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Logic problem - wrong?

In my logic text there's a problem:

Alice, Bill, and Carol are considering going to a party. Dick says he will go only if Carol does and Bill does not. Ed says he will go if either Dick or Alice or both go. You learn that Ed and Bill do not go. Who did go?

The answer given is that no-one went. But according to my reckoning, this is wrong.

Let A = "Alice goes", B = "Bill goes", etc.

I translated the premises as follows-

1. If D, then C and not-B.
2. if A or D, then E.
3. not-E
4. not-B

From these I generated the following-

5. not-(A or D) from 2 and 3.
6. not-A and not-D from 5.

I don't think anything else can be concluded, but it's not the case that no-one went, because whether C (Carol) went is indeterminate.

Any help appreciated!

Re: Logic problem - wrong?

Hi Greg. No, you're wrong.

The answer is indeed that neither A nor B nor C nor D go.

Here's how.

Given:

If (C + notB), then D
and If not(C + notB), then notD

If (D or A), then E

notE and notD

It follows that:

Since notE, then neither D nor A, i.e. both notD and notA.

But notD implies not(C + notB)

and notB + not(C + notB) = notC.

Therefore, notE + notD + notA + notC is true.

Something about you (optional) Logician, philosopher

Re: Logic problem - wrong?

Hi Avi,

Thanks for the reply. But surely

"Dick will go ONLY IF Carol does and Bill doesn't" means

"If D, then (C + notB)"

and not the converse "if (C + notB), then D", as you have translated it?

"ONLY if" is not the same as "if" is it? otherwise what would be the meaning of the common expression "if and only if"? it would just mean "if".

When I say something like "I'll go out only if it's sunny" that means "If it's not sunny, I won't go out". But this is not the same as "if it's sunny, I'll go out", because that doesn't say anything about what I'll do if it's NOT sunny.

And by contraposition, "If it's not sunny, I won't go out" means "If I go out, then it's sunny", which is the converse of "If it's sunny, then I'll go out".

Or am I hopelessly confused...

Re: Logic problem - wrong?

Hi Greg - I took "Dick says he will go only if Carol does and Bill does not." to mean "if Carol does and Bill does not - Dick will go, but if these two conditions are not fulfilled, he won't go." The first implication is implied by "Dick will go" and the second by "ONLY IF" - I think this is clearly the intent of the text (it is not leaving the former issue open).

In any case, it is true, in the actual calculation of the result, only the first half of this compound proposition is actually used. This is the same as your simpler proposition, after contraposition. I put the full interpretation forward in case of need - but it turned out to be unneeded. Notice that I put Dick in the consequents, and not in the antecedent as you do. My purpose was to show you it is a good habit to first put out the givens as they are given, before manipulating them. In fact no contraposition is needed to get the result, since one can equally reason by denying a consequent as by affirming an antecedent.

Regarding the difference between "only if" and "if and only if" in general terms, here it is:
"If and only if X, then Y" means "if X then Y, and if not X, THEN NOT Y".
whereas
"Only if X then Y" means "if X then Y, but if not X, NOT-THEN Y" (this excludes that Y follows notX, but does not imply that notY follows notX).

You are not confused, but these are subtleties worth knowing.

Something about you (optional) Logician, philosopher

Re: Logic problem - wrong?

Hi Avi,

Thanks for the clarification. I must admit I find implication confusing. Most logic books define it as being false when the antecedent is true and the consequent false, and in all other cases true, but this gives counter-intuitive results sometimes. I guess there are a lot of ways in which we use "if then" and no scheme will satisfactorily express all them.

I found your explanation of strict versus material implication helpful ( Future Logic, Chapter 24.3 ). I haven't come across the form "if P, not-then Q" in any modern logic text, but it seems important. Material implication by contrast seems too "rigid", even though it seems that the intention was that it be less so than strict implication ( at least, if I've understood correctly ).

Re: Logic problem - wrong?

Hi Greg.

Understanding hypothetical propositions is very important for both deductive and inductive logic, because they are actually quite complex constructs with many significant details. Do read chapter 24 of FL and indeed the chapters before and after it (23 and 25). (It is also important to know that these concern only logical conditioning, and that de re conditioning has significant differences.)

The difference between material and so-called strict implication lies, as you read, in the modality of the definition of "If P then Q". Authentic implication is that "P without Q is IMPOSSIBLE", whereas the modern logicians' ersatz version that "P without Q is FALSE" is really not implication but simply negation of conjunction. That is why the modern version does not in fact have a negative hypothetical "If P not-then Q" (for this would merely amount to the positive conjunction "P and notQ is true"), whereas the authentic version provides a form for the negation of implication, NOT(if P then Q), which is colloquially said "If P NOT-then Q" (or more wordily, "Given P, it does NOT follow that Q"). This is of course a very important form, as you can see if you read The Logic of Causation" one day.

Something about you (optional) Logician, philosopher

Re: Logic problem - wrong?

Hi Avi,

Thanks again. I've recently purchased a copy of FL and will open a thread here in due course so that I can post any queries that come up as I read it (there are bound to be some).

I'm a little disappointed that the book has no index, although the TOC is quite comprehensive.

Re: Logic problem - wrong?

Hi Greg. Good for you! I hope you got a hardcopy rather than an e-book. It is a big book, best read on paper.

As regards an index, it is a massive very difficult job. I have not to date found software able to help me. A professional is very expensive and anyway he would have to know the subject pretty well to pick the right words.

If you need to find something you can always use the search facility in thelogician.net (you can zero in on FL specifically instead of the whole site).

Something about you (optional) Logician, philosopher

Re: Logic problem - wrong?

Hi Avi,

Yes I have a hardcopy of FL. There are quite a few software packages to help with generating an index, but you're right it's not just a matter of point-and-click.

Re: Logic problem - wrong?

Good, Greg. My recommendation is read/study a chapter a day, or even a section or page a day if you prefer. This is how I study, and I get through very thick books that way. It is surprising how quickly time passes, and then one is halfway through the book or at the end of it.

Do tell me of any book Indexing software you know about.

Something about you (optional) Logician, philosopher

Re: Logic problem - wrong?

Hi Avi,

Regarding index generators, don't know whether you've looked at this one?

http://www.openviewdesign.com/index.php?option=com_content&view=article&id=3&Itemid=124

Re: Logic problem - wrong?

Thank you very much for the link. I'll look into it and thank you again if it works out for me.

... I looked at the program. It is what I always dreamed of. But it would be too big a job for me now, to do this on my 21 titles!

Something about you (optional) Logician, philosopher