Many-valued logics

Infinite-valued Logics

content/many-valued-logic/infinite-valued-logics/infinite-valued-logics.tex

% Part: many-valued-logic% Chapter: three-valued-logics\documentclass[../../../include/open-logic-chapter]{subfiles}\begin{document}\olchapter{mvl}{inf}{Infinite-valued Logics}\olimport{introduction}\olimport{lukasiewicz}\olimport{goedel}\OLEndChapterHook\end{document}

content/many-valued-logic/infinite-valued-logics/introduction.tex

% Part: many-valued-logic% Chapter: infinite-valued-logics% Section: introduction\documentclass[../../../include/open-logic-section]{subfiles}\begin{document}\olfileid{mvl}{inf}{int}\olsection{Introduction}The number of truth values of a matrix need not be finite.  An obviouschoice for a set of infinitely many truth values is the set ofrational numbers between $0$ and~$1$, $V_\infty = [0,1] \cap \Rat$,i.e.,\begin{align*}    V_\infty & = \Setabs{\frac{n}{m}}{n,m \in \Nat \text{ and } n\le m}.\intertext{When considering this infinite truth value set, it is oftenuseful to also consider the subsets}V_m & = \Setabs{\frac{n}{m-1}}{n \in \Nat \text{ and } n\le m}\intertext{For instance, $V_5$ is the set with $5$ evenly spaced truth values,}V_5 & = \{0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1\}.\end{align*}In logics based on these truth value sets, usually only $1$ isdesignated, i.e., $V^+ = \{1\}$.  In other words, we let $1$ play therole of (absolute) truth, $0$ as absolute falsity, but !!{formula}smay take any intermediate value in~$V$.One can also consider the set $V_{[0,1]} = [0,1]$ of all\emph{real} numbers between $0$ and~$1$, or other infinite subsets of$[0,1]$, however. Logics with this truth value set are often called \emph{fuzzy}.\end{document}

content/many-valued-logic/infinite-valued-logics/lukasiewicz.tex

% Part: many-valued-logic% Chapter: three-valued-logics% Section: lukasiewicz\documentclass[../../../include/open-logic-section]{subfiles}\begin{document}\olfileid{mvl}{inf}{luk}\olsection{\L ukasiewicz logic}\begin{editorial}  This is a short ``stub'' of a section on infinite-valued \L  ukasiewicz logic.\end{editorial}\begin{defn}\ollabel{def:lukasiewicz} Infinite-valued \L ukasiewiczlogic~$\LogLuk[\infty]$ is defined using the matrix:\begin{enumerate}  \item The standard propositional language $\Lang L_0$ with  $\lnot$, $\land$, $\lor$, $\lif$.  \item The set of truth values $V_\infty$.  \item $1$ is the only designated value, i.e., $V^+ = \{1\}$.  \item Truth functions are given by the following functions:  \begin{align*}    \tf{\lnot}[\LogLuk](x) & = 1 - x\\    \tf{\land}[\LogLuk](x,y) & = \min(x,y)\\    \tf{\lor}[\LogLuk](x,y) & = \max(x,y)\\    \tf{\lif}[\LogLuk](x,y) & = \min(1,1-(x-y)) = \begin{cases}      1 & \text{if } x \le y\\      1-(x-y) & \text{otherwise.}    \end{cases}    \end{align*}\end{enumerate}$m$-valued \L ukasiewicz logic is defined the same, except $V = V_m$.\end{defn}\begin{prop}  The logic $\LogLuk[3]$ defined by \olref[thr][luk]{def:lukasiewicz}  is the same as $\LogLuk[3]$ defined by \olref{def:lukasiewicz}.\end{prop}\begin{proof}  This can be seen by comparing the truth tables for the connectives  given in \olref[thr][luk]{def:lukasiewicz} with the truth tables  determined by the equations in \olref{def:lukasiewicz}:  \begin{center}    \begin{tabular}{c|c}       $\tf{\lnot}$ & \\       \hline        $1$ & $0$ \\       $1/2$ & $1/2$ \\      $0$ & $1$     \end{tabular}    \quad    \begin{tabular}{c|ccc}       $\tf{\land}[\LogLuk[3]]$ & $1$ & $1/2$ & $0$ \\       \hline       $1$ & $1$ & $1/2$ & $0$ \\       $1/2$ & $1/2$ & $1/2$ & $0$\\       $0$ & $0$ & $0$ & $0$     \end{tabular}    \\[2ex]    \begin{tabular}{c|ccc}       $\tf{\lor}[\LogLuk[3]]$ & $1$ & $1/2$ & $0$ \\       \hline       $1$ & $1$ & $1$ & $1$ \\       $1/2$ & $1$ & $1/2$ & $1/2$ \\      $0$ & $1$ & $1/2$ & $0$     \end{tabular}    \quad    \begin{tabular}{c|ccc}       $\tf{\lif}[\LogLuk[3]]$ & $1$ & $1/2$ & $0$ \\       \hline       $1$ & $1$ & $1/2$ & $0$ \\       $1/2$ & $1$ & $1$ & $1/2$  \\       $0$ & $1$ & $1$ & $1$     \end{tabular}  \end{center} \end{proof}\begin{prop}\ollabel{prop:luk-infty-m}  If $\Gamma \Entails[\LogLuk[\infty]] !B$ then $\Gamma  \Entails[\LogLuk[m]] !B$ for all~$m \ge 2$.\end{prop}\begin{proof}  Exercise.\end{proof}\begin{prob}  Prove \olref[mvl][inf][luk]{prop:luk-infty-m}.\end{prob}In fact, the converse holds as well.Infinite-valued \L ukasiewicz logic is the most popular fuzzy logic.In the fuzzy logic literature, the conditional is often defined as$\lnot !A \lor !B$. The result would be an infinite-valued strongKleene logic.\begin{prob}  Show that $(p \lif q) \lor (q \lif p)$ is a  tautology of~$\LogLuk[\infty]$.\end{prob}\end{document}

content/many-valued-logic/infinite-valued-logics/goedel.tex

% Part: many-valued-logic% Chapter: three-valued-logics% Section: lukasiewicz\documentclass[../../../include/open-logic-section]{subfiles}\begin{document}\olfileid{mvl}{inf}{god}\olsection{G\"odel logics}\begin{editorial}  This is a short ``stub'' of a section on infinite-valued G\"odel logic.\end{editorial}\begin{defn}\ollabel{def:goedel} Infinite-valued G\"odellogic~$\LogGod[\infty]$ is defined using the matrix:\begin{enumerate}  \item The standard propositional language $\Lang L_0$ with  $\lfalse$, $\lnot$, $\land$, $\lor$, $\lif$.  \item The set of truth values $V_\infty$.  \item $1$ is the only designated value, i.e., $V^+ = \{1\}$.  \item Truth functions are given by the following functions:  \begin{align*}    \tf{\lfalse} & = 0\\    \tf{\lnot}[\LogGod](x) & = \begin{cases}      $1$ & \text{if } x =0\\      $0$ & \text{otherwise}    \end{cases}\\    \tf{\land}[\LogGod](x,y) & = \min(x,y)\\    \tf{\lor}[\LogGod](x,y) & = \max(x,y)\\    \tf{\lif}[\LogGod](x,y) & = \begin{cases}      1 & \text{if } x \le y\\      y & \text{otherwise.}    \end{cases}    \end{align*}\end{enumerate}$m$-valued G\"odel logic is defined the same, except $V = V_m$.\end{defn}\begin{prop}  The logic $\LogGod[3]$ defined by \olref[thr][god]{defn:goedel}  is the same as $\LogGod[3]$ defined by \olref{def:goedel}.\end{prop}\begin{proof}  This can be seen by comparing the truth tables for the connectives  given in \olref[thr][god]{defn:goedel} with the truth tables  determined by the equations in \olref{def:goedel}:  \begin{center}    \begin{tabular}{c|c}       $\tf{\lnot}[\LogGod[3]]$ & \\       \hline        $1$ & $0$ \\       $1/2$ & $0$ \\      $0$ & $1$     \end{tabular}    \quad    \begin{tabular}{c|ccc}       $\tf{\land}[\LogGod]$ & $1$ & $1/2$ & $0$ \\       \hline       $1$ & $1$ & $1/2$ & $0$ \\       $1/2$ & $1/2$ & $1/2$ & $0$\\       $0$ & $0$ & $0$ & $0$     \end{tabular}    \\[2ex]    \begin{tabular}{c|ccc}       $\tf{\lor}[\LogGod]$ & $1$ & $1/2$ & $0$ \\       \hline       $1$ & $1$ & $1$ & $1$ \\       $1/2$ & $1$ & $1/2$ & $1/2$ \\      $0$ & $1$ & $1/2$ & $0$     \end{tabular}    \quad    \begin{tabular}{c|ccc}       $\tf{\lif}[\LogGod]$ & $1$ & $1/2$ & $0$ \\       \hline       $1$ & $1$ & $1/2$ & $0$ \\       $1/2$ & $1$ & $1$ & $0$  \\       $0$ & $1$ & $1$ & $1$     \end{tabular}  \end{center} \end{proof}\begin{prop}\ollabel{prop:god-infty-m}  If $\Gamma \Entails[\LogGod[\infty]] !B$ then $\Gamma  \Entails[\LogGod[m]] !B$ for all~$m \ge 2$.\end{prop}\begin{proof}  Exercise.\end{proof}\begin{prob}  Prove \olref[mvl][inf][god]{prop:god-infty-m}.\end{prob}In fact, the converse holds as well.Like $\LogGod[3]$, $\LogGod[\infty]$ has all intuitionistically valid!!{formula}s as tautologies, and the same examples of non-tautologiesare non-tautologies of~$\LogGod[\infty]$:\begin{align*}  & p \lor \lnot p && (p \lif q) \lif (\lnot p \lor q) \\  & \lnot\lnot p \lif p && \lnot(\lnot p \land \lnot q) \lif (p \lor q) \\  & ((p \lif q) \lif p) \lif p && \lnot(p \lif q) \lif (p \land \lnot q)\end{align*}The example of an intuitionistically invalid !!{formula} that isnevertheless a tautology of~$\LogGod[3]$, $(p \lif q) \lor (q \lifp)$, is also a tautology in~$\LogGod[\infty]$. In fact,$\LogGod[\infty]$ can be characterized as intuitionistic logic towhich the schema $(!A \lif !B) \lor (!B \lif !A)$ is added. This wasshown by Michael Dummett, and so $\LogGod[\infty]$ is often referred toas G\"odel--Dummett logic~$\Log{LC}$.\begin{prob}  Show that $(p \lif q) \lor (q \lif p)$ is a  tautology of~$\LogGod[\infty]$.\end{prob}\begin{prob}  Show that $(p \lif q) \lor (q \lif r) \lor (r \lif s)$, which is   a tautology of $\LogGod[3]$, is not a tautology of~$\LogGod[\infty]$.\end{prob}\end{document}