aa r X i v : . [ m a t h . L O ] J un ON MATRIX CONSEQUENCE (EXTENDED ABSTRACT)
ALEXEI MURAVITSKY
Louisiana Scholars’ CollegeNorthwestern State UniversityNatchitoches, LA 71497, U.S.A.
Email : [email protected]
Abstract.
These results are a contribution to the model theory of matrixconsequence. We give a semantic characterization of uniform and couniformconsequence relations. The first concept was introduced by Lo´s and Suszko(1958), and the second by W´ojcicki (1970) to characterize structural conse-quence relations having single adequate matrices. In different variants of thesingle-adequate-matrix theorem of Lo´s and Suszko (corrected by W´ojcicki),the uniformity of consequence is used in conjunction with other properties(e.g. finitariness), and in the W´ojcicki theorem uniformity and couniformityof consequence are used together. These properties have never been treatedindividually, at least in a semantic manner. We consider these notions from apurely semantic point of view and separately, introducing the notion of a uni-form bundle/atlas and that of a couniform class of logical matrices. Then, weshow that any uniform bundle defines a uniform consequence; and if a struc-tural consequence is uniform, then its Lindenbaum atlas is uniform. Thus,any structural consequence is uniform if, and only if, it is determined by a uni-form bundle/atlas. On the other hand, any couniform set of matrices definesa couniform structural consequence. Also, the Lindenbaum atlas of a couni-form structural consequence is couniform. Thus, any structural consequenceis couniform if, and only if, it is determined by a couniform bundle/atlas. Wethen apply these observations to compare structural consequence relations thatare defined in different languages when one language is a primitive extension ofanother. We obtain that for any structural consequence defined in a languagehaving (at least) a denumerable set of sentential variables, if this consequenceis uniform and couniform, then it and the
W´ojcicki’s consequence correspond-ing to it, which is defined in any primitive extension of the given language, aredetermined by one and the same atlas which is both uniform and couniform. Preliminaries
We consider a structural consequence relation, not necessarily finitary (or com-pact), in an abstract propositional language L . We will be using a unifying term abstract logic (usually denoted by S ) in order to employ in one and the same con-text the consequence relation ⊢ S and the consequence operator Cn S both associatedwith S .The set of all L -variables is denoted by V L , the set of all L -formulae by Fm L .For any α ∈ Fm L , V ( α ) is the set of all variables occurring in α . Further, if V ⊆ V L ,we denote: Fm L [ V ] := { α ∈ Fm L | V ( α ) ⊆ V} . F L stands for a formula algebra in L . Given a set X ⊆ Fm L and an abstract logic S , X is an S - theory if Cn S ( X ) = X .It is well known (see, e.g., [5]) that any structural consequence is determinedby a class of (logical) matrices and vice versa. We remind that, given a logicalmatrix M = h A , D i , where A is an algebra of the same signature as the signatureof L and D ⊆ | A | (we call such a matrix an L - matrix and D its logical filter ),a matrix consequence relative to M (or M - consequence for short) is thefollowing relation: X | = M α df ⇐⇒ ( v [ X ] ⊆ D = ⇒ v [ α ] ∈ D, for any valuation v in A ) , where X ∪ { α } ⊆ Fm L .The last definition is extended to a (nonempty) class M = { M i } i ∈ I of L -matricesas follows:(1) X ⊢ M α df ⇐⇒ ( X | = M i α, for each i ∈ I ) . We call the last relation a matrix consequence relative to M (or M - consequence for short).Given an abstract logic S , a logical matrix M is an S - model if X ⊢ S α = ⇒ X | = M α. The set of all S -models is denoted by Mod ( S ).In fact, if a consequence relation is determined by a class of L -matrices, it is alsodetermined by a bundle or atlas. Definition 1.1. A ( nonempty ) family B = {h A i , D i i} i ∈ I of L -matrices is called a bundle if for any i, j ∈ I , the algebras A i and A j are isomorphic. Ignoring the dif-ferences between the algebras A i , we write B = {h A , D i i} i ∈ I . A pair h A , { D i } i ∈ I i ,where each D i is a logical filter in ( or of ) A , is called an atlas . By definition, the matrix consequence relative to an atlas h A , { D i } i ∈ I i is the same as the matrixconsequence related the bundle {h A , D i i} i ∈ I , which in turn is defined in the senseof (1) . An atlas Lin S [Σ S ] = h Fm L , Σ S i , where Σ S is the set of all S -theories, iscalled a Lindenbaum atlas ( relative to S ) . The transition from a class M of matrices to a bundle (or atlas), which woulddefine the M -consequence, can be done (at least) in two ways.An indirect way for such a transition, which is based on the fact that M -consequence is structural, uses the Lindenbaum theorem; cf., e.g., W´ojcicki [4],theorem 2.4, [5] or [1], corollary 3.3.5.1.According to a direct method, given M = {h A i , D i i} i ∈ I , we define an atlas M ∗ = h A , { D ∗ i } i ∈ I i as follows: A := Q i ∈ I A i and, given i ∈ I , D ∗ i := Q j ∈ I H j , where H j := ( D i if j = i | A j | if j = i. We claim that X ⊢ M α ⇐⇒ X ⊢ M ∗ α. In [2], the authors used the notion of uniform consequence (a uniform abstractlogic in our terminology) in an attempt to prove a criterion for a single-matrixconsequence , that is when a structural consequence can be determined by a single
N MATRIX CONSEQUENCE (EXTENDED ABSTRACT) 3 matrix. As W´ojcicki (see, e.g., [5]) has shown, in addition to the notion of unifor-mity, the notion of a couniform abstract logic (in our terminology) is neededfor such a criterion.Originally, the concepts of uniformity and couniformity were defined syntacti-cally. Below we treat them semantically.2.
Uniform consequence
For an arbitrary L -matrix M = h A , D i , we define an operator S : M S ( M )where(2) X ∈ S ( M ) df ⇐⇒ there is an L -valuation v in A such that X = { α ∈ Fm L | v [ α ] ∈ D } . We notice that X | = M α ⇐⇒ ∀ Y ∈ S ( M ) . X ⊆ Y = ⇒ α ∈ Y. In case that we have a set M = { M i } i ∈ I of L -matrices, the corresponding M -consequence can be reformulated in terms of S -operator as follows:(3) X ⊢ M α ⇐⇒ ∀ i ∈ I ∀ Y ∈ S ( M i ) . X ⊆ Y = ⇒ α ∈ Y. We continue with the following definition.
Definition 2.1 (uniform bundle/atlas) . Let M = { M i } i ∈ I be a bundle. We saythat M is uniform if for any X ∪ Y ⊆ Fm L with X ⊆ Fm L [ V L \ V ( Y )] , thefollowing condition is fulfilled: ∀ i, j ∈ I ∀ Z i ∈ S ( M i ) ∀ Z j ∈ S ( M j ) ∃ k ∈ I ∃ Z k ∈ S ( M k ) . X ⊆ Z i & Y ⊆ Z j ⊂ Fm L = ⇒ Z i ∩ Fm L [ V L \ V ( Y )] = Z k ∩ Fm L [ V L \ V ( Y )] & Z j ∩ Fm L [ V ( Y )] ⊆ Z k . An atlas is uniform if the corresponding bundle is uniform.
The above definition is justified by the following two propositions.
Proposition 2.1.
Let M be a uniform bundle. Then the M -consequence is uni-form. Proposition 2.2.
If a structural abstract logic S is uniform, then its Lindenbaumatlas Lin S [Σ S ] is uniform. Corollary 2.2.1.
A structural abstract logic is uniform if, and only if, it is deter-mined by a uniform bundle/atlas. Couniform consequence
We start with the following observation.
Proposition 3.1.
Let S be a structural abstract logic that is determined by a set M = { M j } j ∈ J of L -matrices. The logic S is couniform if, and only if, for anycollection { X i } i ∈ I of formula sets with V ( X i ) ∩ V ( X j ) = ∅ , providing that i = j ,and V ( S i ∈ I { X i } ) = V L , the following condition is satisfied: (4) ∀ i ∈ I ∃ j ∈ J ∃ Z j ∈ S ( M j )( X i ⊆ Z j ⊂ Fm L ) = ⇒∃ k ∈ J ∃ Z k ∈ S ( M k )( S i ∈ I { X i } ⊆ Z k ⊂ Fm L ) . The last proposition leads to the following definition.
ALEXEI MURAVITSKY
Definition 3.1.
A nonempty set M = { M j } j ∈ J of L -matrices is called couniform if for any collection { X i } i ∈ I of formula sets with V ( X i ) ∩V ( X j ) = ∅ , providing that i = j , and V ( S i ∈ I { X i } ) = V L , the condition (4) is satisfied. An atlas h A , { D i } i ∈ I i is couniform if the corresponding bundle {h A , D i i} i ∈ I is couniform. Now we obtain
Proposition 3.2.
Let S be a structural couniform logic. Then its Lindenbaumatlas Lin S [Σ S ] is couniform. Corollary 3.2.1.
A structural logic is couniform if, and only if, it is determinedby a couniform bundle/atlas.
Using W´ojcicki’s criterion (see, e.g., [5], theorem 3.2.7, or [1], proposition 4.1.2),we also obtain the following.
Corollary 3.2.2.
A structural abstract logic is determined by a single matrix if,and only if, it can be determined by a uniform bundle/atlas and a couniform bun-dle/atlas. Extensions
We start with the following two definitions.
Definition 4.1 (extension and primitive extension of a language) . Let L and L ′ betwo propositional languages with V L ∪ C L ∪ F L ⊆ V L ′ ∪ C L ′ ∪ F L ′ , then L ′ is calledan extension of L . If C L ∪ F L = C L ′ ∪ F L ′ and V L ⊆ V L ′ , then we say that L ′ isa primitive extension of L , symbolically L ≺ p L + . Definition 4.2.
Let S be an abstract logic in a language L and S ′ be an abstractlogic in a language L ′ which is an extension of L . Then S ′ is called a conservativeextension of S , symbolically S ≺ c S ′ , if for any set X ∪ { α } ⊆ Fm L , X ⊢ S α ⇐⇒ X ⊢ S ′ α. Proposition 4.1.
Let S and S ′ be structural abstract logics in languages L and L ′ , respectively, with L ≺ p L ′ and card ( V L ′ ) = card ( V L ) ≥ ℵ . If S ≺ c S ′ , then Mod ( S ) = Mod ( S ′ ) . Definition 4.3 (consequence relation ⊢ ( M ) + ) . Let M be a set of logical matricesof type L . Assume that L ≺ p L + . A consequence relation in L + determined by theset M , is denoted by ⊢ ( M ) + . Proposition 4.2.
Let M be a set of logical matrices of type L . Assume that L ≺ p L + . The consequence relation ⊢ ( M ) + if finitary if, and only if, ⊢ M is finitary. Definition 4.4 (relation ⊢ ( S ) + ) . Let a language L + be an extension of a language L . Also, let S be an abstract logic in L . We define a relation ⊢ ( S ) + in P ( Fm L + ) × Fm L + as follows: X ⊢ ( S ) + α df ⇐⇒ there is a set Y ∪ { β } ⊆ Fm L and an L + -substitution σ such that σ ( Y ) ⊆ X , α = σ ( β ) and Y ⊢ S β. We borrow some results from [1], collecting them in one lemma.
N MATRIX CONSEQUENCE (EXTENDED ABSTRACT) 5
Lemma 4.1 ([1], Proposition3.5.4, Proposition 3.5.5, Lemma 4.1.2, Lemma 4.1.4) . Let S be a structural logic which is both uniform and couniform in a language L with card ( V L ) ≥ ℵ and L + be a primitive extension of L . Then ( S ) + is a structuralabstract logic which is both uniform and couniform. Based on Lemma 4.1, given an abstract logic S , we call ( S ) + a W´ojcicki logic ,and the consequence related to the latter a
W´ojcicki consequence .Using the last lemma, we obtain the following.
Proposition 4.3.
Let S be a structural logic which is both uniform and couniformin a language L with card ( V L ) ≥ ℵ and L + be a primitive extension of L . Then S and ( S ) + are determined by one and the same atlas which is both uniform andcouniform. Acknowledgment . I am grateful to Alex Citkin for numerous discussions regard-ing the topic of this article.
References [1] A. Citkin and A. Muravitsky. Lindenbaum method: Tutorial — UNILOG 2018. http://arXiv.org/pdf/1901.05411 , 2019.[2] J. Lo´s and R. Suszko. Remarks on sentential logics.
Nederl. Akad. Wetensch. Proc. Ser. A 61= Indag. Math. , 20:177–183, 1958. Reprinted with a list of corrigenda as [3].[3] J. Lo´s and R. Suszko. Remarks on sentential logics [reprint of mr0098670]. In
Universal logic:an anthology , Stud. Univers. Log., pages 177–184. Birkh¨auser/Springer Basel AG, Basel, 2012.Reprint of [2] with a list of corregenda.[4] R. W´ojcicki. Some remarks on the consequence operation in sentential logics.
Fund. Math. ,68:269–279. (loose errata), 1970.[5] R. W´ojcicki.
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