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Chapter 1. Statement Calculus
- Hong Kong University Press, HKU
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CHAPTER 1 STATE 孔1ENT CALCULUS A. Statements By a statement (or a propωit也n or a declarative sentence) we understand a sentcnce of which it is meaningful to say that its content is true or false. Obviously, each of the following sentcnces is a statement: Geography is a science. Confucius was a soldier. Cheung Sam is dead and Lee Sai is in prison. 2 is smaller than 3 and 3 is a prime number. The steering gear was loose or the driver was drunk. If John is here, then the book is not his. Whereas none of the following sentences can be regar ded as a statement in the above sense: The number 3 is stupid. Friends, Romans, countrymen, lend me your ears. Cousin of Exeter, what thinks your lordship? Throughout this chapter, we shall mainly be concerned with statements. Here we shall briefly describe what we propose to do with them. In the statement ca/culus (or propωitiollal calcullls) of this chapter, with the exception of Sections K and L, we shall not concern ourselves with the relation between the subjects and the predicates of thc statements. Instead we shall deal with the statements themselves as entireties, and study the modes ofcompounding them into further statements. Examples might explain this more clearly. Consider the statement (1) Geography is a science. Being a statement, (1) is either true or false. If it is true, then we say that (1) has truth as its truth va/ue; if it is false, we say that (1) h臼 falsehood as its truth value. Now this truth value is a relation between the subject and the predicate of the sentence (1), and opinion concerning the truth of this sentence may be divided. For us thesentence ‘Geography 4 STATEMENT CALCULUS [Chap. 1 is a science' is merely a statement to which either of the truth values can be meaningfully assigned. The statements (2) 2 is smaller than 3 (3) 3 is a prime number are true statements of arithm~tic. By joining (2) and (3) we can form a new statement (4) 2 is smaller than 3 and 3 is a prime number. Qur main concern about (4) is its truth value in relation to the truth values of (2) and (3). As a statement of arithmetic, (4) is true since (2) and (3) are both true statements of arithmetic. This and similar problems will be discussed more thoroughly in the next few sections. B. Conjunctions In ordinary speech, we frequently join two statements by the word and. Let us consider the statement (1) Cheung Sam is dead and Lee Sai is in prison. \Ve say that (1) has as its first ωmponent the statement ‘Cheung Sam is dead' and as its second compollent the statement ‘Lee Sai is in prison', and moreover that (1) is formed byjoining these two components by the connective and. Ordinarily, a statement such as (1) is accepted as true if both of its componen臼 are true; otherwise it is considered false. Corresponding to this mode of composition, which consists in joining two statements by the connective 徊, we have the concept of 仰~iunction in statement calculus. In the notation of mathematical logic, the conjunction of two statements X and Y is denoted by X ^ Y, and read ‘X and Y'. X ^ Y is a statement which is true if both X and Y are true; otherwise it is false. Thus the truth value ofa conjunction X ^Y is uniquely determined by the truth values of its components X and Y. This can be conveniently expressed in a truth table: 1st Component 2nd Component ConXjuAncY tion X Y T T T F T F T F F F F F where ‘T' stands for truth and ‘F' stands for falsehood. [107.22.56.225] Project MUSE (2024-03-29 09:30 GMT) Sec. B, C] DlSjUNCTlONS 5 c. Disjunctions A second and equally familiar mode of composition consists if.l joining two statements, the components, by the connective or. Unfortunately, the meaning of the word or in English is ambiguous; sometimes it is used in the inclusive sense, meaning either one or the other or both; sometimes it is used in the exclusive sense, meaning either one or the other, but not both. Common usage probably favours the inclusive or. For examplc, in a motor-car manual, we may find the following warning: You will damagc the engine if you run it when it is too low on oil...