Logical Inferences

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# Logical Inferences - PowerPoint PPT Presentation

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

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?
• 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
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:

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

• 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
• 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 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 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?)
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