Necessary & Sufficient Conditions

1. Necessary & Sufficient Conditions Law, Science, Life & Logic

2. Necessary Condition • Must be there for the effect, the charge, to be true • If absent, cannot occur. • No oxygen, no combustion. • No seeds, no plants to grow • Car runs only if gas in tank.

3. Sufficient Condition • Whenever A is present, B will follow-decapitation is sufficient for death. • Getting a B in the course is sufficient for passing.

4. Sufficient but---- • Head off, death occurs, but others things can do same- e.g. knife to heart. • One can pass the course also with an A or a C.

5. Necessary but-- • Fire cannot happen with oxygen alone-need a match. • Plants need seeds, but also water. • Car needs gas, but also…

6. Conditional Statements • Conditional is an if-then statement • If the sun shines, we will have a picnic. • Antecedent (sun shines); (we will have a picnic )is the consequent. • Antecedent is the sufficient condition for the consequent.

7. Antecedent=sufficient • If the antecedent is true, then the consequent must be true. • Symbolize as a -› c or S -›P • Sufficient only- sun shining is one sufficient condition for a picnic; celebrating is another.

8. Consequent is necessary • If there is life, there is oxygen= • Life -› Oxygen • Life is a sufficient condition for oxygen and oxygen is a necessary condition for life. (Without oxygen there is no life and if life is present, there must be oxygen. • If there is fire, there is oxygen; oxygen is a necessary condition for fire; fire is sufficient to know there is oxygen present. Fire -› Oxygen

9. Necessary= Consequent One will be able to light a match on the moon only if there were oxygen on the moon. Oxygen is the necessary condition for fire- it is the consequent of a conditional= Light a match -› oxygen

10. Conditional Arguments • If I can create combustion, then there is oxygen. The match lit (combustion occurred), thus there is oxygen here. • C -› O • C • Thus, O Modus Ponens-Valid- • Affirming the Antecedent

11. Denying the Consequent • If the match lights, there is oxygen in this place. There is not oxygen in this place. The match does not light. Denying consequent • M -› O • ̴ O Valid • ̴ M Modus Tollens

12. Necessary & Sufficient • Something can serve as both necessary and sufficient- “You will get Malaria if and only if you are bitten by a mosquito carrying the germ. Malaria ≈ Mosquito (germ) • If you have Malaria you must have been bitten by the mosquito with the germ.

13. What is a Fallacy? • Fallacy is an argument that does not provide support for its conclusion because of faulty reasoning. • A fallacious argument= one in which the premises, even if true, would not justify the conclusion.

14. Conditional Fallacies • Denying the antecedent- denies a sufficient condition and this is not adequate to deny the consequent. • If Barry Bonds plays, the team will win. Bonds did not play, Thus, the team will not win. • B -› W ̴ B Thus, ̴ W Invalid=Fallacy

15. Affirming Consequent • If Barry Bonds plays, the team will win. The team won, thus Barry Bonds played the game. • Consequent is a necessary condition, but not sufficient. • B -› W W Thus, B Invalid=Fallacy • Eg. Oxygen is necessary to have combustion, but it cannot bring it about.

16. Let’s try some examples • What are the necessary and sufficient conditions for each of the following? • Being a mother ; being a father • Being President of the United States • Being a member of the Olympic team • Producing an award winning essay.