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Today’s Topics

Today’s Topics. Dominant Operators and Grouping Truth Functions and Truth Tables. The operator which determines the statement form of a compound statement is called the dominant operator (or main connective ). Grouping and Statement Forms.

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Today’s Topics

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  1. Today’s Topics • Dominant Operators and Grouping • Truth Functions and Truth Tables

  2. The operator which determines the statement form of a compound statement is called the dominant operator (or main connective)

  3. Grouping and Statement Forms • Grouping determines the statement form of a compound statement • Different groupings produce statements with different meanings

  4. Grouping determines the dominant operator in a statement • The binary operator with the fewest groupers around it is the dominant operator UNLESS there is a tilde with even fewer groupers in which case it is dominant. Tilde is the weakest operator. • ~P  Q  R is not a WFF • ~P  (Q  R) is a conditional • ((~P  Q)  R) is a disjunction • ~((P  Q)  R) is the negation of a disjunction

  5. P C Q  R  ~S • (P C Q)  (R  ~S) • P C [(Q  R)  ~S ] • [P C (Q  R)]  ~S

  6. P  Q  R C S  ~ T • P  [(Q  R) C (S  ~ T)] • [(P  Q)  (R C S)]  ~ T • [P  (Q  R)] C (S  ~ T) • {[P  (Q  R)] C S}  ~ T • [(P  Q)  R] C [S  ~ T] • P  {Q  [(R C S)  ~ T]}

  7. ~ P  Q  ~ R C S ? T • ~ { [P  Q]  [~ R C (S ? T)]} • ~ P  { [Q  ~ (R C S)] ? T} • ~ P  [( Q  ~ R) C (S ? T)] • ~ [ (P  Q)  ~ R] C [S ? T]

  8. When Symbolizing an English Sentence, Identify the Dominant Operator First, and Group AWAY from it. • Paraphrasing Inward • Identify the statement forms of the component statements (remembering that the component statements may be simple statements) and repeat

  9. Paraphrasing Inward • If either all Republicans are heartless or all Democrats are brainless, then the Republic is doomed unless a new prophet arises. • (either all Republicans are heartless or all Democrats are brainless)  (the Republic is doomed unless a new prophet arises) • (either all Republicans are heartless  all Democrats are brainless)  (the Republic is doomed  a new prophet arises) • (H  B)  (D  P)

  10. Statement forms are truth-functional

  11. Truth-Functions • A truth function takes one or more truth values as input and returns one truth value as the output • Truth functional operators determine the truth value of a compound statement given the truth values of the simple statements in it

  12. Truth Tables • A truth table is a complete list of all the possible combinations of truth and falsity for a set of simple statements, showing the effect of each combination on the truth value of a compound having those simple statements as components.

  13. Truth Tables • Each combination of truth values constitutes one row of a truth table. • The number of rows in a truth table is 2n where n equals the number of simple statements. • The truth value of any compound statement is determined by the truth values of its component sentences

  14. Key Ideas • Grouping and Meaning • Paraphrasing Inward • Truth Functional Operators • Truth Tables

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