Deduction, Proofs, and Inference Rules

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Deduction, Proofs, and Inference Rules
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Lets Review What we Know

• Take a look at your handout and see if you have
  any questions
• You should know how to translate all of these
  fairly simple sentences into their logical
  components (be able to go from English to logic
  symbolization)

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Formal Proof of Validity Translate the Following

• If Anderson was nominated, the she went to
  Boston.
• If she went to Boston, then she campaigned there.
• If she campaigned there, she met Douglas.
• Anderson did not meet Douglas.
• Either Anderson was nominated or someone more
  eligible was selected.
• Therefore someone more eligible was selected.

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1. Translate

• A B
• B C
• C D
• D
• A v E
• Therefore E

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2. Establish Validity

• It might seem obvious this argument is valid, but
  we want to prove it
• We could use truth tables, but this would require
  us to make a table with 32 rows since there are 5
  different simple statements involved
• So now what?
• Prove validity by deducing its conclusion from
  its premises using already-known, elementary
  valid arguments

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2. Establish Validity (still)

• Well use three of these basic rules of inference
  (there are a total of 9) in this example
• 1. Hypothetical Syllogism (H.S.)
• If p then q
• And if q then r
• Therefore if p then r
• 2. modus tollens (M.T.)
• If p then q
• q
• Therefore p
• 3. Disjunctive Syllogism (D.S.)
• p v q
• p
• Therefore q

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Validity Established

• Looking at the argument we want to prove valid,
  we see that the conclusion can be deduced from
  the five premises of the original argument by
  four elementary valid arguments (2 H.S. 1 M.T.
  1 D.S.)
• This proves that our original argument is valid

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3. Write the Proof

• 1. Write the premises and the statements that we
  deduce from them in a single column to the
  right of this column, for each statement, its
  justification is written (basically the reason
  why we include that statement in the proof)
• 2. List all the premises first, then the logic
  (e.g. inference rules) used to get at the
  conclusion (which will be listed last)

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3. What it Looks Like
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1. A B
2. B C
3. C D
4. D
5. A v E
6. A C 1,2 H.S.
7. A D 6,3 H.S.
8. A 7,4 M.T.
9. E 5,8 D.S.

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The justification for each statement (the right
most column) consists of the numbers of the
preceding statements from which that line is
inferred, together with the abbreviation for the
rule of inference used to get it
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Definitions

• A formal proof that shows an argument is valid is
  a sequence of statements, each of which is either
  a premise of that argument or follows from
  preceding statements of the sequence by an
  elementary valid argument (i.e. our inference
  rules), such that the last statement in the
  sequence is the conclusion of the argument whose
  validity is being proved
• An elementary valid argument is any argument that
  is a substitution instance of an elementary valid
  argument form (e.g. our inference rules)
• We dont have time to prove the validity of each
  one of these statements, so take our word for it
  that they are valid

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More Complex Substitutions
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• (A B) C (D v E)
• A B
• Therefore C (D v E)
• This sequence above is an elementary valid
  argument because it is a substitution instance of
  the elementary valid argument form modus ponens
  (M.P.), another one of our inference rules. See
  if you can see it
• modus ponens (M.P.)
• If p then q
• And p
• Therefore q

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The Nine Rules of Inference (Pt. 1)

• Modus Ponens (M.P.)
• If p then q
• p
• Therefore q
• Modus Tollens (M.T.)
• If p then q
• q
• Therefore p

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The Nine Rules of Inference (Pt. 2)

• 3. Hypothetical Syllogism (H.S.)
• If p then q
• And if q then r
• Therefore if p then r
• 4. Disjunctive Syllogism (D.S.)
• p v q
• p
• Therefore q

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The Nine Rules of Inference (Pt. 3)

• 5. Constructive Dilemma (C.D.)
• (p q) (r s)
• p v r
• Therefore (q v s)
• 6. Absorption (Abs.)
• p q
• Therefore p (p q)

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The Nine Rules of Inference (Pt. 4)

• 7. Simplification (Simp.)
• p q
• Therefore p
• 8. Conjunction (Conj.)
• p
• q
• Therefore (p q)

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The Nine Rules of Inference (Pt. 5)

• 9. Addition (Add.)
• p
• Therefore (p v q)
• These nine rules of inference correspond to
  elementary argument forms whose validity is
  easily established by truth tables. With their
  air, formal proofs of validity can be constructed
  for a wide range of more complicated arguments.

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ExampleProve the following given the premises
(using inference rules)

• W X
• (W Y) (Z v X)
• (W X) Y
• Z
• Therefore X

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Solution(Strategy Hint see what you can
create from the premises using the inference
rules we know. Keep in mind what youre looking
for this will keep you on track)

1. W X
2. (W Y) (Z v X)
3. (W X) Y
4. Z
5. W (W X) 1 Abs.
6. W Y 5,3 H.S.
7. Z v X 2,6 M.P.
8. X 7,4 D.S.

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Line 5 look at line 1. Use our absorption
rule Line 6 A little harder look at line 5 then
3 it follows the H.S. pattern W (W X)(W
X) Y Therefore W Y Line 7 a fairly
simple M.P. form from lines 2 and 6 Line 8 use
D.S. from lines 7 and 4
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Example 2

• I J
• J K
• L M
• I v L
• Therefore K v M

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Solution

1. I J
2. J K
3. L M
4. I v L
5. I K 1,2 H.S.
6. (I K) (L M) 5,3 Conj.
7. K v M 6,4 C.D.

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