Assignment 8. The Liar

For submission

1. Which of the following robots are paradoxical? Explain why they are if they are; or why not if not.

(a) HAL-mark-1 the robot repairs every robot on the spaceship.
(b) HAL-mark-2 the robot repairs exactly every robot on the spaceship that does not repair itself.
(c) HAL-mark-3 the robot repairs every robot on the spaceship that does not repair itself.

2. HAL the robot is a simple reformulation of Russell's barber paradox. It is the same paradox, dressed up in futuristic guise. Propose a reformulation of another paradox of self-reference. How colorful can you make it?

3. Consider the version of the liar paradox:

S1: Sentence S2 is true.
S2: Sentence S1 is false.

How does Tarski's hierarchical notion of truth block the paradox?

For discussion

Not for submission

A. Yablo arrives at a liar like paradox within an infinite set of sentence that refer to each others' truth without any circularities. Is the use of an infinite set essential? Can a liar paradox be set up in a finite set of sentence without using some sort of circularity?

B. If a three-valued logic is prone to a strengthened liar paradox, what about a four-valued logic?

C. Does it matter if Tarski's and Kripke's proposals are in an idealized formal logic? Should not ordinary language aspire to the precision of formal logic?

D. In the paradox of the unexpected exam, what sort of paradox is it? A true contradiction? Or is it merely something intuitively unexpected?