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As homework, I had to translate the following sentence into FOL:

One can travel between any two Canadian cities by airplane, train, or bus.

P(x) - x is a Canadian city;

Q(x, y) - one can travel by airplane between x and y;

R(x, y) - one can travel by train between x and y;

S(x, y) - one can travel by bus between x and y.

My instructors claim that the correct answer is:

∀x.∀y.P (x) ∧ P (y) → (Q(x, y) ∨ R(x, y) ∨ S(x, y))

I deny that and claim that it is :

∀x.∀y.P (x) ∧ P (y) → (Q(x, y) ∧ R(x, y) ∧ S(x, y))

My reasoning is as follows :

I dismiss their solution by saying that it does not fully capture the information given in the given sentence. If we were in the situation that " One can travel between any two Canadian cities by airplane and not by train and not by bus. ", then the sentence given by my instructors is true and I claim that it should not since the information in the two sentences differ.

The way I reason that my solution is the correct one is that I think of Q, R and S as properties of "one". I am trying to incorporate in my solution that "one" has all of the three properties. By my instructors' solution, a case where only one of the property would be available but not the other two would be identical with the a case where all of the properties would be available.

And one more question, from the sentence "Some students respect all professors." do we conclude that at least two students respect all professors or that we only know that at least one student respects all professors?

shooting-squirrel
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  • The $\vee$ is not exclusive. Your instructors solution says that for any two Canadian cities at least one of the travel methods is available, not only one. – alpacahaircut Dec 09 '13 at 03:55
  • @plattnum Exactly, and I'm claiming that it should express the same information, and that is, that "one" must have 3 properties. By their solution, it can have less than 3. – shooting-squirrel Dec 09 '13 at 03:57
  • Let me put it this way: don't you think one could go from one Canadian city to another using public transportation, even though not all Canadian cities have airports? – alpacahaircut Dec 09 '13 at 04:00
  • I think that one could go from one Canadian city to another using public transportation, even though not all Canadian cities have airports. – shooting-squirrel Dec 09 '13 at 04:03
  • So you would say that the English sentence is true. In order for your FOL to be true you'd have to be able to get between any two cities by plane, for which you'd need airports. – alpacahaircut Dec 09 '13 at 04:05
  • I am translating the sentence, I'm not stating a fact about canadian cities or arguing about the validity of the given sentence. – shooting-squirrel Dec 09 '13 at 04:06
  • I understand that. I was trying to get you to see why your translation is incorrect. The English Sentence is probably true, your translation is probably false, therefore something is wrong in the translation. – alpacahaircut Dec 09 '13 at 04:10
  • clearly, a troll... – shooting-squirrel Dec 09 '13 at 04:11
  • I think plattnum is missing the point here. A simple Google search finds many examples of legal English that construe the phrase the way shooting-squirrel wants. For example, I find “any dividend may be paid by cheque or warrant…” which, writing $C(d)$ for "may be paid by cheque" and $W(d)$ for "may be paid by warrant", clearly means $\forall d. C(d)\land W(d)$, consistent with shooting-squirrel's construal. – MJD Dec 09 '13 at 04:20
  • Similarly I find “Any message may be deleted by its author or by the team captain”, which similarly means that for each message $m$, $m$ may be deleted by its author and $m$ may be deleted by the team captain. (You may complain that it's not obvious that it means this, but it is clear to me, and in any event, that is what it means in this context.) – MJD Dec 09 '13 at 04:22
  • In terms of complaining, it is useless, it's about CS 245 from University of Waterloo. It's a computer science course in logical computation taught by persons who would shame mathematics just by spelling the word "mathematics". I was hoping that I was wrong and that I could see the error. That would have helped my sleep. – shooting-squirrel Dec 09 '13 at 04:34
  • Well, then you can go in peace, knowing that you were right. – MJD Dec 09 '13 at 04:36

2 Answers2

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As you say in your title, this is a question of English, not mathematics. I would read the sentence to say "It is possible to travel between any two Canadian cities by (at least one of) airplane, train, or bus." In that view, using the logical connective "or" is correct. I don't think "one" has anything to do with it. You have clearly identified the mathematical distinction between the two readings.

For your other question: In mathematical English, some a is b clearly is meant to say at least one a is b, so you are only promised one. The s on students is English, not mathematics. If you said some student, there would be an implication that it was only one. In a math course I would think a while before transcribing it as exactly one.

Ross Millikan
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  • Thank you for your reply. The instructors are looking for an exact translation, anything else that is not identical to theirs is wrong. So am I wrong, are the professors wrong? What's the case in the end? – shooting-squirrel Dec 09 '13 at 04:01
  • Your reading is replacing my parenthetical (any one of) with (any of), so you insist that you can get there with all of the three. You have transcribed your reading correctly. I stand by my reading, but recognize that English is not always precise. Decades ago, when I graded papers, if you came in with this I would see you got the point and give you all the points. Your grader may have a different agenda, as you have not produced the desired output. If so, this won't help. – Ross Millikan Dec 09 '13 at 04:12
  • Under further review (as the American NFL says) I think my parenthetical (any one of) would be better rendered as (at least one of). Fixed. – Ross Millikan Dec 09 '13 at 05:01
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This is not really a question of logic, or even mathematics. It belongs to the subfield of linguistics called semantics. The question here, as you say, is about the meaning of the original English, and it is indeed ambiguous.

Of the many, many examples I have found so far that are construed your way, the clearest ones I have found so far have the form:

Applications must be filed by telephone, mail, or by visiting the Bursar's office.

You may console yourself by imagining how irritated your professors would be if they attempted to file by visiting the Bursar's office, and found the Bursar there sneering “the regulations say you must file by telephone, mail, or by visiting the Bursar's office. “Or” requires at most one of these to be true in each instance, and in your case, you must file by mail, not in person.”

MJD
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  • I can't understand how the example with the applications relate to this, but I find the examples that you gave above in the same situation. I will vote this as an answer due to those examples. – shooting-squirrel Dec 09 '13 at 04:46
  • I actually interpret that sentence as meaning "Applications cannot be filed except by telephone, by mail, or by visiting the Bursar's office". It doesn't entail that all three approaches are possible in any given case; it merely creates that implicature, which can easily be canceled. (No one would blink if, right after the sentence you quote, the following sentence were "Applications filed less than three days in advance must be filed by visiting the Bursar's office.") – ruakh Dec 09 '13 at 04:46
  • (But, +1 anyway. The original sentence is genuinely ambiguous, because the scope of the disjunction is unclear.) – ruakh Dec 09 '13 at 04:47
  • I get it now, I think, when the Bursar's employee says to the instructor that " in your case, you must file by mail, not in person.", by the instructors logic(even tough the Bursar's policy is "Applications must be filed by telephone, mail, or by visiting the Bursar's office.") the employee treated him correctly.(But probably the instructor would have not been expecting that answer). Making it in the general case: Suppose a statement is coded into "data". There are proper subsets of that "data", and those do not, by my reasoning,constituate a valid translation. – shooting-squirrel Dec 09 '13 at 04:58