The Shadow Problem Peter Suber, Philosophy Department, Earlham College

I want to axiomatize the system of truths about light and shadow. The following propositions seem true and basic, so I tried them as axioms.

1. Objects cast shadows only if they are opaque.
2. Objects cast shadows only if light shines on them.
3. Shadows do not pass through opaque objects.
4. If a light is shining, then an object is in the shade if and only if another opaque object is casting a shadow on it.

But it seems that these propositions imply something untrue about objects and shadows. Imagine a universe in which there is only one source of light and three opaque objects, A, B, and C, which stand in a line with each other and the light, as illustrated below.

(light)	A	B	C

The four axioms seem to imply that C is not in the shade! By axiom 4, object C is in the shade iff A or B casts a shadow on it. But A cannot cast the shadow, by axiom 3, nor can B cast the shadow, by axiom 2.

Your job is twofold.

1. First, create a formal system whose intended interpretation is the logic of shadow-casting. At first, make your only axioms the four initial statements plus whatever other statements you need to describe the A-B-C situation. Does the counter-intuitive statement that C is not in the shade follow as a theorem? If so, provide the proof.

2. Next, fix the system so that —as far as you can tell— only true statements about shadow-casting follow as theorems. Provide five or ten sample theorems with their proofs, including a proof that C is in the shade.

   In your metatheory, take at least some steps toward showing that the fixed version of the system produces only truths of shadow-casting as theorems. If you can, go further to show that the fixed version of your system produces all truths of shadow-casting (for the A-B-C universe) as theorems. Your metatheory may go well beyond these tasks and prove that your system has, or lacks, other interesting, important, or bizarre properties.

Otherwise follow the directions on the formal system assignment sheet.

Optional additions:

• Find another interpretation of your system completely unrelated to shadow-casting.
• Prove for at least one statement false under your intended interpretation that it is not a theorem. For example, the fixed version of your system will have a theorem saying that C is in the shade. Can you prove that the negation of that statement is a non-theorem?
• Discuss the problems of axiomatizing a branch of physical science, the problems of purging a system of an identified "paradox", and/or the problems of building a formal system with an intended interpretation.
• Identify the ultimate source of the paradox in the original system.

Note. This problem is not original with me, except in the narrow form in which it appears here. Sam Todes (professor of philosophy, Northwestern University) told me the gist of it in the mid-1970's, and he made clear that it was not original with him either. I'd appreciate hearing from anyone who has definite information on who might deserve credit for this elegant problem.

This file is an electronic hand-out for the course, Logical Systems.