Translate the following sentences into TFL

Question

If the bear and the fox are on the box, then either the herring is unhealthy or the cat is on the mat

The cat is on the mat if and only if the bear is on the box but the fox is not.

The fox is off the box only if the bear is on the box, unless the herring is unhealthy.

Answer 1

B: The bear is on the box

F: The fox is on the box

H: The herring is unhealthy

C: The cat is on the mat

(B & F) ⊃ (H v C)

C ≡ (B & ~F)

~H ⊃ (~F ⊃ B) OR (~F ⊃ B) v H

I think the most confusing for students is how to translate sentences like the last sentence into TFL. The sentences of the form "A unless B" seem to confuse students because they often seem to believe that such a statement entails "if B, then ~A" or that it entails "if A, then ~B" (these two statements are equivalent). However, this is not the case. What "A unless B" says is that when B is true, it does not follow that A is true. But it is also not the case that the negation of A follows from B. A less formal explanation of "A unless B" is that it is nothing more than saying that A must be true whenever or if B is false. So if B is false, then A is true. If B were true, however, this does not tell us anything about A. "A unless B" tells us that if B is true, it's still an open question whether or not A is true, for it could be that in some cases A is false and other cases A is true when B is true. So it is not true that if B is true, then A is false. So we're just left with "if ~B, then A" which is also equivalent to "either A or B". This explains the translations of the third sentence.

Hope this helps!