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Spring 2018 CS4860 Applied Logic â Review
Abstract
This document reviews key topics in the course. It relates them to each other and discusses the role of constructive type theory in integrating them into a coherent foun- dation for constructive mathematics and computer science. It also includes a few sample questions similar to what might be on the exam.
The document mentions the fact that at Cornell we have provided the first implementa- tion of Brouwerâs intuitionistic type theory in our Nuprl proof assistant. We formalized and implemented the fundamental principles of Brouwerâs intuitionistic mathematics, bar induction and the continuity principle. These required that we implement the no- tion of free choice sequences, briefly discussed in the course.
We did not devote course time to explore details of the fundamental new features of intuitionistic mathematics. It is interesting to know that these principles have been implemented and are now being used in real analysis. As far as we know, Nuprl is the only proof assistant that implements these principles, providing the first implementa- tion of fully intuitionistic mathematics. It is highly likely that other proof assistants will implement intuitionistic principles in due course. In this summary document we mention at most one or two new result in the area, discovered and published while the course was progressing. For instance, we have shown that Markovâs Principle is false in the Cornell implementation of intuitionistic type theory. This important was discovered and confirmed as this course was being taught. You were among the first to know about it when it was mentioned in the final lecture.
1 Review of Important Logics and Topics
This course on Applied Logic covered the major topics in both pure logic, an ancient subject, and in applied logic, a modern subject strongly tied to formal methods and computer science. We used Raymond Smullyanâs book First-Order Logic [7] as a starting point and then added
material from other sources and wrote new notes on topics related to recent Cornell research in this very active area. Raymond Smullyan has written other very interesting and lively books on logic [9, 8, 10, 11]. His student Melvin Fitting also wrote two excellent books and edited others [3, 4, 5, 2].^1
In the final lecture we mentioned some new results in applied logic and type theory including the discovery that Markovâs Principle is false in fully intuitionistic type theory [1]. It is important in this class to know the statement of Markovâs Principle.
2 Main Topics
Here is a list of the lectures and the most key concepts in each one.
- Introductory Lectures 1 and 2: Logic is an ancient subject. It is good to know roughly how old and who the earliest eminent logicians were and who are some of the major contributors since then. Lecture 2 provides a brief historical overview. Students should be familiar with the eighteen names mentioned and what they contributed.
- Propositional logic: Lectures 3 and 4 defined propositional logic from Smullyan using truth functional semantics. It is important to know what consistency, complete- ness, and decidability mean for this simple logic and more generally.
- Lectures 5 and 6: We discussed the meaning of theorems in mathematics of the form, If the generalized continuum hypothesis (GCH) is true, then P = NP. Here we have two precise mathematical statements which are both open problems. Thus we donât know their truth values, and we might never know them; yet there is a provable rela- tionship between them. How can that be? We discussed the notion of evidence for a proposition and how that is used in proving assertions such as GCH implies (P = NP), a proven result. We then gave rules of evidence for the logic iPC, the iintuitionistic Propositional Calculus.
- Lectures 7 and 8: We illustrated the tableaux style of proof for PC and discussed Tableaux style rules for iPC.
- Lectures 9 and 10: We introduced First-Order Logic, FOL and intuitionistic First-Order Logic iFOL. These are the most important logics in the course. In mathematics it is consider sufficient to FOL, and students should know why that is the case. (^1) Smullyan was also a magician, and you can see elements of this in his books.
3 Sample Questions
Having this background in logic might help you untangle misleading statements you can find on the web and in survey articles. We just found something lately from people trying to axiomatize relativity theory. They claim that FOL cannot model the real numbers R, so it cannot be used for this formalization. What is misleading about that claim? What might they want to say?
- Prove the following statement in iPC if it is constructively true, otherwise explain why it is not constructively true:
((A&B) â (C&D)) â ÂŹD â (ÂŹA ⨠B).
- Is Markovâs Principle classically true? If so prove it in FOL, if not explain why.
- It is convenient to add a sequencing rule, also called the cut rule, such as the following. H
G by Seq(S) H, S G H ` S
Explain why the rule is valid. There is an important result in proof theory that says that any proof using cut can be done without it. What is a simple way to understand this result? In other words, why do we know that this should be true in the normal way mathematics is done informally. What is a standard way to accomplish what this rule accomplishes if we were writing an informal explanation of why G is true given the hypotheses in H?
References
[1] Mark Bickford, Liron Cohen, Robert Constable, and Vincent Rahli. Computability beyond Church-Turing via choice sequences. In 33nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 2018. To appear.
[2] Robert L. Constable. Formal Systems, Logics, and Programs. In Melvin Fitting and Brian Rayman, editors, Outstanding Contributions to Logic, volume 14, pages 23 â 38.
[3] M. Fitting. Intuitionistic model theory and forcing. North-Holland, Amsterdam, 1969.
[4] M. Fitting. Fundamentals of Generalized Recursion Theory. North-Holland, 1981.
[5] Melvin Fitting and Brian Rayman, editors. Raymond Smullyan on Self Reference. Springer, 2017.
[6] Jean-Louis Krivine. Une preuve formelle et intuitionniste du thorme de compltude de la logique classique. Bulletin of Symbolic Logic, 2:405â421, 12 1996.
[7] R. M. Smullyan. FirstâOrder Logic. Springer-Verlag, New York, 1968.
[8] R. M. Smullyan. What is the name of this book?? Pelican, 1984.
[9] Raymond M. Smullyan. 5000 B.C. and Other Philosophical Fantasies. St. Martinâs Press, NY, 1983.
[10] Raymond M. Smullyan. G¨odelâs Incompleteness Theorems. Oxford University Press, New York, 1992.
[11] Raymond M. Smullyan. Diagonalization and self-reference. Number 27 in Oxford Logic Guides. Clarendon Press, Oxford, 1994.
