# Writing Propositions With Propositional Variables

The puzzle I am working on is:

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

$p$: Grizzly bears have been seen in the area.

$q$: Hiking is safe on the trail.

$r$: Berries are ripe along the trail.

Write these propositions using $p$, $q$, and $r$ and logical connectives (including negations).

a) Berries are ripe along the trail, but grizzly bears have not been seen in the area.

b)Grizzly bears have not been seen in the area and hiking on the trail is safe, but berries are ripe along the trail.

c) If berries are ripe along the trail, hiking is safe if and only if grizzly bears have not been seen in the area.

d)It is not safe to hike on the trail, but grizzly bears have not been seen in the area and the berries along the trail are ripe.

e) 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.

f) Hiking is not safe on the trail whenever grizzly bears have been seen in the area and berries are ripe along the trail.

The only one that I bewilder by is e). The answer I came by was $(\neg p \wedge \neg r) \implies q$. However, the true answer is, "$(q→(¬r∧¬p))∧¬((¬r∧¬p)→q)$". Would someone be so gracious as to explain to me why this is the answer?

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Your $(\neg p \wedge \neg r) \implies q$ expresses that $(\neg p \wedge \neg r)$ is a sufficient condition for $q$, which was exactly what you were asked not to express. But the model answer you're quoting is itself quite problematic.

In think the right answer to (e) should have been $$q \rightarrow (\neg r\land \neg p)$$ This expresses that $(\neg r\land \neg p)$ is a necessary condition for $q$.

However, the part that says that this condition is "not sufficient" cannot be expressed purely propositionally. The given answer tries to express it by a negated implication $$\neg((\neg r \land \neg p) \to q)$$ but this is actually logically equivalent (try truth tables if you're not sure!) to $$\neg r \land \neg p \land \neg q$$ that is, "berries are not ripe and bears have not been seen, but nevertheless it is not safe to hike". Note in particular that it asserts positively that it is not safe; that is not at all what we mean by "such-and-such is not a sufficient condition".

What goes wrong here is that when we speak about necessary and sufficient conditions, this language always implies a quantification over all possible worlds (where it is left doubly implicit what a "possible" world is). Propositional logic itself cannot express such a quantification; in order to do so we need to move either to first-order logic (which has explicit quantifiers), or at least to modal logic where we can express

$A$ is a necessary condition for $B$: $\Box(B\to A)$.

$A$ is a sufficient condition for $B$: $\Box(A\to B)$.

$A$ is not a sufficient condition for $B$: $\neg \Box(A\to B)$ -- or equivalently $\Diamond \neg (A\to B)$.

The $\Box$ represents the implicit quantification over all worlds. The reason why nothing obviously goes wrong when we express the two first sentences without the $\Box$ is that the $\Box$ is in front of everything else, and it is conventional to understand that everything in a formula that is not explicitly specified is universally quantified. So in those cases the propositional formula actually succeeds in expressing the intuitive meaning we want.

In the third case (denying that a condition is sufficient), the quantification is either not at the front or not a universal quantification ($\Diamond$ roughly claims that it is possible that such-and-such). Therefore meaning get lost when we leave it unwritten.

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e) 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.

Recall that given any implication: $a \rightarrow b$, we can translate this as saying

"$a$ is a sufficient condition for $b$" $\quad$ or equivalently: $\quad$"$b$ is a necessary condition for $a$."

$(1)$ For hiking to be safe ($a:=q$), the necessary condition ($b:= \lnot r \land \lnot p$) in your case is that berries not be ripe and for grizzly bears to not to have been seen in the area. So we have

$q\rightarrow(\lnot r \land \lnot p).$

$(2)$ However, you are explicitly told that this condition: ($\lnot r \land \lnot p$) is not sufficient, so you have to negate the converse of $(1)$: you need to negate $(\lnot r \land \lnot p) \rightarrow q$. This gives us:

$\lnot[(\lnot r \land \lnot p) \rightarrow q]$.

To write the complete statement, you need the connective $\land$ in between $(1)$ and $(2)$:

$$[q\rightarrow(\lnot r \land \lnot p)]\land \lnot[(\lnot r \land \lnot p) \rightarrow q].$$

I agree with all that the translation used by the text is terrible, as it oversimplifies matters. But what the text was trying to convey is described above, in $(2)$. (No wonder you're confused by the negated implication.)

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Actually, I don't think simply negating the implication properly expresses "not sufficient". See my answer. – Henning Makholm Dec 12 '12 at 21:04
@Henning: your answer (and comments) go far beyond the material that the OP has covered. May I suggest that your comments and answer are more confusing than helpful at this stage of the game? – amWhy Dec 12 '12 at 23:08
Congrats on topping $500$ rep, EMACK! – amWhy Dec 12 '12 at 23:40
@amWhy Thank you very much! – Mack Dec 13 '12 at 3:20
that's correct, EMACK. I think the consensus is that your text did not "do justice" to the complexity involved in the statement. Peter and I outlined "how" the text arrived where it did, in terms of the negated conjunct in the answer key. – amWhy Dec 16 '12 at 16:48

Treat the two parts, "it is necessary ..." and "it is not sufficient ..." separately.

So consider: "For hiking on the trail to be safe, it is necessary that berries not be ripe along the trail and for grizzly bears not to have been seen in the area." That says that if hiking on the trail is safe, then it needs also to be the case that (berries are not ripe along the trail and grizzly bears have not been seen in the area). So [if all you've got to work with is non-modal propositional logic] that part goes to

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

And now consider "For hiking on the trail to be safe, it is sufficient that berries not be ripe along the trail and for grizzly bears not to have been seen in the area." This says that if (berries are not ripe along the trail and grizzly bears have not been seen in the area) then [that is enough for it to be case that] hiking on the trail is safe. In symbols [if, again, all you've got to work with is non-modal propositional logic]

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

But you are told the condition is not sufficient, so it seems that you were intended -- given the proposed answer -- to negate that, to give

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

And then you are supposed to conjoin those to get the desired answer. That explains the answer you were given.

Whether the given answer is a good one is a different matter as Henning Makholm's answer makes clear. I quite agree that that involves a totally inadeqaute translation of "is not sufficient", for the reasons he gives.

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Actually, I don't think simply negating the implication properly expresses "not sufficient". See my answer. – Henning Makholm Dec 12 '12 at 21:03
@HenningMakholm You are absolutely right of course. I was trying to explain how the "official" answer was reached, and stupidly didn't stop to think/say more. So, thanks, I've edited just a bit. – Peter Smith Dec 12 '12 at 23:07