Richard L. Epstein

From the Introduction:

This series of volumes is meant to extend the scope of what we can formalize in classical predicate logic, and in doing so see the real limitations of what can be done. In the first section the standard of modern formal logic, classical predicate logic with equality, is set out, drawing on the full development in *An Introduction to Formal Logic*. In the second section classical predicate logic is extended to formalize reasoning that involves adverbs and relative adjectives by viewing those as modifiers of simpler predicates. What is normally taken to be atomic predicates, such as “barking loudly”, can then have internal structure. Reasoning that involves conjunctions of terms, as in “Tom and Dick lifted the table”, conjunctions of modifiers, conjunctions of predicates, and disjunctions of predicates can also be formalized by viewing them as part of the internal structure of atomic predicates. The internal structure of names is the topic of the third and last section. Names for functions are used in classical predicate logic to form complex names are presented first. In our ordinary reasoning we use descriptions to form functions, such as “the wife of”, and descriptions to form names, such as “the cat that scratched Zoe”. To reason with those we can take account of their internal structure by dropping the assumption that every name must refer to a specific thing. The formal systems that are developed here are not just formalisms but are meant to help us understand how to reason well. Many worked examples show how to use them. They also uncover limitations of the formal work.

The analyses in the examples are tentative, presented with the hope of stimulating you to deeper and clearer analyses. The work here proceeds by abstracting and creating formal models to formalize reasoning. By paying attention to the process of abstracting we gain insight into why we consider some reasoning to be good and some reasoning bad, and insight also into the deeper assumptions we make about the world on which our judgments rely. Questions about the metaphysics we assume for modern formal logic and the nature of formalizing have to be faced, most particularly the assumption that the world is made up of objects that we can name. This work extends the scope of classical predicate logic by showing how to formalize reasoning that involves adverbs, relative adjectives, conjunctions of terms, conjunctions of modifiers, and conjunctions of predicates as part of the internal structure of atomic predicates. Descriptive names functions and non-referring names are also analyzed.

**Contents**

INTRODUCTION

**BACKGROUND**

1 Formal Logic

2 Classical Propositional Logic

3 Formal Theories of Reasoning Well and Limitations of Propositional Logic

4 The Language of Predicate Logic

5 Semantics for Classical Predicate Logic

6 An Axiomatization of Classical Predicate Logic

7 Classical Predicate Logic with Equality

8 Formalizing in Classical Predicate Logic

**THE INTERNAL STRUCTURE OF PREDICATES**

RESTRICTORS of UNARY PREDICATES

9 Adverbs as Predicate Restrictors

10 Adjectives as Predicate Restrictors

11 A Formal Logic of Simple Predicate Restrictors for Unary Predicates

12 Examples of Formalizing

13 Are Predicate Restrictors Extensional?

14 Multiple Predicate Restrictors

15 Variable Predicate Restrictors

16 Classical Predicate Logic with Predicate Restrictors of Unary Predicates

17 Examples of Formalizing

OTHER PREDICATE MODIFIERS

18 Predicate Negators

19 Other Kinds of Predicate Modifiers?

20 Modifiers of Modifiers21 The Pure Negator “Not”

22 Examples of Formalizing

INTERNAL CONJUNCTIONS and DISJUNCTIONS

23 “And” Joining Terms

24 “And” Joining Predicates

25 “And” Joining Modifiers

26 “Or” Joining Predicates

27 Examples of Formalizing RELATIONS

28 Modifiers of Relations

29 Internal Conjunctions and Disjunctions with Relations

30 Examples of Formalizing

A FORMAL THEORY of CLASSICAL PREDICATE LOGIC with PREDICATE MODIFIERS, INTERNAL CONJUNCTIONS and INTERNAL DISJUNCTIONS

31 The Formal Theory

PREDICATES USED AS RESTRICTORS

32 Predicates Restricting Predicates

33 Examples of Formalizing

SUMMARY

**THE INTERNAL STRUCTURE OF NAMES**

FUNCTIONS and DESCRIPTIVE FUNCTIONS

34 Functions

35 Classical Predicate Logic with Function Names

36 Functions and Descriptive Names

37 The Syntax of Descriptive Names and Descriptive Functions

38 Semantics for Descriptive Names and Descriptive Functions

39 An Axiomatization of Classical Predicate Logic with Descriptive Names and Descriptive Functions

40 Examples of Formalizing

NON-REFERRING SIMPLE NAMES

41 Names that Don’t Refer

42 Classical Predicate Logic with Non-Referring Simple Names

43 Examples of Formalizing

44 Non-Referring Simple Names in Mathematics

45 Classical Predicate Logic with Non-Referring Simple Names and Names for Partial Functions

46 Examples of Formalizing Mathematics

47 Classical Predicate Logic with Non-Referring Simple Names, Descriptive Names, and Descriptive Functions

SUMMARY

**APPENDICES**

1 Minimal Metaphysics

2 Events in the Metaphysics of Predicate Logic

3 The Dynamic and the Static

4 Propositional Operators

5 A Mathematical Abstraction of the Semantics

6 Parts of Things

7 Completeness

Proofs

Bibliography

Index of Notation

Index of Examples

Index