### 1.5.prop

```Logical Inferences
Goals for propositional logic
1. Introduce notion of a valid argument & rules of inference.
2. Use inference rules to build correct arguments.
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).
Modus ponens
• A rule of inference is a tautological implication.
• Modus ponens: ( p  (p  q) )  q
p q pq p( pq) (p( pq)) q
T
T
F
F
T
F
T
F
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.
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
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.
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.
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.
Socrates is a man.
Therefore, Socrates is mortal.
• The argument’s form is what matters.
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.
( (h  c)  (~m  ~c)  (m  r) )  (h  r)
r
m
c
h
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?)