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Both sides previous revision Previous revision Next revision | Previous revision | ||
sav08:proof_theory_for_propositional_logic [2008/03/09 19:43] vkuncak |
sav08:proof_theory_for_propositional_logic [2015/04/21 17:30] (current) |
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System $S$ is sound if for every formula $F$, | System $S$ is sound if for every formula $F$, | ||
- | \[ | + | \begin{equation*} |
(\vdash_S F) \rightarrow (\models F) | (\vdash_S F) \rightarrow (\models F) | ||
- | \] | + | \end{equation*} |
We can only prove true formulas. | We can only prove true formulas. | ||
Line 23: | Line 23: | ||
System $S$ is complete if for every formula $F$, | System $S$ is complete if for every formula $F$, | ||
- | \[ | + | \begin{equation*} |
(\models F) \rightarrow (\vdash F) | (\models F) \rightarrow (\vdash F) | ||
- | \] | + | \end{equation*} |
We can prove all valid formulas. | We can prove all valid formulas. | ||
- | ===== Case analysis proof system ===== | + | ===== Some Example Proof Systems ===== |
- | A sound and complete proof system. | + | === Case analysis proof system === |
- | Case analysis rule: | + | A simple A sound and complete proof system. |
- | \[ | + | |
+ | **Rule 1:** Case analysis rule: | ||
+ | \begin{equation*} | ||
\frac{P\{p \mapsto {\it true}\}\ ;\ P\{p \mapsto {\it false}\}} | \frac{P\{p \mapsto {\it true}\}\ ;\ P\{p \mapsto {\it false}\}} | ||
{P} | {P} | ||
- | \] | + | \end{equation*} |
- | Evaluation rule: derive $P$ if $P$ has no variables and it evaluates to //true//. | + | **Rule 2:** Evaluation rule: derive $P$ if $P$ has no variables and it evaluates to //true//. |
- | Adding simplification rules. Adding sound rules preserves soundness and completeness. | + | We can also add simplification rules. Adding sound rules preserves soundness and completeness. |
- | ===== Gentzen proof system ===== | + | === Gentzen proof system === |
Recall Gentzen's proof system from [[proofs and induction]] part of [[lecture02]], ignoring rules for predicates and induction. | Recall Gentzen's proof system from [[proofs and induction]] part of [[lecture02]], ignoring rules for predicates and induction. | ||
- | ===== Resolution ===== | + | === Resolution === |
- | + | ||
- | Definition and soundness. Completeness in next lecture. | + | |
+ | [[Definition of Propositional Resolution]] | ||