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Society: Computer IEEE , ulem Badia Guillermo iz Austria Linz, egtdstructures weighted So h Czech the of 0S s Guillermo ts. rteMarie the er ineFund cience ct). nMath- in turated fthe of and ) char- hose tisfy oa to ms, rst- ary ce 1] to in ar ], e s s . esnn ihgae rdctsi 1] hl tretains it while commutati and [11], literature associativity for in as the framework such in predicates suitable properties, a found important graded as be with proposed can recently reasoning that been has logic and parti fuzzy well-studied the of of most systems contains uninorm-based class residuated This of [25]. ics class the setting, first-order 2). and 1 Chapters Mathematical (e.g. of [8] handbook the Logic consult Fuzzy may informatio one f comprehensive subject, For the notions paper. on related the of necessary development several the and models, first-order eto nstepprwt oecnldn remarks. concluding some with paper the fuzzy Fina ends models. a V saturated for Section paper: theorem elementa existence the the of of and unions chains for results an theorem Tarski-Vaught main satisfy the of should the version element contains an and as each that falsify) types speak that (roughly properties defines variable proves free the IV one and expressing in Section formulas formulas of model. sets of of a pairs sets has of tableau pairs treatm consistent our as for types) (necessary tableaux of of Section notion notions. the model-theoretic introduces basic t some of and extensions algebras, of logic counterpart by uninorm algebraic need fu the mathematical we namely, from logic, preliminaries notions necessary semantical several the recalling presents II Section of means by logics construction. graded elementary first-order existenc an the oth for show to models by is saturated obtained paper present of is the [ of structures [24], goal The [22], saturated Howevermethods. as of [5]. such construction in references the standard found classical T tradition other constructions. classical in ultraproduct the of followed application idea an as co logics built wher fuzzy [15], of be models in in saturated formulated that addressed suggested only been Dellunde but yet known logic, not well fuzzy has is problem mathematical models the such However, [1]). of (cf. construction the conti theory In r model properties. structures expressible of many construction satisfying the elements is, that models, saturated ecos,a h neligpooiinlbssfrthe for basis propositional underlying the as choose, We fuzzy study, our of object the introduce we section this In h ae sognzda olw:atrti introduction, this after follows: as organized is paper The of that is agenda classical the in item important Another nttt fIfrainTer n Automation and Theory Information of Institute UL zc cdm fSciences of Academy Czech rge zc Republic Czech Prague, uz rtodrmdl ae nsuch on based models first-order fuzzy , [email protected] I P II. alsNoguera Carles RELIMINARIES nuous c in ich cular log- vity ing, zzy 28] uld lly, ent his III he or ry er d n e e , of the residuated conjunction, that will be used to obtain the terms and the truth values of the formulas as (where for ◦ results of this paper. stands for any n-ary connective in L): The algebraic semantics of such logics is based on UL-algebras, that is, algebraic structures in the lan- M kxkv = v(x), guage L = {∧, ∨, &, →, 0, 1, ⊥, ⊤} of the form A = M M M A A A A A A A A kF (t1,...,tn)kv = FM(kt1kv ,..., ktnkv ), hA, ∧ , ∨ , & , → , 0 , 1 , ⊥ , ⊤ i such that M M M kP (t1,...,tn)k = PM(kt1k ,..., ktnk ), A A A A v v v • hA, ∧ , ∨ , ⊥ , ⊤ i is a bounded lattice, M A M M A A k◦(ϕ1,...,ϕn)kv = ◦ (kϕ1kv ,..., kϕnkv ), • hA, & , 1 i is a commutative monoid, M M k(∀x)ϕk = inf A {kϕk | m ∈ M}, • for each a,b,c ∈ A, we have: v ≤ v[x→m] M M k(∃x)ϕk = sup≤A {kϕk | m ∈ M}. A A v v[x→m] a & b ≤ c iff b ≤ a → c, (res) If the infimum or supremum does not exist, the corresponding A A A A A A A M M ((a → b) ∧ 1 ) ∨ ((b → a) ∧ 1 )= 1 (lin) value is undefined. We say that is a safe if kϕkv is defined for each P-formula ϕ and each M-evaluation v. A is called a UL-chain if its underlying lattice is linearly Formulas without free variables are called sentences and a set ordered. Standard UL-chains are those define over the real unit of sentences is called a theory. Observe that if ϕ is a sentence, interval [0, 1] with its usual order; in that case the operation then its value does not depend on a particular M-evaluation; A A & is a residuated uninorm, that is, a left-continuous binary we denote its value as kϕkM. If ϕ has free variables among associative commutative monotonic operation with a neutral {x1,...,xn} we will denote it as ϕ(x1,...,xn); then the A element 1 (which need not coincide with the value 1). value of the formula under a certain evaluation v depends only Let Fm denote the set of propositional formulas written on the values given to the free variables; if v(xi) = di ∈ M L M A M in the language of UL-algebras with a denumerable set of we denote kϕkv as kϕ(d1,...,dn)kM. We say that is a variables and let Fm be the absolutely free algebra de- model of a theory T , in symbols M |= T , if it is safe and for L A A fined on such set. Given a UL-algebra A, we say that an each ϕ ∈ T , kϕkM ≥ 1 . A Fm A -evaluation is a homomorphism from L to . The Observe that we allow arbitrary UL-chains and we do not logic of all UL-algebras is defined by establishing, for each focus in any kind of standard completeness properties. Γ ∪{ϕ}⊆ Fm , Γ |= ϕ if and only if, for each UL-algebra Using the semantics just defined, the notion of semantical L A A and each A-evaluation e, we have e(ϕ) ≥ 1 , whenever consequence is lifted from the propositional to the first-order A e(ψ) ≥ 1 for each ψ ∈ Γ. The logic UL is, hence, defined level in the obvious way. Such first-order logics satisfy two as preservation of truth over all UL-algebras, where the notion important properties that we will use in the paper (see e.g. [9]), of truth is given by the set of designated elements, or filter, for each theory T ∪{ϕ,ψ,χ} (inductively defining for each A 0 n+1 n F A = {a ∈ A | a ≥ 1 }. The standard completeness theorem formula α: α = 1, and for each natural n, α = α & α): of UL proves that the logic is also complete with respect to its 1) Local deduction theorem: T, ϕ |= ψ if, and only if, there intended semantics: the class of UL-chains defined over [0, 1] is a natural number n such that T |= (ϕ ∧ 1)n → ψ. by residuated uninorms (the standard UL-chains); this justifies 2) Proof by cases: If T, ϕ |= χ and T, ψ |= χ, then the name of UL (uninorm logic). T, ϕ ∨ ψ |= χ. Most well-known propositional fuzzy logics can be obtained 3) Consequence compactness: If T ϕ, then for some by extending UL with additional axioms and rules (in a finite T0 ⊆ T , T0 ϕ. possibly expanded language). Important examples are G¨odel– Observe that alternatively we could have introduced calculi Dummett logic G and Łukasiewicz logic Ł. and a corresponding notion of deduction ⊢ for these logics, A predicate language P is a triple hP, F, ari, where P is a but we prefer to keep the focus of the paper on the semantics. non-empty set of predicate symbols, F is a set of function III. TABLEAUX symbols, and ar is a function assigning to each symbol a natural number called the arity of the symbol. Let us A tableau is a pair hT,Ui such that T and U are sets of further fix a denumerable set V whose elements are called formulas. A tableau hT0,U0i is called a subtableau of hT,Ui object variables. The sets of P-terms, atomic P-formulas, if T0 ⊆ T and U0 ⊆ U. hT,Ui is satisfied by a model M = and hL, Pi-formulas are defined as in classical logic. A P- hA, Mi, if there is an M-evaluation v such that for each ϕ ∈ M A M A M A M A structure is a pair h , i where is a UL-chain and T , kϕkv ≥ 1 , and for all ψ ∈ U, kψkv < 1 . Also, we M = hM, hPMiP ∈P , hFMiF ∈Fi, where M is a non-empty write hT,Ui |= ϕ meaning that for any model and evaluation n domain; PM is a function M → A, for each n-ary predicate that satisfies hT,Ui, the model and the evaluation must make n symbol P ∈ P; and FM is a function M → M for each ϕ true as well. A tableau hT,Ui is said to be consistent if n-ary function symbol F ∈ F. An M-evaluation of the object there is no finite U0 ⊆ U such that T |= W U0. In the extreme variables is a mapping v: V → M; by v[x→a] we denote the case, we define W ∅ as ⊥. Our choice of terminology here M-evaluation where v[x→a](x) = a and v[x→a](y) = v(y) comes from [4], where such tableaux are introduced for the for each object variable y 6= x. We define the values of the intuitionistic setting, where Boolean negation is also absent. ′ The intuitive idea is that in a semantic tableau as we go along (W Us) ∨ ((∀x)χ(x)), which contradicts the fact that we are we try to make everything on the left true while falsifying considering case (ii). everything on the right. STAGE s +1=3i + 2 : At this stage we make sure that Following [21], we say that a set of sentences T is a ∃- we will eventually obtain an ∃-Henkin theory. If ϕi is not Henkin theory if, whenever T |= (∃x)ϕ(x), there is a constant of the form (∃x)χ(x), then let Ts+1 = Ts and Us+1 = Us. c such that T |= ϕ(c). T is a Henkin theory if T 6|= (∀x)ϕ(x) Otherwise, as in Lemma 2 (2) from [21], we have two cases implies that there is a constant c such that T 6|= ϕ(c). T is to consider: ′ ′ doubly Henkin if it is both ∃-Henkin and Henkin. T is a linear (i) There is a finite Us ⊆ Us such that Ts ∪{ϕi} |= W Us, theory if for any pair of sentences ϕ, ψ either T |= ϕ → ψ or then we define Ts+1 = Ts and Us+1 = Us. T |= ψ → ϕ. (ii) Otherwise, define Ts+1 = Ts ∪{χ(c)} (where c is the The following result shows that each consistent tableau has first unused constant from C) and Us+1 = Us. a model, which will be necessary in the next section. Again, in both cases hTs+1,Us+1i is consistent (check the proof of Lemma 2 (2) from [21]). THEOREM 1. (Model Existence Theorem) Let hT,Ui be a STAGE s +1=3i + 3 : At this stage we work to ensure consistent tableau. Then there is a model that satisfies hT,Ui. that our final theory will be linear. So given the pair hθi, ψii Proof. We will prove this for countable languages, though proceed as in Lemma 2 (3) from [21]. That is, we start from the the generalization to arbitrary cardinals is straightforward. We assumption that hTs,Usi is consistent and letting Us+1 = Us start by adding a countable set C of new constants to the we look to add one of θi → ψi or ψi → θi to Ts to obtain Ts+1 language. We enumerate as ϕ0, ϕ1, ϕ2,... all the formulas of while making the resulting tableau hTs+1,Us+1i consistent. ′ the expanded language, and as hθ0, ψ0i, hθ1, ψ1i, hθ2, ψ2i,... Note that if Ts ∪{θi → ψi} |= W Us+1 and Ts ∪{ψi → ′′ ′ ′′ all pairs of such formulas. We modify the proofs of Theorem θi} |= W Us+1, then Ts ∪{θi → ψi} |= (W Us+1) ∨ (W Us+1) ′ ′′ 4 and Lemma 2 from [21] by building two chains of theories and Ts ∪{ψi → θi} |= (W Us+1) ∨ (W Us+1). Hence, Ts ∪ ′ ′′ T0 ⊆ ··· ⊆ Tn ⊆ ... and U0 ⊆ ··· ⊆ Un ⊆ ... such {(ψi → θi) ∨ (θi → ψi)} |= (W Us+1) ∨ (W Us+1) by proof by cases, and since |= (ψ → θ ) ∨ (θ → ψ ), we obtain that that hSi<ω Ti, Si<ω Uii is a consistent tableau (checking that i i i i ′ ′′ at every stage we obtain a consistent tableau hTi,Uii), plus Ts |= (W Us+1) ∨ (W Us+1), a contradiction. Si<ω Ti is a linear doubly Henkin theory. Then, we will We can already introduce the general notion of type with simply construct the canonical model as in Lemma 3 from respect to a given tableau. [21]. We proceed by induction: DEFINITION 1. A pair of sets of formulas hp,p′i is a type of STAGE 0 : Define T0 = T and U0 = U. ′ STAGE s+1=3i+1 : At this stage, we make sure that our a tableau hT,Ui if the tableau hT ∪ p,U ∪ p i is satisfiable.
final theory will be Henkin. To this end we follow the proof Let Sn(T,U) be the collection of all complete n-types (that of Lemma 2 (1) from [21]. If ϕi is not of the form (∀x)χ(x), is, pairs hp, qi in n-many free variables such that for any φ, then define Ts+1 = Ts and Us+1 = Us. Assume now that either φ ∈ p or φ ∈ q) of the tableau hT,Ui. This is the ϕi = (∀x)χ(x). Then, we consider the following two cases: space of prime filter-ideal pairs of the n-Lindebaum algebra of ′ ′ our logic with the quotient algebra constructed by the relation (i) There is a finite Us ⊆ Us such that Ts |= (W Us) ∨ (∀x)χ(x). Then, we define Ts+1 = Ts ∪{(∀x)χ(x)} and φ ≡ ψ iff hT,Ui φ ↔ ψ. ′ Us+1 = Us. Given formulas σ and θ, we define [hσ, θi] = {hp,p i ∈ ′ (ii) Otherwise, let Ts+1 = Ts and Us+1 = Us∪{χ(c)} (where Sn(T,U) | σ ∈ p,θ ∈ p }. Consider now the collection c is the first unused constant from C up to this stage). B = {[hφ, ψi] | φ, ψ are formulas}. Intuitively, this simply contains all the sets of pairs of theories such that φ is expected We have to check that hT ,U i is consistent in both s+1 s+1 to be true while ψ is expected to fail, for any two formulas cases. Suppose that (i) holds and that T ∪{(∀x)χ(x)} |= W U ′ s s φ, ψ. B is the base for a topology on S (T,U) since given for some finite U ′ ⊆ U . By construction, we must have n s s [hφ, ψi], [hφ′, ψ′i] ∈ B, we have that [hφ ∧ φ′, ψ ∨ ψ′i] ⊆ that T |= (W U ′′) ∨ (∀x)χ(x) for some finite U ′′ ⊆ U . s s s s [hφ, ψi] ∩ [hφ′, ψ′i] and [hφ, ψi] ∩ [hφ ∧ φ′, ψ ∨ ψ′i] ∈ B. Take the finite set U = U ′ ∪ U ′′; clearly we also have s s s Then, there is a topology on S (T,U) such that every open T |= (W U ) ∨ (∀x)χ(x). Now, by the local deduction n s s set of T is just the union of a collection of sets from B. A theorem, T |= ((∀x)χ(x) ∧ 1)¯ n → W U for some n. On s s topological space is said to be strongly S-closed if every family the other hand, T |= W U → W U . Now, recall that s s s of open sets with the finite intersection property has a non- (∀x)χ(x) |= ((∀x)χ(x) ∧ 1)¯ n (this follows from the rules empty intersection [19]. Moreover, we will say that a space is ϕ |= ϕ ∧ 1¯ and ϕ, ψ |= ϕ & ψ). So, by proof by cases, almost strongly S-closed if every family of basic open sets with we have that T ∪ {(W U ) ∨ (∀x)χ(x)} |= W U , which s s s the finite intersection property has a non-empty intersection. means that Ts |= W Us, a contradiction since by induction hypothesis hTs,Usi is consistent. If (ii) holds, suppose that COROLLARY 2. (Tableaux almost strong S-closedness) Let ′ hTs,Us ∪{χ(c)}i is not consistent; then, Ts |= (W Us) ∨ χ(c) hT,Ui be a tableau. If every hT0,U0i, with |T0|, |U0| finite ′ for some finite Us ⊆ Us. Quantifying away the new constant and T0 ⊆ T and U0 ⊆ U, is satisfiable, then hT,Ui is satisfied ′ c, we must have Ts |= (∀x)((W Us) ∨ χ(x)), so Ts |= in some model. Proof. It suffices to show that hT,Ui is consistent. Suppose Let ϕ = (∃x)ψ (the case of ϕ = (∀x)ψ is analogous). A n otherwise, that is, there is a finite U0 ⊆ U such that T |= W U0. Consider kψ(a, x)kM for b ∈ M . Take j > i suffi- ¯ n But then for some finite T0 ⊆ T , T0 |= W U0. Moreover, this ciently large such that b ∈ Mj . By induction hypothesis, A A implies that hT0, {W U0}i cannot be satisfiable, but this is a ψ a,b j ψ a,b . By the elementarity of the chain, k ( )kMj = k ( )kM contradiction with the fact that hT ,U i has a model. A Aj A A 0 0 k(∃x)ψ(a)k i = k(∃x)ψ(a)k . Hence, kψ(a,b)k ≤ Mi Mj M A A IV. MAIN RESULTS x ψ a i . Then x ψ a i is an upper bound for k(∃ ) ( )kMi k(∃ ) ( )kMi Let us start by recalling the notion of (elementary) substruc- A n ture (see e.g. [17]). hA, Mi is a substructure of hB, Ni if the {kψ(a, x)kM | b ∈ M } following conditions are satisfied: in A. Moreover, suppose that u is another such upper bound 1) M ⊆ N. in A. This means that we can find j > i such that u ∈ Aj . 2) For each n-ary function symbol F ∈ F, and elements Then u is an upper bound in Aj of d1,...,dn ∈ M, Aj n {kψ(a, x)kM | b ∈ Mj }, FM(d1,...,dn)= FN(d1,...,dn). j 3) A is a subalgebra of B. which means that 4) For every quantifier-free formula ϕ x ,...,x , and A A ( 1 n) i j Aj k(∃x)ψ(a)kM = k(∃x)ψ(a)kM ≤ u, d1,...,dn ∈ M, i j A B so kϕ(d1,...,dn)kM = kϕ(d1,...,dn)kN . Ai A k(∃x)ψ(a)kM ≤ u. Moreover, hA, Mi is an elementary substructure of hB, Ni i if condition 4 holds for arbitrary formulas. In this case, we also So A A say that B, N is an elementary extension of A, M . When i h i h i k(∃x)ψ(a)kM = k(∃x)ψ(a)kM . instead of subalgebra and subset we have a pair of injections i hg,fi satisfying the corresponding conditions above, we have This establishes as well that the union is this chain of models an embedding. is a safe structure, and hence, a model. A sequence {hAi, Mii | i< γ} of models is called a A structure hA, Mi is said to be exhaustive if every element chain when for all i
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