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chp03+-+Introduction+to+Sentential+Calculus

chp03+-+Introduction+to+Sentential+Calculus - From Logic...

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Sentential Calculus Logic is the science of correct, or as we shall say, valid , argument. A valid argument is one in which it is not possible for the premisses to be true and the conclusion false, and, moreover, you can tell just by looking at the structure of the argument that it is not possible for the premises to be true and the conclusion false. So, to do logic, we have to look at the structure of arguments. An argument is made up of sentences and the sentences are made up of words, and we want to see how the words that appear in a correct argument fit together so as to preclude the possibility that the premises are true and the conclusion false. Since the arguments we are interested in are mostly formulated in English, the most natural way to proceed would be to examine the way English sentences are built up out of English words, trying to determine what the structure of an English sentence contributes to fixing the conditions under which the sentence is true. Regrettably, we can’t proceed that way, because we understand altogether too little about what makes an English sentence true. Natural languages are just too complicated. Instead, we proceed indirectly, working not with natural languages but with formal languages that are vastly simpler than English, but that are still rich enough to display a lot of the features we find in English. We can straightforwardly determine which arguments in the formal language are valid. Next, we’ll learn how to translate from English into the formal language. The plan is to do the translations is such a way that, if the translated argument is valid in the formal language, then the English argument must have been valid as well. Thus we can show an English argument valid by providing a translation into a demonstrably valid argument in the formal language. The formal languages are designed to exhibit the logical structure of English arguments in an especially stark form. We’ll begin with sentential calculus ( SC ), which provides a model of the way compound English sentences are built up out of simple English sentences. Later, we’ll deepen the analysis by examining the internal structure of simple English sentences. The sentences of the formal language of sentential calculus are built up out of simple sentence, called atomic sentences, by putting sentences together using what grammarians call "conjunctions" and what logicians call "connectives." Thus the formal language symbol " " corresponds to the English word "and" or "but." If the formal-language sentence "A" means the same as the English sentence "Jack went up the

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