Chapter Five

Conditional and Indirect Proofs

1. Conditional Proofs

A conditional proof is a proof in which we assume the truth of one of the premises to show that if that premise is true

then the argument displayed is valid.

In a conditional proof the conclusion depends only on the original premise, and not on the assumed premise.

When the scope of the assumed premise ends it has been discharged.

Conditional Proofs, continued

Every correct application of Conditional Proof (CP)

incorporates:• The sentence justified by CP must be a conditional.• The antecedent of that conditional must be the assumed

premise.• The consequent of that conditional must be the sentence

from the preceding line.• Lines are drawn indicating the scope of the assumed

premise.

Conditional Proofs, continued

• All you gain from a conditional proof is one line, which will be the first line below the horizontal line in your proof.

• When using CP, always assume the antecedent of the conditional you hope to justify.

• In deciding what to assume, be guided by the conclusion or the intermediate step you hope to reach.

2. Indirect Proofs

• A contradiction is any sentence that is inconsistent.

• An explicit contradiction is of the form “P” and “not-P”.

Indirect Proofs, continued

The main idea behind the rule of indirect proof (IP) is to see

if we can derive a contradiction from the combination of the set of premises of the argument that we are assessing

for validity and the negation of its conclusion.

This type of proof is also known as the reductio ad absurdum proof

3. Strategy Hints for Using CP and IP

• Use CP if your conclusion is a conditional• Use CP if your conclusion is equivalent to a conditional• Every proof can be solved using IP. So, if all else fails, try

IP.• Note that trying with IP first can sometimes make the

proof more difficult.• When using IP, try to break complex formulas into simpler

units.• IP is especially useful when the conclusion is either

atomic or a negated sentence.

4. Zero-Premise Deductions

• Every truth table tautology can be proved by a zero-premise deduction.

• Tautologies are sometimes termed theorems of logic.• A tautology will follow from any premises whatever.• This is because the negation of a tautology is a

contradiction, so if we use IP by assuming the negation of a tautology, we can derive a contradiction independently of other premises. This is why this process is called a zero-premise deduction.

5. Proving Premises Inconsistent

• If the premises of an argument are inconsistent, then at least one must be false.

• To prove that an argument has inconsistent premises we use the eighteen valid forms.

6. Adding Valid Argument Forms

It is convenient to combine two or more rules into one step.

Logical candidates for such combinations are rules that are often used together—such as DeM and

DN, DN and Impl., and the two uses of DN.

7. An Alternative to Conditional Proof?

Let us adopt a rule, call it TADD, in which a tautology can be added at any time to the premises of an argument in a

deductive sentential proof.

BUT

TADD mixes syntax and semantics in philosophically and logically problematic ways.

8. The Completeness and Soundness of Sentential Logic

We now have two different conceptions of logical truths—tautologies and theorems.

Logicians draw a distinction between the syntax and semantics of a system of logic.

The semantics of a system of logic includes those aspects of it having to do with meaning and truth (e.g., tautologies).

The syntax of a system of logic have to do with its form or structure (e.g., theorems).

The Completeness and Soundness of

Sentential Logic, continued

• A system of logic is complete if every argument that is semantically valid is syntactically valid.

• A system of logic is sound if every argument that is syntactically valid is semantically valid.

• The proof that a system of logic is both sound and complete is part of metalogic.

9. Introduction and Elimination Rules

• Conjunction Introduction • Conjunction Elimination • Disjunction Introduction • Disjunction Elimination• Conditional Introduction• Conditional Elimination• Negation Introduction• Negation Elimination• Equivalence Introduction• Equivalence Elimination• Reiteration

Key Terms

• Absorption• Assumed premise• Complete• Contradiction• Discharged premise• Explicit contradiction• Indirect proof

Key Terms, continued

• Metalogic• Reductio ad absurdum proof• Sound• Theorem• Zero-premise deduction