Über die Autorin bzw. den Autor

Richard L. Epstein With contributions by Leslaw W. Szczerba

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Classical Mathematical Logic

Princeton University Press

Contents

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

Chapter One

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...