\documentstyle{bsl} %%%%%%%%%%%% \pagestyle{plain} \setcounter{page}{1} \input tcilatex \QQQ{Language}{ American English } \begin{document} \noindent Grzegorz Malinowski\vspace{.3in} \begin{enumerate} \begin{description} \item \begin{center} \noindent {\large MANY-VALUED POST LOGICS}\vspace{.3in} \end{center} \end{description} \end{enumerate} \noindent Emil Post's doctoral dissertation appeared in print as Post (1921)% \footnote{% Abstracted as Post (1920).}is one of the major logic works of the first half of the 20th Century. It's contribution to the semantic and metalogic of the classical propositional calculus can hardly be overestimated. Besides of that one of the parts of the dissertation contained the description of the n-valued (n $\geq 2$, n finite) functionally complete algera of logic and its interpretation in terms of the (n-1)\ dimensional Euclidean space. Emil Post independently from \L ukasiewicz introduced his (finitely) many-valued logics in 1920\footnote{% The priority of \L ukasiewicz is unquestioned, though it concerns merely the introduction of the three-valued logic in 1918, widely published as \L ukasiewicz (1920), and thus the first construction many-valued logic at all. In \L ukasiewicz (1929) one finds the generalization of the original concept onto the case of finite and infinite (countable and continuum) number of values.}. The main motivation of the actual consideration of truthvalues next to the truth and falsity stemmed from his work on functional completeness properties of the classical propositional logic. Though apparently tailored algebraically, the fuctionally complete Post logic and algebras have been playing an important role in the area of philosophical logic as well as in advanced applications in Computer Science. The aim of the paper is a concise presentation of Post work on many-valued logic and to show evolution of the original ideas.\vspace{.2in}\ . \noindent {\bf 1. Post matrices}\vspace{.2in}\ \noindent The fundamental, many--valued constructions of Post are connected with two, primitive in the Principia Mathematica, propositional connectives: negation ($\neg $) and disjunction ($\vee $). For any natural $n\geq 2$ Post builds an $n$--valued logical algebra on the linearly ordered set of objects \vspace{3mm} $P_{n} = \{t_{1}, \, t_{2}, \ldots , t_{n}\}$ \vspace{3mm} \noindent $(t_{1} < t_{j} \,$ iff $\, i < j)$ equipped with two operations: unary cyclic {\em rotation}\/ $\neg$ ({\em cyclic negation}) and binary disjunction $\vee$, defined in the following way: \vspace{3mm} $\neg t_{i} = \left\{ \begin{array}{ll} \mbox{$t_{i+1}$} & \mbox{if $\,\,i \neq n$} \\ \mbox{$t_{1}$} & \mbox{if $\,\,i = n$} \end{array} ; \hspace{.5in} \begin{array}{l} t_{i} \vee t_{j} = t_{max(i,j)}. \end{array} \right.$ \vspace{3mm} \noindent The disjunction function fixes on a natural and entirely intuitive meaning of the disjunction connective, typical at least for the most known \vspace{1mm} many-- \linebreak \parbox{3.3in}{valued (including infinite) constructions. In plain terms, the logical value of disjunctive proposition equals the greater of the values of its components. The function of cyclic rotation permutes, in some specified manner, the set $P_{n}$ and the negation corresponding to it is, the case $n = 2$ being excluded, quite special -- compare the beside table. It is just the fact of combining of the latter with an appropriate binary function \linebreak} \ \hspace{.4in} \ \parbox{.9in}{\begin{tabular}{@{}c|c} $a$ & $\neg a$ \\ \cline{1-1} \hline $t_{1}$ & $t_{2}$ \\ $t_{2}$ & $t_{3}$ \\ $:$ & $:$ \\ $:$ & $:$ \\ $t_{n}$ & $t_{1}$ \end{tabular}} \noindent of algebra on $P_n$ that warrants the functional completeness of that algebra, i.e. it ensures that by means of the primitive functions, there can be defined every finitely--argument function on $P_n$, including constant functions and hence the objects $t_1,t_2,\ldots ,t_n$. For a given finite $% n\geq 2$ the algebraic structure:\vspace{3mm} ${\cal P}_n=(\{t_1,t_2,\ldots ,t_n\},\;\neg ,\;\vee )$\vspace{3mm} \noindent will be called $n$--{\em valued Post algebra}. The matrix $P_n$ naturally associated with the algebra ${\cal P}_n$: \vspace{3mm} $P_n=(\{t_1,t_2,\ldots ,t_n\},\;\neg ,\;\vee ,\;\{t_n\})$ \vspace{3mm} \noindent will, in the sequel, be referred to as the (basic) $n$--{\em % valued Post matrix}.\/ It is easily seen that the two--valued Post matrix is isomorphic to the negation--disjunction matrix for the classical propositional calculus. To check it, one must replace $t_{1}$ in $P_{2}$ by the falsity $(0)$ and $t_{2}$ by the symbol of truth $1$. Simultaneously, however, the matrices $P_{n}$ for $n > 2$ are totally incompatible to the mentioned classical matrix, which is the result of nonstandard mode of the negation connective. Hence, for instance, for $n = 3 \; t_{3}$ could be the only counterpart of ``truth'' respective the adopted interpretation of disjunction but then $t_{1}$ would have to correspond to ``falsity'' as $% \neg t_{3} = t_{1}$, which should not take place because $\neg \neg t_{3} = \neg t_{1} = t_{2} \neq t_{3}$. A contradiction. It is remarkable that among the laws of $n$--valued logics determined by Post matrices $P_{n}$ there are many--valued counterparts of some significant classical tautologies expressed in terms of negation and disjunction connectives, including the ``generalized law of the excluded middle'': \vspace{3mm} $p \, \vee \, \neg p \, \vee \, \neg \neg p \, \vee \ldots \vee \underbrace{% \neg \neg \ldots \neg}_{(n-1)\,\,times}p$. \vspace{3mm} \noindent On the other hand, however, the connectives of implication, conjunction and equivalence introduced up to the two--valued pattern \vspace{3mm} $\alpha \rightarrow \beta = \neg \alpha \vee \beta$, $\alpha \wedge \beta = \neg (\neg \alpha \vee \neg \beta)\,\,\,\,\,\,$ and $% \alpha \equiv \beta = (\alpha \rightarrow \beta) \wedge (\beta \rightarrow \alpha)$ \vspace{3mm} \noindent evidently stray away from their classical counterparts. Thus, e.g. the connective $\rightarrow $ resembles no ''reasonable'' connective of implication and $\wedge $ is neither associative nor symmetric. Accordingly, due to the second remark the formula \vspace{3mm} $\neg (p\,\wedge \,\neg p\,\wedge \,\neg \neg p\,\wedge \ldots \wedge \underbrace{\neg \neg \ldots \neg }_{(n-1)\,\,times}p)$ \\ \noindent is not among the laws of n-valued Post logic..One may also check that this formula as it stands, without internal parentheses, is even not well formed. The property of functional completeness of Post logic algebras, i.e. the property that every finitary mapping $f:P_n^m\rightarrow P_n$ ($m\geq 0,m$ finite) can be represented as a composition of the operations $\neg $ and $% \vee $, warrants that all counterparts of classical connectives are definable in ${\cal P}_n$. This implies that the whole classical logic may be interpreted within any many-valued system of Post logic. \vspace{.2in} \noindent {\bf 2. Interpretation}\vspace{.2in}\ \noindent Post did manage to present a semantical interpretation for his nonstandard matrices $P_n$ providing the following, Euclidean in its spirit, construction of the ``spaces'' $E^{n-1}$: \begin{itemize} \item[(1)] Elements $P\in E^{n-1}$ are $(n-1)$--element tuples of ordinary two--valued propositions (represented by small letters), $P=(p_1,p_2,\ldots ,p_{n-1})$ subject to the condition that the true propositions are listed before the false. \item[(2)] $\neg P$ is formed by replacing the first false element of $P$ by its denial, $\neg P=(\neg (p_1\wedge p_2\wedge \ldots \wedge p_n),\neg (p_1\wedge p_2\wedge \ldots \wedge p_{n-1})\wedge (p_1\vee p_2),\ldots ,\neg (p_1\wedge p_2\wedge \ldots \wedge p_{n-1})\wedge (p_{n-2}\vee p_{n-1}))$, the connectives on the right--hand side are the usual (classical) connectives; but if there is no false element in $P$, then all of them are to be denied, in which case $\neg P$ is a sequence of false propositions. \item[(3)] When $P=(p_1,p_2,\ldots ,p_{n-1})$ and $Q=(q_1,q_2,\ldots ,q_{n-1})$, then $P\vee Q=(p_1\vee q_1,p_2\vee q_2,\ldots ,p_{n-1}\vee q_{n-1})$ (with the right--hand side $\vee $ as above). \end{itemize} \noindent The mapping $\underline{i} : E^{n-1} \rightarrow P_{n}$: \vspace{3mm} $\underline{i}(P) = t_{i}\,\,$ iff $P$ contains exactly $(i-1)$ true propositions \vspace{3mm} \noindent establishes an isomorphism of $(E^{n-1}, \vee, \neg)$ onto the Post algebra $P_{n}$. The discussed interpretation shows, among others, that similarly stated logical values appearing in diverse matrices $P_n$ are, regarding the author's intention, objects different from each other. Therefore, for the sake of precision the symbols of logical values $t_i$ should be always indexed by a parameter assigning them to a given matrix.\vspace{3mm} $Example.$ Five-valued Post logic based on the set $P_5=\{t_1,\,t_2,t_3,% \,t_4,t_5\}$ of values may be interpreted in $E^4$ consisting of the following elements:\vspace{3mm} ( 0 , 0 , 0 , 0 ) ( 1 , 0 , 0 , 0 ) ( 1 , 1 , 0 , 0 ) ( 1 , 1 , 1 , 0 ) ( 1 , 1 , 1 , 1 )\vspace{3mm} \noindent which correspond to $t$, $\,t_2$, $t_3$, $\,t_4$, $t_5$, respectively. While $\vee $ of $E^4$ ''coincides'' with Boolean sum on the axes, an application of $\neg $ results in descending step down on the array with the exception that $\neg $ ( 1 , 1 , 1 , 1 ) $=$ ( 0 , 0 , 0 , 0 ).% \vspace{3mm} Urquhart (1973) gave an interpretation of Post logics in terms of a Kripke-style semantics\vspace{3mm} $K_n=($ $S_n$ $,$ $\leq $ $,$ $\vdash $ $),$\vspace{3mm} \noindent where $S_n=\{0,1,...,n-2\}$, $\vdash \subseteq S_n\times For$ and the relation $\leq $ is transitive:\vspace{3mm} (Tr) If $x$ $\vdash $ $\alpha $ and $x\leq y\in S_n$, then $y\vdash \alpha .$% \vspace{3mm} \noindent The conditions stating the sense of Post connectives are the following: \vspace{3mm} \begin{tabular}{lll} $x \vdash \neg \alpha$ & iff & $y \vdash \alpha\,\,\,\,$ for no $\,\,\,\, y \in S_{n}\,\,\,\,$ or there is a $\,\,\,\,y \in S_{n}$ \\ & & such that $\,\,\,\,y < x\,\,\,\,$ and $\,\,\,\, y \vdash \alpha$\\ $x \vdash \alpha \vee \beta$ & iff & $x \vdash \alpha\,\,\,\,$ or $\,\,\,\, x \vdash \beta$. \end{tabular} \vspace{3mm} Several meanings may be attached to ``reference points'' $x\in S_n$% ,.Urquhart suggests a temporal interpretation: $0$ being the present moment, $x\neq 0$ a future moment, then $``x\vdash \alpha ^{\prime \prime }$ reads ``% $\alpha $ being true at (the moment) $x$''. It is worth noting that the assumption (Tr) guarantees that any proposition true at $x$ is also true at every moment $y$ future to $x$. That obviously means that in the framework elaborated, propositions are treated as temporally definitive units and, as such, they must not contain any occasional, time--depending expressions (such as e.g. ``now'', ``today'' etc.). A quick reflection upon the $Example$ would lead one to the conclusion that Urquhart's interpretation is entirely compatible with the original interpretation envisaged by Post himself. Apart from the basic matrices $P_n$, Post considered matrices with more designated elements. In turn, he also defined a family of functionally incomplete $n$--{\em valued implicative matrices}\/ with $k$ designated values $(1\leq k j \,\,\,\,$ and $\,\,\,\, i \geq n-k+1$} \\ \mbox{$t_{n-i+j}$} & \mbox{if $ \,\,\,\,\,\,i > j\,\,\,\,$ and $\,\,\,\,i < n-k+1$}. \end{array} \right. $ \vspace{3mm} \noindent The matrices of that family can serve as a tool for description of the implication connectives of other known many--valued logics. So, for instance, $\rightarrow_{n1}$ and $\rightarrow_{n \; n-1}$ are $n$--valued of \L ukasiewicz and G\"{o}del implication respectively (to obtain the implicative G\"{o}del matrix one ought to reduce the set of distinguished values to $\{t_{n}\}$, previously having built the truth--table for implication). The fact that Post has designated many (at a time) logical values induced a significant impulse for a just emerging, in the 1920's, theory of logical matrices. It seems that other originators of many--valued logics ignored that possibility, or, did not attach great importance to it.\vspace{.2in}\ \noindent {\bf 3. Axiomatization of functionally complete systems of }$n$% {\bf --valued logic}\vspace{.2in}\ \noindent The original $(\neg ,\vee )$ systems of Post's logic are not axiomatized so far. However, the problem of their axiomatizability has been for years a foregone matter; hence S\l upecki (1939) has constructed the largest possible class of functionally complete finite logics and gave a general method of their axiomatization. From this it evidently follows that also Post logics are axiomatizable albeit the problem of providing axioms for their original version still remains open. S\l upecki matrix $S_{nk}$ ($n$ being a given natural number, $1 \leq k \leq n)$ is of the form: \vspace{3mm} $S_{nk} = ( \{1, 2, \ldots, n\}, \rightarrow, R, S, \{1, 2, \ldots, k\})$. \vspace{3mm} \noindent where $\rightarrow$ is a binary (implication), and $R,S$ unary operations defined in the following way: \vspace{3mm} $x \rightarrow y = \left\{ \begin{array}{ll} y & \mbox{if $1 \leq x \leq k$} \\ 1 & \mbox{if $k < x \leq n$} \\ & \end{array} , \right. $ \vspace{3mm} $R(x) = \left\{ \begin{array}{ll} \mbox{$x + 1$} & \mbox{if $1 \leq x \leq n-1$} \\ 1 & \mbox{if $x = n$} \\ & \end{array} , \right. $ \vspace{3mm} $S(x) = \left\{ \begin{array}{ll} 2 & \mbox{if $x = 1$} \\ 1 & \mbox{if $x = 2$} \\ x & \mbox{if $3 \leq x \leq n$} \\ & \end{array} . \right.$ \vspace{3mm} \noindent Functional completeness of each of these matrices is based on Picard (1935): $R$ and $S$ are two of the Picard's functions, in order to define the third, it suffices to put: \vspace{3mm} \noindent \begin{tabular}{llll} $Hx = (x \rightarrow R(x \rightarrow x)) \rightarrow Sx $ & for $k = 1,$ & then & $Hx = \left\{ \begin{array}{ll} 1 & \mbox{if $x = 2$}\\ x & \mbox{if $x \neq 2$} \end{array} \right. $\\ & & & \\ $Hx = R(x \rightarrow x) \rightarrow x$ & for $k > 1,$ & then & $Hx = \left\{ \begin{array}{ll} 1 & \mbox{if $x = k$}\\ x & \mbox{if $x \neq k$}. \end{array} \right. $ \end{tabular} \vspace{3mm} S\l upecki produced an effective proof of axiomatizability of every logic determined by the matrix $S_{nk}$ (any pair $(n,k)$ as above) giving a long list of axioms formulated in terms of implication and special one--argument connectives defined through the superpositions of $R,S$, and $H$. The chief line of approach here is to make capital of the stand character of implication, which can be classically axiomatized using MP (the Detachment Rule). S\l upecki extends MP onto the whole language, taking the \L ukasiewicz's formula: $((p\rightarrow q)\rightarrow r))\rightarrow ((r\rightarrow p)\rightarrow (s\rightarrow p))$ as the only axiom for implication and provides an inductive, combinatorial completeness proof In the end the method of axiomatization in Rosser, Turquette (1952) should at least be mentioned here. Rosser and Turquette determine the conditions that make finitely many-valued propositional logic resemble more the Classical Propositional Calculus, CPC. and hence simplified problem of their axiomatization. The heart of the method is the use special $j$ operators, which play a role of identifiers of respective logical values. Since Post matrices are functionally complete all necessary connectives are definable, including the counterparts of all classical connectives and $j$ as well. Consequently, all Post logics are axiomatizable in the Rosser and Turquette style.\footnote{% An overall idea of axioatization given in Rosser and Turquette coincides with that by Supecki since the central is here also the use of a ''classical implication'' and its rule, MP, as base for the axiom system.} \vspace{.2in}. \noindent {\bf 4. Algebraic counterparts and a new formalization} \vspace{.2in}\ \noindent The concept of Post algebra of order $n(n\geq 2)$ was introduced by Rosenbloom (1942) who defined Post algebras by means of the rotation $% \neg $, the disjunction $\vee $ and some auxiliary functions. Subsequently, it has undergone several modifications resulting both from theoretical and practical reasons (see e.g. Dwinger (1977)). A particular importance among them has the lattice--theoretical characterization (Epstein (1960)) fixing a creative direction for the studies (see 14.4). The equational definition of these algebras is due to Traczyk (1964): {\em Post algebra of order n}\/ $% (n\geq 2)$ can be presented as an algebra having two binary $\cup ,\cap ,$ $% n $ unary operations $-,D_1,\ldots ,D_{n-1}$ and constants $e_0,\ldots ,e_{n-1} $ \vspace{3mm} ${\cal L} = (L, \; \cup, \; \cap, \; -, \; D_{1}, \ldots, D_{n-1}, \; e_{0}, \ldots, \; e_{n-1})$, \vspace{3mm} \noindent satisfying the conditions: \vspace{3mm} \noindent \begin{tabular}{ll} (1) & $(L, \cup, \cap)$ is a distributive lattice with zero, $0 = e_{0}$, and unit, $1 = e_{n-1}$ \\ (2) & $- (x \cup y) = - x \cap - y, \,\,\,\,\,\, -- x =x$ \\ (3) & $e_{i} \cap e_{j} = e_{i}\,$ if $\, i \leq j$\\ (4) & $D_{i}(x \cup y) = D_{i}(x) \cup D_{i}(y), \,\,\,\,\,\, D_{i}(x \cap y) = D_{i}(x) \cap D_{i}(y)$\\ (5) & $D_{i}(x) \cup - D_{i}(x) = 1, \,\,\,\,\,\, D_{i}(x) \cap - D_{i}(x) = 0$\\ (6) & $D_{i}(x) \cap D_{j}(x) = D_{i}(x) \,$ if $\, i \leq j$\\ (7) & $D_{i}(-x) = - D_{n-i}(x)$\\ (8) & $D_{i}(e_{j}) = 1\,$ for $\, i \leq j, \,\,\,\,\,\, D_{i}(e_{j}) = 0\, $ for $\, j < i$\\ (9) & $x = (D_{1}(x) \cap e_{1}) \cup (D_{2} \cap e_{2}) \cup \ldots \cup (D_{n-1}(x) \cap e_{n-1})$. \end{tabular} \vspace{3mm} It can be proved that the set $C(L) = \{D_{i}(x) : x \in L,\, i \in \{1, \ldots, n-1\}\}$ is closed under lattice operations and that the structure $% (C(L,n), \cup, \cap, - 1, 0)$ is a Boolean algebra. Apparently, each Post algebra of order $2$ is a Boolean algebra as well. The simplest Post algebra of order $n$ is the structure based on the set of logical values $\{t_{1}, \ldots, t_{n}\}$ having the operations $x \cup y = max\{x,y\}, \, x \cap y = min\{x,y\}$ and \vspace{3mm} $D_{i}(x) = \left\{ \begin{array}{ll} \mbox{$t_{n}$} & \mbox{if $x > t_{i}$} \\ \mbox{$t_{1}$} & \mbox{if $x \leq t_{i}$} \end{array} ; \,\,\,\,\,\, \begin{array}{l} \mbox{$-t_{i} = t_{n-i+1}$}. \end{array} \right. $ \vspace{3mm} \noindent To the end, every Post algebra of order $n$ is isomorphic to some field of sets $P(U,n)$ (Wade (1945)). \ At present the attention is generally focused on the formalization of Post logics basing on Rousseau algebras. Rousseau (1969) noticed that any Post algebra of order $n$ is a pseudo--Boolean algebra (see Rasiowa (1974)). Consequently, he proposed a definition of Post algebra (of order $n$) which turned out to be exceedingly important from the point of view of applications (see e.g. Rasiowa (1977)). A new operation appearing in this definitional version is a binary operation of {\em relative pseudocomplement}% \/ $\rightarrow $, which on the set of constants $\{e_0,\ldots ,e_{n-1}\}$ can be described as follows: \vspace{3mm} $e_{i} \rightarrow e_{j} = \left\{ \begin{array}{ll} \mbox{$e_{n-1}$} & \mbox{when $i \leq j$} \\ \mbox{$e_{j}$} & \mbox{otherwise.} \end{array} \right. $ \vspace{3mm} \noindent The system of $n$--valued propositional calculus corresponding to the Rousseau algebras (given $n$) is determined in the language with connectives $\neg ,\;$ $\rightarrow ,\;$ $\vee ,\;$ $\wedge ,\;$ $\equiv ,\;$ $D_1,\ldots ,\;$ $D_{n-1},\;$ $e_0,\ldots ,\;$ $e_{n-1}$ (for the sake of brevity algebraic symbols used here bear new meaning, e.g. $e_i$ now refers to logical constants i.e. zero--argument connectives). Its axioms are the schemes of complete derivational axiom system of intuitionistic logic% \footnote{% see e.g. Rasiowa (1974). p. 264} and, for every $i=1,\ldots ,n-1$, \vspace{3mm} \noindent \begin{tabular}{ll} (P11) & $D_{i}(\alpha \vee \beta) \equiv (D_{i}\alpha \vee D_{i}\beta)$\\ (P12) & $D_{i}(\alpha \wedge \beta) \equiv (D_{i}\alpha \wedge D_{i}\beta)$\\ (P13) & $D_{i}(\alpha \rightarrow \beta) \equiv ((D_{1}\alpha \rightarrow D_{1}\beta) \wedge (D_{2}\alpha \rightarrow D_{2}\beta) \wedge \ldots \wedge (D_{i}\alpha \rightarrow D_{i}\beta))$\\ (P14) & $D_{i}(\neg \alpha) \equiv \neg D_{1}\alpha$\\ (P15) & $D_{i}D_{j}\alpha \equiv D_{j}\alpha$\\ (P16) & $D_{i}e_{j}\,\,$ when $\,\,i \leq j\,\,$ and $\,\,\neg D_{i}e_{j}\,\,$ when $\,\,j > j$\\ (P17) & $\alpha \equiv (D_{1}\alpha \wedge e_{1}) \vee \ldots \vee (D_{n-1}\alpha \wedge e_{n-1})$\\ (P18) & $D_{1}\alpha \, \vee \, \neg D_{1}\alpha$. \end{tabular} \vspace{3mm} \noindent And, apart from MP, an extra inference rule is \vspace{3mm} \parbox{.1in}{($r_{n}$)} \ \hspace{.1in} \ \parbox{1in}{ \begin{center} $\alpha$ \\ \vspace{.001in} \rule[.02in]{.5in}{.01in} \\ \vspace{.01in} $D_{n-1} \alpha$ \end{center}} \vspace{3mm} The predicate calculi for Post logics are built in a standard way on the basis of propositional calculi. The most systematic studies of them carried out so far are due to Rasiowa (1974).\vspace{.2in} \vspace{.1in} \noindent {\bf Bibliography}\vspace{.2in} \noindent Dunn, J. M. and Epstein, G. (eds.) (1977). {\em Modern uses of multiple--valued logic\/}. D. Reidel, Dordrecht, Holland.\vspace{.1in} \noindent Dwinger, Ph. (1977). A survey of the theory of Post algebras and their generalizations [in:] Dunn and Epstein (eds.) (1977), 53--75% \vspace{.1in}. \noindent Epstein, G. (1960). The lattice theory of Post algebras. {\em % Transactions of the American Mathematical Society\/}, {\bf 95}, 300--317.% \vspace{.1in} \noindent \L ukasiewicz, J. (1920). O logice tr\'ojwarto\'sciowej. {\em Ruch Filozoficzny\/}, {\bf 5}, 170--171. English tr. On three--valued logic [in:] {\em Selected works\/} -- see \L ukasiewicz (1961), 87--88.\vspace{.1in} \noindent \L ukasiewicz, J. (1929). {\em Elementy logiki matematycznej\/}. Skrypt. Warszawa (II ed. Warszawa 1958, PWN); English tr. {\em Elements of Mathematical Logic\/} translated by Wojtasiewicz, O. Oxford, Pergamon Press, 1963.\vspace{.1in} \noindent Picard, S. (1935). Sur les fonctions d\'efinies dans les ensembles finis quelconques. {\em Fundamenta Mathematicae\/}, {\bf 24}, 198--302% \vspace{.1in}. \noindent Post, E. L. (1920). Introduction to a general theory of elementary propositions. {\em Bulletin of the American Mathematical Society\/}, {\bf 26}% , 437.\vspace{.1in} \noindent Post, E. L. (1921). Introduction to a general theory of elementary propositions. {\em American Journal of Mathematics\/}, {\bf 43}, 163--185.% \vspace{.1in} \noindent Rasiowa, H. (1974). {\em An algebraic approach to non--classical logics\/}. North--Holland, PWN, Amsterdam, Warszawa.\vspace{.1in} \noindent Rasiowa, H. (1977). Many--valued algorithmic logic as a tool to investigate programs [in:] Dunn and Epstein (eds.) (1977), 79--102.% \vspace{.1in} \noindent Rosenbloom, P. C. (1942). Post algebra. I. Postulates and general theory. {\em American Journal of Mathematics\/}, {\bf 64}, 167--188.% \vspace{.1in} \noindent Rosser, J. B. and Turquette, A. R. (1952). {\em Many--valued logics\/}. North--Holland, Amsterdam.\vspace{.1in} \noindent Rousseau, G. (1969). Logical systems with finitely many truth--values. {\em Bulletin de l'Acad\'emie Polonaise des Sciences, S\'erie des sciences math\'ematiques, astronomiques et physiques\/}, {\bf 17}, 189--194.\vspace{.1in} \noindent S\l upecki, J. (1939b). Dow\'od aksjomatyzowal\-no\'s\-ci pe\l nych system\'ow wie\-lo\-war\-to\'s\-cio\-wych ra\-chun\-ku zda\'n (Proof of the axiomatizability of full many--valued systems of propositional calculus). {\em Comptes rendus de s\'eances de la Soci\'et\'e des Sciences et des Lettres de Varsovie Cl. III\/}, {\bf 32}, 110--128.\vspace{.1in} \noindent Traczyk, T. (1964). An equational definition of a class of Post algebras. {\em Bulletin de l'Acad\'emie Polonaise des Sciences Cl. III\/}, {\bf 12}, 147--149.\vspace{.1in} \noindent Urquhart, A. (1973). An interpretation of many--valued logic. {\em % Zeitschrift f\"ur Mathematische Logik und Grundlagen der Mathematik\/}, {\bf % 19}, 111--114.\vspace{.1in} \noindent Wade, C.I. (1945). Post algebras and rings. {\em Duke Mathematical Journal\/}, {\bf 12}, 389--395.\vspace{.1in} \noindent Whitehead, A. N. and Russell, B. (1910). {\em Principia Mathematica\/}, vol. {\bf I}. Cambridge U.P.\vspace{.5in} \noindent Department of Logic \noindent University of \L \'od\TeXButton{z}{\'z} \noindent Poland \noindent e-mail:\ gregmal@krysia.uni.lodz.pl \end{document}