Lecture Notes, 1 Oct 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
      &                    (A → B)
      you may infer B

   2. It looks like the proposition:

      (*)        (A ∧ (A → B)) → 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 ((A ∧ B) → 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:
      (A ∨ B)
      ¬A       
      B

   2. Modul Tollens (method of denying (the consequent)):
      (A → B)
      ¬B       
      ¬A

   3. Addition:
      A       
      (A ∨ B)

   4. Simplification:
      (A ∧ B)
      A

   5. Conjunction:
      A
      B         
      (A ∧ B)

   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		A → B A          B		¬A ∨ B A          B
      MT		A → B ¬B          ¬A		¬A ∨ B ¬B          ¬A
      HS		A → B B → C A → C		¬A ∨ B ¬B ∨ C ¬A ∨ C
      DS		A ∨ B ¬A        B		(already uses only ¬,∨)

   3. Generalizing this pattern, we have Resolution:

        A ∨ B₁ ∨ … ∨ B_n
      ¬A ∨ C₁ ∨ … ∨ C_m
                                                           
      (B₁ ∨ … ∨ B_n) ∨ (C₁ ∨ … ∨ C_m)

      or, more simply:

        A ∨ B
      ¬A ∨ C
        B ∨ C

      • 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!

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