PHI-DSC-202 (Logic II): Unit 3: Formal Proof of Validity (Nineteen Rules)

Contact Hours: 60 | Full Marks: 100 (ESE=70/CCA=30)

Table of Contents

1. Formal Proof of Validity (The Nineteen Rules)
2. Direct Proof Construction
3. Indirect Proof (Reductio Ad Absurdum)
4. Conditional Proof Construction

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Formal Proof of Validity (The Nineteen Rules)

A **Formal Proof of Validity** demonstrates that the conclusion of an argument necessarily follows from its premises by constructing a step-by-step deduction, where each new line is justified by applying one of the nineteen accepted elementary argument forms (rules) to previous lines.

The Nineteen Rules of Inference and Replacement

These rules are grouped into two categories:

1. Nine Rules of Inference (Applied to Whole Lines ONLY):

   These are elementary valid argument forms used to deduce a conclusion from its premises.

   • **MP (Modus Ponens):** (P → Q), P / ∴ Q
   • **MT (Modus Tollens):** (P → Q), ¬Q / ∴ ¬P
   • **HS (Hypothetical Syllogism):** (P → Q), (Q → R) / ∴ (P → R)
   • **DS (Disjunctive Syllogism):** (P ∨ Q), ¬P / ∴ Q
   • **CD (Constructive Dilemma):** ((P → Q) ⋅ (R → S)), (P ∨ R) / ∴ (Q ∨ S)
   • **Simp (Simplification):** (P ⋅ Q) / ∴ P
   • **Conj (Conjunction):** P, Q / ∴ (P ⋅ Q)
   • **Add (Addition):** P / ∴ (P ∨ Q)
   • **DD (Destructive Dilemma):** ((P → Q) ⋅ (R → S)), (¬Q ∨ ¬S) / ∴ (¬P ∨ ¬R)
2. Ten Rules of Replacement (Applied to Part or Whole of a Line):

   These rules state that equivalent logical forms can be substituted for one another.

   • **DM (De Morgan’s Theorems):** ¬(P ⋅ Q) ≡ (¬P ∨ ¬Q); ¬(P ∨ Q) ≡ (¬P ⋅ ¬Q)
   • **Comm (Commutation):** (P ∨ Q) ≡ (Q ∨ P); (P ⋅ Q) ≡ (Q ⋅ P)
   • **Assoc (Association):** ((P ∨ Q) ∨ R) ≡ (P ∨ (Q ∨ R)); ((P ⋅ Q) ⋅ R) ≡ (P ⋅ (Q ⋅ R))
   • **Dist (Distribution):** (P ⋅ (Q ∨ R)) ≡ ((P ⋅ Q) ∨ (P ⋅ R)); (P ∨ (Q ⋅ R)) ≡ ((P ∨ Q) ⋅ (P ∨ R))
   • **DN (Double Negation):** P ≡ ¬¬P
   • **Trans (Transposition):** (P → Q) ≡ (¬Q → ¬P)
   • **Impl (Material Implication):** (P → Q) ≡ (¬P ∨ Q)
   • **Equiv (Material Equivalence):** (P ≡ Q) ≡ ((P → Q) ⋅ (Q → P)); (P ≡ Q) ≡ ((P ⋅ Q) ∨ (¬P ⋅ ¬Q))
   • **Exp (Exportation):** ((P ⋅ Q) → R) ≡ (P → (Q → R))
   • **Taut (Tautology):** P ≡ (P ∨ P); P ≡ (P ⋅ P)

Direct Proof Construction

A **Direct Proof** involves beginning with the premises and deriving the conclusion step-by-step by applying the Nineteen Rules of Inference and Replacement to prior lines.

1. List the premises numerically.
2. The goal is always the conclusion.
3. Apply rules to the premises until the conclusion is reached.

Example Line (Demonstration):
1. (A → B) ⋅ (A → C) (Premise)
2. A (Premise)
3. A → B (1, Simp)
4. B (3, 2, MP)

Indirect Proof (Reductio Ad Absurdum)

The **Indirect Proof (IP)**, or **Reductio Ad Absurdum**, is a method where you assume the negation of the conclusion as an added premise. If this assumption leads to a contradiction (R ⋅ ¬R), then the assumption must be false, and the original conclusion must be true.

1. List the premises.
2. Assume the negation of the conclusion (IP Assumption).
3. Derive a **Contradiction** (e.g., Q ⋅ ¬Q).
4. Conclude the original conclusion, justifying the step by the contradiction and the IP method.

Conditional Proof Construction

The **Conditional Proof (CP)** is used specifically when the conclusion is a conditional statement (P → Q). Instead of deriving the entire conditional, you assume the antecedent (P) as an additional premise and then derive the consequent (Q).

1. List the premises.
2. Assume the antecedent (P) of the conclusion (CP Assumption).
3. Derive the consequent (Q) using the Nineteen Rules.
4. Conclude the full conditional (P → Q), citing all steps from the assumption to the consequent as the justification for the CP method.

Exam Focus: Strategy Choice

Use **Conditional Proof (CP)** if the conclusion is a conditional (P → Q). Use **Indirect Proof (IP)** if the conclusion is not a conditional, or if you get stuck with a Direct Proof, as IP is highly flexible.

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