Question

Explain the difference between a necessary and a sufficient condition, and their relation to conditional statements. Provide an example, and give it in both standard form and symbolic form.

philosophy critical thinking class

Answer 1

• In logic, necessity and sufficiency are implicational relationships between statements.The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true or simultaneously false.
• In the conditional statement, "if S, then N", the expression represented by S is called the antecedent and the expression represented by N is called the consequent.
• In the above situation, we also say that N is a necessary condition for S. In common language this is saying that if the conditional statement is a true statement, then the consequent N must be true if S may at all be true (see "truth table" immediately below). Phrased differently, the antecedent S can not be true without N being true. For example, in order for someone to be called Socrates, it is necessary for that someone to be Named.
• S is a sufficient condition for N.If the conditional statement is true, then if S is true, N must be true; whereas if the conditional statement is true and N is true, then S may be true or be false. In common terms, "S guarantees N". Continuing the example, knowing that someone is called Socrates is sufficient to know that someone has a Name.
• Necessity example-For the whole numbers greater than two, being odd is necessary to being prime, since two is the only whole number that is both even and prime.
  
• Sufficient example-"John is a king" implies that John is male. So knowing that it is true that John is a king is sufficient to know that he is a male.
  
• A condition can be either necessary or sufficient without being the other. For instance, being a mammal (N) is necessary but not sufficient to being human (S),
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