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EDN: HOXOPX 510.643; 517.11
Criterion of Global Admissibility for Logic IPC
Vitaliy V. Rimatskiy*
Siberian Federal University Krasnoyarsk, Russian Federation
Received 29.04.2023, received in revised form 16.06.2023, accepted 01.8.2023 Abstract. We describe globally admissible inference rules for logic IPC. Keywords: modal logic, frame and model Kripke, admissible and globally admissible inferc
Citation: V.V. Rimatskiy, Criterion of Global Admissibility for Logic IPC, J. Sib. Fed. Univ. Math. Phys., 2023, 16(5), 620-627. EDN: HOXOPX.
Introduction
Setting the basic rules of inference is fundamental to logic and its deductive system. The most general variant of possible inference rules is the admissible inference rules introduced by Lorenzen in 1955. An inference rule is admissible in a logic L if the set of theorems L is closed with respect to this rule. Directly from the definition it follows that the set of all admissible rules is the most general concept of rules compatible with the logic L that can be added to the logic without changing the set of theorems provable in L. For the majority of basic non-classical logics (IPC; KC; K4; 54; 55; 54.3, etc.), the problem of decidability with respect to the admissibility of inference rules (the Friedman problem) was solved by V.V.Rybakov in 1980s (see, for example, [1,2]).
To the Kuznetsov's problem (1975) goes back to another way of describing all rules admissible in logic: the assignment of some (finite, explicit) set of admissible rules, from which all other rules admissible in logic will be derived as consequences, i.e. setting a (finite, explicit) basis. It turned out that most basic logics (IPC; KC; K4; 54; Grz, etc.) do not have a finite basis for admissible inference rules, i.e. the Kuznetsov problem for them was solved by V. V. Rybakov in 1980s in the negative (see [1]). For a wide class of logics (including most basic and some tabular logics), an explicit basis of admissible rules was obtained at the beginning 2000s (see [3-5]). The question arose about the further development of the theory of admissible rules.
In addition to using admissible inference rules to describe non-trivial semantic properties of non-classical logics (see [6,7]), one can also propose the following approach. The next step in the study of admissible inference rules for non-classical logics was a globally admissible inference rule. The concept of a globally admissible inference rule was introduced in 2005 (see [8]). Globally admissible rules in the logic L are those inference rules that are admissible simultaneously in all (finitely approximable) extensions of this logic. Such rules develop and generalize the concept of an admissible inference rule.
As for admissible rules when studying a new individual logic or a whole class of logics, for globally admissible rules of inference, questions of resolvability arise (an algorithm for recognizing
* Gemmenty@rambler.ru © Siberian Federal University. All rights reserved
the global admissibility of a given rule), the existence of a basis for them (or an anti-basis from which all rules that are not globally admissible are derived), its finiteness and explicit description (the Kuznetsov problem). Ideally, both a certain criterion for the global admissibility of a rule in a given logic and an explicit description (construction) of an explicit (or finite) basis for such rules is desirable. Whether this problem has a solution is not yet clear.
To date the author is aware of relatively few results devoted to the study of globally admissible inference rules. In a short note [8] the reduction of global admissibility to tabular admissibility was proved: a rule is globally admissible in the logic L if and only if it is admissible in all tabular extensions of the logic L. The (recursive) basis of globally admissible rules in semi-reduced form for the logic IPC was described in [9]. In [10] an explicit (infinite) basis of inference rules that are globally admissible in modal pretabular logics PT2, PT3 was obtained. In [11] the conditions for global admissibility in the logic 54 were obtained, the (recursive) basis and anti-basis of such inference rules globally admissible in 54 were described.
The presented work is devoted to the study of globally admissible rules of intuitionistic logic IPC. A necessary and sufficient condition for global admissibility in the logic IPC is obtained.
1. Denotation, preliminary facts
We assume the reader to be aware of algebraic and Kripke semantics for superintuitionistic logics and also assume a certain initial knowledge concerning basic facts on inference rules and their admissibility (though we recall below briefly all necessary facts). As a good entry point to the subject in whole we would recommend Rybakov [1] for advanced technique concerning su-perintuitionistic logics and inference rules. According to modern trends by a logic we understand the set of all theorem provable in a given axiomatic system. In particular, a superintuitionistic logic A is a set of theorems of a logical axiomatic system, where A includes all theorems of the intuitionistic propositional calculus IPC. In definitions which follow we mean under a prepositional logic an algebraic propositional logic (cf. [1]) though the reader can understand A as a superintuitionistic logic (s.i.) which is enough for our purposes.
Since we are dealing below with only superintuitionistic logics, a Kripke frame, or merely a frame, is a partially ordered set F := (F, <} (a poset for short). As usually, we understand by intuitionistic Kripke model a frame F with a certain valuation V of a set of propositional letters P, where V is upwards stable: Vp e P and Vx,y e F(x e V(p)&(x < y)^y e V(p)). For any propositional formula a with variables from the domain of V and any a e F, a =V a is the denotation for a is valid (or true) at a in the model M := (F, <,V} under the valuation V. If we need to distinguish in which namely model the truth takes place we write (M, a) =V a. Well known fact is that the intuitionistic truth of formulas is stable upwards (Va, b e F, if a =V a and a < b then b =V a) in Kripke models. For definition of open subframes and open submodels we refer to, for example, [1]. If Mi and M2 are frames or models Mi Q M2 is the abbreviation for M1 is open subframe, or respectively submodel, of M2. A useful fact concerning this is that, if M1 is an open submodel of M2, and a e M1 then (M1,a) =V a)^(M2,a) =V a, i.e. the truth is the same at the open submodel as in the model itself.
For any subset X of a fame F, X? := {a | 3b e X(b < a)}, that is X? is the upwards cone generated by X, and X?+ := {a | 3b e X(b ^ a)&Vc e X(—(a ^ c))}. For any antichain y of elements from F, an element c from F is a co-cover for y if and only if c?+ = Uaey (c?). A frame F is rooted, or sharp, if 3a e F such that Vb e F, a < b, then we say a is the root of F and denote that element by root(F). Sm(F) denotes the set of all elements of F with depth not
exceeding m, and Slm(F) is the set of all elements of F with depth m, i.e. — m-slice of F.
All preliminary information concerning inference rules and their admissibility can be found in [1]. We recall briefly below necessary definitions and facts. Let ai,..., an, 3 be some formulas. We understand the figure r, where
ai,... ,an r :=-,
3 '
as the (structural) inference rule, which derives s(3) from s(ai),..., s(an) for every substitution s. We say r is derivable in a logic A if there is a derivation ¡3 in A from the set of assumptions {ai,..., an}. And r is called admissible in A if, for every substitution s, s(3) G A whenever s(ai) G A, ..., s(an) G A. Clearly any derivable rule is admissible, but not conversely in general. Also immediately from the definition we see that the set of all rules admissible in a logic A is the greatest class of inference rules by which we can extend axiomatic system of the logic A preserving theorems of A (which is a particular interest for studying these rules). Derivable rules may replace some fragments of fixed length in derivations, thereby shortening them linearly. Admissible rules, which are not derivable, in principle may reduce derivations even more drastically.
Algebraic description of admissible rules came from Polish Logical School. A rule r is admissible in A iff the quasi-identity q(r) := ai = T&... &an = T ^ 3 = T is valid in the free algebra of countable rank F\(w) from the variety Var(A) of all algebras on which all theorems of A are valid (cf. [1] for details).
Given a propositional logic A, a rule r := ai,..., an/3 is a consequence of a family of rules F in A (denotation F r) if there is a derivation of 3 from ai,... ,an as assumptions in the axiomatic system of A extended by adding F as new rules.
A set of inference rules S admissible in a propositional logic A is a bases for all rules admissible in A if, for any admissible rule r, S r, i.e. r is a consequence of S in A.
It is clear now why we are interested to describe a bases for admissible rules of IPC in precise — doing that we will have exhaustive collection of rules compatible with derivability in IPC, all others will be their consequences.
The admissibility of inference rules in IPC can be described through their validness in certain special «.-characterizing Kripke models. Description of these models ChIPC(n) and criteria for recognizing admissibility in IPC by means of them are given, for instance, in [1]. Since we will strongly occupy these techniques in our paper, we recall briefly the construction of ChIPC (n) and the semantic criterion for recognizing admissibility.
Given a set Pn := {pi,...,pn} of propositional letters, we construct the first slice Si(ChIPC(n) as follows. It consists of the collection of all elements with all possible valuations V of letters from Pn which does not have doubling - elements with the same valuation. Recall that, for any element a of a Kripke model M with a valuation V, V(a) is the set of all propositional letters which are valid under V at a. Assuming Sm(ChiPC(m)) to be constructed, we put in Slm+i(Ch\(m)) the elements as follows. We take arbitrary antichain Y of elements from Sm(ChiPC(m)) having at least one element of depth m and put in Slm+i(ChIPC(n)) all elements c from Si(ChIPC(n)), assuming any c to be immediate predecessor for all elements from Y, such that
(i) V(c) Ç noey V(a); and
(ii) if Y := {a} then V(c) C V(a).
Iterating this procedure we get as the result the model ChIPC (n). Recall a model M is n-characterizing for a logic A if, for any formula a, which is built up out of letters from Pn, a G A iff M = a. We need the following facts.
Theorem 1.1 ([1], Theorem 3.3.11). The model ChIPC(n) is n-characterizing for the logic IPC.
For a given frame F, a given valuation V and a given inference rule r := a1,..., an/3, we say r is valid at F under V, and write F =V r, if as soon as Vx e F and Vi (x =V ai) holds, we have Vx e F(x =V ¡3). A rule r is valid at an intuitionistic frame F if r is valid at F under any intuitionistic valuation, we write then F = r.
Theorem 1.2 ([1], Theorem 3.5.8, Lemma 3.4.2). For any inference rule r, r is admissible in IPC iff r is valid in the frame of ChiPC(n) under any intuitionistic valuation for any given n.
We say that the inference rule r is globally admissible in the logic A0 if r is admissible in any finitely approximate (tabular) logic A, extending logic A0 and denote r e TAd(A0). The set of globally admissible rules B is called the basis of the globally admissible over logic A0, if any globally admissible rule r is consequence from B in any tabular logic A that extends the logic A0, i.e. Vr e TAd(A0) VA D A0 B r.
The main result of [8] was the reduction of the global admissibility of the rule in logic S4(Int) to the admissibility in all tabular extensions of this logic:
Theorem 1.3 (Th.3, [8]). The inference rule r is admissible in all finitely approximated logics, expanding S4(Int) ^^ r is admissuble in all tabular logics (including those generated by root S4(Int)-frames), expanding S4(Int).
We also need for our research a certain reduction of any intuitionistic inference rule to the most simple form as it is possible. These forms — semi-reduced forms — are defined below as in [12] or [9]. Given an inference rule r := a1,..., an/fi. To transform r into semi-reduced form sr(r) we are doing the following steps. First, we take the rule r1 := a1 A ... A an/3, and then the rule
a1 A ■ ■ ■ A an A (3 = xo)
r2 := -,
x0
where x0 is a new variable having no occurrences in r1. Evidently that all rules r, r1 and r2 are equivalent with respect to admissibility in any superintuitionistic logic and with respect to validness at any pseudo-boolean algebra, and at any Kripke frame as well, i.e. they are equivalent in any semantic sense. Our rule r2 has the form a/x0, and now we will disclose (or, maybe better to say, to decompose) the premise of r2 introducing new variables with the aim to make formulas from the premise possibly most simple. What we are doing first, we introduce the rule
rs := x,x ~ a, x0
where x is a new variable not occurring in r2; evidently rs is equivalent to r2 in any sense mentioned above. We call the variable x standing alone at the premise of rs the main variable of the premise and denote it by mv(rs).
Assume we already have a rule r4 equivalent to r in the form
x,x1 = Y1,...,xk = Yk
rA :=-,
x0
where any variable xi does not have occurrences in the formula y%. the premise which is not a simple term, i.e. (i) Yi = ◦ S2, where o not a variable, or (ii) Yi = —S, where S is not a variable. In the case
Take first formula Yi from e {V, A, ^} and ¿i or S2 is (i) we take the rule
:_ x, {xj = Yj | 1 < j <i,i <j < k],yi = Si,y2 = 52,Xi = yi o y2 r5 • ?
xo
where yi and y2 are new variables having no occurrences in r4. For the case (ii) we put
x,xi = Yi, - ■ ■,Xi-i = Yi-i, Xi+i = Yi+i, ■■■,Xk = Yk ,y = 5,Xi =-y r5 :_-,
xo
where y is a new variable having no occurrences in r4. Continuing the described procedure till up all formulas Yi to become simple terms we get the rule sr(r) which we call semi-reduced form of the rule r. Again mv(sr(r)) is the main variable of sr(r) — the variable x having occurrence in the premise of sr(r) as the formula x.
Lemma 1.1 ([12]). The rules sr(r) and r are equipotent w.r.t. admissibility in any superintu-itionistic logic and w.r.t. semantic validness at any pseudo-boolean algebra and at any Kripke frame.
Lemma 1.2 ([12]). Any rule r is a consequence of the rule sr(r) in any superintuitionistic logic X, i.e. sr(r) r.
2. Main result
Next, we consider the intuitionistic rules of inference in a semi-reduced form. The following two statements are quite obvious.
Lemma 2.1. If the rule r is valid on any finite IPC(54)-model M for any valuation, then it is globally admissible in 54(Int).
Proof. Indeed, the n-characteristic model of an arbitrary tabular logic over Int(54) is finite. Consequently, for arbitrary formulaic valuation, the rule will be true on all n-characteristic models, i.e. admissible in all tabular logics over IPC(54). □
Lemma 2.2. If the rule r is globally admissible in IPC(54), then this rule is true on the intuitionistic (modal) model E _ ({e},R, 5) generated by the only reflexive element e for any valuation 5.
Proof by contradiction. Indeed, consider the tabular logic Xe _ X(e) generated by the singleton cluster e. For any n, the frame of the n-characteristic model of this logic is a direct union of one-element clusters (a finite number of copies of the element e). Under valuation S, the rule r is refuted on this n-characteristic model (due to the finiteness of the model, the valuation is formulaic). Hence we conclude that the rule r is not admissible in tabular logic Xe _ X(e), which contradicts the original assumption. □
Let's define the root frame T :_ ({a, b, c}, _ c^, where c ^ a, c ^ b and elements {a, b} forms an antichain (i.e. the element c is a co-cover of the antichain {a, b} and the root of the given frame).
Lemma 2.3. If the rule sr(r) in the semi-reduced form is not globally admissible in the logic IPC, then this rule is refuted on some finite model, and in particular, on the frame T under some valuation.
Proof. Let us assume that the rule sr(r) is not globally admissible in the logic IPC, i.e. is not admissible in some tabular superintuitionistic logic L. So the rule sr(r) is refuted on some n-characteristic model ChL(n) (for some suitable n) with valuation V:
ChL(n) =v a, Va e Pr(sr(r)), 3 y e ChL(n) : y =v xo. (1)
Consider the model F := (yR, V) which is a finite open submodel of the model ChL(n). By the property of an open submodel we have
Va e Pr(sr(r)) Ve e F e =V a, y =V x0. (2)
This proves the first part of the assertion: the rule sr(r) is refuted on some finite model.
Let us now show that this rule is also refuted on the frame T for some valuation. Let us define the valuation of the variables of the rule sr(r) on T as follows:
c =W P ^^ y =v p; b =W P ^^ 3e e y< e =v P,
a =W P ^^ Ve e y< e =v p.
Note that W(c) = V(y), W(b) = Ue£y< V(e), W(a) = ne£y< V(e). In particular, c =w xo. Let us show that the premise Pr(sr(r)) is true on T for the given valuation.
Proposition 1. Va e Pr(sr(r)) c =W a holds in model (T,W) by force of y =V a in the model F.
Proof. (1) Consider first the case when a = (x = x1 V x2) e Pr(sr(r)) or a = (x = x1 A x2) e Pr(sr(r)).
Let a = (x = x1 A x2) e Pr(sr(r)) and y =v (x ^ x1 A x2) (i) & y =v (x1 A x2 ^ x) (ii) holds. If c =W x, then y =V x. In force of (i) we get Vu y ^ u =^ u =V x1 A x2, in particular y =v x1 A x2. Hence y =v x1 & y =v x2 =^ c =w x1 & c =v x2 =^ c =w x1 A x2.
If c =w x1 A x2 holds then c =w x1 & c=v x2 y =v x1 & y =v x2 y =v x1 A x2. In force of truth of (ii) we have y =V x. From where should c =W x. In this way, we proove c =W a .
The case a = (x = x1 Vx2) e Pr(sr(r)) proves analogously. Let a = (x = x1 Vx2) e Pr(sr(r)) and y =V (x ^ x1 V x2) (i) & y =V (x1 V x2 ^ x) (ii). If c =W x, then y =V x. In force of (i) we have Vu y < u =^ u =V x1 V x2, in particular y =V x1 V x2. From this we get y =v x1 V y =v x2 =^ c =w x1 V c =v x2 =^ c =w x1 V x2.
If c =w x1 V x2 c =w x1 V c =v x2 y =v x1 V y =v x2 y =v x1 V x2.
In force of (ii) we have y =V x. From where it follows c =W x. So, we get c \=w a .
(2) Let now a = (x = —x1) e Pr(sr(r)) and y =V (x ^ —x1) (i) & y =V (—x1 ^ x) (ii). If c =W x, then y =V x. In force of truth a e Pr(sr(r)) in the model F, by (i) we conclude U =V —x1, i.e. Vu(y < u =^ u =V x1). From this we have x1 g V(y) & x1 g p|eEy< V(e) & x1 g Ueey< V(e). This entails a, b, c =W x1 Therefore, c =W —x1, what did we need to show.
Let now c =W —x1 holds, therefore, by definition of W we have x1 e V(y) & x1 e Ueey< V(e) & x1 e p|eEy< V(e). Hence, for all e e y^ we get e =V x1 in the model F, i.e. y =V —x1.
From the force of truth a e Pr(sr(r)) in the model F by (ii) we conclude y =V x in model F, what does it entail c =W x. In this way, is true T =W a.
(3) Consider now the case a = (x = x1 ^ x2) e Pr(sr(r)) and y =V (x ^ (x1 ^ x2)) (i) & y =V ((x1 ^ x2) ^ x) (ii). And let performed c =W x, what entails by definition of
W y _V x in model F. By force of a in the model F and by (i) we conclude y _V xi ^ x2. If c _W xi, then y _V xi. From this and y _V xi ^ x2 we get y _V x2 in the model F. Again, according to the definition of the valuation of W we have c x2. So, we prove
c xi c _W x2 ye € T(c ^ e _^ e x2), i.e. c xi ^ x2 in the model T
holds.
Let c _W xi ^ x2 in the model T holds. And let c _W x is true, hence y _V x in the model F. In force of a in the model F and by (ii) we conclude y _V xi ^ x2 in the model F, what does it entail 3u : y < u & y _V xi; u _V x2. It means it's running c _V xi, x2 € f)e£y< V(e). From this we receive c xi, c < a & a _W x2, which contradicts the original assumption c _W xi ^ x2. Hence it is true c a in the model T. □
Lemma 2.4. If the rule sr(r) is refuted on the frame T for some valuation W, then sr(r) is not globally admissible in the logic IPC.
Proof. Let for some valuation W the rule sr(r) is refuted on T:
ya € Pr(sr(r)) ye €T e _v a, 3z € Tz _v xo- (3)
We define a tabular logic L :_ L(T) and show that the rule sr(r) is not admissible in it. Recall that, by constructing the n-characteristic model, the frame Chn(L) of this model has the following structure: the first slice consists of 2n reflexive elements, the second slice consists of co-covers of antichains of at most 2 elements of the first slice.
Note also: (1) frame T is an open subframe of frame Chn(L) for some n; (2) there is a p-morphism f of the frame Chn(L) onto the subframe T' E Chn(L) isomorphic to the frame T. Transferring the valuation from T to the frame Chn(L) with the help of f, we obtain a p-morphism of models f : (Chn(L), f-i(V)) ^ (T, V), which preserves the truth of the formulas. Therefore, the rule sr(r) is refuted on the n-characteristic model Chn(L) under the valuation f-i(V). And that means it will not be admissible in tabular logic L, i.e. is not globally admissible in the logic IPC. □
From the proved theorems it follows:
Theorem 2.1. The rule sr(r) in semi-reduced form is not globally admissible in the logic IPC ^^ this rule is refuted on some finite model, in particular, is refuted on the frame T for some valuation.
The research was financially supported by the Russian Scientific Foundation (Project no. 2321-00213 ).
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Критерий глобальной допустимости в логике IPC
Виталий В. Римацкий
Сибирский федеральный университет Красноярск, Российская Федерация
Аннотация. В работе исследуется глобальная допустимость правил вывода в интуиционистской
пропозициональной логике IPC.
Ключевые слова: модальная логика, фрейм и модель Крипке, допустимое правило вывода, глобально допустимые правила вывода.
