Chapter Five

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Chapter Five

• Conditional and Indirect Proofs

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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.

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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.

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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.

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2. Indirect Proofs

• A contradiction is any sentence that is
  inconsistent.
• An explicit contradiction is of the form P and
  not-P.

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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

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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.

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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.

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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.

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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 togethersuch as DeM
  and DN, DN and Impl., and the two uses of DN.

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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.

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8. The Completeness and Soundness of Sentential
Logic

• We now have two different conceptions of logical
  truthstautologies 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).

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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.

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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

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Key Terms

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

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Key Terms, continued

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