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Proof techniques. CHAPTER TWO. Formal Logic vs. Real-world Arguments. Real-world arguments, unlike the formal proofs of Chapter 1, are normally dependent on context, not the structure of the argument.

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Proof techniques

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Proof techniques

Proof techniques


Formal logic vs real world arguments

Formal Logic vs. Real-world Arguments

  • Real-world arguments, unlike the formal proofs of Chapter 1, are normally dependent on context, not the structure of the argument.

  • In other words, a real-world argument may not be universally valid, though it be valid in some important context.

  • Terminology: What is a conjecture?

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Argument context

Argument Context

  • In real-world situations, we often are interested only in the truth of an argument in a particular context.

  • Example:“If Mary Beth (or some other student) makes an A in CSC 333, then she must be a bright, hard worker.”

    • Call this the Mary Beth Theorem.

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Examination of the mary beth theorem

Examination of the Mary Beth Theorem

  • Can we state the M.B. Theorem formally?

    • Yes. Let “Mary Beth makes an A in CSC 333” be proposition P, and “Mary Beth is a bright, hard worker” be Q.

      • We can state the M.B. Theorem as P -> Q.

        • (Or perhaps more properly, let R be “Mary Beth is bright”, and let S be “Mary Beth is a hard worker”; thus, Q can be decomposed as R ^ S

      • So, we can state the M.B. Theorem as P-> (R ^ S).

    • We can easily establish a truth table for this.

  • As stated in the text, if we can’t translate a real-world argument into a formal proof, we should look askance at the argument.

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Attacking the m b theorem

Attacking the M.B. Theorem

  • Disproving by counterexample:

    • Assume that Mary Beth can be shown to be an imbecile, although she has an A in CSC 333.

    • This would be a case where R (Mary Beth is bright) is false, making (R ^ S) false.

    • So, in at least one case P does NOT imply Q.

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Which is easier

Which is easier?

  • Proving a conjecture using a formal proof, i.e., showing that for all truth values of the propositions, the theorem holds.OR

  • Disproving a conjecture by showing one instance in which the theorem “folds” (does not hold), i.e., a counterexample.

  • Note that showing one example in which it holds is insufficient as a proof.

    • Aside: How does this apply to software testing?

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Exhaustive proofs

Exhaustive Proofs

  • If we have a finite population to which we are applying the M.B. Theorem,

    • say, the students in CSC 333 in spring of 2010,

    • And we can show the truth of the M.B. Theorem for all those students,

  • then we have proved the M.B. Theorem by exhaustion.

  • Aside: How does this apply to software testing?

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Direct proof

Direct Proof

  • If we want to prove the M.B. Theorem for all students who ever enroll in CSC 333, we might attempt a direct proof:

    • Assume P is true.

    • Show that the conjecture is universally true because R ^ S inevitably follows from P.

    • For the M.B. Theorem, we can’t show this.

  • See text for an example of such a proof (p. 92).

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  • Proving by contraposition:

    • We already know that P -> Q is logically equivalent to ~Q -> ~P.

    • So, if we prove, for example, that if Mary Beth is NOT both bright [R] and a hard worker [S], then Mary Beth will NOT get an A in CSC 333, we have proved the original conjecture.

      • Note that the if the contrapositive can be shown to be false in at least one case, this disproves the original conjecture.

        • What if Mary Beth is lazy but cheats cleverly?

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  • Proving by contradiction:

    • Show that P is true and Q is false is a contradiction, i.e., it is always false.

    • In other words, show that it cannot ever be true that P is true and Q is false.

  • See example 10, p. 95.

  • Proving that something is not true is usually more difficult than assuming it is true and then showing a contradiction.

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  • Conjecture

  • Inductive reasoning

  • Deductive reasoning

  • Counterexample

  • Direct proof

  • Contraposition

  • Contradiction

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