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Copyright © Peter Cappello
Logical Inferences
Goals for propositional logic
1. Introduce notion of a valid argument & rules of inference.
2. Use inference rules to build correct arguments.
Copyright © Peter Cappello
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).
p q p qq p (( p qq)) (p (( p qq)))) qq
T T T F F T
F F
Copyright © Peter Cappello
Modus ponens
• A rule of inference is a tautological implication.• Modus ponens: ( p (p q) ) q
Copyright © Peter Cappello
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.
Copyright © Peter Cappello
Other rules of inferenceOther 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
Copyright © Peter Cappello
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.
Copyright © Peter Cappello
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.
Copyright © Peter Cappello
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.
Copyright © Peter Cappello
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.
Copyright © Peter Cappello
( (h c) (~m ~c) (m r) ) (h r)
r
m
c
h
Copyright © Peter Cappello
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?)
