Classical Mathematical Logic – The Semantic Foundations of Logic - Hardcover

Epstein, Richard L.; Szczerba, Leslaw W.

Hardcover

ISBN 10: 0691123004 ISBN 13: 9780691123004

Publisher: Princeton University Press, 2006

Synopsis

In Classical Mathematical Logic, Richard L. Epstein relates the systems of mathematical logic to their original motivations to formalize reasoning in mathematics. The book also shows how mathematical logic can be used to formalize particular systems of mathematics. It sets out the formalization not only of arithmetic, but also of group theory, field theory, and linear orderings. These lead to the formalization of the real numbers and Euclidean plane geometry. The scope and limitations of modern logic are made clear in these formalizations.


The book provides detailed explanations of all proofs and the insights behind the proofs, as well as detailed and nontrivial examples and problems. The book has more than 550 exercises. It can be used in advanced undergraduate or graduate courses and for self-study and reference.



Classical Mathematical Logic presents a unified treatment of material that until now has been available only by consulting many different books and research articles, written with various notation systems and axiomatizations.

About this title

About the Author

Richard L. Epstein received his doctorate in mathematics from the University of California, Berkeley. He is the author of eleven books, including two others in the series "The Semantic Foundations of Logic (Propositional Logics and Predicate Logic), Five Ways of Saying "Therefore," Critical Thinking", and, with Walter Carnielli, "Computability". He is head of the Advanced Reasoning Forum in Socorro, New Mexico.

Excerpt. © Reprinted by permission. All rights reserved.

Classical Mathematical Logic

The Semantic Foundations of LogicBy Richard L. Epstein

Princeton University Press

Copyright © 2011 Princeton University Press
All right reserved.
ISBN: 978-0-691-12300-4

Contents

Preface...................................................................................................xviiAcknowledgments...........................................................................................xixIntroduction..............................................................................................xxiBibliography..............................................................................................487Index of Notation.........................................................................................495Index.....................................................................................................499

Chapter One

Classical Propositional Logic

A. Propositions 1 Other views of propositions 2 B. Types 3 ? Exercises for Sections A and B 4 C. The Connectives of Propositional Logic 5 ? Exercises for Section C 6 D. A Formal Language for Propositional Logic 1. Defining the formal language 7 A platonist definition of the formal language 8 2. The unique readability of wffs 8 3. Realizations 11 ? Exercises for Section D 12 E. Classical Propositional Logic 1. The classical abstraction and truth-functions 13 2. Models 17 ? Exercises for Sections E.1 and E.2 17 3. Validity and semantic consequence 18 ? Exercises for Section E.3 20 F. Formalizing Reasoning 20 ? Exercises for Section F 24 Proof by induction 25

To begin our analysis of reasoning we need to be clear about what kind of thing is true or false. Then we will look at ways to reason with combinations of those things.

A. Propositions

When we argue, when we prove, we do so in a language. And we seem to be able to confine ourselves to declarative sentences in our reasoning.

I will assume that what a sentence is and what a declarative sentence is are well enough understood by us to be taken as primitive, that is, undefined in terms of any other fundamental notions or concepts. Disagreements about some particular examples may arise and need to be resolved, but our common understanding of what a declarative sentence is will generally suffice.

So we begin with sentences, written (or uttered) concatenations of inscriptions (or sounds). To study these we may ignore certain aspects, such as what color ink they are written in, leaving ourselves only certain features of sentences to consider in reasoning. The most fundamental is whether they are true or false.

In general we understand well enough what it means for a simple sentence such as 'Ralph is a dog' to be true or to be false. For such sentences we can regard truth as a primitive notion, one we understand how to use in most applications, while falsity we can understand as the opposite of truth, the not-true. Our goal is to formalize truth and falsity for more complex and controversial sentences.

Which declarative sentences are true or false, that is, have a truth-value? It is sufficient for our purposes in logic to ask whether we can agree that a particular sentence, or class of sentences as in a formal language, is declarative and whether it is appropriate for us to assume it has a truth-value. If we cannot agree that a certain sentence such as 'The King of France is bald' has a truth-value, then we cannot reason together using it. That does not mean that we adopt different logics or that logic is psychological; it only means that we differ on certain cases.

Propositions A proposition is a written or uttered declarative sentence used in such a way that it is true or false, but not both.

Other views of propositions

There are other views of what propositions are. Some say that what is true or false is not the sentence, but the meaning or thought expressed by the sentence. Thus 'Ralph is a dog' is not a proposition; it expresses one, the very same one expressed by 'Ralph is a domestic canine'.

Platonists take this one step further. A platonist, as I use the term, is someone who believes that there are abstract objects not perceptible to our senses that exist independently of us. Such objects can be perceived by us only through our intellect. The independence and timeless existence of such objects account for objectivity in logic and mathematics. In particular, propositions are abstract objects, and a proposition is true or is false, though not both, independently of our even knowing of its existence.

But a platonist, as a much as a person who thinks a proposition is the meaning of a sentence or a thought, reasons in language, using declarative sentences that, they say, represent, or express, or point to propositions. To reason with a platonist it is not necessary that I believe in abstract propositions or thoughts or meanings. It is enough that we can agree that certain sentences are, or from the platonist's viewpoint represent, propositions. Whether for the platonist such a sentence expresses a true proposition or a false proposition is much the same question as whether, from my point of view, it is true or is false.

B. Types

When we reason together, we assume that words will continue to be used in the same way. That assumption is so embedded in our use of language that it's hard to think of a word except as a type, that is, as a representative of inscriptions that look the same and utterances that sound the same. I don't know how to make precise what we mean by 'look the same' or 'sound the same'. But we know well enough in writing and conversation what it means for two inscriptions or utterances to be equiform.

Words are types We will assume that throughout any particular discussion equiform words will have the same properties of interest to logic. We therefore identify them and treat them as the same word. Briefly, a word is a type.

This assumption, while useful, rules out many sentences we can and do reason with quite well. Consider 'Rose rose and picked a rose.' If words are types, we have to distinguish the three equiform inscriptions in this sentence, perhaps as 'Rose_name rose_verb and picked a rose_noun'.

Further, if we accept this agreement, we must avoid words such as 'I', 'my', 'now', or 'this', whose meaning or reference depends on the circumstances of their use. Such words, called indexicals, play an important role in reasoning, yet our demand that words be types requires that they be replaced by words that we can treat as uniform in meaning or reference throughout a discussion.

Suppose now that I write down a sentence that we take to be a proposition:

All dogs bark.

Later I want to use that sentence in an argument, say:

If all dogs bark, then Ralph barks. All dogs bark.

Therefore, Ralph barks.

But we haven't used the same sentence twice, for sentences are inscriptions.

Since words are types, we can argue that these two equiform sentences should both be true or both false. It doesn't matter to us where they're placed on the paper, or who said them, or when they were uttered. Their properties for logic depend only on what words (and punctuation) appear in them in what order. Perhaps it is a simplification, but let's agree that any property that differentiates them won't be of concern to our reasoning.

Propositions are types In any discussion in which we use logic we'll consider a sentence to be a proposition only if any other sentence or phrase that is composed of the same words in the same order can be assumed to have the same properties of concern to logic during that discussion. We therefore identify equiform sentences or phrases and treat them as the same sentence. Briefly, a proposition is a type.

It is important to identify both sentences and phrases, for in the inference above we want to identify the phrase 'all dogs bark' in 'If all dogs bark, then Ralph barks.' with 'All dogs bark.'.

The device I just used of putting single quotation marks around a word or phrase is a way of naming that word or phrase or any linguistic unit. We need some such convention because confusion can arise if it's not clear whether a word or phrase is being used or referred to as a word or phrase. For example, when I write:

The Taj Mahal has eleven letters

I don't mean that the building has eleven letters, but that the phrase does, and I should write:

'The Taj Mahal' has eleven letters

When we do this we say that we mention the word or phrase in quotation marks and that the entire inscription including the quotes is a quotation name of the word or phrase. Otherwise, we simply use the word or phrase, as we normally do. Note that the assumption that words are types is essential in using quotation names. We can also indicate we are mentioning a linguistic unit by italicizing it or putting it in display format. And single quotation marks are used for quoting direct speech, too.

The device of enclosing a word or phrase in double quotation marks is equivalent to a wink or a nod in conversation, indicating that we're not to be taken literally or that we don't really subscribe to what we're saying. Double quotes are called scare quotes, and they allow us to get away with "murder".

Exercises for Sections A and B

1. Of the following, (i) Which are declarative sentences? (ii) Which contain indexicals? (iii) Which are propositions?

a. Ralph is a dog.

b. I am 2 meters tall.

c. Is any politician not corrupt?

d. Feed Ralph.

e. Ralph didn't see George.

f. Whenever Juney barks, Ralph gets mad.

g. If anyone should say that cats are nice, then he is confused.

h. If Ralph should say that cats are nice, then he is confused.

i. If Ralph should say that cats are nice, then Ralph is confused.

j. Dogs can contract myxomatosis.

k. 2 + 2 = 4 .

l. d e^x/dx = e^x

2. Explain why we cannot take sentence types as propositions if we allow the use of indexicals in our reasoning.

C. The Connectives of Propositional Logic

We begin with propositions that are sentences. There are two features of such sentences that contribute to our reasoning: their syntax, by which we mean the analysis of their form or grammar, and their semantics, by which we mean an analysis of their truth-values and meaning or content. These are inextricably linked. The choice of what forms of propositions we'll study leads to what and how we can mean, and the meaning of the forms leads to which of those forms are acceptable.

There are so many properties of propositions we could take into account in our reasoning that we must begin by restricting our attention to only a few. Our starting point is propositional logic where we ignore the internal structure of propositions except as they are built from other propositions in specified ways.

There are many ways to connect propositions to form a new proposition. Some are easy to recognize and use. For example, 'Ralph is a dog and dogs bark' can be viewed as two sentences joined by the connective 'and'.

Some other common connectives are: 'but', 'or', 'although', 'while', 'only if' 'if ... then ...', 'neither ... nor ...'. We want to strike a balance between choosing as few as possible to concentrate on in order to simplify our semantic analyses, and as many as possible so that our analyses will be broadly applicable.

Our starting point will be the four traditional basic connectives of logic: 'and', 'or', 'if ... then ... ', 'not'. In English 'not' is used in many different ways. Here we take it as the connective that precedes a sentence as in It's not the case that ..., or other uses that can be assimilated to that understanding. These four connectives will give us a rich enough grammatical basis to begin our logical investigation.

These English connectives have many connotations and properties, some of which may be of no concern to us in logic. For example, in American English 'not' is usually said more loudly than the surrounding words in the sentence. We'll replace these connectives with formal symbols to which we will give fairly explicit and precise meanings based on our understanding of the English words.

symbol what it will be an abstraction of

^ and

[??] or

[??] it's not the case that

-> if ... then ...

Thus, a complex sentence we might study is 'Ralph is a dog ??dogs bark', corresponding to the earlier example.

Parentheses are a further formal device important for clarity. In ordinary speech we might say, 'If George is a duck then Ralph is a dog and Dusty is a horse'. But it's not clear which of the following is meant:

If George is a duck, then: Ralph is a dog and Dusty is a horse.

If George is a duck, then Ralph is a dog; and Dusty is a horse.

Such ambiguity should have no place in our reasoning. We can require the use of parentheses to enforce one of those two readings:

George is a duck -> (Ralph is a dog ^ Dusty is a horse) (George is a duck -> Ralph is a dog) ^ Dusty is a horse

To lessen the proliferation of quotation marks in naming linguistic items, we can assume that a formal symbol names itself when confusion seems unlikely. Thus, we can say that ^ is a formal connective. Here is some more terminology.

The sentence formed by joining two sentences by ^ is a conjunction of them; each of the original propositions is a conjunct we conjoin with the other. When [??] joins two sentences it is a disjunction formed by disjoining the disjuncts. The proposition 'Ralph is a dog or Ralph does not bark' is not a disjunction, because the formal symbol '[??]' does not appear in it.

The sentence formed by putting [??] in front of another sentence is the negation of it; it is a negation. For example, the negation of 'Ralph is a dog' is '[??](Ralph is a dog)', where the parentheses are used for clarity.

The symbol -> is called the arrow or the conditional. The result of joining two sentences with -> is a conditional; the proposition on the left is called the antecedent, and the one on the right is the consequent.

Exercises for Section C

1. Classify each of the following as a conjunction, disjunction, negation, conditional, or none of those. If a conditional, identify the antecedent and consequent.

a. Ralph is a dog ^ dogs bark.

b. Ralph is a dog -> dogs bark.

c. [??] (cats bark).

d. Cats bark [??] dogs bark.

e. Cats are mammals and dogs are mammals.

f. Either Ralph is a dog or Ralph isn't a dog.

g. [??] cats bark -> ([??] (cats are dogs)).

h. Cats aren't nice.

i. Dogs bark [??] ([??] dogs bark).

j. Ralph is a dog.

k. It is possible that Ralph is a dog.

l. Some dogs are not white.

2. a. Write a sentence that is a negation of a conditional, the antecedent of which is a conjunction. b. Write a sentence that is a conjunction of disjunctions, each of whose disjuncts is either a negation or has no formal symbols in it.

3. Write a sentence that might occur in daily speech that is ambiguous but which can be made precise by the use of parentheses, indicating at least two ways to parse it.

4. List at least three words or phrases in English not discussed in the text that are used to form a proposition from one or more propositions and that you believe are important in a study of reasoning.

D. A Formal Language for Propositional Logic

1. Defining the formal language

We have no dictionary, no list of all propositions in English, nor do we have a method for generating all propositions, for English is not a fixed, formal, static language. But by using variables we can introduce a rigid formal language to make precise the syntax of the propositions we will study.

(Continues...)

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Edition details

• Publisher: Princeton University Press
• Publication date: 2006
• Language: English
• ISBN 10: 0691123004
• ISBN 13: 9780691123004
• Binding: Hardcover
• Number of pages: 544