Valid Arguments with False Conclusions

P1) All dogs are purple

P2) Fido is a dog

C) Therefore, Fido is purple

P1) Heart disease is caused by either too much exercise or eating too much oatmeal

P2) Heart disease is not caused by eating too much oatmeal

C) Therefore, heart disease is caused by excess exercise

P1) Anything that can swim is a fish

P2) I can swim

C) Therefore, I am a fish

Examples that Didn’t Work

P1) Male pattern baldness causes hair loss among men

P2) There are many bald men in the world

C) All bald men have male pattern baldness

P1) When water freezes, it becomes solid

C) Therefore, water is a solid

P1) Clouds started rolling in Sunday night

P2) On Sunday night the weatherman said it was going to rain on Monday

C) Therefore, it rained on Monday

P1) My dog hates getting a bath

P2) My friend’s dog hates getting a bath

C) Therefore, All dogs hate getting a bath

P1) Cats and dogs speak by meowing or barking

P2) Cats bark

C) Therefore, Dogs meow

P1) 90% of people in Tennessee drink their beverages out of straws

P2) Enjoying salt water taffy is a common pastime in Georgia

P3) The best way to treat the bite from a southern copperhead snake is to such out the poisen immediately

P4) The most popular way to eat spaghetti in Mississippi is to such it rather than cut it.

C) Therefore, people in the south suck

P1) Everyone I know likes Family Guy

P2) I like Family Guy

C) Therefore, everyone must like Family Guy

P1) There are weapons of mass destruction in Iraq, we just haven’t found them yet.

P2) The war in Iraq is justified.

Not - Negation

P or ~P is a tautology

Statement	Contrary	Contradictory
It is hot	It is cold	It is not hot
MSU is a great school	MSU is a terrible school	MSU is not a great school

And - Conjunction

If P & Q, then P, and Q. Both P and Q must be true for P & Q to be true. P & Q needn’t be related to each other

Or - Disjunction

Inclusive Disjunction: True when both disjuncts are true

True when only one disjunt is true

Examples: You can have either pumpkin pie or pecan pie

Either mommy or daddy will pick you up

Exclusive Disjunction: False when both disjuncts are true

True when one disjunct is true

Examples: P or not-P

Either this key or that key opens the door

Your meal comes with tea or coffee

If then - Conditionals

Asserts that if the antecedent is true, then the consequent is true. It is false only when the antecedent is true and the consequent is false (minimal truth value connection).

The following are true conditionals

· If the Patriots won the superbowl then Janet Jackson’s top malfunctioned

· If the Panthers won the superbowl then then Janet Jackson’s top malfunctioned

· If the Panthers won the superbowl then Janet Jackson’s top did not malfunction

This conditional is not true

· If the Patriots won the superbowl then Janet Jackson’s top did not malfunction

Counterfactuals

Cannot be represented in propositional logic. In propositional logic both of the following counterfactuals come out true (given someone who didn’t study much for the exam).

· If I had studied all week, then I would have received a 4.0 on my exam

· If I had studied all week, then I would have failed my exam

If cigarette advertising leads teenagers to smoke and advertisers lie about their aims, then either cigarette advertising should not be allowed or it should be strictly regulated.

(C & L) implies (~A or R)

Valid Argument Forms

Modus Ponens

P implies Q P ---- Q

P P implies Q ----- Q

Modus Tollens

P implies Q ~Q ----- ~P

~P implies ~Q Q ----- P

Hypothetical Syllogism

P implies Q

Q implies R

-----

P implies R

Disjunctive Syllogism

P or Q ~Q ----- P

P or Q ~P ----- Q

Constructive Dilemma

P or Q

P implies R

Q implies S

-----

R or S

Equivalent Forms

Double Negation

P is equivalent to ~~P

Commutative

(P & Q) is equivalent to (Q & P)

Contraposition

(P implies Q) is equivalent to (~Q implies ~P)

Definition of Implication

(P or Q) is equivalent to (~P implies Q)

Invalid Argument Forms

Denying the Antecedent

P implies Q

~P

-----

~Q

If Haley shares needles with strangers he is at considerable risk of contracting the AIDS virus. Haley does not share needles with strangers. Therefore, Haley is not at considerable risk of contracting AIDS.

Affirming the Consequent

P implies Q

Q

-----

P

If Petrie is an atheist, she opposes compulsory prayer in public schools. She does oppose compulsory prayer. Therefore, Petrie is an atheist.