Today’s Topics Logical Syntax Logical Syntax o Well-Formed Formulas o Dominant Operator (Main...

Today’s Topics

• Logical Syntaxo Well-Formed Formulaso Dominant Operator (Main Connective)

• Putting words into symbols

Logical Syntax

• Language Operates at 3 Levelso SYNTAXo SEMANTICSo PRAGMATICS

Syntax

• Rules which govern the possibility of meaningful expressions.

• Syntactically correct strings of symbols are called Well-Formed Formulas (WFF’S, pronounced “woofs”)

• '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

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.

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.

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.

Grouping and Statement Forms• Grouping determines the statement form of

a compound statement• Different groupings produce statements

with different meanings

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 ()

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

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.

Conjunction

• A conjunction is composed of two component statements called conjunctso 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

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

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.

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

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

Biconditional

• A biconditional is composed of two elements

• A biconditional is true when the elements agree in truth value (both true or both false)

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

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)

Identifying Main Connectives

• Download the handout on Main Connectives and try the exercises.

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.

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

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.

• 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!

Practice some on your own

• Download the Handout on Symbolization Exercises and work the problems.

Key Ideas

• Logical Syntax• WFF’s• Substitution Instance• Dominant Operator