Visualizing Polymorphisms and Counter-Polymorphisms in S5 Modal Logic

Abstract

We present a graphical representation that allows us to easily determine if a certain modal function is or is not a polymorphism of a given relation. While doing so, we provide a comparison between two ways (a calculative and a diagrammatic one) to analyze a claim about the Sheferness criterion in the theory of clones of (S5) modal functions.

1 Introduction

We exhibit a rather complex logical/mathematical problem involving calculations that are pretty laborious when done by ordinary means, but which can be readily seen using diagrams.

In his excellent paper ‘On functional completeness in the modal logic S5’ [8] the Moldavian logician M. F. Ratsa commits a slight imprecision: he claims that a certain formula ($f_{21}$) is an example of an exclusive polymorphism (in a sense to be defined precisely) of a certain relation ($R_{21}$). We use an extension of the technique presented in an earlier paper [5] in order to show that his claim is incorrect (the technique is not necessary but, as we expect to show, useful), and we provide an alternative formula.

We start by giving an interpretation of S5 formulas as operations on n-dimensional cubes (we will focus on $n \le 4$); then we define the relation expressed by a formula. Next, we define the notion of polymorphism of a relation, after giving a list of relations whose polymorphisms are maximal clones of modal operations. All these notions and results can be found in [8].

We then proceed to the elaboration and refutation of the claim about $f_{21}$, and we finish our paper presenting the above-mentioned alternative formula. We try to keep this material self-contained, but acquaintance with [5] can be helpful while interpreting the diagrams presented here.

2 Modal Formulas as Operations on Cubes

Following Ratsa, we will associate formulas of propositional S5 to operations on the structures $A_{1}, A_{2}$, and $A_{3}$ (cf. Fig. 1). We can think of each $A_{n}$ as an n-dimensional cube or (using the familiar notion of a proposition as a set of possible worlds) as the set of all propositions in a model with n possible worlds.

An interesting way to interpret the structures $A_{n}$ is thinking of them as the set of all bitstrings (i.e. sequences of 0’s and 1’s) of length n (cf. [2]). In fact, the bitstrings of length n can be seen as a sort of characteristic function of the propositions in the models with n possible worlds; e.g. in the model with two possible worlds the necessary proposition will be characterized as 11, the contingent propositions as 10 and 01, and the impossible proposition as 00.

One advantage of thinking of propositions as bitstrings is that it is simple to define how boolean (and modal) operations behave on bitstrings, and if we wish we can translate these definitions back into the more philosophical realm of propositional operations.

The boolean operations on bitstrings can be defined in terms of bitwise (usual) boolean operations. Let $B = b_1, ... , b_n$ and $S = s_1, ... , s_n$ be bitstrings of length n. We define the bitstring negation $\lnot B$ as the bitstring whose terms are, respectively, $\lnot b_1, ... , \lnot b_n$; and we define the bitstring conjunction $B \wedge S$ as the bitstring whose terms are, respectively, $b_1 \wedge s_1$, ... , $b_n \wedge s_n$. The modal operator $\square$ has the rule: $\square B = B$ if B has 1 in every bit, otherwise $\square B$ = the bitstring of same length as B which has 0 in every bit.

Fig. 1.

Structures A1, A2, and A3

Fig. 2.

A1, A2, and A3 under the effect of $\lnot$. It is helpful to notice that the complementary elements of each structure (other than 0 and 1) have names that are either graphically similar ($\omega , \varepsilon$ / $\sigma , \rho$) or phonetically similar ($\nu , \mu$).

Fig. 3.

The graph representation of a binary operation on a structure A is a function from the edges of the complete bipartite graph whose parts are copies of A to the elements of A. We ‘abbreviate’ this representation by giving colors to the elements of A and to the edges themselves. Here we see the action of $\wedge$ over $(A_{1})^{2}, (A_{2})^{2}$, and $(A_{3})^{2}$. Much of the work done in this paper uses these graphs; most of the explanations on the graphs are given in the captions following their figures.

Since we are here dealing with Ratsa’s results, we also present the names he uses to refer to the elements of the structures $A_1, A_2,$ and $A_3$. The elements of $A_1$ he calls simply 1 and 0. As for the elements of $A_2$: 1 stands for 11, $\rho$ stands for 10, $\sigma$ stands for 01, and 0 stands for 00. For $A_3$: 1 stands for 111, $\omega$ stands for 110, $\nu$ stands for 101, $\sigma$ stands for 011, $\rho$ stands for 100, $\mu$ stands for 010, $\varepsilon$ stands for 001, and 0 stands for 000. Ratsa also has names for the elements of $A_4$ but, since we will not enter into details about $A_4$ here, we will omit them.

On top of all that, we decided (cf. Fig. 1) to give colors to the elements of each structure! The choice of colors is quite arbitrary, but we tried to organize them. The colorful colors in A3 are arranged almost like a rainbow, going from infrared to ultraviolet. The choice of black for 0 was suggested by the fact that the RGB code for black is (0, 0, 0). This use of colors allows us to represent operations over these structures (cf. Figs. 2 and 3).

3 Modal Formulas as Relations on Cubes

For any formula $\phi (p_{1}, ... , p_{n})$ of propositional S5, and any m-dimensional cube $A_{m}$, we can think of the $A_{m}$-relation expressed by $\phi$ as the set of n-tuples $\langle t_{1}, ... , t_{n}\rangle \in (A_{m})^{n}$ such that $\phi (t_{1}, ... , t_{n}) = 1$. For instance, the formula $p \veebar q$ expresses the $A_{1}$-relation of difference $\{\langle 0, 1 \rangle , \langle 1, 0 \rangle \}$ (it also represents the relation of complementarity for every A – cf. Fig. 8). Also, the formula $p \rightarrow q$ expresses the $A_{1}$-relation less than or equal to $\{\langle 0, 0 \rangle , \langle 0, 1 \rangle , \langle 1, 1 \rangle \}$. To give an example involving modality and a bigger cube, we note that the formula $p \leftrightarrow \square q$ expresses the $A_{2}$-relation $\{\langle 0, 0 \rangle , \langle 0, \rho \rangle , \langle 0, \sigma \rangle \, \langle 1, 1 \rangle \}$. We give plenty of other examples in the next section.

4 Ratsa’s Relations

The relations presented in this section (together with a pair of relations on $A_{4}$, omitted here for the sake of simplicity) constitute a functional completeness criterion for sets of operations of propositional S5. The proof of this fact is beyond the scope of this paper (details can be checked in [8] or in [4]), but some elaboration on it will be found in the next sections.

We start by considering some $A_{1}$-relations. Here $E^{4}(p, q, r, s)$ means: there is an even number of truths among p, q, r, s (a definition of $E^{4}$ in terms of the usual connectives is: $E^{4}(p, q, r, s) =_{df} (p \leftrightarrow q) \leftrightarrow (r \leftrightarrow s)$). When defining a relation, we simply state a formula that expresses it. There is a correspondence between relations and matrices, to be clarified in the next section.

$$R_{0}=_{df} \lnot p, R_{1}=_{df} p, R_{2}=_{df} p \veebar q, R_{3}=_{df} p \rightarrow q R_{4}=_{df} E^4(p, q, r, s).$$

The corresponding $A_{1}$-matrices are:

$$ M_{0} = \begin{bmatrix} 0 \end{bmatrix} $$

$$ M_{1} = \begin{bmatrix} 1 \end{bmatrix} $$

$$ M_{2} = \begin{bmatrix} 0 \; 1 \\ 1 \; 0 \\ \end{bmatrix} $$

$$ M_{3} = \begin{bmatrix} 0 \; 0 \; 1 \\ 0 \; 1 \; 1 \\ \end{bmatrix} $$

$$ M_{4} = \begin{bmatrix} 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \\ 0 \; 0 \; 1 \; 1 \; 0 \; 0 \; 1 \; 1 \\ 0 \; 1 \; 0 \; 1 \; 0 \; 1 \; 0 \; 1 \\ 0 \; 1 \; 1 \; 0 \; 1 \; 0 \; 0 \; 1 \\ \end{bmatrix} $$

We proceed to consider some $A_{2}$-relations. Here $\bigtriangledown p$ reads ‘it is contingent that p’ and is defined as $\Diamond p \wedge \Diamond \lnot p$; $\triangle p$ reads ‘it is rigid that p’ and is defined as $\lnot \bigtriangledown p$; $\bigtriangledown ^+ p$ reads ‘it is contingently true that p’ and is defined as $\bigtriangledown p \wedge p$; $\bigtriangledown ^- p$ reads ‘it is contingently false that p’ and is defined as $\bigtriangledown p \wedge \lnot p$; $\lnot \bigtriangledown ^- p$ reads ‘it is not contingently false that p’; $\lnot \bigtriangledown ^+ p$ reads ‘it is not contingently true that p’. (Roderick Batchelor devised this notation for the more exotic unary modal functions. See [1].)

$R_{5}=_{df} \bigtriangledown ^{-}p$, $R_{6}=_{df} \lnot \bigtriangledown ^{-}p$, $R_{7}=_{df} \bigtriangledown p$, $R_{8}=_{df} \square (p \leftrightarrow \square q)$, $R_{9}=_{df} \square (p \leftrightarrow \Diamond q)$, $R_{10}=_{df} (\square p \wedge q) \vee (\lnot \Diamond p \wedge \lnot q)$, $R_{11}=_{df}\square (p \leftrightarrow \square q) \vee \square (p \leftrightarrow \Diamond q)$, $R_{12}=_{df}\square (p \leftrightarrow \square q) \vee \square (\lnot p \leftrightarrow \Diamond q)$, $R_{13}=_{df}\square (\lnot p \leftrightarrow \square q) \vee \square (p \leftrightarrow \Diamond q)$, $R_{14}=_{df}\square (\bigtriangledown ^{+}p \leftrightarrow \bigtriangledown ^{+} q)$, $R_{15}=_{df}\triangle p \leftrightarrow \triangle q$, $R_{16}=_{df}(p \leftrightarrow q) \vee (\bigtriangledown p \leftrightarrow \bigtriangledown q)$, $R_{17}=_{df} \triangle p \vee \triangle q$, $R_{18}=_{df} \triangle p \wedge \triangle r \wedge ((p \leftrightarrow r) \vee \triangle q)$, $R_{19}=_{df} \triangle p \wedge \triangle r \wedge ((p \leftrightarrow r) \vee \bigtriangledown q)$.

The corresponding $A_{2}$-matrices are:

$$ M_{5} = \begin{bmatrix} \rho \end{bmatrix} $$

$$ M_{6} = \begin{bmatrix} 0 \; \sigma \; 1 \end{bmatrix} $$

$$ M_{7} = \begin{bmatrix} \rho \; \sigma \end{bmatrix} $$

$$ M_{8} = \begin{bmatrix} 0 \; 0 \; 0 \; 1 \\ 0 \; \rho \; \sigma \; 1 \\ \end{bmatrix} $$

$$ M_{9} = \begin{bmatrix} 0 \; 1 \; 1 \; 1 \\ 0 \; \rho \; \sigma \; 1 \\ \end{bmatrix} $$

$$ M_{10} = \begin{bmatrix} 0 \; 0 \; 1 \; 1 \\ 0 \; \rho \; \sigma \; 1 \\ \end{bmatrix} $$

$$ M_{11} = \begin{bmatrix} 0 \; 0 \; 0 \; 1 \; 1 \; 1 \\ 0 \; \rho \; \sigma \; \rho \; \sigma \; 1 \\ \end{bmatrix} $$

$$ M_{12} = \begin{bmatrix} 0 \; 0 \; 0 \; 0 \; 1 \; 1 \\ 0 \; \rho \; \sigma \; 1 \; 0 \; 1 \\ \end{bmatrix} $$

$$ M_{13} = \begin{bmatrix} 0 \; 0 \; 1 \; 1 \; 1 \; 1 \\ 0 \; 1 \; 0 \; \rho \; \sigma \; 1 \\ \end{bmatrix} $$

$$ M_{14} = \begin{bmatrix} 0 \; 0 \; \rho \; \sigma \; 1 \; 1 \\ 0 \; 1 \; \rho \; \sigma \; 0 \; 1 \\ \end{bmatrix} $$

$$ M_{15} = \begin{bmatrix} 0 \; 0 \; \rho \; \rho \; \sigma \; \sigma \; \sigma \; 1 \; 1 \\ 0 \; 1 \; \rho \; \sigma \; \rho \; \rho \; \sigma \; 0 \; 1 \\ \end{bmatrix} $$

$$ M_{16} = \begin{bmatrix} 0 \; \rho \; \rho \; \sigma \; \sigma \; 1 \\ 0 \; \rho \; \sigma \; \rho \; \sigma \; 1 \\ \end{bmatrix} $$

$$ M_{17} = \begin{bmatrix} 0 \; 0 \; 0 \; 0 \; \rho \; \rho \; \sigma \; \sigma \; 1 \; 1 \; 1 \; 1 \\ 0 \; \rho \; \sigma \; 1 \; 0 \; 1 \; 0 \; 1 \; 0 \; \rho \; \sigma \; 1 \\ \end{bmatrix} $$

$$ M_{18} = \begin{bmatrix} 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 1 \; 1 \\ 0 \; 0 \; \rho \; \sigma \; 1 \; 1 \; 0 \; 0 \; \rho \; \sigma \; 1 \; 1 \\ 0 \; 1 \; 0 \; 0 \; 0 \; 1 \; 0 \; 1 \; 1 \; 1 \; 0 \; 1 \\ \end{bmatrix} $$

$$ M_{19} = \begin{bmatrix} 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 1 \; 1 \\ 0 \; \rho \; \rho \; \sigma \; \sigma \; 1 \; 0 \; \rho \; \rho \; \sigma \; \sigma \; 1 \\ 0 \; 0 \; 1 \; 0 \; 1 \; 0 \; 1 \; 0 \; 1 \; 0 \; 1 \; 1 \\ \end{bmatrix} $$

Finally, we consider the $A_{3}$-relations corresponding to the following matrices. We do not have S5 formulas expressing these relations.

$$ M_{20} = \begin{bmatrix} 0 \; \rho \; \mu \; \nu \; \sigma \; 1 \end{bmatrix} $$

$$ M_{21} = \begin{bmatrix} 0 \; \rho \; \mu \; \varepsilon \; \omega \; \nu \; \sigma \; 1 \\ 0 \; \sigma \; \sigma \; \sigma \; \rho \; \rho \; \rho \; 1 \\ \end{bmatrix} $$

$$ M_{22} = \begin{bmatrix} 0 \; \rho \; \mu \; \varepsilon \; \omega \; \nu \; \sigma \; 1 \\ 0 \; \rho \; \nu \; \omega \; \varepsilon \; \mu \; \sigma \; 1 \\ \end{bmatrix} $$

$$ M_{23} = \begin{bmatrix} 0 \; \rho \; \mu \; \varepsilon \; \omega \; \nu \; \sigma \; 1 \\ 0 \; \sigma \; \omega \; \nu \; \mu \; \varepsilon \; \rho \; 1 \\ \end{bmatrix} $$

5 Polymorphisms and Counter-Polymorphisms

We say that an n-ary operation $f(p_{1}, ... , p_{n})$ is a polymorphism of an m-ary A-relation R if for every $\alpha _{i j} (i = 1, ... , m; j = 1, ..., n) \in A$:

if

$$R(\alpha _{1 1}, \alpha _{2 1}, ... , \alpha _{m 1}) \wedge R(\alpha _{1 2}, \alpha _{2 2}, ... , \alpha _{m 2}) \wedge ... \ \wedge R(\alpha _{1 n}, \alpha _{2 n}, ... , \alpha _{m n})$$

then

$$R(f(\alpha _{1 1}, \alpha _{1 2}, ... , \alpha _{1 n}), f(\alpha _{2 1}, \alpha _{2 2}, ... , \alpha _{2 n}), ... \ , f(\alpha _{m 1}, \alpha _{m 2}, ... , \alpha _{m n})).$$

In this definition, the relation R can be replaced by a matrix M whose columns are the m-sequences of elements of A satisfying R (say, arranged in the ‘alphabetical’ order induced by the order: $0, \rho , \mu , \varepsilon , \omega , \nu , \sigma , 1$). We say that a matrix $M'$ is a submatrix of a matrix M if all columns of $M'$ are columns of M. If $M'$ is a submatrix of M we may write $M' \subseteq M$. Given an n-ary formula f and a matrix M with n columns, by f(M) we mean the column generated applying f in each row of M. If c is a column of matrix M we may write $c \in M$ (or, if that is not the case, $c \notin M$). Using these notions, the above definition can be restated (equivalently, but perhaps more clearly) as follows:

A formula f is a counter-polymorphism of matrix M if there is an $M' \subseteq M$ such that $f(M') \notin M$.

A formula f is a polymorphism of matrix M if f is not a counter-polymorphism of M. If f is a polymorphism of M, we may write $f \in Pol(M)$.

Given a formula f and a family of relations $R^{*} = \langle R_{1}, ... , R_{k}\rangle$, the polymorphic profile of f w.r.t. $R^{*}$ is the k-tuple whose i-th term is 1 if $f \in Pol(R_{i})$, and 0 otherwise. We say that f is an exclusive polymorphism of $R_{i}$ (w.r.t. $R^{*}$) if the polymorphic profile of f has a single occurrence of 1, in its i-th place.

Let $R^{*}$ be the family of the relations in Sect. 4 supplemented by the omitted $A_{4}$-relations $R_{24}$ and $R_{25}$. Ratsa established that a set of modal operations F is functionally complete (i.e. sufficient to define every modal operation) if, for every relation $r \in R^*$ there is an operation $f \in F$ such that $f \notin Pol(r)$.

In the reminder of this paper we only consider the family of relations presented in Sect. 4, so when we say the polymorphic profile of f we mean the polymorphic profile of f w.r.t. the family of relations in Sect. 4.

6 Diagrams for Polymorphisms and Counter-Polymorphisms on $A_1$

In this section we consider a simpler version of the diagrams that will be presented in the end of this paper. In Figs. 4 and 5 we consider the polymorphic profile of the functions $\wedge$ and $\downarrow$ w.r.t. the relations $R_0$ – $R_4$. These are the relations whose sets of polymorphisms are precisely the pre-complete systems of two-valued functions, determined by Emil Post in [7].

Fig. 4.

The action of $\wedge$ on $(A_1)^2$ (on the left) and its polymorphic profile (on the right). We can see that $\wedge$ is a polymorphism of $R_0$ since the line connecting the black nodes in the left part is black; it is also a polymorphism of $R_1$ since the line connecting the white nodes in the left part is white; it is a counter-polymorphism of $R_2$ since, as the figure indicates $0 \wedge 1$ = 0 and $1 \wedge 0$ = 0, i.e. we can use arguments which are different to get values that are equal; it is a polymorphism of $R_3$, as the absence of lines connecting the copies of $M_3$ indicates, and is a counter-polymorphism of $R_4$, since (as indicated) with $\wedge$ we can construct, using arguments in $M_4$, a column of values that is not in $M_4$.

Fig. 5.

$\downarrow$ and its polymorphic profile. It is well known that Peirce’s arrow is a function in terms of which every other truth-function can be defined. This follows immediately from the fact that it is a counter-polymorphism of all relations $R_0$ – $R_4$, which characterize the maximal pre-complete systems of truth-functions.

7 Ratsa’s Alleged Exclusive Polymorphism

Ratsa claims that a certain formula (which we call $f_{21}$) is an exclusive polymorphism of the relation $R_{21}$ (or, what is the same, of the matrix $M_{21}$). He is interested in such a formula because it helps him to prove that his criterion for determining if a single function is functionally complete (i.e. if it is a Sheffer function for S5) is as good as it can be (cf. [8], p. 278).

To properly present Ratsa’s formula, we introduce some preliminary notions (which are interesting in themselves). We start by defining the straightforward propositional relations of independence, connection, compatibility, and incompatibility:

$$\begin{aligned}&Ind(p, q) =_{df}\Diamond (p \wedge q) \wedge \Diamond (p \wedge \lnot q) \wedge \Diamond (\lnot p \wedge q) \wedge \Diamond (\lnot p \wedge \lnot q). \\&Con(p, q) =_{df} \lnot Ind(p, q). \\&Comp(p, q) =_{df} \Diamond (p \wedge q). \\&Incomp(p, q) =_{df} \lnot \Diamond (p \wedge q). \end{aligned}$$

It is interesting to notice that for $A_{i} (i \in \{1, 2, 3\})$ the $A_{i}$-relation expressed by $Ind(p, q) = \varnothing$. In order to find a pair of independent propositions, we need to resort to $A_{4}$ (this fact is noted w.r.t. bitstrings in [2], except that what we call independence is there called unconnectedness. In their terminology: ‘unconnectedness requires bitstrings of length at least 4’).

Since the compatibility relation will be significant in our next definition, we give an explicit characterization of its $A_{3}$-instances, from which the other instances may be derived. We start the characterization by listing some compatible elements of $(A_{3}) ^{2}$: $\langle \mu , \omega \rangle , \langle \mu , \sigma \rangle , \langle \varepsilon , \nu \rangle , \langle \varepsilon , \sigma \rangle , \langle \nu , \omega \rangle , \langle \nu , \rho \rangle , \langle \nu , \sigma \rangle , \langle \omega , \rho \rangle , \langle \omega , \sigma \rangle$ and we finish it by noticing that everything different from 0 is compatible with 1 and with itself, and that compatibility is a symmetric relation.

The modal profile of a pair of propositions p, q is the 4-tuple $Modpro(p, q) =_{df} \langle Comp(p, q), Comp(p, \lnot q), Comp(\lnot p, q), Comp(\lnot p, \lnot q) \rangle$.

To present $f_{21}$ we need to introduce some formulas used in its definition.

$$\begin{aligned}&S(p, q) =_{df} \square (p \vee q) \vee \square (p \rightarrow q) \vee \square (q \rightarrow p) . \\&V(p, q) =_{df} S(p , q) \wedge S(p, \lnot q) \wedge S(\lnot p, q) \wedge S(\lnot p, \lnot q) . \end{aligned}$$

S ‘says’ that p and q are connected even if we disregard its (possible) incompatibility (or equivalently: there is at least one 0 in the last three entries of Modpro(p, q)), while V ‘says’ that p and q are strongly connected, i.e., either (at least) one of them is rigid, or they are both contingent but then either $\square (p \leftrightarrow q)$ or $\square (p \veebar q)$ (this is equivalent to say that sum of the terms of Modpro(p, q) is less than 3).

Ratsa’s formula is:

$$f_{21} = (V(p, q)\rightarrow ((p \rightarrow q) \wedge \lnot \square q)) \wedge (((p \leftrightarrow S(p, q)) \wedge (q \rightarrow S(p, q))) \vee V(p, q)).$$

To see that this is not an exclusive polymorphism of $R_{21}$ it is enough to notice that it is not a polymorphism of $R_{21}$. This is obvious given that $\{\langle \rho , \sigma \rangle , \langle \mu , \sigma \rangle \} \subseteq R_{21}$ and $f_{21}(\rho , \mu ) = \varepsilon$, $f_{21}(\sigma , \sigma ) = 1$ and that $\langle \varepsilon , 1 \rangle \notin R_{21}$. This last claim can perhaps be more easily checked by considering Fig. 10, where we present the action of $f_{21}$ over $A_{1}$, $A_{2}$, and $A_{3}$, and its polymorphic profile.

Fig. 6.

The projection of the first argument ($\pi ^2_1$) is a universal polymorphism. We take advantage of the space left by the absence of counter-polymorphisms of this operation to present the framework we are working with. On the right side of this figure you can see (pairs of) the translation into colors (following Fig. 1) of the relations presented in Sect. 6.

Fig. 7.

The negation of the second argument $\lnot \pi ^2_2$ over $A_{1}$, $A_{2}$ and $A_{3}$ and its polymorphic profile. Notice that the counter-polymorphisms are indicated by horizontal lines connecting relevant columns of the matrices. Notice also that the polymorphic profile of $\lnot \pi ^2_2$ w.r.t. $R_0$ – $R_4$ (0, 0, 1, 0, 1) is complementary of that of $\wedge$ (1, 1, 0, 1, 0) (cf. Fig. 4).

Fig. 8.

This graph represents the action of $\veebar$. It is interesting that the white lines on the left side represent precisely the relation of complementarity in the structures $A_{1}$, $A_{2}$ and $A_{3}$.

Fig. 9.

This is the representation of the very well known (boolean) operation $\uparrow$ (the Sheffer stroke). Notice that it is an exclusive polymorphism of $R_{10}$.

Fig. 10.

$f_{21}$ over $A_{1}$, $A_{2}$ and $A_{3}$ and its polymorphic profile.

Fig. 11.

$\langle \top , \wedge , \leftrightarrow , \uparrow , \wedge , \downarrow , \rightarrow \rangle$ is an exclusive polymorphism of $M_{21}$.

8 Moody Truth-Functions

The definition in this section is essentially the same found in [3], p. 35. Recall the definition of Modpro, given in the last section.

The moody truth-functional representation of a binary modal operation f is a sequence of eight binary truth-functions $\langle f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6}, f_{7}, f_{8} \rangle$, together with the proviso:

• if $Modpro(p, q) = \langle 1, 1, 1, 1 \rangle$, apply $f_{1}$;

• if $Modpro(p, q) = \langle 1, 1, 1, 0 \rangle$ or $\langle 0, 0, 0, 1\rangle$, apply $f_{2}$;

• if $Modpro(p, q) = \langle 1, 1, 0, 1 \rangle$ or $\langle 0, 0, 1, 0\rangle$, apply $f_{3}$;

• if $Modpro(p, q) = \langle 1, 0, 1, 1 \rangle$ or $\langle 0, 1, 0, 0\rangle$, apply $f_{4}$;

• if $Modpro(p, q) = \langle 0, 1, 1, 1 \rangle$ or $\langle 1, 0, 0, 0\rangle$, apply $f_{5}$;

• if $Modpro(p, q) = \langle 1, 1, 0, 0 \rangle$ or $\langle 0, 0, 1, 1\rangle$, apply $f_{6}$;

• if $Modpro(p, q) = \langle 1, 0, 1, 0 \rangle$ or $\langle 0, 1, 0, 1\rangle$, apply $f_{7}$;

• if $Modpro(p, q) = \langle 1, 0, 0, 1 \rangle$ or $\langle 0, 1, 1, 0\rangle$, apply $f_{8}$.

Since we are here ignoring $A_{4}$, when using moody truth-functions, we will restrict ourselves to the 7-tuples corresponding to $f_{2}-f_{8}$.

We claim that the operation expressed by $\langle \top , \wedge , \leftrightarrow , \uparrow , \wedge , \downarrow , \rightarrow \rangle$ is an exclusive polymorphism of $R_{21}$. We support our claim with Fig. 11 and with the captions of the figures preceding it.

9 Conclusion

We are glad to give an exoteric presentation of a somewhat esoteric result, and we hope that this paper is not too enigmatic. We believe that the techniques presented here are also useful in the investigations on clones of k-valued functions (cf. [6]) and we expect to give some new results on this matter soon.