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THE LOGIC BOOK, 4 TH EDITION: CHAPTER TWO Sentential Logic (SL): Symbolization and Syntax. Sentential Connectives : terms used to generate further sentences from existing English sentences. Examples: “and”, “or”, “unless”, “after”, “before”, “it is not the case that” Truth-Functional Usage of Sentential Connectives: a sentential connective is used truth-functionally if and only if it is used to generate a compound sentence from one or more sentences in such a way that the truth-value of the generated compound is wholly determined by the truth-values of those one or more sentences from which the compound is generated, no matter what those truth-values may be . Ex: Jim is in Toronto and Phil is in Ottawa. Jim is in Toronto or Phil is in Ottawa. Sentence Abbreviation : In SL, we will abbreviate entire sentences using capital Roman letters. Jim is in Toronto = X Phil is in Ottawa = Y We can also use subscripts: Nathan is in Tucson = X 1 Catherine is in Oxford= X 2 - Sentences represented by single capital letters will be called atomic sentences . - Sentences made up of one more atomic sentences and one or more sentential connectives will be called molecular sentences . We are now ready to learn the 5 basic sentential connectives of SL: conjunction, disjunction, negation, material conditional, and material biconditional. Conjunction (… and …): Consider the following sentences: Plato is wise. Socrates is ugly. We will abbreviate them as follows: W U We can make a conjunction of these: Both Plato is wise and Socrates is ugly. We will abbreviate this as follows: W & U (Remember that letters abbreviate entire sentences , not just the words “wise” and “ugly”!) We now have a truth-functional , molecular sentence: W & U 1
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We will call this new sentence a conjunction , and each of its elements conjuncts . Note that our new sentence is true if both of its component sentences are true and is false otherwise. This is summarized in the following truth-table : W U W & U T T T T F F F T F F F F Sometimes an English sentence that is not itself a compound sentence can be paraphrased as a compound sentence: (1) Fred and Nancy passed their driving examinations. (2) Both Fred passed his driving examination and Nancy passed her driving Examination. (3) F & N Warning : Not every English use of ‘and’ indicates conjunction. Two shots of vodka and 6 ounces of orange juice make a drink called a screwdriver. But this cannot be sensibly paraphrased as Both two shots of vodka make a drink called a screwdriver and 6 ounces of orange juice make a drink called a screwdriver. So, the original sentence should be paraphrased in SL with a single Roman letter. Other words besides ‘and’ can also indicate conjunction: ‘but’, ‘however’, ‘although’, ‘nonetheless’, ‘moreover’.
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