Lawrence J. Kramer, for the Grice Club.
I wonder if there isn’t some grist for Grice in my favorite time and distance problem. It’s more challenging than difficult, i.e., it seems to rely on some misdirection rather than complexity for its fun. But see what you think. The riddle is credited to Marilyn vos Savant.
Two boats cross a lake leaving from opposite sides. They move at constant speed, pass each other and continue to the opposite shore. They reverse and pass each other again and continue until each has completed a round trip. Make the usual friction-free assumptions about instantaneous acceleration and deceleration and the boats being able to pass through each other as they follow the same course.
When the boats first meet, they are 900 meters from the eastern shore. When they next meet, they are 600 meters from the western shore. How wide is the lake?
I will put the solution in a comment.
Posts
The solution is very simple to understand:
ReplyDelete1. When the boats meet, they have traveled, in the aggregate, one lake width at the first meeting and three lake widths at the second. Thus, the total trip to the second meeting takes three times as long as the trip to the first.
2. Boat A (the originally westbound boat) travels 900 meters from the eastern shore to the first meeting, and (per #1) 2700 meters to the second meeting.
3. Boat A also travels one lake width plus 600 meters to the second meeting.
4. The lake is 2100 meters wide (2700-600)
I'm not sure why I like this problem, but I think it has to do with the need to create a unit of time called the "Aggregate Lake Transit" - the time it takes these two boats, in the aggregate, to travel one lake width. In most time and distance problems, we are given two of speed, time, or distance and asked to figure the third. Here, we get only distances and must concoct our own time and speed units. Boat A moves at 900 meters/ALT, and events happen 1 ALT and 3 ALts from the beginning. If the problem were restated with hours instead of ALTs, we would have too much information unless we were also being asked to compute the time it took to cross the lake (which we do not know in the actual version). But, to ask for the additional information of how long it would take, the problem would have to provide a "real" time unit, and it would fall apart as a problem.
My question, which may or may not be interesting to Griceans, is what makes a problem so hard to solve and so easy to understand when solved? In a way, that's how selfish gene theory strikes me: it seems to have eluded a lot of people, but when it is explained, it is found not really to say anything we didn't already "know."
No idea! But she was a genius, Marylin vos Savant! Thanks for sharing!
ReplyDeleteIs a genia, rather!
ReplyDeleteI see she has The Power of Logical Thinking: Easy Lessons in the Art of Reasoning (1996), which should have interested Grice, and that she studied philo at Washington.
ReplyDeleteI love the idea of an art of reasoning and I'm sure Grice too.
What Kramer proposes may have to do with that. And I'll reflect (if not necessarily reason) about it and hopefully get back.
I was going to ask Kramer (and I am) to perhaps (if you'll excuse me the split -- I always add that, however clumsy!) provide a slight differen formalisation!?
ReplyDeleteI'm going to deal with this in terms of Grice's erotetics -- provided I find some stuff I once copied from one section on his "Aspects of Reason" -- precis, THIS BLOG, and elsewhere.
This because:
a. As Kramer notes, the 'mathematical' problems are indeed questions ("how wide is the lake?"). In fact, they are, most usually, x-questions, rather than yes/no questions.
The fact that Grice does use x-question is good because 'x' is used by mathematicians in dealing with such puzzles. For they are indeed, as Kramer notes, 'equations', i.e. identities.
I'm VERY GOOD at solving equations, but there ARE alternative approaches. So we may need a more abstract formalisation of the 'puzzle', simple as Kramer finds it!
So that we can go axiomlike as in the Pythagoras theorem. Usually these things have a definite number of steps. I think Kramer considers three, only. Which is good. The fewer steps the better.
Etc.
But I should go back to this. And the other comment deals with something different, next.
I'm not sure if Kramer means, but I think he does, the question,
ReplyDelete"This is to me similar to Dawkins's TSG".
-- as rhetorical. I.e.
Something which looks complex (by the look of it) but ain't.
But I see his point!
Not yet able to trace my notes on xy-questions, etc. but this from wiki, 'problem-solving' may be of general Gricean interest. Grice was _fascinated_ by problem-solving:
ReplyDeleteCharacteristics of difficult problems
"As elucidated by Dietrich Dörner and later expanded upon by Joachim Funke, difficult problems have some typical characteristics that can be summarized as follows"
"Intransparency (lack of clarity of the situation)"
---- cfr Grice, "Be perpicuous"
"commencement opacity"
"continuation opacity"
"Polytely (multiple goals)"
"inexpressiveness"
"opposition"
"transience"
"Complexity (large numbers of items, interrelations and decisions)"
"enumerability"
"connectivity (hierarchy relation, communication relation, allocation relation)"
"heterogeneity"
"Dynamics (time considerations)"
"temporal constraints"
"temporal sensitivity"
"phase effects"
"dynamic unpredictability"
etc!
Who formulate these problems? You are a genius, L. J.!
ReplyDelete--- It is indeed a gem of Gricean economy. Because, as I understood it, via your exegesis and all,
_as stated_ as per your post, rather than comment, --
one gets the 'formal' statement (with variables, as it were):
"Boat A moves
at 900m/ALT, and events happen
1 ALT + 3 ALT from the beginning."
And you add, masterfully:
"If the problem were restated
with hours instead of ALTs,
we would have too much information."
which we don't want via
Qual1
-- be as informative as is required
Qual2
do not be more informative than is required.
BUT SOME puzzlers do that intentionally, i.e. flout Grice! And it's part of the gist if not grist to desgrice them!
---
Oddly, when summarising in "Farewell to all that", my precis on Grice on rationality in "Valedictory Essay", THIS BLOG, I omitted a few comments, but Grice was wondering: Why is it that I feel I was wrong when I suggested two maxims of Quantity?
His words, strictly, in WoW:
He has this as one of FOUR points to consider. This is the fourth and last. He states it briefly:
"(4) While it is perhaps NOT
too difficult to envisage
the impacct upon implicature
of a real or apparent
undersupply of information,
the impact of a real or
apparent OVERSUPPLY
--- [Kramer's "we would have
too much information, unless...". JLS]
is much more problematic" (Gr89:372).
So, further grist to the grice:
ReplyDeleteNote Kramer's "unless" -- as per my comment above.
Kramer:
"we would have too much
information, unless..."
i.e. it seems as if Kramer, in trying to see what it is that appeals to him of this brain-teaser, is trusting:
the poser of the riddle (let's call her "Sphynx")
is being cooperative (abiding by the CP and the two informativeness or quantitative, as Kramer charmingly puts it, maxims).
But surely in some problem-solving of that type, it may seem (especially those under time-constraints, e.g.), that can be licensed?
Kramer redux?
ReplyDeleteI.e. reductio ad absurdum (I see Marilyn v. S. seems to have a few typos in her official webpage as to name of some fallacies: 'misericordium', etc., incidentally).
In this case, 'reductio ad absurdum' of the problem-poser NOT abiding by the CP (or at least the Quantitative maxim(s)):
Kramer expresses it as follows:
"If the problem were restated
with hours instead of
Aggregate Lake Transits,
we would have
too much information unless
[emphasis mine. JLS]
"we were also being asked
to compute the time it
took to cross the lake
(which we do not know in
the actual version)."
And here comes the redux:
"But, to ask for the
additional information
of how long it would take,
the problem would have
to provide a "real" time
unit,"
--- this in the specific 'jargon' of 'equations', vide wiki.
Kramer continues:
"and it would fall
apart as a problem."
So the Sphinx is basically collaborating with herself. Ni-ice!
I may write a comment on Marilyn's dropping out of philo at Washington -- my running commentary on her Latin typos in her official page -- 'misericordium', etc. -- and title
ReplyDelete"Riddles of the Sphynx"
"A Gricean Reply to Marilyn von Savant's
Art of Reasoning --"
"in three reverse easy steps -- and back"
or something!
Cu-ute!