Logical Inferences

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

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

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

Modus ponens: An example Suppose the following 2 statements are true: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.By modus ponens, we infer that it is 8am in Santa Barbara.

Other rules of inference Other tautological implications include: p  (p  q)p  (p  q) (p  q)  p(p  q)  p [~q  (p  q)]  ~p[~q  (p  q)]  ~p [(p  q)  ~p]  q[(p  q)  ~p]  q [(p  q)  (q  r)]  (p  r) hypothetical syllogism[(p  q)  (q  r)]  (p  r) hypothetical syllogism [(p  q)  (r  s)  (p  r) ]  (q  s)[(p  q)  (r  s)  (p  r) ]  (q  s) [(p  q)  (r  s)  (~q  ~s) ]  (~p  ~r)[(p  q)  (r  s)  (~q  ~s) ]  (~p  ~r)

Memorize & understand De Morgan’s lawsDe Morgan’s laws The law of the contrapositiveThe law of the contrapositive Modus ponensModus ponens Hypothetical syllogismHypothetical syllogism

Common fallacies 3 fallacies are common: Affirming the converse: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: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:Non sequitur: p  q Socrates is a man. Therefore, Socrates is mortal. On the other hand (OTOH), this is valid: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.The form of the argument is what counts.

Examples of arguments Given an argument whose form isn’t obvious: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: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 arguments... Assertions:Assertions: h: a baby is hungry c: a baby cries m: a baby is mad r: a baby has a red face Argument:Argument: ((h  c)  (~m  ~c)  (m  r))  (h  r) Valid?

Examples of arguments... Argument: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:Assertions: g: Gore will be elected c: California votes for Gore b: California keeps its air base Argument: [(g  c)  (b  g)]  g (valid?)Argument: [(g  c)  (b  g)]  g (valid?)

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