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CHAPTER 1 STATEMENT CALCULU S A. Statement s By a statement (o r a proposition o r a declarative sentence) we understand a sentence of whic h it is meaningful t o say that its content is true or false . Obviously, eac h o f th e followin g sentence s i s a statement : Geography i s a science . Confucius wa s a soldier . Cheung Sa m i s dead an d Le e Sa i i s i n prison . 2 i s smajler tha n 3 and 3 is a prime number . The steerin g gea r wa s loos e or th e drive r wa s drunk . If Joh n i s here, the n th e boo k i s not his . Whereas non e of the following sentence s can be regarded a s a statemen t in th e abov e sense : The numbe r 3 is stupid . Friends, Romans , countrymen , len d m e you r ears . Cousin o f Exeter , wha t think s you r lordship ? Throughout thi s chapter, we shall mainly be concerned wit h statements . Here we shall briefly describ e wha t w e propose t o d o wit h them . I n th e statement calculus (o r propositional calculus) o f thi s chapter , wit h th e exception o f Section s K an d L , w e shall not concer n ourselve s with th e relation betwee n th e subject s an d th e predicate s o f th e statements . Instead w e shal l dea l wit h th e statement s themselve s a s entireties , an d study the modes of compounding them into further statements . Examples might explai n thi s mor e clearly . Consider th e statemen t (1) Geograph y i s a science. Being a statement , (1 ) i s eithe r tru e o r false . I f i t i s true , the n w e sa y that (1 ) ha s truth a s it s ttuth value ; i f i t i s false , w e sa y tha t (1 ) ha s falsehood a s it s trut h value . Now thi s trut h valu e i s a relatio n betwee n the subject and the predicate of the sentence (1), and opinion concernin g the truth of this sentence may be divided. For us the sentence * Geography 4฀ STATEMENT CALCULU S [Chap. 1 is a science* is merely a statement t o which eithe r o f the trut h value s can be meaningfully assigned . The statement s (2) 2 is smaller than 3 (3) 3 is a prime number are true statement s o f arithmetic. B y joining (2 ) and (3 ) we can for m a new statement (4) 2 is smaller than 3 and 3 is a prime number. Our mai n concer n abou t (4 ) is its truth valu e in relatio n t o the trut h values of (2) and (3). As a statement of arithmetic, (4) is true since (2) and (3 ) ar e bot h tru e statement s o f arithmetic . This and similar problem s will be discussed mor e thoroughly i n the next few sections. B. Conjunction s In ordinar y speech , we frequently joi n tw o statements b y the wor d and. Let u s consider the statemen t (1) Cheun g Sam is dead and Lee Sai is in prison. We say that (1 ) has as its first component the statemen t * Cheung Sa m is dead* and as its second component the statement 'Lee Sai is in prison', and moreover that (1) is formed by joining these two components by the connective and. Ordinarily, a statement such a s (1) is accepted a s tru e if bot h o f it s component s ar e true ; otherwis e i t i s considere d false . Corresponding t o this mode of composition, whic h consist s in joining two statements by the connective and, we have the concept of conjunction in statement calculus. In th e notatio n o f mathematical logic , the conjunction o f two statements X an d Y i s denote d b y Z A 7 , an d rea d 'X an d Y \ XA Y i s a statement whic h i s true i...

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