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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