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Logical Inferences
Goals for propositional logic
Introduce notion of a valid argument & rules of inference.
Use inference rules to build correct arguments.
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2. What is a rule of inference?
A rule of inference allows us to specify which conclusions may be inferred from assertions known, assumed, or previously established.
A tautology is a propositional function that is true for all values of the propositional variables (e.g., p  ~p).
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Modus ponens
A rule of inference is a tautological implication.
Modus ponens: ( p  (p  q) )  q
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4. Modus ponens: An example
Suppose the following 2 statements are true:
If it is 11am in Miami then it is 8am in Santa Barbara.
It is 11am in Miami.
By modus ponens, we infer that it is 8am in Santa Barbara.
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5. Other rules of inference
Other tautological implications include: (Is there a finite number of rules of inference?)
p  (p  q)
(p  q)  p
[~q  (p  q)]  ~p
[(p  q)  ~p]  q
[(p  q)  (q  r)]  (p  r) hypothetical syllogism
[(p  q)  (r  s)  (p  r) ]  (q  s)
[(p  q)  (r  s)  (~q  ~s) ]  (~p  ~r)
[ (p  q)  (~p  r) ]  (q  r ) resolution
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Common fallacies
3 fallacies are common:
Affirming the converse:
[(p  q)  q]  p
If Socrates is a man then Socrates is mortal.
Socrates is mortal.
Therefore, Socrates is a man.
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Common fallacies ...
Assuming the antecedent:
[(p  q)  ~p]  ~q
If Socrates is a man then Socrates is mortal.
Socrates is not a man.
Therefore, Socrates is not mortal.
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Common fallacies ...
Non sequitur:
p  q
Socrates is a man.
Therefore, Socrates is mortal.
The following is valid:
If Socrates is a man then Socrates is mortal.
The argument’s form is what matters.
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Examples of arguments
Given an argument whose form isn’t obvious:
Decompose the argument into premise assertions
Connect the premises according to the argument
Check to see that the inference is valid.
Example argument:
If a baby is hungry, it cries.
If a baby is not mad, it doesn’t cry.
If a baby is mad, it has a red face.
Therefore, if a baby is hungry, it has a red face.
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10. Copyright © Peter Cappello 2016
If a baby is hungry, it cries.
If a baby is not mad, it doesn’t cry.
If a baby is mad, it has a red face.
Therefore, if a baby is hungry, it has a red face.
______________________________________
h: a baby is hungry
c: a baby cries
m: a baby is mad
r: a baby has a red face.
Argument:
( (h  c)  (~m  ~c)  (m  r) )  (h  r)
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11. ( (h  c)  (~m  ~c)  (m  r) )  (h  r)
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Examples of arguments ...
Argument:
McCain will be elected if and only if California votes for him.
If California keeps its air base, McCain will be elected.
Therefore, McCain will be elected.
Assertions:
m: McCain will be elected
c: California votes for McCain
b: California keeps its air base
Argument: [(m c)  (b  m)]  m (valid?)
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