p-CONSEQUENCE VERSUS q-CONSEQUENCE OPERATIONS

Bulletin of the Section of Logic Volume 33/4 (2004), pp. 197–207

Szymon Frankowski

p-CONSEQUENCE VERSUS q-CONSEQUENCE OPERATIONS

Abstract Among different generalisations of the ordinary propositional consequence operation at last the following two are a special interest: the so-called quasi-consequence (q-consequence) and plausible consequence (p-consequence) operations, cf.[2],[3] and [1], respectively. The paper shows the special kind of symetry between two classes of operations and takes advantage of syntactic apparatus employed for p-consequences to characterize q-consequences.

1 Let L = (L, f1 , . . . , fn ) be a propositional language with the connectives f1 , . . . , fn , freely generated by the set of propositional variables V ar = {p1 , p2 . . .}. By a p − consequence operation on the language L (cf.[1]) we mean any function Z : 2L −→ 2L satisfying the following conditions (for every sets of formulas X, Y ⊆ L): (i) X ⊆ Z(X); (ii) Z(X) ⊆ Z(Y ) whenever X ⊆ Y . Analogously as in the case of the ordinary consequence operation one may → − → also consider the the conditions of structurality − e Z(X) ⊆ Z( S e X) for every substitution e of the language L, and finitarity: Z(X) = {Z(Xf ) : Xf is finite subset of X}. Let us recall (cf. [2]) that a q − consequence on the language L is a map W : 2L −→ 2L such that W is monotonic (that the conditions (ii) rewritten for W holds) and for every X, Y ⊆ L, W (X ∪ W (X)) = W (X).

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Corollary 1. N : 2L −→ 2L is a consequence operation iff it is both p- and q-consequence operation. 2 Corollary 2. For every monotonic operation N : 2L −→ 2L : (i) N is p-consequence iff ∀X,Y ⊆L (N (X ∪ Y ) = N (X) ⇒ Y ⊆ N (X)); (ii) N is q-consequence iff ∀X,Y ⊆L (Y ⊆ N (X) ⇒ N (X ∪ Y ) = N (X)); (iii) N is consequence iff ∀X,Y ⊆L (N (X ∪ Y ) = N (X) if f Y ⊆ N (X)). 2 Now let us introduce some notions useful in syntactic description of p-consequence (see [1]). Definition 1. By p −S inf erence for the language L we shall understand any element of the set n∈N (L × {∗, 1})n . The intended meaning of the signs *,1 is as follows: for any α ∈ L, when the couple hα, 1i occurs in any place in a p-inference, the formula α is justified with maximal degree of certainty, while the case of the couple hα, ∗i occuring in a p-inference, α is not rejected. In p-inference (hαi , xi ini=1 ) (where (hαi , xi ini=1 ) means (hα1 , x1 i, . . . , hαn , xn i)), (n = 1, 2 . . .), for every i such that 1 ≤ i < n, αi is conceived as a premise and αn is the conclusion. Any non empty set of p-inferences will be said to be a p − rule for L. For example, the set {(hα, 1i, hβ, ∗i, hα ∧ β, ∗i) : α, β ∈ L∧ } is a p-rule for the language L∧ = (L∧ , ∧, . . .). By a p-proof of a formula α from a set X ⊆ L based on a given set R of p-rules we mean the following p-inference (a1 , . . . , ak ): (i) pr1 (ak ) = α; (ii) for all i = 1, 2, . . . , k either pr1 (ai ) ∈ X and pr2 (ai ) = 1 or there is a p-rule r ∈ R and a p-inference (b1 , . . . , bj , ai ) ∈ r such that {b1 , . . . , bj } ⊆ {a1 , . . . , ai−1 }. (pr1 , pr2 are the first and the second projections on the product L × {∗, 1}, respectively). We also say that a formula α is p − derivable from a set of formulae X by p-rules from R (X k–R α in symbols) iff there exists a p-proof of α from the set X based on R. These syntactic notions play the similar role as the ordinary concepts of rules of inference, proof and derivability relation in the theory of consequence relation. For example, we can show that any finitary p-consequence

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operation is determined by a set of p-rules, i.e. it coincides with the derivability relation k–R determined by a set of p-rules R. In order to formulate this result more precisely, for any fixed p-inference (a1 , . . . , an ) and i ∈ {1, . . . , n} we put A1 (i) = {pr1 (aj ) : 1 ≤ j ≤ i & pr2 (aj ) = 1} and A∗ (i) = {pr1 (aj ) : 1 ≤ j ≤ i & pr2 (aj ) = ∗}. Now, given a p-consequence Z, let R(Z) be the set of p-rules, defined as follows: r ∈ R(Z) iff for every X ⊆ L and any p-inference (a1 , . . . , an ) ∈ r, if A∗ (n − 1) ⊆ Z(X) & Z(X ∪ A1 (n − 1)) = Z(X) then pr1 (an ) ∈ Z(X) whenever pr2 (an ) = ∗ and Z(X, pr1 (an )) = Z(X) whenever pr2 (an ) = 1. Theorem 1. (cf.[1]) For every finitary p-consequence operation Z and X ⊆ L, α ∈ L : α ∈ Z(X) if f X k–R(Z) α. 2 In [2] and [3] Malinowski introduced a notion of q-proof, used to syntactic characterization of q-consequence operation. A sequence of formulae α1 , . . . , αn is said to be a q-proof of a formula α from a set of formulae X on the basis of a set of (ordinary) rules of inference R iff αn is identical with α and for each i ∈ {1, . . . , n} there exist r ∈ R and a set Y ⊆ X ∪ {α1 , . . . , αi−1 } such that hY, αi i ∈ r. (Notice, that a premise from X need not to be treated as a possible conclusion). Now we would like to provide some syntactic p-consequence methods to characterize qconsequence operation.

2 Let ≤ be the partial ordering on the set {∗, 1} defined: ∗ < 1. Given a non-empty set of p-rules of inference R we define three notions of q-proof in terms of p-rules. (1) A p-inference (a1 , . . . , an ) is said to be a q1 − proof of a formula α from a set of formulae X on the basis of R iff pr1 (an ) = α and pr2 (an ) = 1, and for every i ∈ {1, . . . , n} either (a) pr1 (ai ) ∈ X and pr2 (ai ) = ∗ or (b) there exist a set {b1 , . . . , bm } ⊆ {a1 , . . . , ai−1 } and a p-rule r ∈ R such that (c1 , . . . , cm , c) ∈ r, where the p-inference (c1 , . . . , cm ) differs from (b1 , . . . , bm ) at most by that pr2 (cj ) ≤ pr2 (bj ) for each j ∈ {1, . . . , m}

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(thus pr1 (cj ) = pr1 (bj ) for every j ∈ {1, . . . , m}), pr1 (ai ) = pr1 (c) and pr2 (ai ) ≤ pr2 (c). For example, the sequence (hp → p, ∗i (by r→ ), hq → q, 1i (by r→ ), h(p → p) ∧ (q → q), 1i (by r∧ ) is a q1 -proof of a formula (p → p) ∧ (q → q) from the empty set of formulae on the basis R = {r→ , r∧ }, where the prules r→ , r∧ for the language (L, →, ∧) are as follows: r→ = {(hα → α, 1i) : α ∈ L}, r∧ = {(hα, ∗i, hβ, ∗i, hα ∧ β, 1i) : α, β ∈ L}. (2) We say that a p-inference (a1 , . . . , an ) is a q2 −proof of a formula α from a set of formulae X on the basis of R iff pr1 (an ) = α and pr2 (an ) = 1, and for every i ∈ {1, . . . , n} either (a) pr1 (ai ) ∈ X and pr2 (ai ) = ∗ or (b) there exist a set {b1 , . . . , bm } ⊆ {a1 , . . . , ai−1 } and r ∈ R such that (b1 , . . . , bm , c) ∈ r, where the pair c is just as in definition (1) (that is, pr1 (ai ) = pr1 (c) and pr2 (ai ) ≤ pr2 (c)), or (c) hpr1 (ai ), 1i ∈ {a1 , . . . , ai−1 }. Notice that the p-inference (hp → p, ∗i (by r→ ), hq → q, 1i (by r→ ), hq → q, ∗i (by (c)), h(p → p) ∧ (q → q), 1i (by r∧ ) is a q2 -proof. (3) A p-inference (a1 , . . . , an ) is said to be a q3 − proof of a formula α from a set of formulae X on the basis of R iff pr1 (an ) = α and pr2 (an ) = 1, and for every i ∈ {1, . . . , n} either (a) pr1 (ai ) ∈ X and pr2 (ai ) = ∗ or (b) there exist a set {b1 , . . . , bm } ⊆ {a1 , . . . , ai−1 } and a p-rule r ∈ R such that (c1 , . . . , cm , ai ) ∈ r, where the p-inference (c1 , . . . , cm ) is the same as in definition (1), i.e., for each j ∈ {1, . . . , m} pr1 (cj ) = pr1 (bj ) and pr2 (cj ) ≤ pr2 (bj ). An example is as follows: the sequence (hp → p, 1i (by r→ ), hq → q, 1i (by r→ ), h(p → p) ∧ (q → q), 1i (by r∧ ) is a q3 -proof. Now for i = 1, 2, 3 we say that a formula α is qi − derivable from a set of formulae X on a basis of a set of p-rules R (X k=iR α in symbols) iff there exists a qi -proof of α from X on the basis R. Lemma 1. The three notions of q-derivability coincide, i.e., for any set of formulas X, a formula α and any non-empty set of p-rules R the following conditions are equivalent:


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(i) X k=1R α; (ii) X k=2R α; (iii) X k=3R α. Proof: (i) ⇒ (ii): Any q1 -proof (a1 , . . . , an ) may be changed into a q2 -proof in the following way: first, consider all ai such that pr2 (ai ) = 1 and take the p-rule which was applied to the ai in such a way that the hpr1 (ai ), ∗i is a premise of a p- inference belonging to that p-rule. Secondly, in each place of occurrence of such the pairs ai put in the q1 -proof the pair hpr1 (ai ), ∗i instead of ai . The resulting sequence is just a q2 -proof. (ii) ⇒ (i): Any q2 -proof (a1 , . . . , an ) becomes a q1 -proof by omitting every ai in the q2 -proof obtained by application of (c) from the definition of a q2 -proof. (i) ⇒ (iii): In order to get a q3 -proof from a given q1 -proof (a1 , . . . , an ) first consider only these ai introduced by application of a p-rule with a pinference such that pr2 (ai ) = ∗ whose conclusion is of the form: hpr1 (ai ), 1i (notice that i 6= n). Next change all such elements in the q1 -proof into hpr1 (ai ), 1i. Then a q3 -proof arises. (iii) ⇒ (i): Obviously, any q3 -proof is a q1 -proof. 2 Following Lemma 1, the only symbol k= will be used for the relation of q-derivability, unless it is not necessary or convenient otherwise. Hereafter any qi -proof, i = 1,2,3, will be called now a q-proof. Corollary 3. For every non-empty set R of p-rules a function WR : 2L −→ 2L defined for any X ⊆ L and α ∈ L by the following expression: α ∈ WR (X) iff X k=R α, is a finitary q-consequence operation. Proof. It is obvious that WR is a monotonic and finitary. In order to show the inclusion WR (X ∪ WR (X)) ⊆ WR (X) notice that any q-proof of α from the set X ∪ WR (X) based on R may be changed into a q-proof of α from the set X. To this aims: consider first all the occurrences of the pairs hβ, ∗i, where β ∈ WR (X), introduced to the q-proof of α from the set X ∪ WR (X) due to (a) of the definition of q-proof. Next, before every occurrence of hβ, ∗i put a q-proof of β from the set X. Now, after each last element hβ, 1i of a q-proof, the pair hβ, ∗i occurs due to (c). Just a q-proof of α from X based on R arised. 2 Now we are about to formulate and to prove a counterpart of Theorem 1 for q-consequences. Still, however so we need a notion of characteristic

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set Rq (W ), of p-rules of inference of W . Rq (W ) plays a similar role for a q-consequence W as the set R(Z) had for a p-consequence Z. So, for a given W on the language L = (L, f1 , . . . , fn ), define the set Rq (W ) as follows: r ∈ Rq (W ) iff for every X ⊆ L and any p-inference (a1 , . . . , an ) ∈ r : (A1 (n − 1) ⊆ W (X) & W (X ∪ A∗ (n − 1)) = W (X) implies that A1 (n) ⊆ W (X) & W (X ∪ A∗ (n)) = W (X)). Theorem 2. For every q-consequence operation W, a formula α and a set of formulae X: α ∈ W (X) whenever X k=Rq (W ) α. Moreover, if W is finitary, then α ∈ W (X) iff X k=Rq (W ) α . Proof: Assume that X k=Rq (W ) α. Consider any q3 -proof (a1 , . . . , an−1 , an ) (where an = hα, 1i) of α from X on the basis Rq (W ). By induction we prove that for every i ∈ {1, . . . , n} : A1 (i) ⊆ W (X) and W (X ∪ A∗ (i)) = W (X). When i = 1, pr1 (a1 ) ∈ X and S pr2 (a1 ) = ∗, or a1 was introduced to the q-proof by a p-rule that is (a1 ) ∈ Rq (W ). In the first case, A1 (1) = ∅ and A∗ (i) ⊆ X, so A1 (1) ⊆ W (X) and W (X ∪ A∗ (1)) = W (X). In the second case, since the set of premises in the p-inference (a1 ) is empty, so from the definition of Rq (W ) (applied for r = {(a1 )}) it follows directly that (A1 (1) ⊆ W (X)) & W (X ∪ A∗ (1)) = W (X). Now assume that A1 (i) ⊆ W (X) and W (X ∪A∗ (i)) = W (X) for some i ∈ {1, . . . , n−1}. We show that A1 (i+1) ⊆ W (X) and W (X ∪A∗ (i+1)) = W (X). When pr1 (ai+1 ) ∈ X and pr2 (ai+1 ) = ∗, we have: A1 (i+1) = A1 (i) and X∪A∗ (i+1) = X∪A∗ (i). Accordingly due to the inductive assumption, the condition A1 (i+1) ⊆ W (X) and W (X ∪A∗ (i+1)) = W (X) holds true. It remains to consider the case when ai+1 is introduced to the q-proof by a p-rule r ∈ Rq (W ): then there is a p-inference (c1 , . . . , cm , ai+1 ) ∈ r such that (1) ∀j∈{1,...m} (pr1 (cj ) = pr1 (bj ) and pr2 (cj ) ≤ pr2 (bj )) for some pairs b1 , . . . , bm such that (2) {b1 , . . . , bm } ⊆ {a1 , . . . , ai }. For the p-inference (c1 , . . . , cm ) let distinguish the sets: C1 (m), C∗ (m) in a similar way as the sets A1 (i), A∗ (i) were considered for the p-inference


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(a1 , . . . , ai ) : C1 (m) = {pr1 (cj ) : 1 ≤ j ≤ m & pr2 (cj ) = 1} and C∗ (m) = {pr1 (cj ) : 1 ≤ j ≤ m & pr2 (cj ) = ∗}. Then from the definition of Rq (W ) and the fact that (c1 , . . . , cm , ai+1 ) ∈ r ∈ Rq (W ) we obtain: ∀Y ⊆L [C1 (m) ⊆ W (Y ) & W (Y ∪C∗ (m)) = W (Y ) implies that (pr2 (an+1 ) = 1 ⇒ C1 (m) ∪ {pr1 (an+1 )} ⊆ W (Y )) & (pr2 (an+1 ) = ∗ ⇒ W (Y ∪ C∗ (m) ∪ {pr1 (an+1 )}) = W (Y ))]. In the sequel we will apply this condition for the set X ∪ A∗ (i), that is we have: (3) (C1 (m) ⊆ W (X ∪ A∗ (i)) & W (X ∪ A∗ (i) ∪ C∗ (m)) = W (X ∪ A∗ (i)) implies that (pr2 (an+1 ) = 1 ⇒ C1 (m) ∪ {pr1 (an+1 )} ⊆ W (X ∪ A∗ (i))) & (pr2 (an+1 ) = ∗ ⇒ W (X ∪ A∗ (i) ∪ C∗ (m) ∪ {pr1 (an+1 )}) = W (X ∪ A∗ (i)))]. Now from (1) and (2) it follows immediately that C1 (m) ⊆ A1 (i), so by the inductive assumption: C1 (m) ⊆ W (X), therefore (4) C1 (m) ⊆ W (X ∪ A∗ (i)). Now we show that (5) W (X ∪ A∗ (i) ∪ C∗ (m)) = W (X ∪ A∗ (i)). From (1) and (2) it follows that C∗ (m) ⊆ A∗ (i) ∪ A1 (i), so on the basis of the inductive assumption we have: C∗ (m) ⊆ A∗ (i) ∪ W (X), which by the monotonicity of W leads to the result: W (X ∪ A∗ (i) ∪ C∗ (m)) ⊆ W (X ∪ A∗ (i) ∪ W (X)). However, W (X ∪ A∗ (i) ∪ W (X)) = W (X) (since due to the inductive assumption: W (X ∪ A∗ (i)) = W (X) and from the fact that for any sets X, Y : W (X ∪ Y ) = W (X) ⇒ W (X ∪ Y ∪ W (X)) = W (X)), hence and from the inductive assumption (5) follows. Now, we have two cases: either pr2 (an+1 ) = 1, or pr2 (an+1 ) = ∗. In the former case from (4), (5) and (3) it follows that C1 (m) ∪ {pr1 (an+1 )} ⊆ W (X ∪ A∗ (i)), so due to the inductive assumption: pr1 (an+1 ) ∈ W (X). Obviously A∗ (i + 1) = A∗ (i). Therefore A1 (i + 1) ⊆ W (X) and W (X ∪ A∗ (i + 1)) = W (X). Now, in the second case, from (4), (5) and (3) it follows that W (X ∪ A∗ (i) ∪ C∗ (m) ∪ {pr1 (an+1 )}) = W (X ∪ A∗ (i)), that is W (X ∪ A∗ (i + 1) ∪ C∗ (m)) = W (X ∪ A∗ (i)). Hence, by the inductive assumption one can show that: W (X ∪ A∗ (i + 1)) = W (X). Obviously, A1 (i + 1) ⊆ W (X) due to inductive assumption, since A1 (i + 1) = A1 (i). In order to prove the second part of the theorem assume that W is finitary and that α ∈ W (X). Then α ∈ W (β1 , . . . , βn ) for some {β1 , . . . , βn } ⊆ X. Consider a p-rule r0 = {(hβ1 , ∗i, . . . , hβn , ∗i, hα, 1i)}. Let for Y ⊆ L, W (Y, β1 , . . . , βn ) = W (Y ). As α ∈ W (Y ) so r0 ∈ Rq (W ). We have just proved that X k=Rq (W ) α. 2

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3 Now we focus the attention on a special syntactic representation of finitary matrix q-consequences. First, we remind the notions of p- and q- matrix used in [1] and [2] respectively, and the operations determined by these matrices and consider result concerning a syntactic representation of finitary matrix p-consequences (cf. [1]). Let A = (M, F1 , . . . , Fn ) be an algebra similar to L. By a p-matrix for L we mean a structure M = (A, D1 , D∗ ), where D1 ⊆ D∗ ⊆ M . Every structure A = (M, D, D), where D, D ⊆ M and D ∩ D = ∅, is called a q-matrix for the language L. Every p-matrix (q-matrix) M defines a pconsequence (q-consequence) operation ZM (WM ) by the clause: for every X ⊆ L, α ∈ L : α ∈ ZM (X) iff for every homomorphism h from L into − → − → M, hα ∈ D∗ whenever h (X) ⊆ D1 (α ∈ WM (X) iff h (X) ∩ D = ∅ implies hα ∈ D). In the paper, for the sake of convenience, we apply the very notion of p-matrix M = (A, D1 , D∗ ) to determine the same concept of q-consequence: α ∈ WM (X) iff for every homomorphism h from L into − → M ( h (X) ⊆ D∗ ⇒ hα ∈ D1 ). One may show that the class of all p-consequences defined on given language L as well as the class of all the q- consequences form complete lattices such that for any family {ZiT: i ∈ I} of p-consequences on L and any X ⊆ L : (inf{Zi : i ∈ I})(X) = {Zi (X) : i ∈ I}, and similarly for a family of q-consequences. In particular, for a given class M of p-matrices, the meet inf{ZM : M ∈ M} will be denoted by ZM . Similarly for a class of matrix q-consequences. A p-rule r is said to be valid for the p-consequence ZM iff for every p-matrix M = (A, D1 , D∗ ) ∈ M, a homomorphism h : L → M and for each − → → − p-inference (a1 , . . . , an ) ∈ r : h (A1 (n − 1)) ⊆ D1 & h (A∗ (n − 1)) ⊆ D∗ implies that h(pr1 (an )) ∈ Dpr2 (an ) . In the sequel, the set of all valid p-rules for a p-consequence ZM will be denoted by R(M). In [1] we have established the following Theorem 3. (cf. [1]) For any class of p-matrices M for L and any X ⊆ L, α ∈ L : X k–R(M) α ⇒ α ∈ ZM (X). Moreover, if ZM is finitary, then X k–R(M) α iff α ∈ ZM (X).2


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Now, we will prove a counterpart of Theorem 3 for the matrix pconsequences is as follows: Theorem 4. For any class of p-matrices M for L and any X ⊆ L, α ∈ L : X k=R(M) α ⇒ α ∈ WM (X). Moreover, if WM is finitary, then X k=R(M) α iff α ∈ WM (X). Proof: Assume that X k=R(M) α, so there is a q3 -proof (a1 , . . . , an ) of α from the set X on the basis R(M). In order to show that α ∈ WM (X), consider a p-matrix M = (A, D1 , D∗ ) from the class M and any homomorphism h : L → M, fulfilling the condition − → (1) h (X) ⊆ D∗ . We show by induction, that for every i ∈ {1, . . . , n} : h(pr1 (ai )) ∈ Dpr2 (ai ) , which applied for i = n leads to the result: hα ∈ D1 . So first consider the case i = 1. When pr1 (a1 ) ∈ X and pr2 (a1 ) = ∗,Sit is obvious that h(pr1 (a1 )) ∈ Dpr2 (a1 ) , due to (1). In turn, if (a1 ) ∈ R(M), then h(pr1 (a1 )) ∈ Dpr2 (a1 ) due to the very definition of the set R(M). Now let us put the inductive assumption that for an i ∈ {1, . . . , n − 1}: (2) for each k ≤ i : h(pr1 (ak )) ∈ Dpr2 (ak ) . It is enough to show that h(pr1 (ai+1 )) ∈ Dpr2 (ai+1 ) . In order to do it, let us suppose at once that the pair ai+1 is introduced to the q-proof (a1 , . . . , an ) by application of a p-rule from R(M) (the case when it is introduced due to point (a) of the definition of q3 -proof S is obvious by (1)), that is there exists a p-inference (c1 , . . . , cm , ai+1 ) ∈ R(M) such that (3) ∀j∈{1,...,m} (pr1 (cj ) = pr1 (bj ) and pr2 (cj ) ≤ pr2 (bj )) for some pairs b1 , . . . , bm such that (4) {b1 , . . . , bm } ⊆ {a1 , . . . , ai }. From (4) and (2) it follows that ∀j∈{1,...,m} : h(pr1 (bj )) ∈ Dpr2 (bj ) . Therefore ∀j∈{1,...,m} : h(pr1 (cj )) ∈ Dpr2 (bj ) ⊆ Dpr2 (cj ) due to (3). Finally, from the definition of R(M), we have h(pr1 (ai+1 )) ∈ Dpr2 (ai+1 ) . In order to prove the second part of the theorem let us consider such a class of p-matrices M that the q-consequence WM is finitary and assume that α ∈ WM (X). Then for some finite Xf ⊆ X: (5) α ∈ WM (Xf ).

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Put X = {β1 , . . . , βn }. Then the p-inference (hβ1 , ∗i, . . . , hβn , ∗i, hα, 1i) is a q-proof of α from X on the S basis of R(M) (that is X k=R(M) α) since (hβ1 , ∗i, . . . , hβn , ∗i, hα, 1i) ∈ R(M) due to (5). 2.

4 In the last section we dealt with the finitary matrix p- and q-consequences. Now we would like to provide a useful criterion of finitarity for matrix p- and q-consequence operations. In fact, it is the same criterion as for ordinary matrix consequences: a finite p-matrix determines a finitary pand q-consequence. The following two theorems are in the fact adaptations of statements from [4]. Lemma 2. Let for any natural k ≥ 1, L(k) = (L(k) , f1 , . . . , fn ) be a pseudo-language freely generated by the set of variables {p1 , . . . , pk } and M = (A, D1 , D∗ ) a p-matrix for L. If card(M ) = n for some natural number n ≥ 1, then there exists a natural number k such that for all α ∈ L, X ⊆ L the following conditions are valid: → (i) α ∈ ZM (X) iff for every substitution e : L −→ L(k) : eα ∈ ZM (− e X), (k) − → α ∈ WM (X) iff for every substitution e : L −→ L : eα ∈ WM ( e X), (ii) the quotient set L(k) / ≈ is finite, where ≈ is a binary relation on L(k) of the form: for all α, β ∈ L : α ≈ β iff for each homomorphism h : L(k) −→ M, hα = hβ. Proof: (i)(⇒): Obvious due to the structurality of the operations ZM and WM . (i)(⇐): Suppose that M = {m1 , . . . , mn }. Assume that α 6∈ ZM (X), → − hence for some homomorphism h : h (X) ⊆ D1 and hα 6∈ D∗ . Let us define substitution eh : L −→ L(k) by the following condition: for each p ∈ V AR(L), if hp = mi , then eh p = pi . Consider any homomorphism h0 : L −→ M such that h0 pi = mi for i = 1, 2, . . . , n. Therefore we have: −− → h0 eh (X) ⊆ D1 , but h0 eh α 6∈ D∗ . Proof of statement for q-consequence is analogous. (ii) number or possible valuations e : L(k) −→ M equals nk , hence set L(k) / ≈ is finite. 2


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Theorem 5. If p-matrix M is finite, then the p-consequence operation ZM and q-consequence operation WM are finitary. The proof is identical with the proof of theorem 3 in [4], p. 61 by defining a congruence relation ≈ as in Lemma 2(ii) 2. Corollary 4. If M is a finite class of finite p-matrices, then ZM and WM are finitary. Proof: Obvious due to Theorem 5 since inf{Zi : i ∈ I} is finitary p-consequence operation whenever for each i ∈ I, Zi is a finitary p-consequence. The same for q-consequences. 2

References [1] Szymon Frankowski, Formalization of Plausible Inference, Bulletin of the Section of Logic 33/1 (2004), pp. 41–52. [2] Grzegorz Malinowski, Q-Consequence Operation, Reports on Mathematical Logic 24 (1990), pp. 49–59. [3] Grzegorz Malinowski, Lattice Properties of a Protologic Inference, Studies in Logic, Grammar and Rhetoric 4 (17) (2001), pp. 51–58. [4] Ryszard W´ojcicki, Strongly finite sentential calculi, [in:] Selected Papers on L Ã ukasiewicz sentential calculi, WrocÃlaw, 1977, pp. 53–77.

Department of Logic University of L Ã ´od´z Poland