) iff for each denumerable C-theory Γ ∪ {ϕ}, such that Γ 6`C ϕ, there exists a non-standard C-algebra [0, 1]?C , and a [0, 1]?C -evaluation e? such that e? (γ) = 1 for each γ ∈ Γ, but e? (ϕ) < 1. Theorem 3.2 Let C be any logic in the sense of Notation 1. If C satisfies FSSC, then C also satisfies SNSC. Proof. Let Γ ∪ {ϕ} be a denumerable C-theory. We can assume, without loss of generality, Γ to be closed under ∧ (it means that, if ρ and ψ belong to Γ, then also ρ ∧ ψ ∈ Γ). If Γ 6`C ϕ, then for each finite Γi ⊂ Γ, one has Γi 6`C ϕ. By hypothesis C satisfies FSSC, therefore for each i ∈ N, there exists a standard C-algebra [0, 1]C,i and a [0, 1]C,i -evaluation ei such that ei (γij ) = 1 for each γij ∈ Γi and ei (ϕ) < 1. Let us now consider the following construction: let X ⊆ P(N) be defined in the following way: For each γ ∈ Γ, put {i ∈ N | ei (γ) = 1} ∈ X. In other words, if for every γ ∈ Γ we call Yγ = {i ∈ N | ei (γ) = 1}, then X = {Yγ | γ ∈ Γ}. Now we need to prove the following: Claim 1 (a) ∅ 6∈ X (b) X satisfies f.i.p. (finite intersection property). Proof. of Claim. (a) If ∅ ∈ X, then it means that there is a formula γ ∈ Γ such that ei (γ) < 1 for each i ∈ N. On the other hand {γ} ⊂f in Γ and therefore there exists a i0 ∈ N such that ei0 (γ) = 1. Thus ∅ 6∈ X. (b) Let Ya , Yb ⊆ X. More precisely let γa , γb ∈ Γ and Yx = {i ∈ N | ei (γx ) = 1} for x = a, b. Then Ya ∩ Yb = {i ∈ N : ei (γa ) = 1 and ei (γb ) = 1} = {i ∈ N : ei (γa ∧ γb ) = 1} ∈ X being Γ closed under ∧. That X satisfies f.i.p. follows by the associative property of ∩ and ∧. The claim is now completely proved. The above Claim, together with Lemma 2.1, ensures that X can be extended to an ultrafilter. Let therefore U be an ultrafilter extending X and let 10
[0, 1]?C =
Q
i∈N [0, 1]C,i /U.
[0, 1]?C
Los Theorem ([3]) ensures to be an C-algebra. Moreover let e? be the ? [0, 1]C -evaluation defined as: for each C-formula ψ, e? (ψ) = he1 (ψ), e2 (ψ), . . .iU . Clearly e? (ψ) = 1 iff {i ∈ I | ei (ψ) = 1} ∈ U. Analogously e? (ψ) < 1 iff {i ∈ I | ei (ψ) < 1} ∈ U. Now we are going to show that e? is a model of Γ, and e? (ϕ) < 1: (1) For each γ ∈ Γ, {i ∈ N | ei (γ) = 1} ∈ X ⊆ U. Therefore e? (γ) = 1. In other words e? is a model of Γ. (2) Recall that e? (ϕ) < 1 iff {i ∈ N | ei (ϕ) < 1} ∈ U. Now {i ∈ N | ei (ϕ) < 1} = N ∈ U, and hence the claim follows. This concludes the proof. Grigolia proved in [19] that Lk enjoys SC. It is easy to see that Lk also enjoys FSSC. In fact if Γ∪{ϕ} is a finite Lk -theory and Γ |=Sk ϕ, then also Γ |=[0,1]M V ϕ because every evaluation of Lukasiewicz language into the standard chain Sk clearly is a [0, 1]M V -evaluation. By the finite strong standard completeness of Lukasiewicz logic, one has Γ `L ϕ and hence Γ `Lk ϕ. Thus also FSSC holds. Now, always using the ultraproduct construction, we are going to provide an easy proof of SSC for Lk . Theorem 3.3 Let Γ ∪ {ϕ} be any arbitrary Lk -theory. If Γ 6`Lk ϕ, then there is an evaluation e of Lk on the standard Lk -algebra Sk being a model of Γ, but such that e(ϕ) < 1. In other words Lk enjoys SSC. Proof. As in the proof of Theorem 3.2 let Γ ∪ {ϕ} be a denumerable theory over Lk such that Γ 6`Lk ϕ. As in the above construction consider all the finite sub-theories Γi of Γ. Once again it is trivial to notice that Γi 6`Lk ϕ. This means that, modulo the FSSC of Lk , for each Γi there exists an evaluation ei on the standard Lk -algebra Sk such that ei is a model of Γi , but ei (ϕ) < 1. As in the proof of the above theorem we can define an ultrafilter U over N (for each i ∈ N, Γi 6`Lk ϕ), and the ultraproduct construction leads to a structure Sk? = SkI /U such that Sk? |= γ for each γ ∈ Γ, but Sk? 6|= ϕ. Now, by Lemma 2.4 we know that Sk? is isomorphic to Sk . In other words there is an Sk -model for Γ which is not a model for ϕ. Thus Lk fulfills SSC. Notice that the main point in the previous proof is Lemma 2.4. Therefore the previous theorem can be further improved as follows: Corollary 3.4 Let C be any logic fulfilling the following properties: (i) The class of standard C-algebras is finite, (ii) Each standard C-algebra is finite. Then C enjoys SSC. 11
4
Application to modal probabilistic fuzzy logics
In [21] H´ ajek, Godo and Esteva introduced the logic F P (L) to deal with simple probability of crisp events in a fuzzy logical setting. The main idea underlying this approach can be summarized as follows: (1) Probabilistic events are equivalence classes (modulo equi-provability) of propositional formulas of classical logics. This means that the logic of events is classical logic. (2) The logic for reasoning about probabilistic sentences is Lukasiewicz logic (or its rational expansion RPL, Rational Pavelka Logic, cf [20, 34]). This because Lukasiewicz logic is equipped with a connective (⊕) behaving as a truncated sum: for each x, y ∈ [0, 1], x ⊕ y = min{1, x + y}. Therefore in such a setting we can express the finite additivity law of a probability by If ϕ, ψ are incompatible, then P (ϕ ∨ ψ) = P (ϕ) ⊕ P (ψ). (3) Extend the language of Lukasiewicz logic by means of a unary modality P standing for probable, and introduce axioms and rules ensuring P to behave as a probability measure. Then, in those last years, many improvements of this approach have been proposed (see for instance [11, 14, 15, 16, 17, 20, 29]). In particular in [16], we introduced a modal fuzzy logic to reason about the probability of fuzzy events. In that paper we defined the logic F P (Lk , L) where the finite-valued Lukasiewicz logic Lk was used as to deal with finite-valued events (i.e. events are considered as equivalence classes of formulas of Lk ), while L was the ground logic allowing a treatment of simple probability. This logic is so defined: Definition 4.1 The language of the logic F P (Lk , L) is built up over a countable set of propositional variables V = {p1 , p2 , . . .}, the truth-constant 0, the connective → of Lukasiewicz logic, and a symbol P for the modality probably. Formulas of F P (Lk , L) split into two classes: (F m(V )) The set F m(V ) of non-modal formulas: these will be formulas of Lk . Non-modal formulas will be denoted by lower case Greek letters ϕ, ψ . . . (M F ) The set M F of modal formulas, built from atomic modal formulas P (ϕ), with ϕ ∈ F m(V ), using the connective → and the truth-constant 0. We shall denote them by upper case Greek letters Φ, Ψ . . . Axioms and rules of F P (Lk , L) are as follows: - Axioms of L for modal and non-modal formulas. - Axioms (K1) and (K2) (see Definition 1.7) restricted to non-modal formulas. The following axiom schemata for the modality P : 12
(F P 1) P (¬ϕ) ≡ ¬P (ϕ), (F P 2) P (ϕ → ψ) → (P (ϕ) → P (ψ)), (F P 3) P (ϕ ⊕ ψ) ≡ [(P (ϕ) → P (ϕ&ψ)) → P (ψ)]. The rule of modus ponens (for modal and non-modal formulas) The rule of necessitation: from ϕ derive P ϕ which only applies to non-modal formulas. The relation of logical consequence in F P (Lk , L) is defined as usual and it is denoted by `F P . For instance, given a modal theory Γ we will write Γ `F P Φ to denote that Φ is provable from Γ in F P (Lk , L). Models are constituted by the class of probabilistic Kripke systems. These are defined as follows: Definition 4.2 A probabilistic Kripke model for F P (Lk , L) is a system M = hW, e, Ii where: - W is a non-empty set whose elements are called nodes, - e : W × V → {0, 1/k, . . . , (k − 1)/k, 1} is such that, for each w ∈ W , e(w, ·) : V → {0, 1/k, . . . , (k − 1)/k, 1} is an evaluation of propositional variables which extends to an Sk -evaluation of (non-modal) formulas of F m(V ) in the usual way. - For each ϕ ∈ F m(V ), define the course of values of ϕ as the function # ϕ# W : W → [0, 1] by putting ϕW (w) = e(w, ϕ). The set of courses of values F mW = {ϕ# W | ϕ ∈ F m(V )} is a clan over W . I is a state over the clan F mW , i.e. I : F mW → [0, 1] satisfies: #
(i) I(1W ) = 1, # (ii) I(¬ϕ# W ) = 1 − I(ϕW ), # # # # # (iii) I(ϕ# W ⊕ ψW ) = I(ϕW ) + I(ψW ) − I(ϕW ψW ).
where ¬, ⊕ are taken as the point-wise extensions of the Lukasiewicz operations in [0, 1]M V , while ϕ# ψ # stands for ¬(¬ϕ# ⊕ ¬ψ # ). Given a probabilistic Kripke model M for F P (Lk , L), a formula Φ and a w ∈ W , the truth value of Φ in M at the node w (kΦkM,w ) is inductively defined as follows: If Φ is a non-modal formula ϕ, then kϕkM,w = e(w, ϕ), # If Φ is an atomic modal formula P (ψ), then kP (ψ)kM = I(ψW ),
13
If Φ is a non-atomic modal formula, then its truth value is computed by evaluating its atomic modal sub-formulas, and then by using the truth functions associated to the L-connectives occurring in Φ. Notice that if Φ is a modal formula, then its truth value in a probabilistic Kripke model is independent from w, for this reason we omitted the subscript w in kP (ψ)kM . The notions of model and validity of a formula in a theory are defined as usual. Notice that, given that the class of models above introduced are based on standard measures, that is measures taking value in the unit interval [0, 1], it make sense to speak about (strong-finite strong) standard completeness also in this case. In [16] we proved F P (Lk , L) to be finite strong standard complete. For what follows it is important to recall the main points of that proof: the standard strategy consists in translating each modal formula of F P (Lk , L) in a propositional formula of Lukasiewicz logic. The translation will be denoted by • and it works as follow: - For each atomic modal formula P (ϕ), introduce a new propositional variable into the language of L, and call it pϕ . - Then inductively define: – (P (ϕ))• = pϕ , – (0)• = 0, – (Φ ◦ Ψ)• = Φ• ◦ Ψ• , for ◦ ∈ {→, &}. Let now Γ be a finite modal theory over F P (Lk , L). In accordance with • , define Γ• = {Ψ• | Ψ ∈ Γ} and F P • = {Θ• | Θ is an instance of (Pi ), i = 1, 2, 3} ∪ {pϕ |`Lk ϕ}. Now the following holds: Lemma 4.3 ([16]) Let Φ be any modal formula of F P (Lk , L) and let Γ be a finite modal theory over F P (Lk , L). Then Γ `F P Φ iff Γ• ∪ F P • `L Φ• . Therefore, the translation • preserves the deductions even by a finite theory, but unfortunately, even if we start with a finite modal theory over F P (Lk , L), the translation • leads to an infinite theory over L, actually F P • is infinite. As we know, infinite theories do not fit well with Lukasiewicz logic because L enjoys FSSC, but it does not enjoy SSC. For this reason we will adopt now the following further translation which allows to reduce F P • to a finite (but equivalent) set of formulas of L. As above assume Γ ∪ {Φ} to be a finite modal theory over F P (Lk , L), and let V0 be the following set of propositional variables: 14
V0 = {vi | vi occurs in some ϕ, P (ϕ) v Ψ, Ψ ∈ Γ ∪ {Φ}}, i.e. V0 is the set of all the propositional variables occurring in all the Lk -formulas occurring in some the modal formula of Γ ∪ {Φ}. Clearly V0 is finite. Let now F m(V0 ) be the set of all Lk -formulas which can be defined starting from V0 . Then notice that the Lindenbaum algebra F m(V0 )/∼k , where ∼k denotes the relation of provable equivalence of k + 1-valued Lukasiewicz logic Lk , is finite (see [6] for more details). This means that there are only finitely many different classes [ϕ]∼k = {ψ ∈ F m(V0 ) |`k ϕ ≡ ψ}. For each [ϕ]∼k we can choose a representative of the class, we will denote it by ϕ . Again notice that there are only finitely many ϕ ’s. Let us now adopt the following further translation: - For each modal formula Φ, let Φ be the formula resulting from the substitution of each propositional variable pϕ occurring in Φ• by pϕ ,
- If Φ = Θ ◦ Λ then Φ = Θ ◦ Λ (with ◦ ∈ {&, →}), and 0 = 0. In accordance with such translation, we define Γ and F P as: Γ = {Ψ | Ψ• ∈ Γ• } and F P = {Υ | Υ is an instance of (Pi ), i = 1, 2, 3} ∪ {pϕ |`Lk ϕ}. In particular notice that {pϕ |`Lk ϕ} = {p1 }, and that Γ is now a finite L-theory. The following holds. Lemma 4.4 ([16]) Γ• ∪ F P • `L Φ• iff Γ ∪ F P `L Φ . Proof. See [16], Lemma 3.1. Now we can prove the completeness theorem. Theorem 4.5 F P (Lk , L) is sound and finite strong complete with respect to the class of its probabilistic Kripke models. Proof. (Sketch) As usual soundness is easy. As to prove completeness, assume Γ ∪ {Φ} to be a finite modal theory over F P (Lk , L) such that Γ 6`F P Φ. From Lemma 4.3 and Lemma 4.4 it follows that Γ 6`F P Φ iff Γ ∪ F P ` 6 L Φ .
15
Now Γ ∪ F P is a finite theory over L, and L enjoys FSSC. This means that there exists a [0, 1]M V -evaluation v which is a model of Γ ∪ F P , but v(Φ ) < 1. Let now M = hW, e, µi be a system where: - W is the set of all Lk -evaluation over all non-modal formulas of F P (Lk , L) (hence not only those of F m(V0 )), - e : W × V is defined as: for each w ∈ W and each p ∈ V , w(p) if p ∈ V0 , e(w, p) = 0 otherwise. Then e is extended to an evaluation of non-modal formulas as usual. - For each ϕ ∈ F m(V ), µ(ϕ) = v(pϕ ). Finally it is easy to see that M is a Kripke model for F P (Lk , L) and it is a model of Γ, but not a model of Φ. This concludes the proof.
4.1
Simple probability
Now we want to extend the logic F P (Lk , L) as to deal with events described by formulas of the whole (infinite-valued) Lukasiewicz logic L. The logic allowing this treatment is denoted by F P (L, L) and it is defined as follows: Definition 4.6 The language of F P (L, L) is the same of F P (Lk , L) logic, therefore non-modal and modal formulas are defined in the same way. Axioms and deduction rules are those of Lukasiewicz logic for both non-modal and modal formulas. Also the rule of necessitation (N ) of Definition 4.1, which acts on non-modal formulas, is added. Notice that, from the point of view of the expressive power, the logic F P (L, L) allows to express and reasoning with sentences expressing the probability of those events which can be modeled with the formulas of the whole Lukasiewicz logic. This means that, recalling the McNaughton Theorem we can now reasoning about the probability of those fuzzy events which can be represented by McNaughton functions. On the other hand in F P (Lk , L) we can only deal with those fuzzy events which are finite-valued approximations of a McNaughton function. Definition 4.7 A non-standard model for F P (L, L) is a system M = h[0, 1]? , W, e, I ? i: where - [0, 1]? is a non-trivial ultrapower of the real unit interval [0, 1].
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- W and e are as in the case of models above described, with the only difference that now, for each fixed node w, the evaluation e(w, ·) take value in the whole real unit interval [0, 1]. - I ? is a non-standard state defined over the clan F mW and taking values into [0, 1]? . If M is a non-standard Kripke model for F P (L, L) and if Φ is a modal formula, then the truth value of Φ in M is defined as in the case of F P (Lk , L) (see Definition 4.2). Notice that each atomic modal formula P (ϕ) may be evaluated by a hyperreal number. Non-modal formulas are evaluated as usual. Now we are going to prove that F P (L, L) is strongly complete with respect to the class of its non-standard Kripke models (that is to say that F P (L, L) enjoys SNSC). The strategy we will use in the proof of the following theorem is the same used in the proof of Theorem 4.5. The fact that the above strategy does not allow to prove standard completeness (that is completeness with respect to standard Kripke models) depends on the fact that, unlike the case of MVk algebras, the variety of MV-algebras is not locally finite. This means that the free MV-algebra over a finite number of generators, is not finite. Thus the translation does not allow to reduce the L-theory F P • to a finite one (actually F P ). On the other hand, the fact that does not reduce F P to a finite theory, allows us to prove strong non-standard completeness, that is even with respect to infinite (countable) modal theories. Theorem 4.8 F P (L, L) is sound and strong complete with respect to the class of its non-standard Kripke models. Proof. Let Γ ∪ {Φ} be an arbitrary (finite or denumerable) modal theory over F P (L, L). Assume Γ 6`F P Φ. Let be defined as in the proof of Theorem 4.5. Then it holds that Γ 6`F P Φ iff Γ ∪ F P 6`L Φ . (2) As said before F P is no more finite, then from (2) and Theorem 3.2, there is a non-standard MV-algebra [0, 1]?M V and a [0, 1]?M V -evaluation v such that v satisfies Γ ∪ F P , but such that v(Φ ) < 1. Then define a non-standard Kripke model M = h[0, 1]? , W, e, I ? i as follows: - [0, 1]? is the lattice reduct of [0, 1]?M V , - W and e are defined as in the proof of Theorem 4.5, the only necessary modification consists in saying that, here, W is the set of all the [0, 1]M V evaluation of Lukasiewicz (non-modal) formulas, while in the above proof W was taken as the set of Sk -evaluations of non-modal formulas. # - For each L formula ϕ, I ? (ϕ# W ) = v(pϕ ), where remember ϕW : W → [0, 1] is defined by putting ϕ# W (w) = e(w, ϕ) = w(ϕ).
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Clearly M is a model for Γ, and kΦkM < 1. In order to prove that M is a nonstandard Kripke model for F P (L, L) we have just to show that the following properties for I ? hold: (1) If L ` ϕ ≡ ψ, then I ? (ϕ# ) = I ? (ψ # ): if L ` ϕ ≡ ψ, then ϕ = ψ and thus the claim follows. (2) If L ` ϕ, then I ? (ϕ# ) = 1: if L ` ϕ, then ϕ ∈ [1]∼L and thus the claim follows by the above property (1). (3) I ? (¬ϕ# ) = 1 − I ? (ϕ# ). This instance of axiom F P 1, p(¬ϕ) ≡ ¬pϕ , is in F P , hence we have I ? (¬ϕ# ) = I ? ((¬ϕ)# ) = v(p(¬ϕ) ) = v(¬pϕ ) = 1 − v(pϕ ) = 1 − I ? (ϕ# ). (4) I ? (ϕ# ⊕ ψ # ) = I ? (ϕ# ) + I ? (ψ # ) − I ? (ϕ# &ψ # ). This instance of axiom (F P 3), p(ϕ⊕ψ) ≡ (pϕ → p(ϕ&ψ) ) → pψ , is in F P , hence I ? (ϕ# ⊕ ψ # ) = I ? ((ϕ ⊕ ψ)# ) = v(p(ϕ⊕ψ) ) = v(pϕ ) + v(pψ ) − v(p(ϕ&ψ) ) = I ? (ϕ# ) + I ? (ψ # ) − I ? (ϕ# &ψ # ). Then M is a probabilistic Kripke model for F P (L, L). This ends the proof of the theorem.
4.2
Conditional probability
While there seems to be an agreement on the notion of state as the proper generalization of probability on MV-algebras, the generalization of the notion of conditional probability on MV-algebras is a matter of discussion. Indeed, in the last years, different answers have been provided to the question “what is a conditional state?” (see [13, 18, 27]). Roughly speaking there have been two main approaches: in the first one, exploited in [13, 18], and similarly to what happens e.g. in classical probability theory under de Finetti’s interpretation, a conditional state is introduced as a primitive notion, that is, as a two-place function s(· | ·) satisfying some basic properties. For instance, Gerla generalizes in [18] a notion of conditional state previously proposed by Di Nola et al. in [13]. The definition is as follows, where B(A) denotes the Boolean skeleton of the MV-algebra A, that is remember, B(A) = {x ∈ A | x ⊕ x = x}, which is the largest sub-Boolean algebra of A. Definition 4.9 ([18]) A conditional state of an MV-algebra A is a function s : A × B → [0, 1], where B ⊆ A is an MV-bunch1 , satisfying the following conditions: (i) s(· | y) is a state on A for every y ∈ B, 1 For any MV-algebra A = hA, ⊕, ¬, 0, 1i, B ⊆ A is an MV-bunch if 1 ∈ B, 0 6∈ B, and B is closed under ⊕. In [18], the following example has been presented: Let A be an MV-algebra and let s be a state on A, then the set B = {x ∈ A | s(x) 6= 0} is an MV-bunch. In this case B is also said to be the MV-bunch of s.
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(ii) s(y | y) = 1 for each y ∈ B ∩ B(A) (iii) s(x y | z) = s(y | z) · s(x | y z) for any x ∈ A, y ∈ B(A), z ∈ B ∩ B(A) such that y z ∈ B, (iv) s(x | y) · s(y | 1) = s(y | x) · s(x | 1) for any x, y ∈ B. Following the above definition we can define the logic F CP (L, LΠ 12 ) to deal with conditional states as follows: Definition 4.10 The language of F CP (L, LΠ 12 ) is obtained by extending the language of LΠ 12 (cf [12]) by means of a binary modality P (· | ·). Formulas of F CP (L, LΠ 21 ) again split into two classes: (i) the set of non-modal formulas F m(V ), which are formulas of L like in F P (L, L); and (ii), letting Sat(V ) = {ϕ ∈ F m(V ) | L 6` ¬ϕ}, the set of modal formulas built from atomic modal formulas P (ϕ | ψ), with ϕ ∈ F m(V ) and ψ ∈ Sat(V ), using the connectives and constants of LΠ 12 (&, →, &Π , →Π , 0, 12 ). Axioms and rules of F P (L, LΠ 12 ) are as follows, where B(V ) = {ϕ |`L ϕ∨¬ϕ}: - axioms and rules of L for non-modal formulas, and axioms and rules of LΠ 12 for modal formulas. - the following axioms for the modality P : (CP1) P (ϕ → χ | ψ) → (P (ϕ | ψ) → P (χ | ψ)), (CP2) P (¬ϕ | ψ) ≡ ¬P (ϕ | ψ), (CP3) P (ϕ ⊕ χ | ψ) ≡ [(P (ϕ | ψ) → P (ϕ&χ | ψ)) → P (χ | ψ)], (CP4) P (ψ | ψ), for each ψ ∈ B(V ) (CP5) P (ϕ&χ | ψ) ≡ P (χ | ψ)&Π P (ϕ | χ&ψ), for each χ, ψ ∈ B(V ) (CP6) P (ϕ | ψ)&Π P (ψ | 1) ≡ P (ψ | ϕ)&Π P (ϕ | 1) - the rule of necessitation: from ϕ derive P (ϕ | ψ) - the rule of substitution of equivalents: from ϕ ≡ ψ derive P (χ | ϕ) ≡ P (χ | ψ), for ϕ, ψ ∈ B(V ). A semantic for F CP (L, LΠ 12 ) is constituted by the class of non-standard conditional Kripke models. Definition 4.11 A non-standard conditional Kripke model is a system h[0, 1]? , W, e, Ci, where [0, 1]? , W and e are defined as in the case of non-standard Kripke models # for F P (L, L), while C : F mW × (F mW \ {0W }) → [0, 1]? is a conditional state as in Definition 4.9, and taking value in a non-trivial ultrapower [0, 1]? of the real unit interval [0, 1].
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Let Ψ be an atomic modal formula of F CP (L, LΠ 12 ) and let M be a nonstandard conditional Kripke model, then the evaluation of Ψ in M is defined as: If Ψ = 12 , then k 12 kM = 12 , # If Ψ = P (ϕ | ψ), then kP (ϕ | ψ)kM = C(ϕ# W | ψW ).
Compound modal formulas and non-modal formulas are evaluated as usual. In particular a compound modal formula Φ is evaluated into M by evaluating all its atomic components and then by using the truth functions associated to the LΠ 21 connectives occurring in Φ. Following the same lines used in the proof of Theorem 4.5 it is not difficult to prove that F CP (L, LΠ 12 ) is complete with respect to the class of non-standard conditional Kripke models. Theorem 4.12 Let Γ ∪ {Φ} be an arbitrary modal theory over F CP (L, LΠ 12 ). If Γ 6`F CP Φ, then there is a non-standard conditional Kripke model M such that kΨkM = 1 for all Ψ ∈ Γ, and kΦkM < 1. Proof. Let • be defined as in the proof of Theorem 4.5, but with the following modification: for any atomic modal formula P (ϕ | ψ) enlarge the language of LΠ 12 by means of a fresh propositional variable pϕ|ψ . Then • (P (ϕ | ψ))• = pϕ|ψ , • ( 12 )• = Let
1 2
and (0)• = 0.
defined as in the case of Theorem 4.5. Again Γ 6`F CP Φ iff Γ ∪ F CP ` 6 L Π 1 Φ . 2
(3)
Now, (3) and the strong non-standard completeness of LΠ 12 make the job. In fact, Γ ∪ F CP 6`LΠ 1 Φ ensures the existence of a non-standard LΠ 21 2
? ? algebra [0, 1]L and a [0, 1]L -evaluation e? being a model for Γ ∪ F CP , Π 12 Π 21 but e? (Φ) < 1. Now let M = h[0, 1]? , W, e, Ci, where [0, 1]? , W, e are defined as in the proof of Theorem 4.5, and let C be defined as: for all ϕ ∈ F m(V ) and ψ ∈ Sat(V ), # ? C(ϕ# W | ψW ) = e (p(ϕ |ψ ) ).
Reasoning as in the above proof it is not difficult to prove that M is a nonstandard conditional Kripke model. In particular notice that C is a conditional state because e? satisfies all the ( -translations of the) axioms in FCP. By induction on the complexity of the modal formula Ψ we are dealing with, it can be proved that kΨkM = e? (Ψ ). Moreover, given that e? is a model for Γ , kΨkM = 1 for each Ψ ∈ Γ, but kΦkM < 1. 20
∗ The other approach has been essentially developed by Kroupa in [27] where the notion of conditional state has been introduced as definable from the notion of simple state on a PMV-algebra. Remember that a PMV-algebra (cf [31]) is an algebra hA, ⊕, Π , ¬, 0i such that: (i) Its Π -free reduct hA, ⊕, ¬, 0i is an MV-algebra, (ii) Π is commutative, associative, has 1 has neutral element, and the following holds for any a, b, c ∈ A: (1) (a Π b) (a Π c) = a Π (b c). (2) If a Π a = 0, then a = 0. Due to the fact that (2) cannot be written by means of equations, the class of PMV-algebras forms a quasi-variety. In [24] Horˇc´ık and Cintula proved P L0 to be complete with respect to the quasi-variety of PMV-algebras. Definition 4.13 ([27]) Let A0 = (A, ⊕, Π , ¬, 0) be a PMV-algebra and let s be a state on its MV-reduct A. Then, a non-negative real number s(x | y) is a conditional state of x given y if s(x | y) is any solution of the equation s(y) · s(x | y) = s(x Π y). It is clear that when s(y) > 0, a conditional state is simply defined as s(x | y) =
s(x Π y) s(y)
for all x ∈ A. It is worth noticing here that if we replace in the above definition Π by the MV-algebra conjunction , then s(· | y) might not be a state any longer. It depends on the fact that does not distributes on ⊕. To cope with this second idea, we will define the logic F P (P L0 , LΠ 21 ) where we used the logic P L0 as logic for events, and we will introduce a unary modality P . In order to overcome the limitation of defining a conditional state s(ϕ | ψ) only in those cases where s(ψ) > 0, we will introduce a special axiom for P ensuring the modality to be interpreted in a faithful state, i.e. a state s such that s(ψ) > 0 for all ψ 6= 0. Definition 4.14 The language of F P (P L0 , LΠ 21 ) is obtained by extending the language of LΠ 12 by means of an unary operator P . Formulas are as follows: − The class F m(V ) of non modal formulas is the class of P L0 formulas (where V denotes as usual the set of propositional variables). Similarly to Definition 4.10, call Sat(V ) = {ϕ ∈ F m(V ) | P L0 6` ¬ϕ}.
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− The class M F of modal formulas contains all the atomic modal formulas like P (ϕ) (ϕ being a non-modal formula), 0, and 12 , M F is taken closed under the connective of LΠ 12 . Axioms and rules are all those of P L0 (cf [24]) restricted to non-modal formulas, all the axioms and rules of LΠ 12 restricted to modal formulas plus the necessitation rule (N ) of Definition 4.1. The modality P is axiomatized by the axioms (F P 1) − (F P 3) of Definition 4.1 plus the following: (F P 4) ∇P (ϕ) for each ϕ ∈ Sat(V ), where ∇Φ stands for ¬∆¬Φ. Definition 4.15 A non-standard Kripke model for F P (P L0 , LΠ 21 ) is a system M = h[0, 1]? , W, e, µi where - [0, 1]? is a non trivial ultrapower of the real unit interval, - W is a non empty set nodes, - e : W × F m(V ) → [0, 1] is such that for each w ∈ W , the map e(w, ·) : F m(V ) → [0, 1] is a [0, 1]P M V -evaluation. - µ : F mW → [0, 1]? is a non-standard faithful state2 over the MV-clan # with product F mW defined as F mW = {ϕ# W : W → [0, 1] | ϕW (w) = e(w, ϕ), ϕ ∈ F m(V )}. Notice that |F mW | = |F m(V )| = ℵ0 . Therefore F mW is a countable and semisimple MV-algebra (see [28], indeed Belluce proved in [1] that semisimple MV-algebra are precisely clans of continuous functions). Now Mundici proved in [32] the following: Proposition 4.16 ([32]) If an MV-algebra A is semisimple and countable, then A has a faithful state. If A is not semisimple, then A has no faithful state. Thus the previous definition works. Using the same strategy used in the proofs of Theorem 4.5 and Theorem 4.12 it not difficult to find a proof for the strong completeness of F P (P L0 , LΠ 12 ) with respect to its class of Kripke models. The only thing to add is a proof of the ? fact that, if e? is a [0, 1]L -model for Γ ∪ F P the non-standard simple state Π1 2
? µ defined as µ(ϕ# W ) = e (pϕ ) is faithful. But it directly follows form the fact ? that e (p(∇ϕ) ) = 1 for any ϕ ∈ Sat(V ) (remember in fact that e? satisfies in particular all the translations of the instance of the axiom (F P 4)). Thus, if ϕ ∈ # ? ? ? Sat(V ), then µ(ϕ# W ) > 0. µ(ϕW ) = e (pϕ ), now e (p(∇ϕ) ) = e (∇pϕ ) = 1.
Thus µ(ϕ# W ) = e(pϕ ) > 0. Thus the following holds: 2 Recall that a state s : A → [0, 1] is said faithfull if for each non-zero element a ∈ A, s(a) > 0 (see [32] for further details).
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Theorem 4.17 The logic F P (P L0 , LΠ 12 ) is sound and strong non-standard complete with respect to the class of its Kripke-style models.
5
Conclusion
In this paper we introduced the notion of strong non-standard completeness (SNSC) and we proved that each logic for which standard completeness does not preserve under deduction from infinite theories, enjoys SNSC. This means that if we want to handle deductions from infinite theories, then we have to use a domain of interpretation different from the standard real unit interval, actually its non-Archimedean extension [0, 1]? . Then we applied this result to study the completeness of those modal fuzzy logics to deal with simple and conditional probability of fuzzy (infinite-valued) events. In particular we showed that, if a fuzzy probabilistic logic allows to reason about probabilistic sentences on infinite-valued events, then a class of models based on non-standard measures, are a complete semantic of it. Fortunately we did not succeed (until now) in proving that standard measures are not a complete semantic for those fuzzy probabilistic logic. Thus the problem of showing those logics to be standard complete can be reasonably considered open. From a purely algebraic point of view we would like to use the strong nonstandard completeness to approach to first order fuzzy logics. In fact, as we know, there are first order fuzzy logics which are not complete when we use [0, 1] as domain of interpretation (as for instance first order Lukasiewicz logic, L∀). Our research in this direction will be in providing results linking the SNSC for a propositional logic C and the non-standard completeness for its first order extension.
Acknowledgments The author feels deeply indebted with Carles Noguera for the suggestion of the name strong non-standard completeness, and to the anonymous referee whose remarks really improved the final version of this paper.
References [1] Belluce L. P., Semisimple MV-algebras of infinite-valued logic and bold fuzzy set theory. Canad. J. Math. 38 (6), 1356–1379, (1986) [2] Blok W. J., Pigozzi D., Algebraizable logics. Mem. Amer. Math. Soc. 396, vol 77, (1989). [3] Burrus S., Sankappanavar H. P., A course in Universal Algebra, Springer-Velag, 1980.
23
[4] Chang C.C., Keisler H.J., Model Theory. North-Holland Publishing Company, Amsterdam-New York-Oxford, (1973). [5] Cintula P., Advances in the LΠ and LΠ 12 logics, Archive for Mathematical Logic, 42, 449–468, (2003). [6] Cignoli R., D’Ottaviano I.M.L., Mundici D., Algebraic Foundations of Many-valued Reasoning. Kluwer, 2000. [7] Cignoli R., Esteva F., Godo L., Torrens A., Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Computing, 4, 106–112, (2000). [8] Di Nola A., MV-algebras in the treatment of uncertainty. In Fuzzy Logic, R. Lowen and M. Rubens (eds.), 123–131, Kluwer Academic Press, Dordrecht, 1993. [9] Esteva F., Godo L., Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems, 124, 271–288, (2001). [10] Esteva F., Gispert J., Godo L., Montagna F., On the Standard and Rational Completeness for some Axiomatic Extension of the Monoidal Tnorm Logic. Studia Logica, 71, 199–226, (2002). ´jek P., Reasoning about probability using fuzzy [11] Esteva F., Godo L., Ha logic. Neural Network World, 10, No. 5, 811–824 (2000). [12] Esteva F., Godo L., Montagna F., LΠ and LΠ 21 : two complete fuzzy systems joining Lukasiewicz and Product logics, Archive for Mathematical Logic, 40, 39–67, (2001). [13] Di Nola A., Georgescu G., Lettieri A., Conditional states in finitevalued logic. In E.P. Klement, D. Dubois, H. Prade, editor, Fuzzy Sets, logics, and reasoning about knowledge. 161–174. Kluwer, (1999). [14] Flaminio T., A Zero-Layer Based Fuzzy Probabilistic Logic for Conditional Probability. Lecture Notes in Artificial Intelligence, 3571: 8th European Conference on Symbolic and Quantitaive Approaches on Reasoning under Uncertainty ECSQARU’05, Barcelona, Spain, July 2005. Llu´ıs Godo (Ed.), 714–725. [15] Flaminio T., NP-Containment for the Coherence Tests of Assessment of Conditional Probability: a Fuzzy-Logical Approach, Archive for Mathematical Logic, to appear. [16] Flaminio T., Godo, L, A logic for reasoning about the probability of fuzzy events. Fuzzy Sets and Systems. In print, (2006). [17] Flaminio T., Montagna F., A Logical and Algebraic Treatment of Conditional Probability. Archive for Mathematical Logic, 44, 245–262 (2005). 24
[18] Gerla B., Many-valed Logics of Continuous t-norms and Their Functional Representation. Ph.D. Thesis, University of Milan, (2001). [19] Grigolia R., Algebraic analysis of Lukasiewicz-Tarski n-valued logical systems. In Selected Papers on Lukasiewicz Sentencial Calculi, R. W´ ojcicki and G. Malinowski (Eds), Wroclaw: Polish Academy of Science, Ossolineum. pp 81–91 (1977). ´jek P., Metamathematics fo Fuzzy Logic, Kluwer, 1998. [20] Ha ´jek P., Godo L., Esteva F., Probability and Fuzzy Logic. In Proc. of [21] Ha Uncertainty in Artificial Intelligence UAI’95, (Besnard and Hanks, Eds.) Morgan Kaufmann. San Francisco, 237–244 (1995). ˇ´ık R., Standard Completeness Theorem for ΠMTL Logic. Archive [22] Horc for Mathematical Logic, 44, 413–424. (2005). ˇ´ık R., On the Failure of Standard Completeness in ΠMTL for In[23] Horc finite Theories. Fuzzy Sets and Systems. In print, (2007). ˇ´ık R., Cintula P., Product Lukasiewicz Logic. Archive for Math[24] Horc ematical Logic, 43, 477–503, (2004). [25] Hurd A, E., Loeb P. A., An introduction to Nonstandard Real Analysis, Academic Press, Orlando, 1985. [26] Jenei S. and Montagna F. A proof of standard completeness for Esteva and Godo’s logic MTL. Studia Logica, 70, 183–192, 2002. [27] Kroupa T., Conditional probability on MV-algebras, Fuzzy Sets and Systems, 149, Issue 2, 369–381, (2005). [28] Kroupa T., Every state on semisimple MV-algebra is integral. Fuzzy Sets and Systems, 157, pp. 2771–2782. (2006). [29] Marchioni E., Godo L., A logic for reasoning about coherent conditional probability: a fuzzy modal logic approach. Lecture Notes in Artificial Intelligence, 3229: 9th European Conference on Logic in Artificial Intelligence JELIA’04. Lisbon, Portugal, September 2004. Jo´e J´ ulio Alferes and Joao Leite (Eds). 213–225. ˇ´ık R., On weakly cancellative fuzzy [30] Montagna F., Noguera C., Horc logics. Journal of Logic and Computation, 16, 423–450, 2006. [31] Montagna F., Panti G., Adding structures to MV-algebras, Journal of Pure and Applied Algebra, 164 (3), 365387, (2001). [32] Mundici D., Averaging the truth-value in Lukasiewicz logic. Studia Logica 55 (1), 113–127, (1995).
25
[33] Noguera C., Algebraic study of axiomatic extensions of triangular norm based fuzzy logics. Ph.D. Thesis, Barcelona, (Spain), 2006. [34] Pavelka J., On fuzzy logic I, II, III, Zeitschr. f. Math. Logik und Grundl. der Math, 25, 45–52, 119–134, 447–464, (1979). [35] Robinson A., Non-Standard Analysis, North-Holland, Amsterdam, (1996).
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