MATRICES FOR PREDICATE LOGICS

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Jacek Hawranek and Marek Tokarz. MATRICES FOR PREDICATE LOGICS. This is an abstract of a lecture read at the seminar of the Section of. Logic, Polish ...
Bulletin of the Section of Logic Volume 6/1 (1977), pp. 27–30 reedition 2011 [original edition, pp. 27–31]

Jacek Hawranek and Marek Tokarz

MATRICES FOR PREDICATE LOGICS

This is an abstract of a lecture read at the seminar of the Section of Logic, Polish Academy of Sciences, Wroclaw, October 1976. Let F be the set of all formulas built up by means of individual variables x1 , x2 , . . ., predicate letters P1 , P2 , . . . (Pi being a λ(i)-ary predicate letter), connectives F1 , . . . , Fn , and quantifiers ∀ and ∃, according to the usual definition of a formula. By a (logical) matrix for F we shall mean any sequence M = (M, f1 , . . . , fn , ∩, ∪, B), where M is a non-empty set, B ⊆ M , (M, f1 , . . . , fn ) is an algebra similar to (F , F1 , . . . , Fn ), and both ∩ and ∪ are functions from 2M into M . By a structure connected with M (or simply a structure if M is fixed up) we mean any sequence A = (A, R1 , R2 , . . .) where A is a non-empty set and for all i ∈ ω, Ri is a function from Aλ(i) into M . Let us denote V = {x1 , x2 , . . .}. A valuation of F in A (where A is connected with M ) is any function ϕ from V ∪ F into A ∪ M such that ϕ(V ) ⊆ A ϕ(Pi xji . . . xjλ(i) ) = Ri (ϕxji , . . . , ϕxjλ(i) ) ϕ(Fi α1 . . . αk ) = fi (ϕα1 , . . . , ϕαk ) ϕ(∀xi α) = ∩{ϕ0 α : ϕ0 =i ϕ} ϕ(∃xi α) = ∪{ϕ0 α : ϕ0 =i ϕ} where “ϕ0 =i ϕ” stands for “ϕ0 (xj ) = ϕ(xj ) for all j 6= i”. (Structure in a very similar sense are discussed e.g. in [1], [6].) Let V (α) denote the set of all variables free in α, and let α[xi /xj ] denote the result of substituting xj for all free occurrences of xi in α, with normal restrictions. If α, β are formulas (α, β ∈ F ) then we shall

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Jacek Hawranek and Marek Tokarz

say that α and β are similar, (in symbols α ∼ β), if one of them can be obtained from the other by changing some bound variables. A function e : F → F is said to be a substitution if e is a homomorphism with respect to F1 , . . . , Fn , ∀x1 , ∀x2 , . . . , ∃x1 , . . . and the following conditions are satisfied: (a) V (eα) ⊆ V (α), all α ∈ F (b) for every atomic α ∈ F , and for every i, j ∈ ω, there is some α0 ∈ F of a special form, see [3], such that eα ∼ α0 and e(α[xi /xj ]) ∼ α0 [xi /xj ]. (A detailed discussion of the notion of substitution in languages with quantifiers is contained in [3] and [4].) Any pair L = (F , C), where C : 2F → 2F satisfies the well-known Tarski’s postulates for a consequence operation (cf. [5]), is called a logic. S L is finite (and so is C) provided that α ∈ {C(Y ) : Y ⊆ X and Y < ℵ0 } whenever α ∈ C(X). L is structural (and so is C) provided that eC(X) ⊆ C(eX), for all X ⊆ F and for all substitutions e. C1 6 C2 stands for “C1 (X) ⊆ C2 (X), all X ⊆ F ”. Let F be the set of all formulas in some language, and let M = ∗ (M, f1 , . . . , fn , ∩, ∪, B) be a matrix for F . Define operations CM , CM and F C M on 2 by (A)

α ∈ CM (X)

iff

for every structure A connected with M and for every valuation ϕ of F in A, if ϕX ⊆ B then ϕα ∈ B (comp. [2] for the case of sentential logics).

(B)

∗ α ∈ CM (X)

iff

for every structure A connected with M if for every valuation ϕ of F in A, ϕX ⊆ B, then for every valuation ϕ of F in A, ϕα ∈ B.

(C)

α ∈ C M (X)

iff

ϕα ∈ B for every valuation ϕ of F in any structure, whenever ϕX ⊆ B for every valuation ϕ of F in any structure.

One can easily check it that each one of the operations defined in (A), (B), (C) is a consequence in F . Moreover

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Matrices for Predicate Logics

Theorem 1. For all M : ∗ are structural. a. Both CM and CM

b. C M is Post-complete. ∗ 6 CM . c. CM 6 CM (Note that C M is not in general a structural consequence operation.) ∗ Theorem 2. If M is a finite matrix then CM (CM , C M ) is a finite consequence operation.

Now consider the following new notion of a matrix consequence. Let M and F be as above, and let h be a function from F into M . Then h is said to be a valuation of F in M provided that the following conditions hold: hFi α1 . . . αk = fi (hα1 , . . . , hαk ) h

∃ ∀

xi α =

∪ ∩

{hβ[xi /xj ] : β ∼ α and j ∈ ω}

0 as follows: Define CM

(D)

0 α ∈ CM (X)

iff

for every valuation h of F in M , if hX ⊆ B then hα ∈ B.

0 Problem. Prove (or disprove) that CM = CM .

References [1] C. C. Chang and J. H. Keisler, Continuous model theory, Princeton 1966, Princeton Univ. Press. [2] J. Lo´s and R. Suszko, Remarks on sentential logics, Ind. Math. 20 (1958), pp. 177–183. [3] W. A. Pogorzelski and T. Prucnal, Structural completeness of the first-order predicate calculus, Zeitschr. math. Log. 21 (1975), pp. 315– 320. [4] W. A. Pogorzelski and T. Prucnal, The substitution rule for predicate letters in the first-order predicate calculus, Reports on Mathematical Logic 5 (1975), pp. 77–90.

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Jacek Hawranek and Marek Tokarz

¨ [5] A. Tarski, Uber einige fundamentale Begriffe der Metamathematik, Comptes Rendus des s´ eances de la Soci´ et´ e des Sciences et des Lettres de Varsovie 23 (1930), cl. iii, pp. 22–29. [6] J. Waszkiewicz and B. W¸eglorz, On products of structures for generalized logics, Studia Logica 25 (1969), pp. 7–15.

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