**1 - Modus Ponens (MP)**

It states that if the propositions A → B and A are true, then B is also true.

Modus Ponens is also referred to as the implication elimination because it

eliminates the implication A → B and results in only the proposition B. It also

affirms the truthfulness of antecedent. It is written as:

**α_1 → α_2, α_1 =>α_2**

**2 - Modus Tollens (MT)**It states that if A → B and ¬ B are true then ¬ A is also true. Modus Tollens is

also referred to as denying the consequent as it denies the truthfulness of the

consequent. The rule is expressed as:

**α_1 → α_2, ~α_2 =>~α_1**

**3 - Disjunctive Syllogism (DS)**Disjunctive Syllogism affirms the truthfulness of the other proposition if one of

the propositions in a disjunction is false.

**Rule 1:α_1 ∨α_2, ~α_1=>α_2 Rule 2:α_1 ∨α_2, ~α_2 =>α_1**

**4 - Addition**The rule states that if a proposition is true, then its disjunction with any other

proposition is also true.

**Rule 1:α_1=>α_1∨α_2 Rule 2:α_2=>α_1 ∨α_2**

**5 - Simplification**Simplification means that if we have a conjunction, then both the constituent

propositions are also true.

**Rule 1:α_1 ∧α_2 =>α_1 Rule 2:α_1 ∧α_2 =>α_2**

**6 - Conjunction**Conjunction states if two propositions are true, then their conjunction is also

true. It is written as:

**α_1, α_2 =>α_1 ∧α_2**

**7 - Hypothetical Syllogism (HS)**The rule says that the conclusion α_1 → α_3 is true, whenever conditional

statements α_1 → α_2 and α_2 → α_3hold the truth values. This rules also

shows the transitive nature of implication operator.

**α_1 → α_2, α_2 → α_3 =>α_1 → α_3**

**8 - Absorption**The rule states that if the literal α_1 conditionally implies another literal α_2

**i.e.,α_1 → α_2 is true, then α_1 → (α_1 ∧α_2)**also holds.

** α_1 → α_2 =>α_1 → (α_1 ∧α_2)**

**9 - Constructive Dilemma (CD)**

According to the rule, if proposition **(α_1 ∨α_3)** and proposition** ((α_1 → α_2)****∧ (α_3 → α_4))**have true values, then the well-formed formula **(α_2 ∨α_4)** also

holds true value.

**(α_1 ∨α_3 ), (α_1 → α_2) ∧ (α_3 → α_4) =>α_2 ∨α_4**

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