Introduction to Logic

Patrick Suppes

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486138053
• Topic: Mathematics
• Subtopic: Logic in Mathematics
• Index: Mathematics

Part I of this coherent, well-organized text deals with formal principles of inference and definition. Part II explores elementary intuitive set theory, with separate chapters on sets, relations, and functions. Ideal for undergraduates.

PART I

PRINCIPLES OF INFERENCE AND DEFINITION

CHAPTER 1

THE SENTENTIAL CONNECTIVES

To begin with, we want to develop a vocabulary which is precise and at the same time adequate for analysis of the problems and concepts of systematic knowledge. We must use vague language to create a precise language. This is not as silly as it seems. The rules of chess, for example, are a good deal more precise than those of English grammar, and yet we use English sentences governed by imprecise rules to state the precise rules of chess. In point of fact, our first step will be rather similar to drawing up the rules of a game. We want to lay down careful rules of usage for certain key words: ‘not’, ‘and’, ‘or’, ‘if ..., then...’, ‘if and only if’, which are called sentential connectives. The rules of usage will not, however, represent the rules of an arbitrary game. They are designed to make explicit the predominant systematic usage of these words; this systematic usage has itself arisen from reflection on the ways in which these words are used in ordinary, everyday contexts. Yet we shall not hesitate to deviate from ordinary usage whenever there are persuasive reasons for so doing.

§ 1.1 Negation and Conjunction. We deny the truth of a sentence by asserting its negation. For example, if we think that the sentence ‘Sugar causes tooth decay’ is false, we assert the sentence ‘Sugar does not cause tooth decay’. The usual method of asserting the negation of a simple sentence is illustrated in this example: we attach the word ‘not’ to the main verb of the sentence. However, the assertion of the negation of a compound sentence is more complicated. For example, we deny the sentence ‘Sugar causes tooth decay and whiskey causes ulcers’ by asserting ‘It is not the case that both sugar causes tooth decay and whiskey causes ulcers’. In spite of the apparent divergence between these two examples, it is convenient to adopt in logic a single sign for forming the negation of a sentence. We shall use the prefix ‘–’, which is placed before the whole sentence. Thus the negation of the first example is written:

-(Sugar causes tooth decay).

The second example illustrates how we may always translate ‘–’; we may always use ‘it is not the case that’.

The main reason for adopting the single sign ‘–’ for negation, regardless of whether the sentence being negated is simple or compound, is that the meaning of the sign is the same in both cases. The negation of a true sentence is false, and the negation of a false sentence is true.

We use the word ‘and’ to conjoin two sentences to make a single sentence which we call the conjunction of the two sentences. For example, the sentence ‘Mary loves John and John loves Mary’ is the conjunction of the sentence ‘Mary loves John’ and the sentence ‘John loves Mary’. We shall use the ampersand sign ‘&’ for conjunction. Thus the conjunction of any two sentences P and Q is written

P & Q.

The rule governing the use of the sigh i & is in close accord with ordinary usage. The conjunction of two sentences is true if and only if both sentences are true. We remark that in logic we may combine any two sentences to form a conjunction. There is no requirement that the two sentences be related in content or subject matter. Any combinations, however absurd, are permitted. Of course, we are usually not interested in sentences like ‘John loves Mary, and 4 is divisible buy 2’. Although it might seem desirable to have an additional rule stating that we may only conjoin two sentences which have a common subject matter, the undesirability of such a rule becomes apparent once we reflect on the vagueness of the notion of common subject matter.

Various words are used as approximate synonyms for ‘not’ and ‘and’ in ordinary language. For example, the word ‘never’ in the sentence:

I will never surrender to your demands

has almost the same meaning as ‘not’ in:

I will not surrender to your demands.

Yet it is true that ‘never’ carries a sense of continuing refusal which ‘not’ does not.

The word ‘but’ has about the sense of ‘and’, and we symbolize it by ‘&’, although in many cases of ordinary usage there are differences of meaning. For example, if a young woman told a young man:

I love you and I love your brother almost as well,

he would probably react differently than if she had said:

I love you but I love your brother almost as well.

In view of such differences in meaning, a natural suggestion is that different symbols be introduced for sentential connectives like ‘never’ and ‘but’. There is, however, a profound argument against such a course of action. The rules of usage agreed upon for negation and conjunction make these two sentential connectives truth-functional; that is, the truth or falsit...