# Axioms of algorithmic logic

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The axioms are grouped in three sets

## Axioms of propositional logic

We are quoting the axiom system of Helena Rasiowa and Roman Sikorski
Any formula of a scheme listed below is an axiom of algorithmic logic. The signs $\alpha, \beta, \delta$ are to be replaced by a formula. $Ax_1 \qquad ((\alpha\Rightarrow\beta )\Rightarrow((\beta\Rightarrow\delta)\Rightarrow(\alpha \Rightarrow\delta)))$

$Ax_2 \qquad(\alpha\Rightarrow(\alpha \vee\beta))$

$Ax_3 \qquad (\beta\Longrightarrow(\alpha\vee\beta))$

$Ax_4 \qquad((\alpha\Longrightarrow\delta)~\Longrightarrow((\beta \Longrightarrow\delta)~\Longrightarrow~((\alpha\vee\beta )\Longrightarrow\delta)))$

$Ax_5 \qquad((\alpha\wedge\beta )\Longrightarrow\alpha)$

$Ax_6 \qquad((\alpha\wedge\beta )\Longrightarrow\beta)$

$Ax_7 \qquad((\delta\Longrightarrow\alpha )\Longrightarrow((\delta\Longrightarrow\beta)\Longrightarrow(\delta \Longrightarrow(\alpha\wedge\beta))))$

$Ax_8 \qquad ((\alpha \Longrightarrow(\beta\Longrightarrow\delta))\equiv((\alpha\wedge\beta )\Longrightarrow\delta))$

$Ax_9 \qquad\ ((\alpha\wedge\lnot\alpha )\Longrightarrow\beta)$

$Ax_{10} \qquad ((\alpha\Longrightarrow (\alpha\wedge\lnot\alpha))\Longrightarrow\lnot\alpha)$

$Ax_{11} \qquad (\alpha\vee\lnot\alpha)$

## Axioms of predicate calculus

$Ax_{12} \qquad ((x:=\tau )true\Longrightarrow((\forall x)\alpha(x)\Longrightarrow(x:=\tau)\alpha (x))) \\ \qquad\qquad {\small where\ term\ }\tau {\ \small\ is\ of\ the\ same\ type\ as\ the\ variable\ x}$ $Ax_{13} \qquad (\forall x)\alpha(x)\equiv\lnot (\exists x)\lnot\alpha(x)$

$Ax_{14} \qquad K((\exists x)\alpha (x))\equiv(\exists y)(K\alpha(x/y))\ \ \ \ \ \ \ \ \ \ \ \ \ \ for\ y\notin V(K)$

## Axioms of algorithmic logic

$Ax_{14} \qquad K((\exists x)\alpha (x))\equiv(\exists y)(K\alpha(x/y))$ for $y\notin V(K)$

$Ax_{15} \qquad K(\alpha\vee\beta)\equiv((K\alpha)\vee(K\beta))$

$Ax_{16} \qquad K(\alpha\wedge\beta)\equiv((K\alpha)\wedge(K\beta))$

$Ax_{17} \qquad K(\lnot\alpha)\Longrightarrow\lnot(K\alpha)$

$Ax_{18} \qquad ((x:=\tau)\gamma\equiv(\gamma(x/\tau)\wedge (x:=\tau)true))~\wedge((q:=\gamma^\prime)\gamma\equiv\gamma(q/\gamma^ \prime))$

$Ax_{19} \qquad \textbf{begin } K;M \textbf{ end}\, \alpha \equiv K(M\alpha)$

$Ax_{20} \qquad \textbf{if}\ \gamma\ \textbf{then}\ K \ \textbf{else } M \textbf{ fi}\, \alpha\equiv((\lnot\gamma\wedge M\alpha )\vee(\gamma\wedge K\alpha))$

$Ax_{21} \qquad \textbf{while }\gamma \textbf{ do }K\textbf{ od}\,\alpha\equiv((\lnot\gamma\wedge\alpha )\vee(\gamma\wedge K(\textbf{while }\gamma\textbf{ do }K\textbf{ od} (\lnot\gamma\wedge\alpha))))$

$Ax_{22} \qquad \bigcap K\alpha\equiv (\alpha\wedge(K\bigcap K\alpha))$

$Ax_{23} \qquad \bigcup K\alpha \equiv(\alpha\vee(K\bigcup K\alpha))$