The Open Logic Text

en · official · sha256:a95730929e9c5413e5000b0a9466c42e213727cc58c451f2b7a38487ded37cf2

% Part: many-valued-logic % Chapter: syntax-and-semantics % Section: matrices \documentclass[../../../include/open-logic-section]{subfiles} \begin{document} \olfileid{mvl}{syn}{mat} \olsection{Matrices} A many-valued logic is defined by its language, its set of truth values~$V$, a subset of designated truth values, and truth functions for its connective. Together, these elements are called a \emph{matrix}. \begin{defn}[Matrix] \ollabel{defn:matrix} A \emph{matrix} for the logic~$\Log L$ consists of: \begin{enumerate} \item a set of connectives making up a language~$\Lang L$; \item a set $V \neq \emptyset$ of truth values; \item a set $V^+ \subseteq V$ of designated truth values; \item for each $n$-place connective $\star$ in $\Lang L$, a truth function~$\tf{\star} : V^n \to V$. If $n = 0$, then $\tf{\star}$ is just an element of~$V$. \end{enumerate} \end{defn} \begin{ex} The matrix for classical logic~\LogCL{} consists of: \begin{enumerate} \item The standard propositional language $\Lang L_0$ with $\lfalse$, $\lnot$, $\land$, $\lor$, $\lif$. \item The set of truth values $V = \{\True, \False\}$. \item $\True$ is the only designated value, i.e., $V^+ = \{\True\}$. \item For $\lfalse$, we have $\tf{\lfalse} = \False$. The other truth functions are given by the usual truth tables (see \olref{fig:tf-CL}). \end{enumerate} \begin{figure} \begin{center} \begin{tabular}{c|c} $\tf{\lnot}$ & \\ \hline $\True$ & $\False$ \\ $\False$ & $\True$ \end{tabular} \quad \begin{tabular}{c|cc} $\tf{\land}$ & $\True$ & $\False$ \\ \hline $\True$ & $\True$ & $\False$ \\ $\False$ & $\False$ & $\False$ \end{tabular} \quad \begin{tabular}{c|cc} $\tf{\lor}$ & $\True$ & $\False$ \\ \hline $\True$ & $\True$ & $\True$ \\ $\False$ & $\True$ & $\False$ \end{tabular} \quad \begin{tabular}{c|cc} $\tf{\lif}$ & $\True$ & $\False$ \\ \hline $\True$ & $\True$ & $\False$ \\ $\False$ & $\True$ & $\True$ \end{tabular} \end{center} \caption{Truth functions for classical logic~$\LogCL$.} \ollabel{fig:tf-CL} \end{figure} \end{ex} \end{document}

Referencing norms: DESIGNATED_VALUES, MATRIX, ValueCount, designated, truth_value, value_count

en · official · sha256:8871b73edeec0e6486e315fb83cd80928738c5c9068f576492cec82c92f5cf97

% Part: many-valued-logic % Chapter: syntax-and-semantics % Section: semantic-notions \documentclass[../../../include/open-logic-section]{subfiles} \begin{document} \olfileid{mvl}{syn}{sem} \olsection{Semantic Notions} Suppose a many-valued logic $\Log L$ is given by a matrix. Then we can define the usual semantic notions for~$\Log L$. \begin{defn} \begin{enumerate} \item !!^a{formula}~$!A$ is \emph{satisfiable} if for some~$\pAssign{v}$, $\pSat{v}{!A}$; it is \emph{unsatisfiable} if for no $\pAssign{v}$, $\pSat{v}{!A}$; \item !!^a{formula}~$!A$ is a \emph{tautology} if $\pSat{v}{!A}$ for all !!{valuation}s~$v$; \item If $\Gamma$ is a set of !!{formula}s, $\Gamma \Entails !A$ (``$\Gamma$ entails $!A$'') if and only if $\pSat{v}{!A}$ for every !!{valuation}~$\pAssign{v}$ for which $\pSat{v}{\Gamma}$. \item If $\Gamma$ is a set of !!{formula}s, $\Gamma$ is \emph{satisfiable} if there is !!a{valuation}~$\pAssign{v}$ for which $\pSat{v}{\Gamma}$, and $\Gamma$ is \emph{unsatisfiable} otherwise. \end{enumerate} \end{defn} We have some of the same facts for these notions as we do for the case of classical logic: \begin{prop} \ollabel{prop:semanticalfacts} \begin{enumerate} \item $!A$ is a tautology if and only if $\emptyset \Entails !A$; \item If $\Gamma$ is satisfiable then every finite subset of $\Gamma$ is also satisfiable; \item\ollabel{def:monotonicity}% Monotonicity: if $\Gamma \subseteq \Delta$ and $\Gamma \Entails !A$ then also $\Delta \Entails !A$; \item\ollabel{def:Cut}% Transitivity: if $\Gamma \Entails !A$ and $\Delta \cup \{ !A\} \Entails !B$ then $\Gamma \cup \Delta \Entails !B$; \end{enumerate} \end{prop} \begin{proof} Exercise. \end{proof} \begin{prob} Prove \olref[mvl][syn][sem]{prop:semanticalfacts} \end{prob} In classical logic we can connect entailment and the conditional. For instance, we have the validity of \emph{modus ponens}: If $\Gamma \Entails !A$ and $\Gamma \Entails !A \lif !B$ then $\Gamma \Entails !B$. Another important relationship between $\Entails$ and $\lif$ in classical logic is the semantic deduction theorem: $\Gamma \Entails !A \lif !B$ if and only if $\Gamma \cup \{!A\} \Entails !B$. These results \emph{do not} always hold in many-valued logics. Whether they do depends on the truth function~$\tf{\lif}$. \end{document}

Referencing norms: LogicDescribed, MV_CONSEQUENCE, consequence_reading, logic_described

en · official · sha256:534d74c0bfc218cc1c0800c37cd9d7c93f2d5e7a18ed0eb45c6fd2e373425078

% Part: many-valued-logic % Chapter: syntax-and-semantics % Section: valuations-sat \documentclass[../../../include/open-logic-section]{subfiles} \begin{document} \olfileid{mvl}{syn}{val} \olsection{\usetoken{P}{valuation} and Satisfaction} \begin{defn}[!!^{valuation}s] Let $V$ be a set of truth values. A \emph{!!{valuation}} for $\Lang{L}$ into~$V$ is a function~$\pAssign{v}$ assigning !!a{element} of~$V$ to the !!{propositional variable}s of the language, i.e., $\pAssign{v} \colon \PVar \to V$. \end{defn} \begin{defn}\ollabel{defn:pValue} Given !!a{valuation}~$\pAssign{v}$ into the set of truth values~$V$ of a many-valued logic~$\Log L$, define the evaluation function $\pValue{v} \colon \Frm[L] \to V$ inductively by: \begin{enumerate} \item $\pValue{v}(\Obj p_n) = \pAssign{v}(\Obj p_n)$; \item If $\star$ is a $0$-place connective, then $\pValue{v}(\star) = \tf{\star}[\Log L]$; \item If $\star$ is an $n$-place connective, then \[ \pValue{v}(\star(!A_1, \dots, !A_n)) = \tf{\star}[\Log L] (\pValue{v}(!A_1), \dots, \pValue{v}(!A_n)). \] \end{enumerate} \end{defn} \begin{defn}[Satisfaction] \ollabel{defn:satisfaction} The !!{formula}~$!A$ is \emph{satisfied} by !!a{valuation}~$\pAssign{v}$, $\pSat{v}{!A}[\Log L]$, iff $\pValue{v}(!A)[\Log L] \in V^+$, where $V^+$ is the set of designated truth values of~$\Log L$. We write $\pSat/{v}{!A}[\Log L]$ to mean ``not $\pSat{v}{!A}[\Log L]$.'' If $\Gamma$ is a set of !!{formula}s, $\pSat{v}{\Gamma}[\Log L]$ iff $\pSat{v}{!A}[\Log L]$ for every~$!A \in \Gamma$. \end{defn} \end{document}

en · official · sha256:8001e86f40c438e5586da54c7f79ace8254d2dd2ce71290a38b7defe51e89020

% Part: many-valued-logic % Chapter: three-valued-logics % Section: introduction \documentclass[../../../include/open-logic-section]{subfiles} \begin{document} \olfileid{mvl}{thr}{int} \olsection{Introduction} If we just add one more value~$\Undef$ to $\True$ and $\False$, we get a three-valued logic. Even though there is only one more truth value, the possibilities for defining the truth-functions for $\lnot$, $\land$, $\lor$, and $\lif$ are quite numerous. Then a logic might use any combination of these truth functions, and you also have a choice of making only $\True$ designated, or both $\True$ and~$\Undef$. We present here a selection of the most well-known three-valued logics, their motivations, and some of their properties. \end{document}

en · official · sha256:0b632552858c410d3670357a0df7071cb5a3c5bd285681e86988ed07263852d7

% Part: many-valued-logic % Chapter: three-valued-logics % Section: kleene \documentclass[../../../include/open-logic-section]{subfiles} \begin{document} \olfileid{mvl}{thr}{skl} \olsection{Kleene logics} Stephen Kleene introduced two three-valued logics motivated by a logic in which truth values are thought of the outcomes of computational procedures: a procedure may yield $\True$ or $\False$, but it may also fail to terminate. In that case the corresponding truth value is undefined, represented by the truth value~$\Undef$. To compute the negation of a proposition~$!A$, you would first compute the value of~$!A$, and then return the opposite of the result. If the computation of $!A$ does not terminate, then the entire procedure does not either: so the negation of $\Undef$ is $\Undef$. To compute a conjunction $!A \land !B$, there are two options: one can first compute~$!A$, then $!B$, and then the result would be $\True$ if the outcome of both is~$\True$, and $\False$ otherwise. If either computation fails to halt, the entire procedure does as well. So in this case, the if one conjunct is undefined, the conjunction is as well. The same goes for disjunction. However, if we can evaluate $!A$ and $!B$ in parallel, we can do better. Then, if one of the two procedures halts and returns $\False$, we can stop, as the answer must be false. So in that case a conjunction with one false conjunct is false, even if the other conjunct is undefined. Similarly, when computing a disjunction in parallel, we can stop once the procedure for one of the two disjuncts has returned true: then the disjunction must be true. So in this case we can know what the outcome of a compound claim is, even if one of the components is undefined. On this interpretation, we might read $\Undef$ as ``unknown'' rather than ``undefined.'' The two interpretations give rise to Kleene's strong and weak logic. The conditional is defined as equivalent to $\lnot !A \lor !B$. \begin{defn} \emph{Strong Kleene logic}~$\LogKs$ 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 $V = \{\True, \Undef, \False\}$. \item $\True$ is the only designated value, i.e., $V^+ = \{\True\}$. \item Truth functions are given by the following tables: \begin{center} \begin{tabular}{c|c} $\tf{\lnot}$ & \\ \hline $\True$ & $\False$ \\ $\Undef$ & $\Undef$ \\ $\False$ & $\True$ \end{tabular} \quad \begin{tabular}{c|ccc} $\tf{\land}[\LogKs]$ & $\True$ & $\Undef$ & $\False$ \\ \hline $\True$ & $\True$ & $\Undef$ & $\False$ \\ $\Undef$ & $\Undef$ & $\Undef$ & $\False$\\ $\False$ & $\False$ & $\False$ & $\False$ \end{tabular} \\[2ex] \begin{tabular}{c|ccc} $\tf{\lor}[\LogKs]$ & $\True$ & $\Undef$ & $\False$ \\ \hline $\True$ & $\True$ & $\True$ & $\True$ \\ $\Undef$ & $\True$ & $\Undef$ & $\Undef$ \\ $\False$ & $\True$ & $\Undef$ & $\False$ \end{tabular} \quad \begin{tabular}{c|ccc} $\tf{\lif}[\LogKs]$ & $\True$ & $\Undef$ & $\False$ \\ \hline $\True$ & $\True$ & $\Undef$ & $\False$ \\ $\Undef$ & $\True$ & $\Undef$ & $\Undef$ \\ $\False$ & $\True$ & $\True$ & $\True$ \end{tabular} \end{center} \end{enumerate} \end{defn} \begin{defn} \emph{Weak Kleene logic}~$\LogKw$ 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 $V = \{\True, \Undef, \False\}$. \item $\True$ is the only designated value, i.e., $V^+ = \{\True\}$. \item Truth functions are given by the following tables: \begin{center} \begin{tabular}{c|c} $\tf{\lnot}$ & \\ \hline $\True$ & $\False$ \\ $\Undef$ & $\Undef$ \\ $\False$ & $\True$ \end{tabular} \quad \begin{tabular}{c|ccc} $\tf{\land}[\LogKw]$ & $\True$ & $\Undef$ & $\False$ \\ \hline $\True$ & $\True$ & $\Undef$ & $\False$ \\ $\Undef$ & $\Undef$ & $\Undef$ & $\Undef$\\ $\False$ & $\False$ & $\Undef$ & $\False$ \end{tabular} \\[2ex] \begin{tabular}{c|ccc} $\tf{\lor}[\LogKw]$ & $\True$ & $\Undef$ & $\False$ \\ \hline $\True$ & $\True$ & $\Undef$ & $\True$ \\ $\Undef$ & $\Undef$ & $\Undef$ & $\Undef$ \\ $\False$ & $\True$ & $\Undef$ & $\False$ \end{tabular} \quad \begin{tabular}{c|ccc} $\tf{\lif}[\LogKw]$ & $\True$ & $\Undef$ & $\False$ \\ \hline $\True$ & $\True$ & $\Undef$ & $\False$ \\ $\Undef$ & $\Undef$ & $\Undef$ & $\Undef$ \\ $\False$ & $\True$ & $\Undef$ & $\True$ \end{tabular} \end{center} \end{enumerate} \end{defn} \begin{prop} $\LogKs$ and $\LogKw$ have no tautologies. \end{prop} \begin{proof} If $\pAssign v(p) = \Undef$ for all !!{propositional variable}s~$p$, then any formula~$!A$ will have truth value~$\pValue v(!A) = \Undef$, since \[ \tf{\lnot}(\Undef) = \tf{\lor}(\Undef, \Undef) = \tf{\land}(\Undef, \Undef) = \tf{\lif}(\Undef, \Undef) = \Undef \] in both logics. As $\Undef \notin V^+$ for either $\LogKs$ or $\LogKw$, on this !!{valuation}, $!A$ will not be designated. \end{proof} Although both weak and strong Kleene logic have no tautologies, they have non-trivial consequence relations. \begin{prob} Which of the following relations hold in (a) strong and (b) weak Kleene logic? Give a truth table for each. \begin{enumerate} \item $p, p \lif q \Entails q$ \item $p \lor q, \lnot p \Entails q$ \item $p \land q \Entails p$ \item $p \Entails p \land p$ \item $p \Entails p \lor q$ \end{enumerate} \end{prob} Dmitry Bochvar interpreted $\Undef$ as ``meaningless'' and attempted to use it to solve paradoxes such as the Liar paradox by stipulating that paradoxical sentences take the value~$\Undef$. He introduced a logic which is essentially weak Kleene logic extended by additional connectives, two of which are ``external negation'' and the ``is undefined'' operator: \begin{center} \begin{tabular}{c|c} $\tf{\sim}$ & \\ \hline $\True$ & $\False$ \\ $\Undef$ & $\True$ \\ $\False$ & $\True$ \end{tabular} \quad \begin{tabular}{c|c} $\tf{+}$ & \\ \hline $\True$ & $\False$ \\ $\Undef$ & $\True$ \\ $\False$ & $\False$ \end{tabular} \end{center} \begin{prob} Can you define $\sim$ in Bochvar's logic in terms of $\lnot$ and~$+$, i.e., find !!a{formula} with only the !!{propositional variable}~$p$ and not involving $\sim$ which always takes the same truth value as~$\mathord{\sim}p$? Give a truth table to show you're right. \end{prob} \end{document}

Referencing norms: K3

en · official · sha256:507bd6c306ed2b506894a6eabdd04729d8ee23c92492465c9bf101383e94a698

% Part: many-valued-logic % Chapter: three-valued-logics % Section: multiple-designation \documentclass[../../../include/open-logic-section]{subfiles} \begin{document} \olfileid{mvl}{thr}{mul} \olsection{Designating not just $\True$} So far the logics we've seen all had the set of designated truth values $V^+ = \{\True\}$, i.e., something counts as true iff its truth value is~$\True$. But one might also count something as true if it's just not~$\False$. Then one would get a logic by stipulating in the matrix, e.g., that $V^+ = \{\True, \Undef\}$. \begin{defn} The \emph{logic of paradox}~$\LogLP$ 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 $V = \{\True, \Undef, \False\}$. \item $\True$ and $\Undef$ are designated, i.e., $V^+ = \{\True, \Undef\}$. \item Truth functions are the same as in strong Kleene logic. \end{enumerate} \end{defn} \begin{defn} Halld\'en's \emph{logic of nonsense}~$\LogHal$ is defined using the matrix: \begin{enumerate} \item The standard propositional language $\Lang L_0$ with $\lnot$, $\land$, $\lor$, $\lif$ and a $1$-place connective~$+$. \item The set of truth values $V = \{\True, \Undef, \False\}$. \item $\True$ and $\Undef$ are designated, i.e., $V^+ = \{\True, \Undef\}$. \item Truth functions are the same as weak Kleene logic, plus the ``is meaningless'' operator: \begin{center} \begin{tabular}{c|c} $\tf{+}$ & \\ \hline $\True$ & $\False$ \\ $\Undef$ & $\True$ \\ $\False$ & $\False$ \end{tabular} \end{center} \end{enumerate} \end{defn} By contrast to the Kleene logics with which they share truth tables, these \emph{do} have tautologies. \begin{prop}\ollabel{prop:LP-taut-CL} The tautologies of $\LogLP$ are the same as the tautologies of classical propositional logic. \end{prop} \begin{proof} By \olref[syn][sub]{prop:mvl-cl}, if $\Entails[\LogLP] !A$ then $\Entails[\LogCL] !A$. To show the reverse, we show that if there is !!a{valuation}~$\pAssign v\colon \PVar \to \{\False, \True, \Undef\}$ such that $\pValue v(!A)[\LogKs] = \False$ then there is !!a{valuation}~$\pAssign {v'}\colon \PVar \to \{\False, \True\}$ such that $\pValue {v'}(!A)[\LogCL] = \False$. This establishes the result for $\LogLP$, since $\LogKs$ and $\LogLP$ have the same characteristic truth functions, and $\False$ is the only truth value of $\LogLP$ that is not designated (that is the only difference between $\LogLP$ and~$\LogKs$). Thus, if $\Entails/[\LogLP] !A$, for some !!{valuation}~$\pAssign v$, $\pValue v(!A)[\LogLP] = \pValue v[\LogKs](!A) = \False$. By the claim we're proving, $\pValue{v'}[\LogCL](!A) = \False$, i.e., $\Entails/[\LogCL] !A$. To establish the claim, we first define $\pAssign {v'}$ as \[ \pAssign {v'}(p) = \begin{cases} \True & \text{if } \pAssign {v}(p) \in \{\True, \Undef\}\\ \False & \text{otherwise} \end{cases} \] We now show by induction on $!A$ that (a)~if $\pValue v(!A)[\LogKs] = \False$ then $\pValue {v'}(!A)[\LogCL] = \False$, and (b)~if $\pValue v(!A)[\LogKs] = \True$ then $\pValue {v'}(!A)[\LogCL] = \True$ \begin{enumerate} \item Induction basis: $!A \ident p$. By \olref[syn][val]{defn:pValue}, $\pValue v(!A)[\LogKs] = \pAssign v(p) = \pValue {v'}(!A)[\LogCL]$, which implies both (a) and~(b). For the induction step, consider the cases: \item $!A \ident \lnot !B$. \begin{enumerate} \item Suppose $\pValue v(\lnot!B)[\LogKs] = \False$. By the definition of $\tf{\lnot}[\LogKs]$, $\pValue v(!B)[\LogKs] = \True$. By inductive hypothesis, case~(b), we get $\pValue {v'}(!B)[\LogCL] = \True$, so $\pValue {v'}(\lnot !B)[\LogCL] = \False$. \item Suppose $\pValue v(\lnot!B)[\LogKs] = \True$. By the definition of $\tf{\lnot}[\LogKs]$, $\pValue v(!B)[\LogKs] = \False$. By inductive hypothesis, case~(a), we get $\pValue {v'}(!B)[\LogCL] = \False$, so $\pValue {v'}(\lnot !B)[\LogCL] = \True$. \end{enumerate} \item $!A \ident (!B \land !C)$. \begin{enumerate} \item Suppose $\pValue v(!B \land !C)[\LogKs] = \False$. By the definition of $\tf{\land}[\LogKs]$, $\pValue v(!B)[\LogKs] = \False$ or $\pValue v(!B)[\LogKs] = \False$. By inductive hypothesis, case~(a), we get $\pValue {v'}(!B)[\LogCL] = \False$ or $\pValue {v'}(!C)[\LogCL] = \False$, so $\pValue {v'}(!B \land !C)[\LogCL] = \False$. \item Suppose $\pValue v(!B \land !C)[\LogKs] = \True$. By the definition of $\tf{\land}[\LogKs]$, $\pValue v(!B)[\LogKs] = \True$ and $\pValue v(!B)[\LogKs] = \True$. By inductive hypothesis, case~(b), we get $\pValue {v'}(!B)[\LogCL] = \True$ and $\pValue {v'}(!C)[\LogCL] = \True$, so $\pValue {v'}(!B \land !C)[\LogCL] = \True$. \end{enumerate} \end{enumerate} The other two cases are similar, and left as exercises. Alternatively, the proof above establishes the result for all !!{formula}s only containing $\lnot$ and~$\land$. One may now appeal to the facts that in both $\LogKs$ and $\LogCL$, for any $\pAssign v$, $\pValue v(!B \lor !C) = \pValue v(\lnot(\lnot!B \land \lnot !C))$ and $\pValue v(!B \lif !C) = \pValue v(\lnot(!B \land \lnot !C))$. \end{proof} \begin{prob} Complete the proof \olref[mvl][thr][mul]{prop:LP-taut-CL}, i.e., establish (a) and~(b) for the cases where $!A \ident (!B \lor !C)$ and $!A \ident (!B \lif !C)$. \end{prob} \begin{prob} Prove that every classical tautology is a tautology in~$\LogHal$. \end{prob} Although they have the same tautologies as classical logic, their consequence relations are different. $\LogLP$, for instance, is \emph{paraconsistent} in that $\lnot p, p \Entails/ q$, and so the principle of explosion $\lnot !A, !A \Entails !B$ does not hold in general. (It holds for some cases of $!A$ and $!B$, e.g., if $!B$~is a tautology.) \begin{prob} Which of the following relations hold in (a)~$\LogLP$ and in (b)~$\LogHal$? Give a truth table for each. \begin{enumerate} \item $p, p \lif q \Entails q$ \item $\lnot q, p \lif q \Entails \lnot p$ \item $p \lor q, \lnot p \Entails q$ \item $\lnot p, p \Entails q$ \item $p \Entails p \lor q$ \item $p \lif q, q\lif r \Entails p \lif r$ \end{enumerate} \end{prob} What if you make $\Undef$ designated in $\LogLuk[3]$? \begin{defn} The logic \emph{3-valued R-Mingle}~$\LogRM[3]$ 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 $V = \{\True, \Undef, \False\}$. \item $\True$ and $\Undef$ are designated, i.e., $V^+ = \{\True, \Undef\}$. \item Truth functions are the same as \L ukasiewicz logic~$\LogLuk[3]$. \end{enumerate} \end{defn} \begin{prob} Which of the following relations hold in $\LogRM[3]$? \begin{enumerate} \item $p, p \lif q \Entails q$ \item $p \lor q, \lnot p \Entails q$ \item $\lnot p, p \Entails q$ \item $p \Entails p \lor q$ \end{enumerate} \end{prob} Different truth tables can sometimes generate the same logic (entailment relation) just by changing the designated values. E.g., this happens if in G\"odel logic we take $V^+ = \{\True, \Undef\}$ instead of~$\{\True\}$. \begin{prop}\ollabel{prop:gl-udes} The matrix with $V = \{\False, \Undef,\True\}$, $V^+=\{\True, \Undef\}$, and the truth functions of $3$-valued G\"odel logic defines classical logic. \end{prop} \begin{proof} Exercise. \end{proof} \begin{prob} Prove \olref[mvl][thr][mul]{prop:gl-udes} by showing that for the logic~$\Log L$ defined just like G\"odel logic but with $V^+=\{\True,\Undef\}$, if $\Gamma \Entails/[\Log L] !B$ then $\Gamma \Entails/[\LogCL] !B$. Use the ideas of \olref[mvl][thr][mul]{prop:LP-taut-CL}, except instead of proving properties (a) and~(b), show that $\pValue v(!A)[\LogGod] = \False$ iff $\pValue {v'}(!A)[\LogCL] = \False$ (and hence that $\pValue v(!A)[\LogGod] \in \{\True,\Undef\}$ iff $\pValue {v'}(!A)[\LogCL] = \True$). Explain why this establishes the proposition. \end{prob} \end{document}

Referencing norms: DesignatedCount, LP, P_EXPLOSION, ParaconsistentHere, contradiction_reading, designated_count, paraconsistent_here