chapter four proofs. 1. argument forms an argument form is a group of sentence forms such that all...
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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
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