UDC 519.95 : 51.01 : 164

MATHEMATICS

M. F. RATSA

A CRITERION OF FUNCTIONAL COMPLETENESS IN THE LOGIC CORRESPONDING TO JAŚKOWSKI’S FIRST MATRIX

(Presented by Academician P. S. Novikov on 4 IX 1965)

1. The simplest of the propositional logics intermediate between classical logic and constructive (i.e., intuitionistic) logic is, as is well known, the logic corresponding to the so-called first Jaśkowski matrix*. This logic, called (at the suggestion of A. A. Markov) Smetanich’s logic**, is simple in the sense that it is constructed tabularly, as a variant of three-valued logic in which the basic operations—conjunction \(\&\), disjunction \(\vee\), implication \(\supset\), and negation \(\neg\)—are given by the tables of Jaśkowski’s first matrix. Smetanich’s logic is intermediate between constructive logic and classical logic not only in the degree of simplicity of its construction, but also in the sense that every logical relation having the form of an equivalence (identity) of formulas, implication, etc., if true in constructive logic, is true also in Smetanich’s logic, and if true in the latter, is true also in classical logic (but not conversely). In this connection, it is natural to try first to solve, for Smetanich’s logic, certain questions that have been solved for classical logic but still present considerable difficulties for constructive logic. Examples of questions of this kind are those of functional completeness, an attempt at a systematic consideration of which for constructive logic was undertaken by A. V. Kuznetsov \((^5)\).

The consideration of questions of functional completeness for Smetanich’s logic begins with the consideration of the notion of a function of Smetanich’s logic as a function that can be expressed by a formula of propositional logic whose operation signs are interpreted according to the tables of Jaśkowski’s first matrix, i.e., in other words, expressed through the basic operations of this logic. Under such a consideration, there arises first of all the question of the relation of this notion to the general notion of a function of three-valued logic (a function from \(P_3\)), since questions of functional completeness for the usual variant of three-valued logic (when the class of all functions that can be specified by tables with three values is considered) were investigated by S. V. Yablonskii \((^{6,7})\). Such a consideration may also be connected with the problem of a more detailed investigation than before of classes of functions of three-valued logic—a problem that naturally arises in connection with the results of Yu. I. Yanov and A. A. Muchnik \((^8)\), who discovered an essential difference between three-valued logic and two-valued (i.e., classical) logic.

* On Jaśkowski matrices see \((^{1,2})\). The corresponding structures, called Jaśkowski structures, were considered by V. A. Yankov \((^3)\).

** Ya. S. Smetanich, in an article \((^4)\) and in other works, was apparently the first to study this logic not only in connection with other logics, but also by singling it out for more special consideration. He also, as V. A. Yankov reports in \((^3)\), proposed constructing a calculus corresponding to this logic by adding to the constructive (intuitionistic) propositional calculus a new axiom

\[ ((x \supset y)\vee(y \supset z))\vee(z \supset u)). \]

A. V. Kuznetsov \((^5)\) defines Smetanich’s logic as the logic of a three-element structure.

1. We shall denote propositional variables by the letters \(x, y, z\), and \(u\), possibly with subscripts. The values of variables are denoted by \(0, \tau, 1\). The class of all such functions, the domain of values of each of whose arguments is the set \(\{0,\tau,1\}\) and whose values belong to the same set, is denoted (see (7)) by the symbol \(P_3\). In what follows, wherever the contrary is not specified, by the term function* we shall mean a function from \(P_3\).

We define the operations \(\&, \vee, \supset\), and \(\neg\) by tables, the totality of which (with the indication that the designated value, i.e. truth, is taken to be \(1\)) is called Jaśkowski’s first matrix:

\[ (x \& y) \]

\[ (x \vee y) \]

\[ (x \supset y) \]

\[ \neg x \]

A function (from \(P_3\)) is called a function of Smetanich logic if it is expressible (by means of superpositions) through the functions

\[ (x \& y);\ (x \vee y),\ (x \supset y),\ \neg x. \tag{\(\Sigma_0\)} \]

We denote the class of all functions of Smetanich logic by the letter \(S\). An example of a function not belonging to the system \(\Sigma_0\) but belonging to the class \(S\) is equivalence \((x \sim y)=((x \supset y)\ \&\ (y \supset x))\).

We say that a function \(f(x_1,\ldots,x_m)\) preserves a predicate \(P(x_1,\ldots,x_n)\) if, for all possible values of the variables \(x_{ij}\) \((i=1,\ldots,m;\ j=1,\ldots,n)\), from the fact that \(P(x_{11},x_{12},\ldots,x_{1n})\), \(P(x_{21},x_{22},\ldots,x_{2n}),\ldots,P(x_{m1},x_{m2},\ldots,x_{mn})\) holds, it follows that
\(P(f(x_{11},x_{21},\ldots,x_{m1}),\ldots,f(x_{1n},x_{2n},\ldots,x_{mn}))\) holds. The set of all functions preserving the predicate \(P\) is called the preservation class of the predicate \(P\) (see (9), pp. 104–105, and also (10)).

1. We denote the preservation class of the predicate \(x \ne \tau\) by the letter \(T\), and the preservation class of the predicate \(\neg x=\neg y\) by the letter \(U\)**. Notice that the constant \(\tau\) belongs to the class \(U\), but does not belong to \(T\). An example of a function belonging to \(T\) but not to \(U\) is the function \(\rfloor x\) (weak negation), defined by the equalities \(\rfloor 0=1,\ \rfloor \tau=1,\ \rfloor 1=0\) (cf. (11)). From consideration of the tables given above it is clear that all functions of the system \(\Sigma_0\) belong to the class \(T \cap U\). It is also not difficult to show that to the latter belongs every such and only such function \(f(x_1,\ldots,x_n)\) for which the identity

\[ \neg\neg f(x_1,\ldots,x_n)=f(\neg\neg x_1,\ldots,\neg\neg x_n) \tag{1} \]

holds.

Theorem 1. \(S=T \cap U\).

Theorem 1′. In order that a function \(f(x_1,\ldots,x_n)\) belong to the class \(S\), it is necessary and sufficient that it satisfy the identity (1).

* The sign \(0\) is convenient for denoting falsehood, the sign \(1\) for denoting truth, especially when simultaneously considering classical, constructive, and intermediate logics, also taking into account the traditions in the algebra of logic going back to Boole and Żegalkin; the sign \(\tau\) is convenient for denoting a third value, intermediate between falsehood and truth. S. V. Yablonskii and a number of other authors denote the variables differently; however, for the convenience of using their constructions and results, in the present work this difference in notation is regarded as inessential and is ignored in what follows.

** This notation for these two classes was chosen because the first of them is one of the classes called in (7) classes of type \(T\) (namely, the class of functions preserving the set \(\{0,1\}\)), and the second of them is one of the classes called in (7) classes of type \(U\) (the class of functions preserving the partition \(\{\{0\},\{\tau,1\}\}\)).

To prove Theorems 1 and \(1'\), in view of what was said above, it is enough to prove that every function \(f\) belonging to \(T\cap U\) is expressible in terms of the functions of the system \(\Sigma_0\). The proof of the latter is based on the fact that every such function \(f\) is either the constant \(0=(x\&\neg x)\), or a disjunction of certain functions \(g\), for each of which the number of such tuples (of argument values) not containing \(\tau\) on which \(g\) differs from \(0\) is equal to one; and each such function \(g\) is equal to the conjunction of certain functions of the form \(\neg x_i\), or of the form \(\neg\neg x_j\), or of the form \((A\supset B)\), or of the form \(B\), where \(A\) is a conjunction of certain variables, and \(B\) is a disjunction of certain variables.

1. A system (class) \(\Sigma\) of functions from a given class \(K\) will be called (functionally) complete in \(K\) if all functions of the class \(K\) are expressible in terms of functions from \(\Sigma\). A system \(\Sigma\) of functions from the class \(K\) will be called precomplete in \(K\) if \(\Sigma\) is not complete in \(K\), but for every function \(f\) from \(K\) not belonging to \(\Sigma\), the system \(\Sigma\cup\{f\}\) is complete in \(K\).

Theorem 2. The class \(S\) is precomplete in the class \(T\).

For the proof it is enough to show the following: 1) if \(f\in T\) and \(f\notin S\), then the function \(\neg x\) is expressible in terms of \(f\) and functions from \(S\); 2) the system \(\Sigma_0\cup\{\neg x\}\) is complete in \(T\).

Theorem 3. The class \(S\) is precomplete in the class \(U\).

It is enough to show the following: 1) if \(f\in U\) and \(f\notin S\), then the constant \(\tau\) is expressible in terms of \(f\) and functions from \(S\); 2) the system \(\Sigma_0\cup\{\tau\}\) is complete in \(U\).

Let us note that the classes \(T\) and \(U\) themselves, as is clear from their definition (taking also into account the footnote to it) and from work \((^7)\), are precomplete in \(P_3\).

For each of the following 10 predicates we shall consider the class of all functions from \(S\) preserving the given predicate

\[ x=0,\qquad x=1,\qquad \neg x\ne \neg y,\qquad \neg x\leqslant \neg y, \]

\[ \neg(x\sim y)=\neg(z\sim u),\qquad (x\&y)\ne\tau,\qquad ((x\vee y)\vee(x\sim y))=1, \]

\[ ((x\subset y)\&(\neg x\supset \neg y)=1,\qquad ((x\vee\neg x\vee y)\&(\neg x\sim\neg y))=1,\tag{П} \]

\[ \neg\neg((x\&y)\vee(x\&z)\vee(y\&z))=(x\sim(y\sim z)); \]

the ten classes thus defined will be denoted respectively by the symbols
\(S_0,S_1,S_2,S_3,S_4,S_5,S_6,S_7,S_8,S_9\).*

For comparing the 10 classes just defined with the known 18 classes precomplete in \(P_3\) \((^7)\), the following two theorems are useful.

Theorem 4. If the class \(K\) coincides either with \(S_0\), or with \(S_1\), or with \(S_6\), then there exists a class \(J\) precomplete in \(P_3\) such that \(K=J\cap S\).

Theorem 5. For every class \(J\) precomplete in \(P_3\), the intersection \(J\cap S\) is equal to the class \(S\), or equal to \(S_0\), or equal to \(S_1\), or equal to \(S_6\), or is strictly contained in at least one of the classes \(S_0,S_1,S_3,S_5,S_6\).

1. Theorem 6 (criterion of functional completeness in Smetanich’s logic). In order that a system \(\Sigma\) of functions from the class \(S\) be complete in \(S\), it is necessary and sufficient that for each \(i=0,1,\ldots,9\) there exist a function belonging to \(\Sigma\) but not belonging to \(S_i\).

* Cf. the first 5 predicates of the list (П) with the 5 predicates (see \((^9)\), p. 105), whose preservation classes are precomplete in \(P_2\) (i.e. in two-valued logic). The role played by the last three of them is, to some extent, analogous to the role of the classes \(S_0,S_1,S_2,S_3,S_4\) in \(S\). This is seen from comparing Lemmas 1, 2, and 3 with the proof of sufficiency of the criterion of functional completeness in \(P_2\) (see \((^7)\), pp. 18–20), and is explained by the existence of such a homomorphism (in the sense of § 8 of work \((^7)\)) of the class \(S\) onto the class \(P_2\), under which the images of the classes \(S_0,S_1,S_2,S_3,S_4\) are classes precomplete in \(P_2\). As for the remaining 5 predicates of the list (П), attention should be paid to the following property of the last three of them, namely that the predicate under consideration entails the predicate \(\neg x=\neg y\), occurring in the definition of the class \(U\). Cf. the first two of these three predicates with the formulas used for the proof of Lemma 2 of work \((^5)\). The last of these two predicates is more simply written as:
\[ \neg\neg(x\&y)=(x\vee y). \]

The necessity of the criterion follows from the fact that the classes \(S_0, S_1,\ldots,S_9\) are closed with respect to superpositions and are strictly contained in \(S\). The proof of sufficiency is based on the fact that, denoting by the symbols \(f_0, f_1,\ldots,f_9\) functions (from \(\Sigma\)) that do not belong respectively to \(S_0, S_1,\ldots,S_9\), one can prove the following lemmas.

Lemma 1. Through the functions \(f_0, f_1\), and \(f_2\) the constants \(0\) and \(1\) are expressible.

Lemma 2. Through the functions \(0, 1\), and \(f_3\) the function \(\neg x\) is expressible.

Lemma 3. Through the functions \(0, 1, \neg x\), and \(f_4\) every function of the class \(S\) not taking the value \(\tau\) is expressible.

Lemma 4. Through the functions \(f_5, f_6, f_7, f_8, f_9\) and functions of the class \(S\) not taking the value \(\tau\), the functions \((x \& y)\) and \((x \supset y)\) are expressible.

Lemma 5.
\[ (x \vee y)=(((x \supset y)\supset y)\&((y \supset x)\supset x)). \]

1. The subsequent theorems are, in essence, consequences of Theorem 6.

Theorem 7. There exist exactly 10 precomplete classes of functions in \(S\), namely the classes \(S_0, S_1,\ldots,S_9\).

We call a function \(f(x_1,\ldots,x_n)\) trivial if there exists a variable \(x_i\) such that \(f(x_1,\ldots,x_n)=x_i\) for all values of the variables; otherwise we call it nontrivial.

Theorem 8. There exists a nontrivial function belonging simultaneously to all precomplete classes in \(S\).

For the proof it suffices to give an example: the function \(\neg\neg x\) or the function \((x \vee \neg(x \supset y))\).

Theorem 9. In every complete system of functions in \(S\) there exists a subsystem, complete in \(S\), containing no more than 9 functions.

We call a function of a class \(K\) Sheffer in \(K\) if all functions of the class \(K\) are expressible through it (cf. \((^7)\), p. 77, and also \((^5)\)).

Theorem 10. In order that a function of the class \(S\) be Sheffer in \(S\), it is necessary and sufficient that it belong to none of the following six classes: \(S_0, S_1, S_2, S_7, S_8, S_9\). A system \(\Sigma\) of functions from \(K\) is called weakly complete in \(K\) if the system consisting of the functions belonging to \(\Sigma\) and the constants belonging to \(K\) is complete in \(K\) (cf. \((^9)\), p. 104).

Theorem 11. In order that a system \(\Sigma\) of functions of the class \(S\) be weakly complete in \(S\), it is necessary and sufficient that for each \(i=3,4,\ldots,9\) there exist a function belonging to \(\Sigma\) but not belonging to \(S_i\).

I express my deep gratitude to A. V. Kuznetsov for posing the problems and for valuable advice, and to E. A. Pavlova for a number of consultations.

Institute of Mathematics with Computing Center
Academy of Sciences of the Moldavian SSR

Received
2 IX 1965

REFERENCES

\(^1\) S. Jaśkowski, Actes du Congrès International de Philosophie Scientifique, 6, Paris, 1936, p. 58.
\(^2\) B. Yu. Pilchak, Ukrainian Math. Journal, 4, No. 2, 174 (1952).
\(^3\) V. A. Yankov, DAN, 151, No. 4, 796 (1963).
\(^4\) Ya. S. Smetanich, Trudy Moskov. Mat. Obshch., 9, 357 (1960).
\(^5\) A. V. Kuznetsov, DAN, 160, No. 2, 274 (1965).
\(^6\) S. V. Yablonskii, DAN, 95, No. 6, 1153 (1954).
\(^7\) S. V. Yablonskii, Trudy Mat. Inst. im. V. A. Steklova, AN SSSR, 51, 5 (1958).
\(^8\) Yu. I. Yanov, A. A. Muchnik, DAN, 127, No. 1, 44 (1959).
\(^9\) S. A. Yanovskaya, Mathematics in the USSR over 40 Years (1917–1957), 1, Moscow, 1959, p. 13.
\(^10\) A. V. Kuznetsov, UMN, 16, issue 2 (98), 201 (1961).
\(^11\) A. V. Kuznetsov, All-Union Colloquium on General Algebra, summaries of communications and reports, Novosibirsk, 1963, p. 32.