Saturday, August 28, 2010

Jim's Logic Proof for driving as he does.

Jim had studied enough logic to know he must keep going,
but not enough to know he was supposed to drive down the
center of the road.
01. L v R     In the past, Jim had noted that either the
              left lane is closed, or the right lane is
02. R > Ml    And he knew that if right lane is closed, 
              then one must take the left lane
03. L > Mr    And also he knew that if the left lane is
              closed, then one must take the right lane.
04. L & R     But Jim can now see that both lanes are
              closed ahead.
05. L         Which means, the left one is closed 
              (4, Simplification) 
06. R         And, too, that the right one is closed.
              (4, Simplification) 
07. Mr        So, Jim deduces he must take the right lane.
              (3,5, Modus Ponens) 
08. Ml v Mr   That, or he must take the left one. 
              (7, Addition)

Jim has correctly reasoned that he must take either of 
the two lanes, and so he yells, "Onward!" and punches the 
gas, as the above picture shows. Yet with a bit more study,
Jim would have reasoned as follows:

09. Ml         "But I must likewise take the left lane."
               (2,6 Modus Ponens)
10. Ml & Mr    "Therefore, I must take the left land and the
                right lane!" (7 & 9 Conjunction)

Such contingencies of life show the value of studying logic.

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At 3:50 PM, Blogger kyle8425 said...

The problem is, both 2 and 3 are false. If one lane is closed, it is not the case that one "must" take the other lane--for, they might both be closed. In addition, one could always turn back. He has made the false assumption that there is always a lane to use.

At 7:28 PM, Blogger Brint Montgomery said...

This is what I like about the word "must", nice and ambiguous.

Maybe Jim was thinking "must" in the sense of should. I don't know. Still, Jim is apparently very literally minded; and, unlike you, he doesn't add further premises--like, "they might both be closed." (Jim doesn't like "mights") This account is Jim thinking at the very moment of seeing that they are, in fact, both closed--or, that's what he takes the signs to mean, since they read that way!

Nor would Jim turn back--crazy talk!-- since, as the picture shows, the white line in the center means it's a one-way, for both lanes. So, that's why Jim yelled "onward!".

At 6:00 PM, Blogger kyle8425 said...

If he had concluded "Ahead, the road must widen to include a center lane"--that seems justified given certain background assumptions, and is similar but distinct from "drive in the middle." What would we need to add to justify that conclusion I wonder.


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