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A derivation (or proof) of $\phi$ from $\Gamma$ is a finite sequence of inferences $\Sigma_{i}\vdash\phi_{i}$ such that the last one is $\Gamma\vdash\phi%$ and such that each %$\Sigma_{i}\vdash\phi_{i}$ is shown to be a legitimate inference using only the rules above and $\Sigma_{j}\vdash\phi_{j}$ for $j<i$. A sentence $\phi$ is derivable from $\Gamma$ or is a theorem of $\Gamma$ if there is a proof of $\phi$ from $\Gamma$ or, equivalently, if $\phi$ is a member of the smallest set $\Sigma$ such that $\Gamma\subseteq\Sigma$ and, for any formula $\sigma$ if $$\Sigma\vdash\sigma$, then $\sigma\in\Sigma$.

-- ShaughanLavine - 04 May 2009
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Topic revision: r3 - 02 Nov 2009 - 20:14:41 - TWikiGuest
 
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