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Logical Inferences

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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Logical Inferences

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  1. Logical Inferences

  2. De Morgan’s Laws • ~(p  q)  (~p  ~q) • ~(p  q)  (~p  ~q)

  3. The Law of the Contrapositive (p q)  (~q ~p)

  4. 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).

  5. Modus ponens • A rule of inference is a tautological implication. • Modus ponens: ( p  (p  q) )  q

  6. 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.

  7. 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)

  8. Memorize & understand • De Morgan’s laws • The law of the contrapositive • Modus ponens • Hypothetical syllogism

  9. 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.

  10. 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.

  11. 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.

  12. 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.

  13. 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?

  14. 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?)

  15. Characters •    •        •    •   •      •        

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