User:Arthur Rubin/Rules

From


 * Note: I'm going to skip rules of identity
 * Note: Even if this is a potentional copyright violation, does anyone doubt I can get permission?


 * Tautology:
 * If $$\vDash P_1 \land P_2 \land \cdots \land P_n \rightarrow Q$$ ($$P_1 \land P_2 \land \cdots \land P_n \rightarrow Q$$ is a tautology), then $$P_1,P_2, \cdots, P_n \vdash Q$$
 * (note that n may equal 0.)


 * Tautologically Equivalent Formulas
 * If $$\vDash \psi_1 \leftrightarrow \psi_2$$ and $$\varphi_2$$ is obtained from $$\varphi_1$$ by substituing some occurences of $$\psi_1$$ by $$\psi_2$$, then $$\vDash \varphi_1 \leftrightarrow \varphi_2$$

Specific tautologies:
 * (Implict rule of proof)
 * If
 * $$\Gamma_1 \vdash P_1$$,
 * $$\Gamma_2 \vdash P_2$$,
 * $$\Gamma_n \vdash P_n$$,
 * $$P_1, P_2, \cdots, P_n \vdash Q$$, then
 * $$\Gamma_1 \cup \Gamma_2 \cup \cdots \cup \Gamma_n \vdash Q$$
 * $$\Gamma_n \vdash P_n$$,
 * $$P_1, P_2, \cdots, P_n \vdash Q$$, then
 * $$\Gamma_1 \cup \Gamma_2 \cup \cdots \cup \Gamma_n \vdash Q$$


 * 1) $$P \land (P \rightarrow Q) \rightarrow Q$$ Rule of Detachment, Modus Ponens
 * 2) $$\neg Q \land (P \rightarrow Q) \rightarrow \neg P$$ (Modus Tollendo Tollens)
 * 3) $$\neg P \land (P \lor Q) \rightarrow Q$$ (Modus Tollendo Ponens)
 * 4) $$P \land Q \rightarrow P$$ (Rule of Simplification)
 * 5) $$P \rightarrow P \lor Q$$ (Rule of Addition)
 * 6) Rule of Adjunction
 * 7) *$$P \rightarrow (Q \rightarrow P \land Q)$$
 * 8) *$$(P \rightarrow Q) \land (P \rightarrow R) \rightarrow (P \rightarrow Q \land R)$$
 * 9) $$(P \rightarrow Q) \land (Q \rightarrow R) \rightarrow (P \rightarrow R)$$ (Rule of Hypothetical Syllogism)
 * 10) Rules of Alternative Proof
 * 11) *$$(P \rightarrow Q) \land (\neg P \rightarrow Q) \rightarrow Q$$
 * 12) *$$ (P \lor Q \rightarrow R) \leftrightarrow (P \rightarrow R) \land (Q \rightarrow R)$$
 * 13) Rule of Absurdity
 * 14) *$$(P \rightarrow Q \land \neg Q) \rightarrow \neg P$$
 * 15) *$$P \land \neg P \rightarrow Q$$
 * 16) $$P \lor \neg P$$ (Rule of the Excluded Middle)
 * 17) $$\neg (P \land \neg P)$$ (Rule of Contradiction)
 * 18) Communtative Rules
 * 19) *$$P \lor Q \leftrightarrow Q \lor P$$
 * 20) *$$P \land Q \leftrightarrow Q \land P$$
 * 21) Associative Rules
 * 22) *$$P \lor (Q \lor R) \leftrightarrow (P \lor Q) \lor R$$
 * 23) *$$P \land (Q \land R) \leftrightarrow (P \land Q) \land R$$
 * 24) Distributive Rules
 * 25) *$$P \lor (Q \land R) \leftrightarrow (P \lor Q) \land (P \lor R)$$
 * 26) *$$P \land (Q \lor R) \leftrightarrow (P \land Q) \lor (P \land R)$$
 * 27) De Morgan's Rules
 * 28) *$$\neg (P \lor Q) \leftrightarrow \neg P \land \neg Q$$
 * 29) *$$\neg (P \land Q) \leftrightarrow \neg P \lor \neg Q$$
 * 30) $$\neg \neg P \leftrightarrow P$$ (Rule of Double Negation)
 * 31) Rules for the Conditional
 * 32) *$$(P \rightarrow Q) \leftrightarrow \neg P \lor Q$$
 * 33) *$$\neg (P \rightarrow Q) \leftrightarrow P \land \neg Q$$
 * 34) Rules for the Biconditional
 * 35) *$$(P \leftrightarrow Q) \leftrightarrow (P \rightarrow Q) \land (Q \rightarrow P)$$
 * 36) *$$(P \leftrightarrow Q) \leftrightarrow (P \land Q) \lor (\neg P \land \neg Q)$$
 * 37) *$$(P \leftrightarrow Q) \leftrightarrow (\neg P \lor Q) \land (P \land \neg Q)$$
 * 38) Idempotency Rules
 * 39) *$$P \lor P \leftrightarrow P$$
 * 40) *$$P \land P \leftrightarrow P$$
 * 41) Rules of Contraposition
 * 42) *$$(P \rightarrow Q) \leftrightarrow (\neg Q \rightarrow \neg P)$$
 * 43) *$$(P \rightarrow \neg Q) \leftrightarrow (Q \rightarrow \neg P)$$
 * 44) *$$(\neg P \rightarrow Q) \leftrightarrow (\neg Q \rightarrow P)$$
 * 45) $$(P \rightarrow (Q \rightarrow R)) \leftrightarrow (P \land Q \rightarrow R)$$ (Rule of Exportation-Importation)
 * 46) Rules of Absorption
 * 47) *$$P \lor (P \land Q) \leftrightarrow P$$
 * 48) *$$P \land (P \lor Q) \leftrightarrow P$$

(non-tautological rules)
 * Conditional Proof
 * If $$\Gamma, P \vdash Q$$ then $$\Gamma \vdash P\rightarrow Q$$


 * Indirect proof
 * If $$\neg P, \Gamma \vdash Q \land \neg Q$$, then $$\Gamma \vdash P$$

(non-tautological rules of the predicate calculus)
 * Existential generalization
 * $$\varphi[t|v]\vdash(\exists v)\varphi$$ if the substitution of t for v in $$\varphi$$


 * Existential Proof
 * If $$\Gamma, \psi[v|u]\vdash \varphi$$, then $$\Gamma, (\exists u)\psi \vdash \varphi$$, provided that
 * The substitution of v for u in $$\psi$$ is valid
 * v is not free in $$\varphi$$
 * v is not free in $$\Gamma$$


 * Universal Generalization
 * If $$\Gamma \vdash \varphi$$, then $$\Gamma \vdash (\forall v)\varphi$$,
 * if v is not free in $$\Gamma$$.


 * Universal Specification
 * $$(\forall v)\varphi \vdash \varphi[t|v]$$,
 * if the substitution of t for v is valid.