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