Discrete Structures

Lecture Notes, 4 Oct 2010

Last Update: 4 October 2010

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§1.5: Rules of Inference (cont'd)

  1. Prove that this argument is valid:

    1. Syntax & semantics of representation of atomic propositions:

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

    2. Translation of argument:

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

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

      • without using truth tables

    4. Strategy:

      Want to show:    busy (i.e., C)

        Can derive C from P3 via MP if we can show: plays
          Can derive "plays" from P2 via MT if we can show: pt
            Can derive "pt" from P1 via DS if we can show: ¬ft
              But we have ¬ft = P4

      So, "unwind" this strategy to write the proof:

        start with "¬ft"
          use that & DS on P1 to derive "pt"
            use that & MT on P2 to derive "plays"
              use that & MP on P3 to derive "busy"

    5. Here's the proof:

      2.pt ∨ ft:P1
      3.pt :1,2; DS
      4.¬plays → ¬pt :P2
      5.plays :3,4; MT
      6.plays → busy :P3
      7.busy :5,6; MP

  2. Digression: Was that really a legitimate use of MT?

    1. After all, MT says:

      From:                    AB
      &:                       ¬B
      you may infer:    ¬A

    2. But the application above seems be of the form:

      From:               ¬A¬B
      &:                       B
      you may infer:    A

    3. Reply: We can handle this in several ways:

      1. We could say (as I implicitly did above) that both of these are MT:

        • After all, at their core, they follow the "same" pattern:

          • From a conditional and the "opposite" of its consequent,
            you may infer the "opposite" of its antecedent.

        • But how would a computer know that?

      2. We could say that there are two versions of MT,
        namely, the two in IIA and IIB, above.

        • That's another way to interpret what I did in the example.

      3. We could prove that version B is derivable from version A; here's how:

        1. ¬A¬B : premise
        2. B               : premise
        3. ¬¬B           : b; DN+SL

          • (i.e., line c follows from line b by Double Negation
            (which tells us that they are logically equivalent)
            together with Shakespeare's Law
            (which tells us that we can replace one with the other))

        4. ¬¬A           : a,c; MT (version A)
        5. A               : d; DN+SL

        Now that we have this proof, we can appeal to MT (version B) in the future.

        • It's like a "macro" in a computer program;
          we can "expand" the macro whenever we need it.

  3. Is This Argument Valid?

    1. Syntax & semantics of representation of atomic propositions:

        g = "I play golf"
        t = "I play tennis"
        sun = "It's Sunday"
        sat = "it's Saturday"

    2. Translation of argument:

        P1:  (g ∨ t)
        P2:  (¬sun → (g ∧ t))
        P3:  ((sat ∨ sun) → ¬g)
        C :  ¬g

    3. Strategy:

      1. Try to show ¬g (i.e., C)
          Can derive C from P3 via MP, if we can show (sat ∨ sun)
            Can derive (sat ∨ sun) from "sun" via Addition, if we can show "sun"
              Can derive "sun" from ¬(g ∧ t) via MT, if we can show ¬(g ∧ t)
                Can derive ¬(g ∧ t) from (¬g ∨ ¬t) via DM+SL,
                if we can show (¬g ∨ ¬t)
                  Can derive (¬g ∨ ¬t) either from ¬g
                  or from ¬t via Addition.
                    Can we derive either one of those?

      2. Deriving ¬g won't help; that's what we're trying to do in the first place!
          (This is called "begging the question" or arguing in a circle.)

      3. Can we derive ¬t?

        • P1 contains "t", but there's no rule that would allow us to derive ¬t from P1
        • P2 contains "t", but there's no rule that would allow us to derive ¬t from P2

        • So we probably can't derive ¬t.

      4. So, let's revert to truth tables to see if the argument is valid or not.

        • This exercise is left to the reader :-)
        • Construct a truth table with 24=16 rows
          & find a row with T premises but F conclusion!

          • (The answer is online at the book's website!)

        • Hint: Make
            tval(g) = T
            tval(sat) = tval(sun) = F
            tval(t) = T
          • e.g., it's Monday & I play golf and tennis.

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