Necessary & Sufficient Conditions Law, Science, Life & Logic

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

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

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

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

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

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

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

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

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

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 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 M -› O ̴ O Valid ̴ O Valid ̴ M Modus Tollens ̴ M Modus Tollens

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) 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. If you have Malaria you must have been bitten by the mosquito with the germ.

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

Conditional Fallacies Denying the antecedent- denies a sufficient condition and this is not adequate to deny the consequent. 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. 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 B -› W ̴ B Thus, ̴ W Invalid=Fallacy

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

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