Question

Translate the following sentence into sentential logic.

It is not true that ATHLETICISM is a necessary and sufficient condition for FITNESS.

Answer 1

NECESSARY CONDITIONS:

A is necessary for B

Now, to say that one state of affairs (event) A is necessary for another state of affairs (event) B is just to say that if the first thing does not obtain (happen), then neither does the second. Thus, for example, to say

ATHLETICISM is sufficient for FITNESS    is just to say that if A (i.e., ATHLETICISM) doesn't obtain then neither does B (FITNESS). The sentence is accordingly paraphrased and symbolized as follows.

if not A, then not B

    [~A → ~B]

ATHLETECISM is necessary for FITNESS

A = ATHELETISM

B = FISTNESS

(~A -> ~ B)

SUFFICIENT CONDITIONS:

A is sufficient for B

Now, to say that one state of affairs (event) A is sufficient for another state of affairs (event) B is just to say that B obtains (happens) provided (if) A obtains (happens). So for example, to say that

ATHELETICS is sufficient for FITNESS     is to say that

B happens provided (if) A happens which may be symbolized quite simply as:

A -> B

The general principle is as follows.

A is sufficient for B is paraphrased as

if A, then B

COMBINATIONS OF NECESSITY AND SUFFICIENCY:

ATHLETICISM both necessary and sufficient for FITTNESS

This is quite clearly the conjunction of a necessity statement and a sufficiency statement, as follows.

ATHLETICISM is necessary for FITTNESS, and ATHLETICISM is sufficient for FITTNESS

The latter is symbolized:

(~A→ ~B) & (A→ B)

Now, by negating the statement we can get it as follows:

It is not true that ATHLETICISM is a necessary and sufficient condition for FITNESS

~ [(~A→ ~B) & (A→ B)]

Hence, translation of the sentence into sentential logic is:

It is not true that ATHLETICISM is a necessary and sufficient condition for FITNESS.

~ [(~A→ ~B) & (A→ B)]