Lesson2
- 1. Today’s Topics Logical Syntax Well-Formed Formulas Dominant Operator (Main Connective) Putting words into symbols
- 2. Logical Syntax Language Operates at 3 Levels SYNTAX SEMANTICS PRAGMATICS
- 3. Syntax Rules which govern the possibility of meaningful expressions. Syntactically correct strings of symbols are called Well-Formed Formulas (WFF’S, pronounced “woofs”)
- 4. 'Statement Letter'--capital letter 'Connective'-- tilde, dot, wedge, arrow, double arrow 'Grouper'-- parenthesis, bracket, brace 'Symbol' -- a statement letter, connective, or grouper 'Formula'-- any horizontal string of symbols 'Left-hand grouper' -- a '(', '[', or '{' 'Matching right-hand grouper’-- the mirror image of a left-hand grouper 'Binary Connective' -- any connective other than a tilde
- 5. A WFF is either: (a) a statement letter (b) a tilde followed by a WFF, (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 Note: Every compound WFF (those not covered under (a)) is a substitution instance of a statement form .
- 6. Substitution Instance 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.
- 7. Identifying WFF’s Download the Handout on Well-Formed Formulas and discuss the examples with your classmates via the bulletin board. 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.
- 8. Grouping and Statement Forms Grouping determines the statement form of a compound statement Different groupings produce statements with different meanings
- 9. 5 Logical Operators (Connectives) Name English Symbol Negation not tilde (~) Conjunction and dot ( ) Disjunction or wedge ( ▼ ) Conditional if, then arrow ( ) Biconditional if & only if double arrow ( )
- 10. Our 5 logical operators produce statement forms that are truth-functional Negation ~p Conjunction p q Disjunction p ▼ q Conditional p q Biconditional p q
- 11. In statement forms , the lower case letters are sentential variables, that is, they stand for a complete statements but are not themselves statements The logical operators in a statement form are constant s .
- 12. Conjunction A conjunction is composed of two component statements called conjuncts The component statements may be either simple or compound A conjunction is true only when both of the conjuncts are true Conjunction is commutative and associative
- 13. Disjunction A disjunction is composed of two component statement called disjuncts A disjunction is true whenever either or both of the disjuncts is true Disjunction is commutative and associative
- 14. Negation 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.
- 15. Negation A negation is composed of a tilde and a constituent element A negation is true when the constituent element is false Remember: Negation is a logical operation. ALWAYS represent negation with a tilde
- 16. Conditional A conditional is composed of two elements, the antecedent (the ‘if’ part of an if, then, statement) and the consequent (the ‘then’ part) A conditional is true if either the antecedent is false or the consequent true
- 17. Biconditional A biconditional is composed of two elements A biconditional is true when the elements agree in truth value (both true or both false)
- 18. The connective which determines the statement form of a compound statement is called the dominant operator (or main connective )
- 19. Dominant Operators (Main Connectives) The connective which determines the statement form of a compound statement is called the dominant operator (or main connective ) The dominant operator is the connective with the greatest scope (the fewest groupers around it)
- 20. Identifying Main Connectives Download the handout on Main Connectives and try the exercises.
- 21. Putting Words Into Symbols Statements are either simple (represented by a statement letter ) or compound . A compound statement is any statement containing at least one connective In our language a Capital letter stands for an entire simple statement. A dictionary is used to indicate which letters stand for which statements.
- 22. When Symbolizing an English Sentence, Identify the Dominant Operator First, and Group AWAY from it. Paraphrasing Inward Identify the statement forms of the component sentence(s) and repeat
- 23. How paraphrasing inward works: 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). The sentence is a conditional, so begin by identifying the antecedent and consequent of it. Underline the antecedent and italicize the consequent.
- 24. You get: If Jones wins the nomination or Dexter leaves the party , then Williams is the sure winner. Now, begin symbolizing: (Jones wins the nomination or Dexter leaves the party) Williams is the sure winner The antecedent is a disjunction, so show that (Jones wins the nomination ▼ Dexter leaves the party) Williams is the sure winner Finally, replace statements with statement letters (J ▼ D) W and you are done!
- 25. Practice some on your own Download the Handout on Symbolization Exercises and work the problems.
- 26. Key Ideas Logical Syntax WFF’s Substitution Instance Dominant Operator