MATHEMATICS

S. V. YABLONSKII

ON LIMIT LOGICS

(Presented by Academician M. V. Keldysh on 18 VII 1957)

A natural generalization of \(k\)-valued logics \(P_k\) (1) is the infinitely valued logic \(P_{\aleph_0}\), which is the set of all functions \(\Phi(x_1, x_2, \ldots, x_n)\) whose arguments are defined on the set \(E^{\aleph_0}=\{0, 1, 2, \ldots\}\) and such that \(\Phi(a_1, a_2,\ldots, a_n)\in E^{\aleph_0}\) whenever \(a_i\in E^{\aleph_0}\) \((i=1,2,\ldots,n)\). It is clear that \(P_{\aleph_0}\) contains a continuum of distinct functions depending on the variables \(x_1, x_2,\ldots\). However, in applications, in particular when considering interpretations of deductive calculi constructed with the aid of a finite number of primitive connectives, one has to consider not the whole class \(P_{\aleph_0}\), but its subsets consisting of a countable number of functions. In what follows we shall be interested in closed subclasses \(\mathfrak{F}\) of \(P_{\aleph_0}\), i.e. such subsets of functions that are invariant under the operation of superposition. With respect to these subclasses we shall assume that each of them contains a countable number of functions.

Definition. A system of functions \(\mathfrak{P}=\{f(x_1,\ldots,x_n)\}\) is said to be homomorphically mapped into a system \(\mathfrak{Q}=\{g(y_1,\ldots,y_n)\}\) if there exists a one-to-one correspondence of the arguments \(x_i\leftrightarrow y_i\), and to each function \(f(x_1,\ldots,x_n)\in\mathfrak{P}\) there uniquely corresponds a function \(g(y_1,\ldots,y_n)\in\mathfrak{Q}\) depending on the corresponding arguments, with the property that to every superposition of functions from \(\mathfrak{P}\) that belongs to \(\mathfrak{P}\) there corresponds an analogous superposition of the corresponding functions of the system \(\mathfrak{Q}\), which also belongs to \(\mathfrak{Q}\). If, under this mapping, the homomorphism holds in both directions, then the systems \(\mathfrak{P}\) and \(\mathfrak{Q}\) are said to be isomorphic.

Definition. A closed class \(\mathfrak{F}\subset P_{\aleph_0}\) is called limit (a limit logic) if: 1) \(\mathfrak{F}\) consists of a countable number of functions; 2) \(\mathfrak{F}\) contains homomorphic preimages of the \(k\)-valued logics \(P_k\) \((k=2,3,\ldots)\).

A limit logic is a generalization of \(k\)-valued logics and, obviously, occupies an intermediate position between \(k\)-valued logics and countably valued logic. Here the question naturally arises: how many pairwise nonisomorphic limit logics exist? In the present note an exhaustive answer to the question posed is given. Namely, it is established that the maximal cardinality of a set of pairwise nonisomorphic limit logics is equal to the continuum.

Below we consider functions \(f(x,y)\in P_{\aleph_0}\) having the so-called box structure. We say that a function \(f_0(x,y)\) has box structure if there exists a partition of the set \(E^{\aleph_0}\)

\[ E^{\aleph_0}=\mathcal{E}_0\cup \mathcal{E}_1\cup \mathcal{E}_2\cup\ldots, \]

\[ \mathcal{E}_0=\{0\},\quad \mathcal{E}_1=\{1,\ldots,k_1\},\quad \mathcal{E}_2=\{k_1+1,\ldots,k_2\},\ldots,\quad 0<k_1<k_2<\ldots, \]

such that \(f_0(\mathcal{E}_i,\mathcal{E}_i)\subset \mathcal{E}_i\) for every set \(\mathcal{E}_i\) containing more than one element; \(f_0(\mathcal{E}_i,\mathcal{E}_i)\) is equal either to \(\{0\}\) or to \(\mathcal{E}_i\) if \(\mathcal{E}_i\) contains one element and \(i\geq 1\); and \(f(x,y)=0\) in the remaining cases (Table 1).

Let \(S(x)\) be a function mapping \(E^{\aleph_0}\) onto \(E^{\aleph_0}\). We shall call the func-

tion \(S^{-i}(f_0(S(x), S(y)))\), where \(f_0\) has a box structure, a function with a box structure.

Lemma 1. A function \(h(x_1,\ldots,x_n)\) that is a superposition* of the function with box structure \(f_0(x,y)\) and the constants \(0,1,\ldots\) is either a constant, or a function which on each box \(\underbrace{\mathfrak{E}_i \times \cdots \times \mathfrak{E}_i}_{n\ \text{times}}\) assumes either values from \(\mathfrak{E}_i\) or is identically equal to zero, while outside the boxes it always assumes the value \(0\); moreover, \(h(x_1,\ldots,x_n)\) essentially depends on all its variables \(x_1,\ldots,x_n\).

Table 1

We conduct the proof by induction on superposition. For the initial functions the assertion is obvious. Let
\(h(x_1,\ldots,x_n)=f_0(h_1(x_{i_1},\ldots,x_{i_r}), h_2(x_{j_1},\ldots,x_{j_m}))\). The following cases are possible:

a) \(h_1(x_{i_1},\ldots,x_{i_r}) \equiv c_1\), \(h_2(x_{j_1},\ldots,x_{j_m}) \equiv c_2\). Then \(n=r=m=0\) and \(h(x_1,\ldots,x_n)\equiv f_0(c_1,c_2)=c\);

b) \(h_1(x_{i_1},\ldots,x_{i_r})\equiv c_1\), while \(h_2(x_{j_1},\ldots,x_{j_m})\not\equiv \mathrm{const}\). It follows that \(r=0\) and \(m=n\). Therefore \(h_2(x_{j_1},\ldots,x_{j_m})=h_2^{1}(x_1,\ldots,x_n)\), where \(h_2^{1}\) is the function obtained from \(h_2\) by a permutation of variables. By the inductive hypothesis, the assertion of the lemma is true for \(h_2^{1}(x_1,\ldots,x_n)\). Suppose that on the box \(\underbrace{\mathfrak{E}_i \times \cdots \times \mathfrak{E}_i}_{n\ \text{times}}\) the function \(h_2^{1}\) assumes values from \(\mathfrak{E}_i\). Then on this box \(h(x_1,\ldots,x_n)\) assumes values from the set \(f_0(c_1,\mathfrak{E}_i)\), where \(f_0(c_1,\mathfrak{E}_i)=\{0\}\) when \(c_1=\bar{\mathfrak{E}}_i\), and \(f_0(c_1,\mathfrak{E}_i)\subset \mathfrak{E}_i\) when \(c_1\in\mathfrak{E}_i\). If, however, on the box \(\underbrace{\mathfrak{E}_i \times \cdots \times \mathfrak{E}_i}_{n\ \text{times}}\) \(h_2^{1}(x_1,\ldots,x_n)\equiv 0\), then on it
\(h(x_1,\ldots,x_n)\equiv f_0(c,0)=0\), i.e. \(h\) is identically equal to zero. Obviously, outside the indicated boxes \(h(x_1,\ldots,x_n)=0\).

c) \(h_1(x_{i_1},\ldots,x_{i_r})\not\equiv \mathrm{const}\), \(h_2(x_{j_1},\ldots,x_{j_m})\not\equiv \mathrm{const}\). For the functions \(h_1\) and \(h_2\), by assumption, the assertion of the lemma is true. Suppose that on the box \(\underbrace{\mathfrak{E}_i \times \cdots \times \mathfrak{E}_i}_{n\ \text{times}}\) both functions assume values from \(\mathfrak{E}_i\). Then on it \(h(x_1,\ldots,x_n)\) assumes values from \(f_0(\mathfrak{E}_i,\mathfrak{E}_i)\subset \mathfrak{E}_i\). If at least one of the functions \(h_1\) or \(h_2\) on this box is identically equal to zero, then \(h(x_1,\ldots,x_n)\) will also, under these conditions, be identically equal to zero. Since every point lying outside the boxes \(\underbrace{\mathfrak{E}_i \times \cdots \times \mathfrak{E}_i}_{n\ \text{times}}\), cov-

* Here we assume that, in forming superpositions, the introduction of inessential variables is not allowed.

is necessarily either outside the boxes \(\underbrace{\mathcal{E}_i\times\cdots\times\mathcal{E}_i}_{r\text{ times}}\) (with respect to the variables \(x_{i_1},\ldots,x_{i_r}\)), or outside the boxes \(\underbrace{\mathcal{E}_i\times\cdots\times\mathcal{E}_i}_{m\text{ times}}\) (with respect to the variables \(x_{j_1},\ldots,x_{j_m}\)), then at least one of the functions \(h_1\) or \(h_2\) is equal to zero at this point. Therefore, outside the boxes \(\underbrace{\mathcal{E}_i\times\cdots\times\mathcal{E}_i}_{n\text{ times}}\), \(h(x_1,\ldots,x_n)=0\). Consequently, in all cases \(h(x_1,\ldots,x_n)\) has the required form. In passing we have also established that \(h(x_1,\ldots,x_n)\) depends essentially on all of its variables \(x_1,\ldots,x_n\). Thus, the lemma is proved. An analogous result holds if \(f_0(x,y)\) is replaced by \(S^{-1}(f_0(S(x),S(y)))\).

Let \(h(x,y)\not\equiv\mathrm{const}\) be a superposition of the function \(f(x,y)\) with a box structure and of the constants \(0,1,\ldots\). On the basis of the proved Lemma 1, this function on the box \(\mathcal{E}_i\times\mathcal{E}_i\) either takes values from the set \(\mathcal{E}_i\), or is identically equal to zero on it. In the latter case the box \(\mathcal{E}_i\times\mathcal{E}_i\) is subdivided into one-cell boxes with zero values. Thus the function \(h\) will also have a box structure. To each box function let us assign a sequence consisting of nonnegative integers and the symbol \(\infty\): \(n_1,n_2,\ldots\), where \(n_i\) is the number of boxes of the \(i\)-th order (the corresponding set \(\mathcal{E}_i\) contains \(i\) elements). Obviously, \(0\leq n_i\leq\infty\). This sequence is constructed uniquely from the function \(h(x,y)\). Further, assign to the function \(h(x,y)\) a real number \(\gamma=0,\gamma_1\gamma_2,\ldots\), defined as follows: \(\gamma_i=0\) if \(n_i\leq 1\);

Table 2

\(\gamma_i=1\) if \(n_i>1\). We see that to every function \(h(x,y)\) with a box structure there corresponds a real number.

Lemma 2. Let \(h(x,y)\not\equiv\mathrm{const}\) be a superposition of the function with a box structure \(f(x,y)\) and of the constants \(0,1,\ldots\), and suppose \(f(x,y)\) has at most one box of the 1st order \((n_1=1)\). Let \(\gamma\) and \(\alpha\) be the real numbers corresponding, respectively, to \(h(x,y)\) and \(f(x,y)\). Then \(\gamma\geq\alpha\).

Lemma 3. If \(\mathfrak{P}\) and \(\mathfrak{D}\) are two isomorphic sets of functions, and all constants belong both to \(\mathfrak{P}\) and to \(\mathfrak{D}\), then the corresponding functions \(f(x,y)\) and \(g(x,y)\) with box structures are assigned equal real numbers \(\alpha=\beta\).

The isomorphism of the sets \(\mathfrak{P}\) and \(\mathfrak{D}\) means the existence of a one-to-one correspondence between the constants belonging to these sets. Since \(\mathfrak{P}\) and \(\mathfrak{D}\) contain all constants, it follows that the corresponding functions \(f(x,y)\) and \(g(x,y)\) have an equal number of boxes of each order.

Theorem. The cardinality of a maximal subset of all limit logics in \(P_{\aleph_0}\) that are pairwise nonisomorphic is equal to the continuum.

In one direction the assertion is obvious, since the cardinality of the set

Table 3

of limit logics does not exceed the cardinality of all countable subsets of \(P_{\aleph_0}\).

Now we shall show that the maximal cardinality of a subset of pairwise nonisomorphic limit logics is not less than the continuum. For this purpose we construct a continuous family of pairwise nonisomorphic limit logics. Let \(\alpha\) \((0 \leqslant \alpha \leqslant 1/2)\) be an arbitrary real number. Consider one of its expansions into an infinite binary fraction (taking \(1/2 = 0.011\ldots\)):
\(\alpha = 0,\alpha_1\alpha_2\ldots\). Under the imposed restrictions \(\alpha_1=0\). Define a family of functions \(f_\alpha(x,y)\) in the following way. The function \(f_\alpha(x,y)\) has a box structure \((n_1,n_2,\ldots)\), where \(n_i=\alpha_i+1\), and in the table there first occur \(n_1\) boxes of order 1, then \(n_2\) boxes of order 2, and so on. Let the \(i\)-th box \(\mathcal E_i\times \mathcal E_i\) have order \(k_i \geqslant 2\), and let \(\mathcal E_i=\{s_i,s_i+1,\ldots,s_i+k_i-1\}\); then on these boxes \((i=2,3,\ldots)\) \(f_\alpha(x,y)\) has the form presented in Table 2. Outside these boxes (i.e. outside boxes of order \(k_i \geqslant 2\)) \(f_\alpha(x,y)=0\). We note that on \(\mathcal E_i\times \mathcal E_i\),
\(f_\alpha(x,y)=s_i+\varphi_i(x,y)\), where
\(\varphi_i(x,y)=\max(x,y)+1 \pmod {k_i}\) is Webb’s function \({}^{2}\).

Denote by \(\mathfrak T_\alpha\) the subclass of functions in \(P_{\aleph_0}\) generated by superpositions of the functions \(f_\alpha(x,y)\) and the constants \(0,1,\ldots\). It is easy to see that \(\mathfrak T_\alpha\) is countable and contains homomorphic images of the \(k\)-valued logics \(P_k\) \((k=2,3,\ldots)\). Let \(\alpha \ne \beta\). For definiteness let \(\alpha>\beta\). Suppose that \(\mathfrak T_\alpha\) is isomorphic to \(\mathfrak T_\beta\). Then the function \(f_\beta(x,y)\) from \(\mathfrak T_\beta\) would correspond to a function \(h(x,y)\) from \(\mathfrak T_\alpha\). On the one hand, on the basis of Lemma 3, the real number \(\gamma\) corresponding to \(h(x,y)\) must be equal to \(\beta\), i.e. \(\gamma=\beta\). On the other hand, since \(h(x,y)\) cannot be a constant, by Lemma 2, \(\gamma \geqslant \alpha\). Comparing these results, we arrive at a contradiction. Consequently, \(\mathfrak T_\alpha\) is not isomorphic to \(\mathfrak T_\beta\), and the theorem is completely proved.

In conclusion we give examples of two limit logics. Let the functions \(\psi_k(x,y)\) and \(\psi(x,y)\) be defined by Table 3. It is obvious that the class \(\mathfrak T\), generated by superpositions of the functions \(\psi_2(x,y), \psi_3(x,y),\ldots\), and the class \(Q\), generated by superpositions of the function \(\psi(x,y)\), are limit logics; moreover, \(\mathfrak T\) has a countable basis, while \(Q\) has a basis consisting of one function. Further, \(\mathfrak T\) contains not only homomorphic images of \(P_k\), but also isomorphic copies of \(P_k\).

Received
16 VII 1957

REFERENCES

\({}^{1}\) S. V. Yablonskii, DAN, 95, No. 6 (1954).
\({}^{2}\) D. Webb, Proc. Nat. Acad. Sci. 21, 252 (1935).