Axioms of algorithmic logic: Różnice pomiędzy wersjami
(→Axioms of AL) |
(→Axioms of AL) |
||
| Linia 1: | Linia 1: | ||
| − | == Axioms of | + | The axioms are grouped in three sets |
| − | < | + | == Axioms of propositional logic == |
| − | + | We are quoting the axiom system of Helena Rasiowa and Roman Sikorski<br /> | |
| − | + | ||
| − | \ | + | <span style="color: Blue"><math>Ax_1 \qquad ((\alpha\Rightarrow\beta |
| + | )\Rightarrow((\beta\Rightarrow\delta)\Rightarrow(\alpha | ||
| + | \Rightarrow\delta)))</math> | ||
| + | |||
| + | <span style="color: Blue"><math>Ax_2$] $(\alpha\Longrightarrow(\alpha | ||
\vee\beta))$ | \vee\beta))$ | ||
| − | |||
| − | \ | + | <span style="color: Blue"><math>Ax_3$] $(\beta\Longrightarrow(\alpha\vee\beta))$% |
| + | |||
| + | <span style="color: Blue"><math>Ax_4$] $((\alpha\Longrightarrow\delta)~\Longrightarrow((\beta | ||
\Longrightarrow\delta)~\Longrightarrow~((\alpha\vee\beta | \Longrightarrow\delta)~\Longrightarrow~((\alpha\vee\beta | ||
)\Longrightarrow\delta)))$ | )\Longrightarrow\delta)))$ | ||
| − | + | ||
| + | <span style="color: Blue"><math>Ax_5$] $((\alpha\wedge\beta | ||
)\Longrightarrow\alpha)$ | )\Longrightarrow\alpha)$ | ||
| − | + | ||
| + | <span style="color: Blue"><math>Ax_6$] $((\alpha\wedge\beta | ||
)\Longrightarrow\beta)$ | )\Longrightarrow\beta)$ | ||
| − | + | ||
| + | <span style="color: Blue"><math>Ax_7$] $((\delta\Longrightarrow\alpha | ||
)\Longrightarrow((\delta\Longrightarrow\beta)\Longrightarrow(\delta | )\Longrightarrow((\delta\Longrightarrow\beta)\Longrightarrow(\delta | ||
\Longrightarrow(\alpha\wedge\beta))))$ | \Longrightarrow(\alpha\wedge\beta))))$ | ||
| − | + | ||
| + | <span style="color: Blue"><math>Ax_8$] $((\alpha | ||
\Longrightarrow(\beta\Longrightarrow\delta))\equiv((\alpha\wedge\beta | \Longrightarrow(\beta\Longrightarrow\delta))\equiv((\alpha\wedge\beta | ||
)\Longrightarrow\delta))$ | )\Longrightarrow\delta))$ | ||
| − | + | ||
| + | <span style="color: Blue"><math>Ax_9$] $((\alpha\wedge\lnot\alpha | ||
)\Longrightarrow\beta)$ | )\Longrightarrow\beta)$ | ||
| − | + | ||
| + | <span style="color: Blue"><math>Ax_{10}$] $((\alpha\Longrightarrow | ||
(\alpha\wedge\lnot\alpha))\Longrightarrow\lnot\alpha)$ | (\alpha\wedge\lnot\alpha))\Longrightarrow\lnot\alpha)$ | ||
| − | \item[$Ax_{11}$] $(\alpha\vee\lnot\alpha)$ | + | \item[$Ax_{11}$] $(\alpha\vee\lnot\alpha)$<span> |
| + | == Axioms of predicate calculus == | ||
\item[$Ax_{12}$] $((x:=\tau | \item[$Ax_{12}$] $((x:=\tau | ||
)true\Longrightarrow((\forall x)\alpha(x)\Longrightarrow(x:=\tau)\alpha | )true\Longrightarrow((\forall x)\alpha(x)\Longrightarrow(x:=\tau)\alpha | ||
Wersja z 17:11, 19 lis 2015
The axioms are grouped in three sets
Axioms of propositional logic
We are quoting the axiom system of Helena Rasiowa and Roman Sikorski
[math]Ax_1 \qquad ((\alpha\Rightarrow\beta )\Rightarrow((\beta\Rightarrow\delta)\Rightarrow(\alpha \Rightarrow\delta)))[/math]
[math]Ax_2$] $(\alpha\Longrightarrow(\alpha \vee\beta))$ \ltspan style="color: Blue"\gt\ltmath\gtAx_3$] $(\beta\Longrightarrow(\alpha\vee\beta))$% \ltspan style="color: Blue"\gt\ltmath\gtAx_4$] $((\alpha\Longrightarrow\delta)~\Longrightarrow((\beta \Longrightarrow\delta)~\Longrightarrow~((\alpha\vee\beta )\Longrightarrow\delta)))$ \ltspan style="color: Blue"\gt\ltmath\gtAx_5$] $((\alpha\wedge\beta )\Longrightarrow\alpha)$ \ltspan style="color: Blue"\gt\ltmath\gtAx_6$] $((\alpha\wedge\beta )\Longrightarrow\beta)$ \ltspan style="color: Blue"\gt\ltmath\gtAx_7$] $((\delta\Longrightarrow\alpha )\Longrightarrow((\delta\Longrightarrow\beta)\Longrightarrow(\delta \Longrightarrow(\alpha\wedge\beta))))$ \ltspan style="color: Blue"\gt\ltmath\gtAx_8$] $((\alpha \Longrightarrow(\beta\Longrightarrow\delta))\equiv((\alpha\wedge\beta )\Longrightarrow\delta))$ \ltspan style="color: Blue"\gt\ltmath\gtAx_9$] $((\alpha\wedge\lnot\alpha )\Longrightarrow\beta)$ \ltspan style="color: Blue"\gt\ltmath\gtAx_{10}$] $((\alpha\Longrightarrow (\alpha\wedge\lnot\alpha))\Longrightarrow\lnot\alpha)$ \item[$Ax_{11}$] $(\alpha\vee\lnot\alpha)$\ltspan\gt == Axioms of predicate calculus == \item[$Ax_{12}$] $((x:=\tau )true\Longrightarrow((\forall x)\alpha(x)\Longrightarrow(x:=\tau)\alpha (x)))$ \ \ \newline \qquad\qquad{\small where term }$\tau${\small\ is of the same type as the variable x} \item[$Ax_{13}$] $(\forall x)\alpha(x)\equiv\lnot (\exists x)\lnot\alpha(x)$ \item[$Ax_{14}$] $K((\exists x)\alpha (x))\equiv(\exists y)(K\alpha(x/y))$ \begin{small}for $y\notin V(K)$ \end{small} \item[$Ax_{15}$] $K(\alpha\vee\beta)\equiv((K\alpha)\vee(K\beta))$ \item[$Ax_{16}$] $K(\alpha\wedge\beta)\equiv((K\alpha)\wedge(K\beta))$% \item[$Ax_{17}$] $K(\lnot\alpha)\Longrightarrow\lnot(K\alpha)$% \item[$Ax_{18}$] $((x:=\tau)\gamma\equiv(\gamma(x/\tau)\wedge (x:=\tau)true))~\wedge((q:=\gamma\prime)\gamma\equiv\gamma(q/\gamma \prime))$ \item[$Ax_{19}$] \textbf{begin} $K;M$ \textbf{end} $\alpha \equiv K(M\alpha)$ \item[$Ax_{20}$] \textbf{if} $\gamma$ \textbf{then} $K $ \textbf{else} $M$ \textbf{fi} $\alpha\equiv((\lnot\gamma\wedge M\alpha )\vee(\gamma\wedge K\alpha))$ \item[$Ax_{21}$] \textbf{while}~ $\gamma$~\textbf{do}~$K$~\textbf{od}~$\alpha\equiv((\lnot\gamma\wedge\alpha )\vee(\gamma\wedge K($\textbf{while}~$\gamma$~\textbf{do}~$K$~\textbf{od}$% (\lnot\gamma\wedge\alpha))))$ \item[$Ax_{22}$] $\displaystyle{\bigcap K\alpha\equiv (\alpha\wedge(K\bigcap K\alpha))}$ \item[$Ax_{23}$] $\displaystyle{\bigcup K\alpha \equiv(\alpha\vee(K\bigcup K\alpha))}$ \lt/span\gt[/math]
