Formal System Assignment
Peter Suber, Philosophy Department, Earlham College

Do one of the following.

1. Create a formal system with at least one intended interpretation and do something interesting with it. It may be interesting in its interpretation(s), the way it permits contradiction or self-reference, the methods of proof it permits, the power and simplicity of its axioms or transformation rules, its non-standard features, or in any other way. It may be a serious contribution to logic solving unsolved problems, proving unproved theorems, or exploring unexplored assumptions —or it may be a "toy". Its interesting features may be more logical than metalogical or vice versa. I don't want to be any more specific, or I might cramp your creativity.

Feel free to use the library for ideas on doctrine, technique, variations, unsolved problems, anything; if you do, please provide a bibliography of the sources you consulted.

Your system can be entirely original with you, or an extension or variation of any system you've already seen (in Hunter or elsewhere). You can, for example, take TFPL or PL and replace an independent axiom with its negation, extend the language or rules of inference, add or subtract axioms, permit infinitely long wffs or proofs, and so on. If you choose to vary an existing system, and if the original is in Hunter, you can specify the unchanged parts simply by citing Hunter by page. If you want to use Copi's rules of inference, cite Copi by page.

2. Do the shadow problem, described in a separate hand-out.

I've written a sample formal system that fulfills most of the assignment. It includes tips and reminders to help you fulfill the whole assignment.

Whether you pick #1 or #2, write your formal system with logical rigor, and adequate informal English explanations of what's going on.

By logical rigor, I mean that your system should meet these conditions.

1. It must be a formal system in Hunter's technical sense, not just half a system or an essay about formal systems.
2. All axioms and theorems must be wffs or wff-schemata.
3. Your theorems must be proved in the strict sense of proof. Every step in each proof must be a wff justified by previous theorems and explicit rules of inference.
4. In your exposition, develop the system formally (syntactically) first, and then spell out its interpretation(s).
5. You must frame and prove some metatheorems.

By adequate informal English explanations of what's going on I mean the following.

1. After introducing them syntactically, translate each axiom and theorem into an English-language sentence about the intended domain.
2. After introducing your alphabet and grammar syntactically, explain what sort of statements your wffs will be when interpreted in your intended interpretation.
3. After introducing your rules of inference syntactically, explain what they do in light of your intended interpretation.
4. If a rule of grammar, rule of inference, axiom, theorem, proof, or other feature of your system is particularly important for your intended interpretation, or for the success of your system, point that out and explain why. If you don't, it might go unnoticed by a person (like me) who is not clued in.
5. Err on the side of explaining more rather than less. Don't put your reader in the position of deciphering unknown heiroglyphics.

In your metatheorems, try to prove results that show or tend to show that your system is successful in capturing the domain you intended to capture.

Please limit yourself to notation that your computer can print in order to avoid handwritten symbols. Since you will use a lot of notation, proofread carefully!

Submit two copies so that I can return one with comments and put the other on reserve in the library for the rest of the class to see.

For those of you who have not had Symbolic Logic, this will be the hardest assignment of the course. It is the only time in the semester when you will have to be sharp on translating English into notation, and in performing derivations according to rules. See me early if this will be a problem.

Answer to a frequently asked question:  No one should feel pressure to do an original system instead of the shadow problem. The shadow problem is not a cop-out. It's not even easier than half the original systems I see.

This assignment produces the most interesting work of the course, the most creative thinking, and the most education.

The chief pitfalls with this assignment are the following:

• to underestimate the work it requires
• to delay the work too long to do justice to your idea
• to start with an idea that is too large, or too informal, and refine or abandon it too late
• to fail to turn in the core on time, hence to get my feedback too late to incorporate before the system due date
• to leave one or more parts informal that ought to be formal
• to give too little informal English explanation of what's going on —with the result that a reader with good will (like me) can't tell what's going on

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

Peter Suber, Department of Philosophy, Earlham College, Richmond, Indiana, 47374, U.S.A.