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This is from Discrete Mathematics and Its Applications:

Let $p, q,$ and $r$ be the propositions:

$\quad p:$ Grizzly bears have been seen in the area.
$\quad q:$ Hiking is safe on the trail.
$\quad r:$ Berries are ripe along the trail.

Write these propositions using $p,q,$ and $r$ and logical connectives (including negation):

  • For hiking on the trail to be safe, it is necessary but not sufficient that berries not be ripe along the trail and for grizzly bears not to have been seen in the area.

I read up on necessary and sufficient from here What is the difference between necessary and sufficient conditions?

  • If $p \to q$ ($p$ implies $q$), then $p$ is a sufficient condition for $q$.
  • If $\neg p \to \neg q$ (not $p$ implies not $q$), then $p$ is a necessary condition for $q$.

From these two conditions how would you apply necessary but not sufficient?

The way I expressed this is: $$ (\neg r \land \neg p) \to q$$

I mainly got to this because "if $p$ then $q$" is the same as $q$ is necessary for $p$. $q$ in this case would be two conditions — berries not be ripe along the trails and grizzly bears not to have been seen in the area".

How would the necessary but not sufficient clause affect the answer though? Would it make a difference?

Community
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committedandroider
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2 Answers2

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If we analyse what you wrote, namely $(\neg r \land \neg p) \to q$, then it says "$q$ is necessary for $r$ and $p$ to be false". Equivalently, "$r$ and $p$ being false is sufficient for $q$". That is:

  • It is necessary that hiking be safe if the berries are not ripe and there are no grizzlies.
  • It is sufficient that the berries are not ripe and there are no grizzlies for hiking to be safe.

This is not one of the clauses of the given statement. In fact, the second reading is precisely the negation of the second half of the given statement, viz:

For hiking to be safe, it is not sufficient that the berries are not ripe and there are no grizzlies.

Hence, half of our solution is:

$$\neg((\neg r \land \neg p)\to q)$$

For the other part, i.e.:

For hiking to be safe, it is necessary that the berries are not ripe and there are no grizzlies.

let us take a more structured approach. First, replace the relevant parts of the sentence by $p,q,r$:

For $q$, it is necessary that $\neg r$ and $\neg p$.

Using your definition of a necessary statement, this can be symbolised as:

$$\neg(\neg r \land \neg p) \to \neg q$$

Hence in the end, the (or rather, a) symbolic representation of the full sentence becomes:

$$\neg(\neg r \land \neg p) \to \neg q \land \neg((\neg r \land \neg p)\to q)$$

As you rightly remark, $p \to q$ is also a correct interpretation for "$q$ is necessary for $p$", and we could just as well have used that instead of $\neg q \to \neg p$.

The bottom line is that once we can parse a given linguistic construction (case in point, the phrases necessary and sufficient) in symbolism, then we can also combine these with other known logical phrases, such as "not" and "and". Therefore, we can derive a logical interpretation of the phrase "necessary but not sufficient" (since in this context, "but" is to be read as "yet also").

Lord_Farin
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The statement that the condition is necessary is an implication in one direction, and the statement that this condition is not sufficient is the negation of the implication in the other direction. Thus, we have the structure (safe $\to$ conditions) $\land \neg$ (conditions $\to$ safe). Fleshing this out gives the answer: $$ [p\to(\neg r \land \neg p)]\land \neg [(\neg r \land \neg p)\to q].\tag{1} $$ Of course, you can express $(1)$ with some other logically equivalent answers as well, but this one will do for your specific problem.

Daniel W. Farlow
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