Assignment five

1. Although it is true that Achilles must cover an infinite
number of ever smaller distances in order to catch the Tortoise, just because a
series is infinite does not mean that it diverges to infinity. In this case,
the distance that Achilles must cover is continuously divided in half, and ∑1/2^{n} converges to one, so Achilles must only travel a finite distance
to catch the tortoise. He can accomplish this in a finite amount of time.
Thinking about it slightly differently, as the distances Achilles must travel
get infinitely smaller, so does the amount of time it takes him to cover each
distance. The sum of all those infinitely decreasing segments of time is a
finite number.

2. The third principle, C, is that logical deduction is valid. C explicitly states a principle so intuitive that most people take it for granted: If A implies B, and A is true, B is true. Carroll argues that there could be an infinite sequence of premises similar to C, but none of them would ever truly bridge the gap from A and B to Z. D, for example, would say that if it is true that if A implies B, and A is true, B is true (ie if A, B, and C are all true), Z must be true.

This reminds me of the impossibility
of using deductive reasoning to get from “is” premises to “ought” conclusions. No
matter how many premises there are, it is impossible to use a syllogism to get
all the way to the conclusion without first accepting a principle separate from
the logic itself. In the case of is/ought, that acceptance has to do with basic
ethical goals; in the case of Carroll’s Tortoise, it is an acceptance of
deduction itself.^{1}

1. And yet we feel more comfortable saying, “There are no *absolute* ethical principles” than “There
are no *absolute* logical principles”,
perhaps because we can generally progress comfortably through our daily lives without
assuming the first, but can barely complete a single thought without assuming
the second.