Lecture Notes, 29 Sep 2010

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§1.5: Rules of Inference

1. Artificial intelligence (AI) is the branch of computer science concerned with computational theories of cognition.
   1. Knowledge representation and reasoning (KRR) is the branch of AI concerned with how to:
      1. represent information (what we've done so far)
         • in a formal language that can be handled by a computer

      2. & reason with that information
         (as represented in that language)

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2. Reasoning involves constructing & verifying "arguments" or "proofs"…
   1. …which are used:
      1. to answer questions
      2. & to give reasons for the answers

   2. E.g.: How can you pass this course?
      (Or: Why didn't I pass this course?)

      Possible answer:

      • If you get a passing grade in all assignments,
        then you will pass the course. (premise 1)

      • You get a passing grade. (premise 2)

      • ∴ You pass. (conclusion)

      This is called an "argument"
      (in the sense of a legal argument, not a yelling match)

   3. Let p = "You get a passing grade in all assignments."
      Let q = "You (will) pass the course".

      Then the form of this argument can be shown by the following sequence of propositions:

      (p → q)                      premise 1
      p                                premise 2
      ∴ q                            conclusion

      This "argument form" is also called a "rule of inference"

   4. Consider a truth table for this argument:

      prem 2	conc	prem 1
      p	q	(p→q)
      T	T	T
      T	F	F
      F	T	T
      F	F	T

      Note that row 1 is the only row where both premises are T.
      In that row, the conclusion is also T.

   5. The name of this rule of inference (or basic argument form) is modus ponens (MP) (Latin for "method of affirmation (of the antecedent)")

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3. When is an argument a "good" argument?
   1. There are 2 separate conditions for goodness:
      1. logical goodness
      2. factual goodness

   2. Def:
      An argument from premises P₁, …, P_n to conclusion C
      is_def
      a sequence of propositions ⟨P₁, …, P_n, C⟩,
      where C is alleged to follow logically from the P_i.

   3. Def:
      An argument (or: an argument form) is (semantically) valid (= "logically good")
      =_def
      it is truth-preserving.

      i.e., if all of its premises are T,
      then its conclusion must be T

      or: it is impossible for all of its premises to be simultaneously T, yet its conclusion is F.

   4. Def:
      A proof is_def a valid argument.

   5. How can you tell if an argument is truth-preserving?
      1. An argument is truth-preserving (or: is a proof)
         iff
         it is an argument in which each proposition P_i or conclusion C is either:
         1. a tautology (hence T)
         2. a "premise"
            • (premises can be logically contingent
              but they are assumed to be T)

         3. or follows validly from previous propositions in the sequence
            by one or more rules of inference
            • (and—because they are valid argument forms—each rule of inference is itself truth-preserving)

      2. Note the recursion:
         • Base Case = propositions assumed to be T
         • Recursive Case = "sub-proofs" based on valid (truth-preserving) rules of inference

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4. Another rule of inference: Hypothetical Syllogism (HS)

   (A → B)
   (B → C)
   ∴ (A → C)
   

   You should construct a truth table to convince yourself that this is truth-preserving, hence valid.

   Notes:
   • "syllogism" is just an old-fashioned word for "argument"
   • This can also be called "transitivity of →"

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5. Here is a more complex e.g. of an argument:
   If you pass all assignments, then you pass the course.
   If you pass the course, then you'll be successful in life.
   You pass the course.
   ∴ You'll be successful in life.

   Is this valid?

   We can prove validity by syntactically deriving all "intermediate" conclusions:

   1. First, let's symbolize the argument:

      Let p = You pass all assignments.
      Let q = You pass the course.
      Let r = You'll be successful in life.

      Then:

      P1:   (p → q)
      P2:   (q → r)
      P3:   p
      C :   ∴ r

   2. Proof of validity:
      (compare this to a computer program with line numbers and comments)

      line num	proposition	comment mark	justification
      1.	(p → q)	:	Prem P1
      2.	(q → r)	:	Prem P2
      3.	(p → r)	:	From lines 1,2; by HS
      4.	p	:	Prem P3
      5.	r	:	From lines 3,4; by MP

   3. Alternative proof:
      (Just as you can write more than one program to accomplish the same task
      (have the same input-output behavior),
      you can also write more than one proof of an argument.)
      1. (p → q)  : P1
      2. p            : P3
      3. q            : 1,2; MP
      4. (q → r)   : P2
      5. r             : 3,4; MP

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6. But is this really a good argument?
   • No: Because P2 is probably F

   • ∴ the argument is not "sound":

   • Def:

     An argument is sound
     =_def
     it is valid (logically good)
     & all of its premises are T (it is "factually" good)

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