Fundamental Rules of Logic and Inference

English16 Jul 20255.04 KB

Modus Ponens (MP)

1. If you study, then you will have good grades. (P → Q)
2. You study. (P)
3. Therefore, you will have good grades. (Q) [From MP (1,2)]

Modus Tollens (MT)

1. If you give good tests, then your score will increase. (P → Q)
2. Your score will not increase. (¬Q)
3. Therefore, you do not give good tests. (¬P) [From MT (1,2)]

Adjunction (Conj)

1. You study. (P)
2. The material is clear. (Q)
3. Therefore, you study and the material is clear. (P ∧ Q) [From Adjunction (1,2)]

Simplification (Simp)

Example 1: Deriving P

1. You study and have good grades. (P ∧ Q)
2. Therefore, you study. (P) [From Simplification (1)]

Example 2: Deriving Q

1. You study and have good grades. (P ∧ Q)
2. Therefore, you have good grades. (Q) [From Simplification (1)]

Disjunctive Syllogism (DS)

Example 1: Eliminating ¬P

1. You study or you failed the exam. (P ∨ Q)
2. You do not study. (¬P)
3. Therefore, you failed the exam. (Q) [From DS (1,2)]

Example 2: Eliminating ¬Q

1. You study or you failed the exam. (P ∨ Q)
2. You did not fail the exam. (¬Q)
3. Therefore, you study. (P) [From DS (1,2)]

Addition (Add)

1. You study. (P)
2. Therefore, you study or you work. (P ∨ Q) [From Addition (1)]

Hypothetical Syllogism (HS)

1. If you study, then you will pass the tests. (P → Q)
2. If you pass the tests, then you will finish your degree. (Q → R)
3. Therefore, if you study, then you will finish your degree. (P → R) [From HS (1,2)]

Constructive Dilemma (CD)

1. You study or you fail the tests. (P ∨ Q)
2. If you study, then you will finish your degree. (P → R)
3. If you fail the tests, then you will work. (Q → S)
4. Therefore, you will finish your degree or you will work. (R ∨ S) [From CD (1,2,3)]

Idempotence of Disjunction (ID)

1. You study or you study. (P ∨ P)
2. Therefore, you study. (P) [From ID (1)]

De Morgan's Laws (DM)

Rule 1: Negation of Conjunction

1. It is not true that you are dancing and it is not true that you are having fun. (¬P ∧ ¬Q)
2. Therefore, it is not true that (you are dancing or you are having fun). (¬(P ∨ Q)) [From DM (1)]

Rule 2: Negation of Disjunction

1. It is not true that (you are dancing or you are having fun). (¬(P ∨ Q))
2. Therefore, it is not true that you are dancing and it is not true that you are having fun. (¬P ∧ ¬Q) [From DM (1)]

Rule 3: Negation of Conjunction (Reverse)

1. It is not true that (you are dancing and having fun). (¬(P ∧ Q))
2. Therefore, it is not true that you are dancing or it is not true that you are having fun. (¬P ∨ ¬Q) [From DM (1)]

Rule 4: Negation of Disjunction (Reverse)

1. It is not true that you are dancing or it is not true that you are having fun. (¬P ∨ ¬Q)
2. Therefore, it is not true that (you are dancing and having fun). (¬(P ∧ Q)) [From DM (1)]

Rule 5: Conjunction to Double Negated Disjunction

1. You are dancing and having fun. (P ∧ Q)
2. Therefore, it is not true that (it is not true that you are dancing or it is not true that you are having fun). (¬(¬P ∨ ¬Q)) [From DM, DN (1)]

Rule 6: Negated Disjunction to Negated Conjunction

1. It is not true that (you are dancing or it is not true that you are having fun). (¬(P ∨ ¬Q))
2. Therefore, it is not true that you are dancing and you are having fun. (¬P ∧ Q) [From DM, DN (1)]

Biconditional Equivalence (BE)

1. 2 x 3 = 6 if and only if 3 + 3 = 6. (P ↔ Q)
2. Therefore, if 2 x 3 = 6, then 3 + 3 = 6. (P → Q) [From BE (1)]
3. Therefore, if 3 + 3 = 6, then 2 x 3 = 6. (Q → P) [From BE (1)]

Double Negation (DN)

Example 1: From Negated to Affirmative

1. It is not true that you are not having fun. (¬¬P)
2. Therefore, you are having fun. (P) [From DN (1)]

Example 2: From Affirmative to Negated

1. You are having fun. (P)
2. Therefore, it is not true that you are not having fun. (¬¬P) [From DN (1)]

Commutative Law (CL)

Conjunction

1. You are dancing and having fun. (P ∧ Q)
2. Therefore, you are having fun and dancing. (Q ∧ P) [From CL (1)]

Disjunction

1. You are dancing or having fun. (P ∨ Q)
2. Therefore, you are having fun or dancing. (Q ∨ P) [From CL (1)]

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