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Reflexivity of: $\Gamma\vdash t=t$ , where is a constant symbol or a variable.
Transitivity of: %\Gamma , r=s,s=t\vdash r=t$%, where , , and are constant symbols or variables.
Substitution: If $\phi$ is a formula according to either of the first two clauses of the definition of a formula, and are variables and constant symbols, and $\psi$ is obtained from $\phi$ by replacing some occurrences of by , then $\Gamma ,s=t,\phi\vdash\psi$ .
Thinning: If $\Sigma$ is a set of formulas and $\Gamma\vdash\phi$ , then $\Gamma\cup\Sigma\vdash\phi$ .
Cut: If $\Gamma\vdash\sigma$ for every $\sigma$ in $\Sigma$ and $\Sigma\vdash\phi$ , then $\Gamma\vdash\phi$ .

Proofs and theorems

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 . 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 $\sigma\in\Sigma$ .

Consistency

A set $\Gamma$ of formulas is inconsistent if there is a formula $\psi$ such that $\Gamma\vdash\phi$ and $\Gamma\vdash (\lnot\phi )$ . A set $\Gamma$ of formulas is consistent if it is not inconsistent.

-- ShaughanLavine - 03 May 2009

-- ShaughanLavine - 04 May 2009
Latex rendering error!! dvi file was not created.

Topic revision: r2 - 05 Nov 2009 - 02:10:01 - TWikiGuest

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