Download Conditional and Indirect Proofs

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 zeropremise 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
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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