Lesson2

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Ureddit Symbolic Logic presentation for week 2.

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Lesson2

  1. 1. Today’s Topics <ul><li>Logical Syntax </li></ul><ul><ul><li>Well-Formed Formulas </li></ul></ul><ul><ul><li>Dominant Operator (Main Connective) </li></ul></ul><ul><li>Putting words into symbols </li></ul>
  2. 2. Logical Syntax <ul><li>Language Operates at 3 Levels </li></ul><ul><ul><li>SYNTAX </li></ul></ul><ul><ul><li>SEMANTICS </li></ul></ul><ul><ul><li>PRAGMATICS </li></ul></ul>
  3. 3. Syntax <ul><li>Rules which govern the possibility of meaningful expressions. </li></ul><ul><li>Syntactically correct strings of symbols are called Well-Formed Formulas (WFF’S, pronounced “woofs”) </li></ul>
  4. 4. <ul><li>'Statement Letter'--capital letter </li></ul><ul><li>'Connective'-- tilde, dot, wedge, arrow, double arrow </li></ul><ul><li>'Grouper'-- parenthesis, bracket, brace </li></ul><ul><li>'Symbol' -- a statement letter, connective, or grouper </li></ul><ul><li>'Formula'-- any horizontal string of symbols </li></ul><ul><li>'Left-hand grouper' -- a '(', '[', or '{' </li></ul><ul><li>'Matching right-hand grouper’-- the mirror image of a left-hand grouper </li></ul><ul><li>'Binary Connective' -- any connective other than a tilde </li></ul>
  5. 5. A WFF is either: <ul><li>(a) a statement letter </li></ul><ul><li>(b) a tilde followed by a WFF, </li></ul><ul><li>(c) a left-hand grouper followed by a WFF followed by a binary connective followed by a WFF followed by a matching right-hand group </li></ul><ul><li>Note: Every compound WFF (those not covered under (a)) is a substitution instance of a statement form . </li></ul>
  6. 6. Substitution Instance <ul><li>A compound WFF  is a substitution instance of the statement form  if, but only if,  can be obtained by replacing each sentential variable in  with a WFF, using the same WFF for the same sentential variable throughout. </li></ul>
  7. 7. Identifying WFF’s <ul><li>Download the Handout on Well-Formed Formulas and discuss the examples with your classmates via the bulletin board. </li></ul><ul><li>Go to http://www.poweroflogic.com and go to chapter 7 and try your hand at determining whether or not a formula is a WFF. </li></ul>
  8. 8. Grouping and Statement Forms <ul><li>Grouping determines the statement form of a compound statement </li></ul><ul><li>Different groupings produce statements with different meanings </li></ul>
  9. 9. 5 Logical Operators (Connectives) <ul><li>Name English Symbol </li></ul><ul><li>Negation not tilde (~) </li></ul><ul><li>Conjunction and dot (  ) </li></ul><ul><li>Disjunction or wedge ( ▼ ) </li></ul><ul><li>Conditional if, then arrow (  ) </li></ul><ul><li>Biconditional if & only if double arrow (  ) </li></ul>
  10. 10. Our 5 logical operators produce statement forms that are truth-functional <ul><li>Negation ~p </li></ul><ul><li>Conjunction p  q </li></ul><ul><li>Disjunction p ▼ q </li></ul><ul><li>Conditional p  q </li></ul><ul><li>Biconditional p  q </li></ul>
  11. 11. In statement forms , the lower case letters are sentential variables, that is, they stand for a complete statements but are not themselves statements <ul><li>The logical operators in a statement form are constant s . </li></ul>
  12. 12. Conjunction <ul><li>A conjunction is composed of two component statements called conjuncts </li></ul><ul><ul><li>The component statements may be either simple or compound </li></ul></ul><ul><li>A conjunction is true only when both of the conjuncts are true </li></ul><ul><li>Conjunction is commutative and associative </li></ul>
  13. 13. Disjunction <ul><li>A disjunction is composed of two component statement called disjuncts </li></ul><ul><li>A disjunction is true whenever either or both of the disjuncts is true </li></ul><ul><li>Disjunction is commutative and associative </li></ul>
  14. 14. Negation <ul><li>A negation is composed of a tilde and a constituent element , which may be either a simple statement or a compound statement. To negate a simple statement, put a tilde in front of it. To negate a compound statement, encase it in parentheses and put a tilde outside the parentheses. </li></ul>
  15. 15. Negation <ul><li>A negation is composed of a tilde and a constituent element </li></ul><ul><li>A negation is true when the constituent element is false </li></ul><ul><li>Remember: Negation is a logical operation. ALWAYS represent negation with a tilde </li></ul>
  16. 16. Conditional <ul><li>A conditional is composed of two elements, the antecedent (the ‘if’ part of an if, then, statement) and the consequent (the ‘then’ part) </li></ul><ul><li>A conditional is true if either the antecedent is false or the consequent true </li></ul>
  17. 17. Biconditional <ul><li>A biconditional is composed of two elements </li></ul><ul><li>A biconditional is true when the elements agree in truth value (both true or both false) </li></ul>
  18. 18. The connective which determines the statement form of a compound statement is called the dominant operator (or main connective )
  19. 19. Dominant Operators (Main Connectives) <ul><li>The connective which determines the statement form of a compound statement is called the dominant operator (or main connective ) </li></ul><ul><li>The dominant operator is the connective with the greatest scope (the fewest groupers around it) </li></ul>
  20. 20. Identifying Main Connectives <ul><li>Download the handout on Main Connectives and try the exercises. </li></ul>
  21. 21. Putting Words Into Symbols <ul><li>Statements are either simple (represented by a statement letter ) or compound . </li></ul><ul><li>A compound statement is any statement containing at least one connective </li></ul><ul><li>In our language a Capital letter stands for an entire simple statement. A dictionary is used to indicate which letters stand for which statements. </li></ul>
  22. 22. When Symbolizing an English Sentence, Identify the Dominant Operator First, and Group AWAY from it. <ul><li>Paraphrasing Inward </li></ul><ul><li>Identify the statement forms of the component sentence(s) and repeat </li></ul>
  23. 23. How paraphrasing inward works: <ul><li>If Jones wins the nomination or Dexter leaves the party, then Williams is the sure winner. (J, D, W where J = Jones wins the nomination, D = Dexter leaves the party, W=Williams wins). </li></ul><ul><li>The sentence is a conditional, so begin by identifying the antecedent and consequent of it. </li></ul><ul><li>Underline the antecedent and italicize the consequent. </li></ul>
  24. 24. <ul><li>You get: </li></ul><ul><li>If Jones wins the nomination or Dexter leaves the party , then Williams is the sure winner. </li></ul><ul><li>Now, begin symbolizing: (Jones wins the nomination or Dexter leaves the party)  Williams is the sure winner </li></ul><ul><li>The antecedent is a disjunction, so show that </li></ul><ul><li>(Jones wins the nomination ▼ Dexter leaves the party)  Williams is the sure winner </li></ul><ul><li>Finally, replace statements with statement letters </li></ul><ul><li>(J ▼ D)  W and you are done! </li></ul>
  25. 25. Practice some on your own <ul><li>Download the Handout on Symbolization Exercises and work the problems. </li></ul>
  26. 26. Key Ideas <ul><li>Logical Syntax </li></ul><ul><li>WFF’s </li></ul><ul><li>Substitution Instance </li></ul><ul><li>Dominant Operator </li></ul>

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