Logical InferencesLogical Inferences

De Morgan’s LawsDe Morgan’s Laws

• ~(p ~(p q) q) (~p (~p ~q) ~q)

• ~(p ~(p q) q) (~p (~p ~q) ~q)

The The Law of the ContrapositiveLaw of the Contrapositive

(p (p q) q) (~q (~q ~p) ~p)

What is a rule of inference?What is a rule of inference?

• A A rule of inferencerule of inference allows us to specify allows us to specify

which conclusions may be inferred from which conclusions may be inferred from

assertions known, assumed, or previously assertions known, assumed, or previously

established.established.

• A A tautologytautology is a propositional function that is is a propositional function that is

true for all values of the propositional true for all values of the propositional

variables (e.g., pvariables (e.g., p ~p). ~p).

Modus ponensModus ponens

• A A rule of inferencerule of inference is a tautological implication. is a tautological implication.

• Modus ponensModus ponens: : ( p ( p (p (p q) ) q) ) q q

p q p qq p (( p qq)) (p (( p qq)))) qq

T T T T TT F F F TF T T F T

F F T F T

Modus ponens: An exampleModus 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 By modus ponens, we inferinfer that it is 8am in that it is 8am in

Santa Barbara.Santa Barbara.

Other rules of inferenceOther rules of inference

Other tautological implications include:Other tautological implications include:

• pp (p (p q) q)• (p (p q) q) pp• [~q [~q (p (p q)] q)] ~p~p• [(p [(p q) q) ~p] ~p] qq• [(p [(p q) q) (q (q r)] r)] (p (p r) r) hypothetical syllogismhypothetical syllogism

• [(p [(p q) q) (r (r s) s) (p (p r) ] r) ] (q (q s) s) • [(p [(p q) q) (r (r s) s) (~q (~q ~s) ] ~s) ] (~p (~p ~r) ~r)

Memorize & understandMemorize & understand

• De Morgan’s lawsDe Morgan’s laws

• The law of the contrapositiveThe law of the contrapositive

• Modus ponensModus ponens

• Hypothetical syllogismHypothetical syllogism

Common fallaciesCommon fallacies

3 fallacies are common:3 fallacies are common:

• Affirming the converseAffirming 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 ...Common fallacies ...

• Assuming the antecedentAssuming 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 ...Common fallacies ...

• Non sequiturNon 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 The formform of the argument is what counts. of the argument is what counts.

Examples of argumentsExamples 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 ...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 ...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 Argument: [(g c) c) (b (b g)] g)] g g (valid?)(valid?)

CharactersCharacters