Philosophy DSM 252 Unit 5 | 4th Semester Notes

Formal Proof of Validity

A Formal Proof of Validity is a step-by-step deduction where each step follows logically from previous steps using established rules. While Truth Tables (Unit 4) are effective for small arguments, they become too complex for arguments with many variables. Formal proof offers a more elegant and direct method of demonstration.

A formal proof is a sequence of statements, each of which is either a premise or is derived from preceding statements by a rule of inference, ending with the conclusion.

The Nine Rules of Inference

These rules are the "logical building blocks" of any proof. Each rule represents an elementary argument form that is always valid.

Rule Name	Abbreviation	Logical Structure
1. Modus Ponens	M.P.	p ¹ q, p / ∴ q
2. Modus Tollens	M.T.	p ¹ q, ~q / ∴ ~p
3. Hypothetical Syllogism	H.S.	p ¹ q, q ¹ r / ∴ p ¹ r
4. Disjunctive Syllogism	D.S.	p v q, ~p / ∴ q
5. Constructive Dilemma	C.D.	(p¹q)•(r¹s), p v r / ∴ q v s
6. Absorption	Abs.	p ¹ q / ∴ p ¹ (p • q)
7. Simplification	Simp.	p • q / ∴ p
8. Conjunction	Conj.	p, q / ∴ p • q
9. Addition	Add.	p / ∴ p v q

Constructing a Formal Proof

To solve a problem in Unit 5, you must number each line and state the rule used. For example:

A \supset B (Premise)
A (Premise) / ∴ B \lor C
B (From 1, 2 by M.P.)
B \lor C (From 3 by Add.) - Conclusion Reached!

Deep Dive into Key Patterns

Modus Ponens (M.P.) vs. Modus Tollens (M.T.)

M.P. is forward-moving: if you have the "if" part (antecedent), you get the "then" part (consequent). M.T. is backward-moving: if you deny the "then" part, you must deny the "if" part.

Simplification (Simp.) vs. Addition (Add.)

Simplification: From an "AND" statement (p \bullet q), you can pull out the left side. It reduces information.
Addition: From any statement p, you can add anything else with an "OR" (v). It expands the statement while maintaining truth.

Exam Strategies: The "Proof-Hunter's" Guide

Strategy 1: Start at the End. Look at the conclusion. If the conclusion is a conjunction (X \bullet Y), your goal is to derive X and Y separately and use Conj.
Strategy 2: Look for M.P. first. It is the most common move. If you see a conditional statement, hunt for its antecedent among the other lines.
Common Error: Trying to use Simplification on a Disjunction (p \lor q). NEVER do this. Simplification only works on Conjunctions (\bullet).
Common Error: "Affirming the Consequent." If you have p \supset q and q, you cannot conclude p. This is a fallacy, not a rule.
FAQ: Do I have to use all premises?
In textbooks, yes, usually every premise is needed. In real logic, an argument can be valid even if some premises are redundant.

Unit 5: Formal Proof of Validity

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