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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.

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


(q →(¬r ∧ ¬p)) ∧ ¬((¬r ∧ ¬p) → q)

The confusing part is the second part of the disjunction.

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(q → ( ¬r ∧ ¬p )) ∧ ¬(( ¬r ∧ ¬p ) → q)

((Hiking is safe on the trail) implies ((berries are not ripe) and (grizzlies have not been seen in the area))) and (((berries are not ripe) and (grizzlies have not been seen in the area)) does not imply (hiking is safe on the trail)).

This captures the notion of necessary but not sufficient: in every case where (hiking is safe) we also have ((berries not ripe) and (grizzlies not seen)), but we cannot say that when ((berries not ripe) and (grizzlies not seen)) we always also have (hiking is safe).

  • Yeah, I was thinking "we know it's necessary, but we don't know it's sufficient" rather than "it is necessary but not sufficient". Is there a point to leaving my answer up, then? Seems like OP got there first, even if their answer could use more detail. – Canyon Mar 14 '17 at 5:13
  • I'm not taking your meaning. Of course they're not identical, but neither answer has ((~r & ~p) --> ~q). – Canyon Mar 14 '17 at 5:21
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    No---run a truth table and you'll see that it would be impossible to be safe. This is very likely not what OP wants to say. – Canyon Mar 14 '17 at 5:34
  • Hey Canyon - see my answer, that's what I was getting at :) – Mr. Kennedy Mar 14 '17 at 21:49
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Where: p : Grizzly bears have been seen in the area. q : Hiking is safe on the trail. r : Berries are ripe along the trail

First, one suggestion - it might make things easier to analyze if instead of p, q and r, you use S for safe, B for berries and G for grizzlies such that:

  • S : Hiking is safe on the trail.
  • B : Berries are ripe along the trail
  • G : Grizzly bears have been seen in the area.

...that said, I will continue with your convention.


(q →(¬r ∧ ¬p)) ∧ ¬((¬r ∧ ¬p) → q)

The confusing part is the second part of the disjunction.

What disjunction? You have a conjunction operator (∧, &, •) with two operands (φ∧ψ). The operands are conditional statements (this → that), each of which is constituted by the conjunction of two terms (this & that) implying (→) or implied by (←) a third term (the other thing).

To be clear, a necessary and sufficient conditional statement is a conjunction of (φ→ψ)∧(φ←ψ). This formulation is the same as (φ→ψ)∧(ψ→φ). So, what disjunction (not excluding any resulting equivalencies if that is your intended meaning)?

What you have, however, is a little different and can be formulated as:

  • (φ→ψ)∧¬(ψ→φ)
  • (φ→ψ)∧¬(φ←ψ)
  • (φ→ψ)∧(φ↚ψ)
  • (φ→ψ)∧(ψ↛φ)

It might be easier to think of as a sufficiently bi-conditional material non-implication (i.e. a biconditional with an abjunction) - the negation of implication: ¬(φ→ψ) - i.e. the "not sufficient" in "necessary but not sufficient." Not that for terms of logic, "but" is "and".


As for the first part of your formulation, I concur that (q→(¬r∧¬p)) adequately describes the case such that if it is safe, then it is necessarily so that there are no bears and no berries. To be clear, the truth table demonstrates that where q is true (i.e. when it is safe), the conditional is only true if both r and p are false.

enter image description here

If it helps, you can also think of this as the section of the "B" term in this Venn diagram where (010) corresponds to A=F, B=T, C=F and p=A, q=B, r=C

enter image description here


So what of the insufficiency such that no berries or grizzlies is insufficient for hiking to be safe? A truth table for your formulation of the second half:

enter image description here

...and it must be asked if a truth table for the entire bi-conditional expression clearly demonstrates that your expression conveys the notion that no berries and no bears is sufficient for hiking safety, but hiking is not necessarily safe if and only if there are no berries nor bears (i.e. you might still fall off a cliff or get pummeled by a rock slide!)

enter image description here


Since we can see that your formulation DOES NOT preserve the sufficient case I think you can instead see how this formulation does:

enter image description here

Hence,

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.

Can be also be articulated: "For every case where it may be safe to hike, it is necessary that there be no bears and no berries, but for every case where there are no bears and berries it is not sufficiently safe to hike."

And this is formulated as

(q →(¬r ∧ ¬p)) ∧ ¬((¬r ∧ ¬p) → ¬q)

...or, using Safe, Berries and Grizzlies:

(S→(¬B∧¬G)) ∧ ¬((¬B∧¬G)→¬S)

Q.E.D.

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