SD+ Derivation Rules

All the Derivation Rules of SD and

Rules of Inference

Modus Tollens $(\text{MT})$

	$\mathscr{P}\supset\mathscr{Q}$
	$\sim\mathscr{Q}$
$\rhd$	$\sim\mathscr{P}$

Hypothetical Syllogism $(\text{HS})$

	$\mathscr{P}\supset\mathscr{Q}$
	$\mathscr{Q}\supset\mathscr{R}$
$\rhd$	$\mathscr{P}\supset\mathscr{R}$

Disjunctive Syllogism $(\text{DS})$

	$\mathscr{P}\lor\mathscr{Q}$
	$\sim\mathscr{P}$
$\rhd$	$\mathscr{Q}$

or

	$\mathscr{P}\lor\mathscr{Q}$
	$\sim\mathscr{Q}$
$\rhd$	$\mathscr{P}$

Rules of Replacement

Commutation $(\text{Com})$

$\mathscr{P}\;\&\;\mathscr{Q} \;\lhd\rhd\; \mathscr{Q}\;\&\;\mathscr{P}$

$\mathscr{P}\lor\mathscr{Q} \;\lhd\rhd\; \mathscr{Q}\lor\mathscr{P}$

Association $(\text{Assoc})$

$\mathscr{P}\;\&\;(\mathscr{Q}\;\&\;\mathscr{R}) \;\lhd\rhd\; (\mathscr{P}\;\&\;\mathscr{Q})\;\&\;\mathscr{R}$

$\mathscr{P}\lor(\mathscr{Q}\lor\mathscr{R}) \;\lhd\rhd\; (\mathscr{P}\lor\mathscr{Q})\lor\mathscr{R}$

Implication $(\text{Impl})$

$\mathscr{P}\supset\mathscr{Q} \;\lhd\rhd\; \sim\mathscr{P}\lor\mathscr{Q}$

Double Negation $(\text{DN})$

$\mathscr{P} \;\lhd\rhd\; \sim\sim\mathscr{P}$

De Morgan $(\text{DeM})$

$\sim(\mathscr{P}\;\&\;\mathscr{Q}) \;\lhd\rhd\; \sim\mathscr{P}\,\lor\sim\mathscr{Q}$

$\sim(\mathscr{P}\lor\mathscr{Q}) \;\lhd\rhd\; \sim\mathscr{P}\;\&\;\sim\mathscr{Q}$

Idempotence $(\text{Idem})$

$\mathscr{P} \;\lhd\rhd\; \mathscr{P}\;\&\;\mathscr{P}$

$\mathscr{P} \;\lhd\rhd\; \mathscr{P}\lor\mathscr{P}$

Transposition $(\text{Trans})$

$\mathscr{P}\supset\mathscr{Q} \;\lhd\rhd\; \sim\mathscr{Q}\supset\,\sim\mathscr{P}$

Exportation $(\text{Exp})$

$\mathscr{P}\supset(\mathscr{Q}\supset\mathscr{R}) \;\lhd\rhd\; (\mathscr{P}\;\&\;\mathscr{Q})\supset\mathscr{R}$

Distribution $(\text{Dist})$

$\mathscr{P}\;\&\;(\mathscr{Q}\lor\mathscr{R}) \;\lhd\rhd\; (\mathscr{P}\;\&\;\mathscr{Q}) \lor (\mathscr{P}\;\&\;\mathscr{R})$
$\mathscr{P}\lor(\mathscr{Q}\;\&\;\mathscr{R}) \;\lhd\rhd\; (\mathscr{P}\lor\mathscr{Q}) \;\&\; (\mathscr{P}\lor\mathscr{R})$

Equivalence $(\text{Equiv})$

$\mathscr{P}\equiv\mathscr{Q} \;\lhd\rhd\; (\mathscr{P}\supset\mathscr{Q})\;\&\; (\mathscr{Q}\supset\mathscr{P})$

$\mathscr{P}\equiv\mathscr{Q} \;\lhd\rhd\; (\mathscr{P}\;\&\;\mathscr{Q})\lor (\sim\mathscr{P}\;\&\;\sim\mathscr{Q})$