Unit 5: Formal Truth of Validity

Introduction to Formal Proof of Validity
The Nine Rules of Inference
Application and Step-by-Step Proof Construction
Exam Focus: Tips, Pitfalls, and FAQs

Introduction to Formal Proof of Validity

The Formal Truth of Validity refers to a method of proving that a deductive argument is valid by showing that its conclusion can be derived from its premises through a series of elementary valid arguments. Unlike truth tables, which can become unwieldy with many variables, this method uses a set of established Rules of Inference to construct a step-by-step proof.

A formal proof of validity is a sequence of statements, each of which is either a premise or is derived from preceding statements by a rule of inference, such that the last statement in the sequence is the conclusion of the argument.

The Nine Rules of Inference

These nine rules are the "building blocks" of deductive proof. They are elementary argument forms that are always valid.

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

Application and Step-by-Step Proof Construction

To construct a formal proof, you list the premises, number them, and then derive intermediate steps until you reach the conclusion. Each step must cite the rule used and the previous line numbers it was applied to.

Example Proof

Argument: If it rains (R), the ground is wet (W). It is raining. Therefore, the ground is wet and it is cloudy (C) or not cloudy.

1. R ¹ W (Premise)
2. R (Premise)
3. W (From 1, 2 by M.P.)
4. W v (C • ~C) (From 3 by Addition)

Deep Explanation of Key Rules

Modus Ponens: Often called "affirming the antecedent." If the "if" part is true, the "then" part must follow.
Modus Tollens: Often called "denying the consequent." If the "then" part is false, the "if" part must have been false.
Simplification: Since a conjunction (AND) is only true if both parts are true, if you have "p • q", you are logically allowed to extract "p".
Addition: A disjunction (OR) is true if at least one part is true. If you know "p" is true, you can add anything to it with an "OR" and the statement remains true.

Exam Focus: Tips, Pitfalls, and FAQs

Exam Tip: When starting a complex proof, look at the Conclusion. Work backward to see which rules could result in that specific form. For example, if the conclusion is a conjunction (p • q), you will likely need to use the Rule of Conjunction as your final step.

Common Pitfall: Do not use "Simplification" on a disjunction (v). You can only simplify a conjunction (•). Example: From "p v q", you cannot conclude "p". This is a common error in exams.

Frequently Asked Questions

Q: Can I use these rules in any order?
A: Yes, as long as the line you are using as a premise for a rule has already been stated or derived previously in the proof.

Q: What is the difference between these rules and the rules for Syllogisms in Unit 3?
A: Unit 3 rules (Copi's 6 rules) are for testing if a categorical syllogism is valid. These 9 rules are for constructing a proof to show that a statement follows from others.