Chapter_03 - THE LOGIC BOOK, 4TH EDITION: CHAPTER THREE...

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THE LOGIC BOOK, 4 TH EDITION: CHAPTER THREE Recall the characteristic truth-tables for the connectives we have learned: A B A B C D C D T T T T T T T T T T T F T F F T F T T F F T F F T F T F T T F F F F F F F F F F X ~ X T F T F T F A B A B C D C D T T T T T T T T T T T F T F F T F T F F F T F T T F T F F T F F F T F F F F T F 1
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A sentence’s truth-table tells you the truth value of the sentence for every possible truth-value assignment (TVA). How do I set up a truth-table? 1. Draw a horizontal line. 2.  List the atomic sentences in alphabetical order, then the  compound sentence (or sentences). 3. Figure out how many rows the truth table will have. Each row represents one TVA. Every atomic sentence has two possible TVA’s. Thus, the number of possible TVA’s is 2 n , where n is the number of atomic sentences. 4. Draw a vertical line to the right of the atomic sentences, going down that number of rows (2 n ). 5. Assign truth-values for the atomic sentences Atomic sentence #1 will assigned ‘T’ for the 1 st half of the TVA’s and ‘F’ for the 2 nd half. Atomic sentence #2 will be assigned ‘T’ for the 1 st and 3 rd quarters of the TVA’s and ‘F’ for the 2 nd and 4 th quarters. Atomic sentence #3 will be assigned ‘T’ for the 1 st , 3 rd , 5 th and 7 th eighth of the TVA’s and ‘F’ for the 2 nd , 4 th , 6 th and 8 th eighths of the TVA’s. 2
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Let’s make a truth table for the sentence: ~R T Here I have listed the two atomic sentences in alphabetical order and the compound sentence above the horizontal line. R T ~ R T Since there are 2 atomic sentences, there will be 2 2 possible TVA’s. That’s 2 × 2 = 4 rows. I draw a vertical line going down 4 rows and start assigning truth-values. R T ~ R T T T T F F T F F Now I have to figure out what the compound sentence’s truth-value is for every row, or for each possible TVA. 3
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THE COMPOUND SENTENCE: 1. Assign truth-values to each occurrence of an atomic sentence on the right side of the vertical line in accordance to the TVA’s on the left side of the vertical line. 2. Using the truth-values assigned to the atomic sentences and the truth-tables for the five connectives, determine truth-values for the compound sentence, working from the atomic sentences up to bigger and bigger molecular sentences. Put a ‘T’ or ‘F’ in a column directly below the connective for every row of the truth- table. 3. The last column you complete should be below the main connective of the compound sentence. Mark that connective with an arrow . The truth-value under the main connective is the truth- value of the compound sentence for each truth- value assignment. 4
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Chapter_03 - THE LOGIC BOOK, 4TH EDITION: CHAPTER THREE...

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