HW2

Homework 2: A Paradox, Induction, and Research Basics

Due Wednesday Sep 23 at 22:00

Reading:

• See the reading list (textbooks Ladyman and Walliman), for lecture 2.
• Lecture notes from lecture 2.
• Zeno's paradoxes (Links to an external site.) (Wikipedia)
• Raven paradox (Links to an external site.) (Wikipedia)

Optional reading:

Jorge Luis Borges, The Perpetual Race of Achilles and the Tortoise, see  AchillesTortoise.pdf Download AchillesTortoise.pdf

1. Achilles and the Tortoise

"The implications of the word jewel—precious little thing, delicate though not necessarily fragile, easy to transport, translucency that can also be impenetrable, ageless flower—make it pertinent here. I know of no better qualification for Achilles's paradox, so indifferent to the definitive refutations which have been nullifying it for over twenty-three centuries that we can already declare it immortal. The repeated tours of the mystery proposed by such endurance, the fine ignorance it has visited upon humanity, are gifts we have no choice but to accept gratefully. Let us revive it once more, if only to convince ourselves of perplexity and arcane intimations. I intend to devote a few pages—a few moments—to its presentation and most noteworthy revisions. Its inventor, as is well known, was Zeno of Elea, disciple of Parmenides, who denied that anything could happen in the universe."

(from Jorge Luis Borges, The Perpetual Race of Achilles and the Tortoise, essay from 1929).

Read the Wikipedia page on Zeno’s paradoxes, in particular that of Achilles and the Tortoise.

The paradox of Achilles and the Tortoise is said to have been invented by Zeno of Elea, a disciple of Parmenides, almost 2500 years ago. Zeno's paradoxes have been the food of discussion for millennia.

A paradox can be viewed as consisting of three parts: A premise consisting of facts and established truths, an argument which is a logical derivation from the premise, and a conclusion which seems to be false.

In other words, a paradox can be defined as an unacceptable conclusion derived by apparently acceptable reasoning from apparently acceptable premises.

Describe your own resolution of the paradox of Achilles and the Tortoise!  For example by stating the three parts above clearly, and identifying problems with one or more of them.

2. When does induction work?

The Raven Paradox was introduced in the 1940s by Carl Gustav Hempel (1905-1997), a German writer and philosopher. It questions the notion that a hypothesis H is supported by an observation that concurs with H, i.e., the basis for scientific induction.

Hempel's hypothesis was: All ravens are black.  An observation of a non-black raven would falsify the hypothesis. But does each observation of a black raven strengthen it? And how do observations of non-ravens affect the hypothesis?

The hypothesis can be expressed in predicate logic as :

H1: ∀x R(x) ⇒ B(x)

Another, logically equivalent, way of writing this is:

H2: ∀x ¬B(x) ⇒ ¬R(x)

i.e., All non-black objects are non-ravens. 

If we considered observations relating to this statement, an observation of a non-black raven would still falsify the hypothesis, but now any observation of a non-black non-raven, such as a yellow banana or pink elephant, would strengthen it! And sightings of black objects do not seem to matter at all.

a) A number of attempts at explaining the raven paradox may be found in the Wikipedia article Raven paradox (Links to an external site.). Which explanation do you find the most convincing? Motivate your answer!

b) Read chapter 2.4, Theory and observation, in the book by Ladyman, and the lecture notes from lecture 2.  Goodman's paradox is one example that illustrates the problems of inductive inference. Give another example of when induction does not work, i.e., when induction could lead you (or someone else, or an algorithm) to incorrect or inappropriate answers. 

3. Research basics

a) In chapter 1 of the textbook by Walliman, eight different ways of using research to obtain new knowledge are listed: categorize, describe, explain, evaluate, compare, correlate, predict, control. Find examples of at least two of these in computer science research. Examples should be specific with at least one reference. Some research projects may illustrate more than one of these concepts.

b) Also, read chapter 2 of the book by Walliman, and discuss which of the approaches and philosophical viewpoints mentioned that are used in computer science research? Give examples.

New - a note on references

Please make sure that you include references in your submission in the same way as in a research report.

If you use an explicit quotation from a source you should give a reference, and if you refer to an idea or a result from the research literature,  for example in Part 3, but also when referring to an explanation of the Raven paradox. References should preferably be given at the end in a reference list (with numbered references in the text), and should be in some standard reference format.

Handing in your solution

Save your solution as a pdf file and hand it in both in Canvas (HW2a) and Peergrade  (HW2). Do not write your name in the pdf file.

Peer grading

You will be asked to review the homework of three other students. Your own solution will also be reviewed in this way.

Feedback from your TA

Your teacher will grade your submission and report the result in Canvas

Complete means you have passed this assignment.

Incomplete means you have to hand in a new version.