**Equivalence Laws**

1.- Commutative CONM

This law can be applied with three of the four dyadic connectives: conjunction, disjunction, and biconditional. With the only connective that this law cannot be applied is with the connective of the conditional. Change the order of the propositions without changing the connective.

**(P^Q) ⇆ (Q^P)**

**(P v Q) ⇆ (Q v P)**

2.- Double negation DN.

A doubly negated proposition is equal to its affirmation and vice versa.

**P ⇆ (~~P)**

3.- Morgan MORGAN

It has two ways in which the law can be applied. From a proposition in negated conjunction, the negation of each of the conjunctives can be obtained but changing the connective to disjunction, but changing the connective to conjunction.

**[~( P ^ Q)] ⇆ (~P v ~Q)**

O well

**[~(P v Q)] ⇆ (~P ^ ~Q)**

4.- ASOC Association

This law orders in various ways without altering the products, when you have the same logical connective, be it the conjunction or disjunction.

**[(P v Q) v R] ⇆ [P v (Q v R)]**

O well

**[(P^Q)^R] ⇆ [P^(Q^R)]**

5.- DISTR distribution

They are applied when there are two different connectives: conjunction-disjunction or disjunction-conjunction. Distribute the statement outside the parentheses with those inside it.

**[(P v Q) ^ R] ⇆ [(P ^ R) v (Q ^ R)**

**[(P ^ Q) v R)] ⇆ [(P v R) ^ (Q v R)**

6.- Trivial Tautology TAU

Two equal propositions joined by a conjunction or a disjunction is equivalent to having the same proposition.

**(P v P) ⇆ P**

**(P^P) ⇆ P**

7.- IM material implication

This law allows changing the main connective of the “conditional” proposition by “disjunction”, but denying the antecedent.

**(P → Q) ⇆ (~P v Q)**

O well

**(P → Q) ⇆ ~(P ^ ~Q)**

8.- TRANSPOSE AFTER

The transposition of a conditional preposition is a proposition with the same conditional connective changing the antecedent and consequent prepositions and negating them respectively.

**(P → Q) ⇆ (~Q → ~P)**

9.- EXP Export

It changes from conjunction connective to conditional when the antecedent is a conjunction, and groups them differently by leaving the first conjunctive as the antecedent of the entire preposition and passing the second conjunctive to the consequent of the proposition as part of another conditional.

**[(P^Q) → R] ⇆ [P → (Q → R)]**

10.- EM material equivalence

The biconditional proposition is equivalent to the conjunction of the conditional propositions that are part of the biconditional in both directions.

**(P ⇆ Q) ⇆ [(P → Q) v (~Q → ~P)**

**(P ⇆ Q) ⇆ [(P → Q) ^ (Q → P)**