Title: Deduction, Proofs, and Inference Rules
1Deduction, Proofs, and Inference Rules
2Lets 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)
3Formal 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.
41. Translate
- A B
- B C
- C D
- D
- A v E
- Therefore E
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52. 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
62. 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
7Validity 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
83. 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)
93. What it Looks Like
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- A B
- B C
- C D
- D
- A v E
- A C 1,2 H.S.
- A D 6,3 H.S.
- A 7,4 M.T.
- 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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10Definitions
- 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
11More 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
12The 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
13The 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
14The 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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15The Nine Rules of Inference (Pt. 4)
- 7. Simplification (Simp.)
- p q
- Therefore p
- 8. Conjunction (Conj.)
- p
- q
- Therefore (p q)
16The 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.
17ExampleProve 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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18Solution(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)
- W X
- (W Y) (Z v X)
- (W X) Y
- Z
- W (W X) 1 Abs.
- W Y 5,3 H.S.
- Z v X 2,6 M.P.
- 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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19Example 2
- I J
- J K
- L M
- I v L
- Therefore K v M
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20Solution
- I J
- J K
- L M
- I v L
- I K 1,2 H.S.
- (I K) (L M) 5,3 Conj.
- K v M 6,4 C.D.
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