Discrete Structures

Lecture Notes, 1 Oct 2010

Last Update: 1 October 2010

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§1.5: Rules of Inference (cont'd)

  1. Rule of Inference vs. Tautologies:

    1. Recall that MP is the rule of inference:

        From                A
        &                    (AB)
        you may infer B

    2. It looks like the proposition:

      (*)        (A ∧ (AB)) → B)

      But it isn't!

    3. Differences:

      1. MP is a sequence of 3 propositions;
        it is a valid argument form

      2. (*) is a single proposition;
        it is a tautology

    4. But they are related:

        MP is valid iff (*) is a tautology

    5. In general, a rule of inference

        From A
        &      B
        infer C

      is valid iff ((AB) → C) is a tautology

      • This is called the "Deduction Theorem", and requires proof
        (but we won't prove it)

  2. More Rules of Inference:

    1. Disjunctive Syllogism:


    2. Modul Tollens (method of denying (the consequent)):


    3. Addition:


    4. Simplification:


    5. Conjunction:


    6. There is a chart of some rules of inference on p. 66 (Table 1).

      1. In general, for each connective, there are "introduction" and "elimination" rules,
      2. some of which require nested subproofs.

        • An "introduction" rule for a connective "*" tells you how to "introduce" as a line of a proof a new proposition whose principal connective is "*".
        • An "elimination" rule for a connective "*" tells you how to "eliminate" a connective to create a new proposition for a line of a proof.

      3. Rules of inference are "atomic" valid argument forms.

        • Longer arguments are valid if their "sub-arguments" are valid;
          a sub-argument is valid if its sub-arguments are valid;
          and so on, stopping at "atomic" arguments that are rules of inference.
        • This is another example of what I have been calling "recursion"

  3. The Resolution Rule of Inference:

    1. Resolution is a single rule of inference that can do the work of all of the others.

      • ∴, it's useful for computational implementations of FOL.
      • The programming language Prolog is based on FOL and resolution.

    2. Consider the following chart of rules of inference.
      • The familiar rules are in the first column.
      • The second column rewrites the rule using only ¬ & ∨
          (which we know are functionally complete!)

      MP      AB
      MT      AB
      HS      AB
      DS      AB
           (already uses only ¬,∨)

    3. Generalizing this pattern, we have Resolution:

        AB1 ∨ … ∨ Bn
      ¬AC1 ∨ … ∨ Cm
      (B1 ∨ … ∨ Bn) ∨ (C1 ∨ … ∨ Cm)

      or, more simply:


      • To use this as the only rule,
        we need to represent all propositions in "clause form"
        (or "conjunctive normal form")
          i.e., using only ¬, ∧, & ∨

        • Take CSE 463 to learn more.

  4. More examples of proofs:

    1. Prove that this argument is valid:

      P1:   Lynn works part time or full time.
      P2:   If Lynn doesn't play on the team,
                  then she doesn't work part time.
      P3:   If Lynn plays on the team,
                  then she's busy.
      P4:   Lynn doesn't work full time.
      C :   ∴ Lynn is busy.

    2. Syntax & Semantics of Representation:

      Let pt = Lynn works part time.
      Let ft = Lynn works full time.
      Let plays = Lynn plays on the team.
      Let busy = Lynn is busy.

    3. Translation of argument:

      P1:    pt &or ft
      P2:    ¬plays → ¬pt
      P3:    plays → busy
      P4:    ¬ft
      C :    ∴ busy

    4. We want to prove that this is a valid argument

      • without using truth tables

    5. How? Next time!

Next lecture…

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