Three Views of Logic: Mathematics, Philosophy, and Computer Science

DONALD W. LOVELAND
RICHARD E. HODEL
S. G. STERRETT
Edition: STU - Student edition
Pages: 376
https://www.jstor.org/stable/j.ctt5hhqc8

1. Front Matter
(pp. i-iv)
(pp. v-viii)
3. Preface
(pp. ix-xii)
4. Acknowledgments
(pp. xiii-xvi)
Donald W. Loveland, Richard E. Hodel and S. G. Sterrett
5. PART 1. Proof Theory
• 1 Propositional Logic
(pp. 3-30)
DONALD W. LOVELAND

There are several reasons that one studies computer-oriented deductive logics. Such logics have played an important role in natural language understanding systems, in intelligent database theory, in expert systems, and in automated reasoning systems, for example. Deductive logics have even been used as the basis of programming languages, a topic we consider in this book. Of course, deductive logics have been important long before the invention of computers, being key to the foundations of mathematics as well as a guide in the general realm of rational discourse.

What is a deductive logic? It has two major components, a formal language...

• 2 Predicate Logic
(pp. 31-60)
DONALD W. LOVELAND

First-order logicis an enrichment of propositional logic that allows for added capabilities of expression and representation of logical notions. The termspredicate logicandpredicate calculusare also used for this logic although the latter term can represent slightly different notions of logic. We will use the terms interchangeably.

The reader is already familiar with the notion of a predicate, as it is used to represent the induction statement in the employment of induction in proofs of the preceding section. Technically, it is a mapping from a set of objects within an interpretation to true or false . Thus...

• 3 An Application: Linear Resolution and Prolog
(pp. 61-80)
DONALD W. LOVELAND

As an application of the resolution proof system we present a restriction of the proof system closely tied to the programming language Prolog (for PROgramming in LOGic). With our focus on logic in this book we do not pursue a general introduction to Prolog, but only suggest the correspondence between the resolution restriction and the core structure of Prolog. The primary point of this chapter is to illustrate that resolution logic has a concrete application outside of logic itself. We do give two illustrations of simple Prolog programs in an appendix for those interested in a further sense of the...

• APPENDIX A: The Induction Principle
(pp. 81-81)
• APPENDIX B: First-Order Valuation
(pp. 82-83)
• APPENDIX C: A Commentary on Prolog
(pp. 84-91)
• References
(pp. 91-92)
6. PART 2. Computability Theory
• 4 Overview of Computability
(pp. 95-122)
RICHARD E. HODEL

The second part of this book is on computability theory, a fundamental branch of mathematical logic. Other branches include proof theory, discussed by Loveland in Part 1 of this book, nonclassical logic, discussed by Sterrett in Part 3, model theory, and set theory. Computability theory has two major goals: clarify the notion of an algorithm; and develop a methodology for proving that certain problems arenotsolvable by an algorithm. There are two problems from Hilbert’s Program and his related research that explicitly ask for analgorithmicsolution to the problem; these areHilbert’s Decision ProblemandHilbert’s Tenth Problem....

• 5 A Machine Model of Computability
(pp. 123-164)
RICHARD E. HODEL

By definition, an n-ary function$F:{\mathbb{N}^n} \to \mathbb{N}$iscomputableif there is an algorithm that calculates F. However, this is not a precisely defined concept since it depends on the informal notion of an algorithm. This chapter is our first attempt to give a precise definition of computability that captures the above informal approach. The model of computability is a machine model and is described in terms of an ideal computer called aregister machine. Given an n-ary function F, we say that F is RM-computableif there is a program for the register machine with the following property: Given${{\text{a}}_{\text{1}}}{\text{, }}{\text{. }}{\text{. }}{\text{. , }}{{\text{a}}_{\text{n}}}{\text{ }} \in {\text{ }}\mathbb{N}$...

• 6 A Mathematical Model of Computability
(pp. 165-218)
RICHARD E. HODEL

In this chapter we describe a model of computability that will give us a precise mathematical counterpart to the informal notions of a computable function, a decidable relation, and a semi-decidable relation. These new concepts are:recursive function,recursive relation, and RE (recursively enumerable)relation. A major result states that these three classes coincide respectively with the RM-computable functions, the RM-decidable relations, and the RM-semi-decidable relations....

• List of Symbols
(pp. 219-220)
• References
(pp. 220-220)
7. PART 3. Philosophical Logic
• 7 Non-Classical Logics
(pp. 223-242)
S. G. STERRETT

So far, we’ve studied some of the powerful possibilities — and some of the surprising limitations — of formal systems of logic. We’ve seen that it’s possible to automate some deductions, so that some proofs can be carried out by computer programs. In Part 1, we looked at the system of logical deduction behind the algorithms used in a core part of the Artificial Intelligence language PROLOG; that system of logical deduction is known as resolution logic. Algorithms based on resolution logic have been devised, implemented, and used to carry out proofs. Some automated question-answering systems make use of resolution logic, too....

• 8 Natural Deduction: Classical and Non-Classical
(pp. 243-287)
S. G. STERRETT

In Part 1, it was stated that a deductive logic has two components: a formal language and a notion of deduction. The language of sentence logic, or propositional logic, which consisted of an alphabet and rules for forming well-formed formulas, was presented there, as follows.

The alphabet of the formal language for propositional logic consisted of

1. statement letters (for which we use uppercase letters, which may be subscripted):P,Q,R,P1,Q1,R1, . . .;

2. logical connectives: ∧, ∨, ¬, →, ↔; and

3. punctuation signs, which were the signs for parentheses, i.e.,) and (.

The...

• 9 Semantics for Relevance Logic: A Useful Four-Valued Logic
(pp. 288-314)
S. G. STERRETT

In this chapter we look at one semantics for the logic of entailment.¹ There are many possible semantics for a given logic, and a variety of semantics has been developed for E and fragments of E. Different semantics are useful for different purposes. The four-valued semantics presented here is especially useful for certain types of automated reasoning applications.

As explained in Part 1 of this book, the truth value of awff is determined by a valuation function. The valuation function comes into play after an interpretation of the atomic statements has been made. An interpretation associates a truth value with...

• 10 Some Concluding Remarks on the Logic of Entailment
(pp. 315-316)
S. G. STERRETT

Any of the numerous proof systems of the logic of entailment can be used to define the system E; the system FE presented in Part 3 of this book was only one such proof system. Likewise, the semantics presented here is only one of numerous semantics that have been developed for E. As we mentioned, fragments of E can also be defined, and proof systems and logic semantics for some of these fragments have been developed as well. We presented only the propositional calculus for E; the reader may be interested to know that a quantifier logic for E has...

• References
(pp. 316-318)
8. Index
(pp. 319-322)