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Logical Inferences. De Morgan’s Laws. ~(p  q)  (~p  ~q) ~(p  q)  (~p  ~q). The Law of the Contrapositive. (p q)  (~q ~p). What is a rule of inference?.

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de morgan s laws
De Morgan’s Laws
  • ~(p  q)  (~p  ~q)
  • ~(p  q)  (~p  ~q)
the law of the contrapositive
The Law of the Contrapositive

(p q)  (~q ~p)

what is a rule of inference
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
Modus ponens
  • A rule of inference is a tautological implication.
  • Modus ponens: ( p  (p  q) )  q
modus ponens an example
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 rules of inference

Other tautological implications include:

  • 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)
memorize understand
Memorize & understand
  • De Morgan’s laws
  • The law of the contrapositive
  • Modus ponens
  • Hypothetical syllogism
common fallacies
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 fallacies1
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 fallacies2
Common fallacies ...
  • Non sequitur:

p  q

Socrates is a man.

Therefore, Socrates is mortal.

  • On the other hand (OTOH), this is valid:

If Socrates is a man then Socrates is mortal.

Socrates is a man.

Therefore, Socrates is mortal.

  • The form of the argument is what counts.
examples of arguments
Examples of arguments
  • Given an argument whose form isn’t obvious:
    • Decompose the argument into assertions
    • Connect the assertions according to the argument
    • Check to see that the inferences are valid.
  • Example argument:

If a baby is hungry then it cries.

If a baby is not mad, then it doesn’t cry.

If a baby is mad, then it has a red face.

Therefore, if a baby is hungry, it has a red face.

examples of arguments1
Examples of arguments ...
  • Assertions:
    • 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)

Valid?

examples of arguments2
Examples of arguments ...
  • Argument:

Gore will be elected iff California votes for him.

If California keeps its air base, Gore will be elected.

Therefore, Gore will be elected.

  • Assertions:
    • g: Gore will be elected
    • c: California votes for Gore
    • b: California keeps its air base
  • Argument: [(g c)  (b  g)]  g (valid?)
characters
Characters
  •   
  •       
  •   
  •  
  •     
  •        