Axioms of algorithmic logic: Różnice pomiędzy wersjami

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(Axioms of propositional logic)
(Axioms of propositional logic)
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\vee\beta))</math>
 
\vee\beta))</math>
  
<span style="color: Blue"><math>Ax_3$] $(\beta\Longrightarrow(\alpha\vee\beta))$%
+
<span style="color: Blue"><math>Ax_3 \qquad (\beta\Longrightarrow(\alpha\vee\beta))</math>
  
<span style="color: Blue"><math>Ax_4$] $((\alpha\Longrightarrow\delta)~\Longrightarrow((\beta
+
<span style="color: Blue"><math>Ax_4 \qquad((\alpha\Longrightarrow\delta)~\Longrightarrow((\beta
 
\Longrightarrow\delta)~\Longrightarrow~((\alpha\vee\beta
 
\Longrightarrow\delta)~\Longrightarrow~((\alpha\vee\beta
)\Longrightarrow\delta)))$
+
)\Longrightarrow\delta)))</math>
  
<span style="color: Blue"><math>Ax_5$] $((\alpha\wedge\beta
+
<span style="color: Blue"><math>Ax_5 \qquad((\alpha\wedge\beta
)\Longrightarrow\alpha)$
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)\Longrightarrow\alpha)</math>
  
<span style="color: Blue"><math>Ax_6$] $((\alpha\wedge\beta
+
<span style="color: Blue"><math>Ax_6 \qquad((\alpha\wedge\beta
)\Longrightarrow\beta)$
+
)\Longrightarrow\beta)</math>
  
<span style="color: Blue"><math>Ax_7$] $((\delta\Longrightarrow\alpha
+
<span style="color: Blue"><math>Ax_7 \qquad((\delta\Longrightarrow\alpha
 
)\Longrightarrow((\delta\Longrightarrow\beta)\Longrightarrow(\delta
 
)\Longrightarrow((\delta\Longrightarrow\beta)\Longrightarrow(\delta
\Longrightarrow(\alpha\wedge\beta))))$
+
\Longrightarrow(\alpha\wedge\beta))))</math>
  
 
<span style="color: Blue"><math>Ax_8$] $((\alpha
 
<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))</math>
  
<span style="color: Blue"><math>Ax_9$] $((\alpha\wedge\lnot\alpha
+
<span style="color: Blue"><math>Ax_9 \qquad ((\alpha\wedge\lnot\alpha
)\Longrightarrow\beta)$
+
)\Longrightarrow\beta)</math>
  
<span style="color: Blue"><math>Ax_{10}$] $((\alpha\Longrightarrow
+
<span style="color: Blue"><math>Ax_{10} \qquad ((\alpha\Longrightarrow
(\alpha\wedge\lnot\alpha))\Longrightarrow\lnot\alpha)$
+
(\alpha\wedge\lnot\alpha))\Longrightarrow\lnot\alpha)</math>
\item[$Ax_{11}$] $(\alpha\vee\lnot\alpha)$<span>
+
 
 +
\item[$Ax_{11} \qquad (\alpha\vee\lnot\alpha)</math><span>
 
== Axioms of predicate calculus ==
 
== Axioms of predicate calculus ==
 
\item[$Ax_{12}$] $((x:=\tau
 
\item[$Ax_{12}$] $((x:=\tau
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(\exists x)\lnot\alpha(x)$
 
(\exists x)\lnot\alpha(x)$
 
\item[$Ax_{14}$] $K((\exists x)\alpha
 
\item[$Ax_{14}$] $K((\exists x)\alpha
(x))\equiv(\exists y)(K\alpha(x/y))$ \begin{small}for $y\notin V(K)$                                                       \end{small}
+
(x))\equiv(\exists y)(K\alpha(x/y))$ \begin{small}for $y\notin V(K)$ \end{small}
 +
== Axioms of algorithmic logic ==                                                 
 
\item[$Ax_{15}$] $K(\alpha\vee\beta)\equiv((K\alpha)\vee(K\beta))$
 
\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[$Ax_{16}$] $K(\alpha\wedge\beta)\equiv((K\alpha)\wedge(K\beta))$%

Wersja z 17:19, 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 \qquad(\alpha\Rightarrow(\alpha \vee\beta))[/math]

[math]Ax_3 \qquad (\beta\Longrightarrow(\alpha\vee\beta))[/math]

[math]Ax_4 \qquad((\alpha\Longrightarrow\delta)~\Longrightarrow((\beta \Longrightarrow\delta)~\Longrightarrow~((\alpha\vee\beta )\Longrightarrow\delta)))[/math]

[math]Ax_5 \qquad((\alpha\wedge\beta )\Longrightarrow\alpha)[/math]

[math]Ax_6 \qquad((\alpha\wedge\beta )\Longrightarrow\beta)[/math]

[math]Ax_7 \qquad((\delta\Longrightarrow\alpha )\Longrightarrow((\delta\Longrightarrow\beta)\Longrightarrow(\delta \Longrightarrow(\alpha\wedge\beta))))[/math]

[math]Ax_8$] $((\alpha \Longrightarrow(\beta\Longrightarrow\delta))\equiv((\alpha\wedge\beta )\Longrightarrow\delta))[/math]

[math]Ax_9 \qquad ((\alpha\wedge\lnot\alpha )\Longrightarrow\beta)[/math]

[math]Ax_{10} \qquad ((\alpha\Longrightarrow (\alpha\wedge\lnot\alpha))\Longrightarrow\lnot\alpha)[/math]

\item[$Ax_{11} \qquad (\alpha\vee\lnot\alpha)</math>

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}

Axioms of algorithmic logic

\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[$Ax_{23}$] $\displaystyle{\bigcup K\alpha \equiv(\alpha\vee(K\bigcup K\alpha))}$