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150 - handout 14 - 9 march 2010 - expanding truth-tables, again

150 - handout 14 - 9 march 2010 - expanding truth-tables, again

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P HILOSOPHY 150: I NTRODUCTION TO L OGIC H ANDOUT 14: 9 M ARCH 2010 Alison Duncan Kerr E XPANDING O UR U SE OF T RUTH -T ABLES , A GAIN . I. B ACK TO THE B ASICS . A. S IMPLE vs. C OMPOUND Sentences. A sentence is COMPOUND if it logically contains another complete sentence as a component. A sentence is SIMPLE if and only if it is not compound. E XERCISE (i) : State whether the following sentences are simple or compound, and for those that are compound, state all the simple components. E.g., Either Mike or John will have to clean up the kitchen after dinner. Answer: Compound Mike will have to clean up the kitchen after dinner. John will have to clean up the kitchen after dinner. 1. Mary hit a home run and a triple. 2. John enjoys a baseball game if he can have some popcorn. 3. Dogs with fleas make poor house pets. 4. Dogs don’t like bumblebees. 5. Mary told John that he was gaining weight. 6. Neither John nor Mary likes gooseberries. B. Determining the M AJOR O PERATOR . The MAJOR OPERATOR of a formula is one that determines the overall form of the sentence and is the operator introduced last in the process of constructing the formula from its more elementary components. Thus, the major operator of (((A • B) ( C D)) ((E F) • ~G)) is the second wedge, and so the formula is a disjunction , even though it contains all the other operators. The major operator of ((A • B) (C D)) is the arrow, which makes the formula a conditional even though a conjunction and a disjunction occur as subformulas. When we ask “What kind of sentence is this?” we are referring to the major operator of the formula. E.g., (A B): is a biconditional because the major operator is the ‘ ’. ~(A B): is a negation because the major operator is the ‘~’. ((A B) • (C D)): is a conjunction because the major operator is the ‘•’. With simple formulas, one can determine the major operator at a glance, but with more complicated formulas it is a bit more difficult. One way to determine the major operator is to pair parentheses, starting with the smallest subformulas and working your way up to the largest. The last operator you cover in this process will be the major operator. E.g., ( ( ( ( A • B ) ( C D ) ) ( ( E F ) • ~ G ) ) ( ( A B ) • ~ ( C D ) ) ) E XERCISE (ii) : Identify the major operator in the following formulas. (You might have to specify “the second dot”, “the third arrow”, and so on, in very complex formulas. 1. ((A B) • C) 2. ((A • B) ~ (B • D)) 3. (~A ~ (C D)) 4. (((A ~B ) (C D)) ~E) 5. ((((A B) • C) D) • (E (F • D))) 6. (A (B (C (D (E (F (G H))))))) 7. (((((((A B) C) D) E) F) G) H) 8. ((((A B) • (C D)) ((A • B) (C • D))) (((G H) (P Q)) • R))
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P HILOSOPHY 150: I NTRODUCTION TO L OGIC H ANDOUT 14: 9 M ARCH 2010 Alison Duncan Kerr 2 II. T ESTING THE V ALIDITY OF AN A RGUMENT F ORM —N ON -F ULL T RUTH -T ABLE M ETHOD .
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