# Logic Rules Cheat Sheet

When working with logic in discrete math appliations there are a plethora of rules you can use for working with the well formed formulas. Remembering them all can be a daunting task, which is why I like to have a cheat sheet available. As such, here’s a simple one that I like to use when working with these problems.

# Equivalence Rules

ExpressionEquivalent ToName of the Rule
$P \lor Q$$Q \lor P$Commutative - comm.
$P \land Q$$Q \land P$Commutative - comm.
$\left(P\lor Q\right)\lor R$$P \lor\left(Q\lor R\right)$Associative - ass.
$\left(P\land Q\right)\land R$$P\land\left(Q\land R\right)$Associative - ass.
$\left(P\lor Q\right)'$$P'\land Q'$DeMorgan’s Law
$\left(P\land Q\right)'$$P'\lor Q'$DeMorgan’s Law
$P\rightarrow Q$$P'\lor Q$Implication - imp.
$P$$\left(P'\right)'$Double Negation - dn.
$P\iff Q$$\left(P\rightarrow Q\right)\land\left(Q\rightarrow P\right)$Definition of Equivalence - equ.

# Inference Rules

ExpressionCan DeriveName of the Rule
$P,\ P\rightarrow Q$$Q$Modus Ponens - mp.
$P\rightarrow Q,\ Q'$$P'$Modus Tollens - mt.
$P,\ Q$$P\land Q$Conjunction - con.
$P\land Q$$P,\ Q$Simplification - simp.
$P$$P\lor Q$Addition - add.
$P\rightarrow Q,\ Q\rightarrow R$$P\rightarrow R$Hypothetical Syllogism - hs
$P\lor Q,\ P'$$Q$Disjunctive Syllogism - ds.
$P\rightarrow Q$$Q'\rightarrow P'$Contraposition - cont.
$Q'\rightarrow P'$$P\rightarrow Q$Contraposition - cont.
$P$$P\land P$Self Reference - self.
$P\lor P$$P$Self Reference - self.
$\left(P\land Q\right)\rightarrow R$$P\rightarrow\left(Q\rightarrow R\right)$Exportation - exp.
$P,\ P'$$Q$Inconsistency - inc.

# Derivation Rules

The general workflow for using derivation rules is:

• Strip off the quantifiers
• Work with the independent well formed formulas
• Insert the quantifiers back in

Universal Instantiation - ui.: From $\left(\forall x\right)P\left(x\right)$ we can deduce $P\left(t\right)$.

Note: t must not already appear as a variable in the expression for $P\left(x\right)$.

Existential Instantiation - ei.: From $\left(\exists x\right)P\left(x\right)$ we can deduce $P\left(t\right)$.

Note: t must be introduced for the first time. As such, you will want to do these early in proofs.

Universal Generalization - ug.: From $P\left(x\right)$ we can deduce $\left(\forall x\right)P\left(x\right)$.

Notes:

• $P\left(x\right)$ hasn’t been deduced by existential instantiation from any hypothesis in which x was a free variable.
• $P\left(x\right)$ hasn’t been deduced by existential instantiation from another wff in which x was a free variable.

Existential Generalization - eg.: From $P\left(a\right)$ we can deduce $\left(\exists x\right)P\left(x\right)$.

Note: x must not appear in $P\left(a\right)$.