Today’s Topics

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 F is a substitution instance of the statement form Y if, but only if, F can be obtained by replacing each sentential variable in Y 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 constants.

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 Handouton Symbolization Exercises and work the problems.

26. Key Ideas • Logical Syntax • WFF’s • Substitution Instance • Dominant Operator