## Advice on Chapter 3 Problems

The following are excerpts from my email replies to students who have asked for help on problems in past years. If you're having trouble with one, you might see whether there are any comments about it here. I list problem number first, then in most cases a line or two from the student email prefixed by '>' (usually an answer they thought was correct), and then my own comments. Sometimes I comment on several questions about the same problem.

Very important for several of these problems: remember that "Neither P nor Q" gets translated by ~(P v Q), or equivalently by ~P & ~Q.

3.7

The DeMorgan's equivalences are helpful in thinking about this problem (or vice versa -- the problem can help to see why the equivalences are true). In particular, consider statement 10:

(using ~ for negation). This means that it's not the case that ~Adjoins(b,f) and SameRow(b,f) are both true -- i.e. one or the other must be false. Symbolically, using DeMorgan's laws, statement 10 is equivalent to

Now, ~~Adjoins(b,f) says the same thing as Adjoins(b,f). (If it's not true that b does not adjoin f, then it must be the case that b does adjoin f.) So statement 10 means the same as:

Now, b and f must be in the same row. This is implied by statement 1. So we cannot make the second disjunct true. But the above disjunction will be true if either disjunct is true, so all we need to do is make the other disjunct true, i.e. let b adjoin f. Voila!

3.18.2

> (LeftOf(a,b) v RightOf(a,b)) & ~FrontOf(a,b) & ~BackOf(a,b)

All you need is ~FrontOf(a,b) & ~BackOf(a,b). Adding that a is left of or right of b is not only unnecessary, it actually makes the translation incorrect in a subtle way, since it rules out the possibility that a = b, which is not ruled out by the sentence being translated.

3.18.10

> (~Tet(a) v Tet(b)) & (~Cube(a) v Cube(b)) & (~Dodec(a) v Dodec(b))

Hmm, I can't quite figure out what you were thinking on this one. Oh, wait, maybe I see. I think that if we were using exclusive or, your answer would be OK. However, "v" is inclusive: "P v Q" is true if either or both disjuncts are true. So e.g. "~Tet(a) v Tet(b)" is compatible with b being a tetrahedron and a not being a tetrahedron. Same thing for the cube and dodec parts, so your sentence doesn't actually imply that the objects are the same shape.

3.20.6

> ~Larger(e, d) v ~ Smaller(e,d)

The sentence is supposed to say that e is neither larger nor smaller than d (i.e. it's not larger and it's not smaller). Your sentence says instead that it's either not larger or not smaller.

3.21.4

> (Cube(c) & Cube(d)) & ~(Small(c) & Small(d))

the problem's the second conjunct. "neither of them is small" means they're both not small: ~Small(c) & ~Small(d) or, equivalently, ~(Small(c) v Small(d)). What you wrote, "~(Small(c) & Small(d))", means that it's not the case that both of them are small, i.e. that one or the other is not small. If c is large and d is small, your sentence is true but the English is not.

3.21.5

This sentence is grammatically a bit complicated. "Neither e nor a is to the right of c and to the left of b." It helps to rephrase this to make it clearer what the "neither" and "nor" parts are: "Neither (e is to the right of c and to the left of b) nor (a is to the right of c and to the left of b)." After that it's not too bad: the parenthesized parts are straightforward to translate, and then those translations can be plugged in for P and Q in the standard translation of "Neither P nor Q."

3.21.7

> ~(Between(c,a,b) & FrontOf(c,a) & FrontOf(c,b))

The original sentence implies that c isn't between a and b, isn't in front of a, and isn't in front of b. Your sentence will be true if any of these three things is true, so your sentence isn't equivalent to the original. Changing the "&"s to "v"s would fix the problem. Or you could write it as a conjunction of negations:

~Between(c,a,b) & ~FrontOf(c,a) & ~FrontOf(c,b)

> ~Between(c, a, b) & ~(FrontOf(c, a) & FrontOf(c, b))

The English sentence says that c is neither between a and b nor in front of either of them. You have the "neither" part right. The "nor" part should say c is not in front of either a or b. Your version will be true if c is in front of one but not the other; it needs to say instead that it's in front of neither.

> ~Between(c, a, b) & ~(FrontOf(c, a) & FrontOf(c, b))

The English sentence says that c is neither between a and b nor in front of either of them. You have the "neither" part right, so let's consider the "nor" part. This part needs to say that it's not the case that: c is in front of either a or b. This could be symbolized ~(FrontOf(c,a) v FrontOf(c,b)) or (by a DeMorgan's equivalence) as ~FrontOf(c,a) & ~FrontOf(c,b).

However, your version says only that c is not in front of *both* a and b, i.e. that it is either not in front of a or not in front of b. This isn't strong enough to capture what the English sentence is saying.

3.21.8

> I have a quick question. On 3.21 #8 it says, "Either both a
> and e are tetrahedra or both a and f are." Doesn't this
> imply an exclusive or? I don't think we have really covered
> that in class (and the grading program didn't accept that as
> an answer), but I was just wondering if that is what would
> technically be correct.

Good question. We'll talk a little more about this in class at some point. My own view (shared by the text) is that "or" in English is almost always inclusive, not exclusive -- sometimes there is a pragmatic suggestion of exclusivity, but it's not actually logically implied. (Compare: I am asked for a letter of recommendation; I write: "Mr. X is a very nice person, and attends class regularly. Sincerely, CB". I have certainly managed to suggest that Mr. X is not a very good student, but I haven't actually said that. I think usually when we read an "or" sentence and think it's exclusive, it's really that sort of pragmatic implication. Compare section 7.3 of the text, which starts on p. 187.)

So: I see how the exclusive reading can seem natural here, but I don't actually think it's right. if it turned out that a, e, and f were all tetrahedra, I don't think we would regard the original sentence as false, although it might have been misleading.

> (Tet(a) v Tet(e)) v ((Tet(a) & Tet(f)) & ~(Tet(a) v Tet(e)))

Well, the last conjunct of your sentence says that neither a nor e is a tetrahedron. This isn't implied by the original sentence.
The English says either a and e are both tetrahedra or a and f are both tetrahedra. If you try to stay close to the English, this should fall into place fairly neatly.

3.21.9

> ~(FrontOf(d, c) & FrontOf(d, b)) & ~ FrontOf(c, b)

That's not equivalent to "neither d nor c is in front of either c or b", is it? Your first conjunct is equivalent to "either d is not in front of c or d is not in front of b", but you need to say that it isn't in front of either of them. The second conjunct says that c is not in front of b, but you also need to say that c is not in front of c. (This may seem to go without saying, since it *has* to be true, but it's in the original sentence so it should be in the translation too.)

> ~(FrontOf(d, c) & FrontOf(d, b)) & ~(FrontOf(c, c) & FrontOf(c, b))
> [GG hint:] Think about other sentences with the same form, for
> example, "Neither Clinton nor Gore is taller than either Michael
> Jordan or Magic Johnson." How would you say that in FOL?

The GG's hint is very helpful here. The sentence needs to say that neither of the first two is in front of either of the second two. Your sentence will be true if d and c are in front of one but not in front of the other; it needs to imply that d isn't in front of either c or b and c isn't in front of either c or b. ("Not in front of either" means "not in front of the first and not in front of the second" -- this is related to the DeMorgan's equivalences.)

> ~(FrontOf(d, c) & FrontOf(c, b)) & ~(FrontOf(d, b) & FrontOf(c, c))

You can think about it this way. English: "Neither d nor c is in front of either c or b."

This means: it is the case neither that d is in front of either c or b, nor that c is in front of either c or b.

Now, "Neither A nor B" means ~(A v B) or, equivalently, ~A & ~B. In this case, A is "d is in front of either c or b" and B is "c is in front of either c or b".

Last update: January 27, 2011.
Curtis Brown  |  Symbolic Logic   |  Philosophy Department  |   Trinity University
cbrown@trinity.edu