Chapter Four

Proofs

1. Argument Forms

An argument form is a group of sentence forms such that all of its substitution instances are arguments.

Argument Forms, continued

• If an argument form has no substitution instances that are invalid, it is said to be a valid argument form.

• An argument form that has even one invalid argument as a substitution instance is called an invalid argument form.

2. The Method of Proof: Modus Ponens and Modus Tollens

• Truth tables give us a decision procedure for any sentential argument.

• There is another method available to demonstrate validity of sentential arguments: the method of proof, or natural deduction.

The Method of Proof, continued

A proof of an argument is a series of steps that starts with premises; each step beyond the premises is derived from a valid argument form by being a substitution instance of it;

the last step is the conclusion.

Method of Proof, continued

Modus Ponens (MP):

p q⊃

p

Therefore, q.

Method of Proof, continued

Modus Tollens (MT):

p q⊃

˜q

Therefore, ˜p

Method of Proof, continued

Do not confuse either MP or MT with the invalid arguments that resemble them:

Affirming the Consequent:

p q⊃qTherefore, p

Method of Proof, continued

Denying the Antecedent:

p q⊃

˜p

Therefore, ˜q.

3. Disjunctive Syllogism (DS) and Hypothetical Syllogism (HS)

Another valid argument form is the Disjunctive Syllogism (DS). This has two forms:

p q∨˜pTherefore, q

Andp q∨˜qTherefore, q

DS and HS, continued

Another valid argument form is the Hypothetical Syllogism

(HS):

p q⊃q r⊃Therefore, p r⊃

4. Simplification and Conjunction

Another valid argument form is Simplification (Simp.), which has two forms:

p.qTherefore, p

And

p.qTherefore, q

Simplification and conjunction, continued

Conjunction (Conj.) is another valid argument form:

p

q

Therefore, p.q

5. Addition and Constructive Dilemma

Another valid argument form is Addition (Add.):

p

Therefore, p q∨

Addition and constructive dilemma, continued

Another valid argument form is Constructive Dilemma (CD):

p q∨p r⊃q r⊃Therefore, r s ∨

6. Principles of Strategy

• Look for forms that correspond to valid rules of inference• Remember: small sentences are your friends!• Once you have mastered the rules of inference, you will

find completing many proofs much easier by working backwards from the conclusion.

• Trace the connections between the letters in the argument, starting with those in the conclusion.

• Begin the proof with the letter most distant from those in the conclusion.

7. Double Negation (DN) and DeMorgan’s Theorem (DeM)

Double Negation (DN) is an equivalence argument form:

p

Therefore, ˜˜p

And

˜˜p

Therefore, p

DN and DeM, continued

DeMorgan’s Theorem (DeM):

˜(p . q) is equivalent to ˜p ˜q∨

And

˜(p q) is equivalent to ˜p . ˜q∨

8. Commutation (Comm.), Association (Assoc.), and

Distribution (Dist.)There are three more valid equivalence argument forms:

Commutation (Comm.):

p q is equivalent to q p∨ ∨

p . q is equivalent to q . p

Comm., Assoc., and Dist., continued

Association (Assoc.):

p (q r) is equivalent to (p q) r∨ ∨ ∨ ∨p . (q . r) is equivalent to (p . q) . R

Comm., Assoc., and Dist., continued

Distribution (Dist):

p . (q r) is equivalent to (p . q) (p . r)∨ ∨

p (q . r) is equivalent to (p q) . (p r)∨ ∨ ∨

9. Contraposition, Implication, and Exportation

Contraposition (Contra.):

p q is equivalent to ˜q ˜p⊃ ⊃

Implication (Impl.):

p q is equivalent to ˜p q⊃ ∨

Contraposition, Implication, and Exportation, continued

Exportation (Exp.):

(p . q) r is equivalent to p (q r)⊃ ⊃ ⊃

10. Tautology and Equivalence

Another valid equivalence form is Tautology

(Taut.):

p is equivalent to p . p

p is equivalent to p p∨

Tautology and Equivalence, continued

There are two valid argument forms called

Equivalence (Equiv.):

p ≡ q is equivalent to (p q) . (q p )⊃ ⊃

p ≡ q is equivalent to (p . q) (˜p . ˜q)∨

11. More Principles of Strategy

• Break down complex sentences with DeM, Simp, and Equiv.

• Use DeM and Dist to isolate “excess baggage”• Use Impl when you have a mix of conditionals

and disjunctions• Work backward from the conclusion

12. Common Errors in Problem Solving

• Using implicational forms on parts of lines• Reluctance to use Addition• Reluctance to use Distribution• Trying to Prove What Cannot be Proved• Failure to Notice the Scope of a Negation Sign

Key Terms

• Argument form• Completeness• Decision procedure• Equivalence argument form• Expressive completeness• Implicational argument form• Invalid argument form• Valid argument form