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Today’s Topics Logical Syntax Logical Syntax o Well-Formed Formulas o Dominant Operator (Main Connective) Putting words into symbols Putting words into symbols

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Logical Syntax Language Operates at 3 Levels Language Operates at 3 Levels o SYNTAX o SEMANTICS o PRAGMATICS

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Syntax Rules which govern the possibility of meaningful expressions. Rules which govern the possibility of meaningful expressions. Syntactically correct strings of symbols are called Well-Formed Formulas (WFF’S, pronounced “woofs”) Syntactically correct strings of symbols are called Well-Formed Formulas (WFF’S, pronounced “woofs”)

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'Statement Letter'--capital letter 'Statement Letter'--capital letter 'Connective'-- tilde, dot, wedge, arrow, double arrow 'Connective'-- tilde, dot, wedge, arrow, double arrow 'Grouper'-- parenthesis, bracket, brace 'Grouper'-- parenthesis, bracket, brace 'Symbol' -- a statement letter, connective, or grouper 'Symbol' -- a statement letter, connective, or grouper 'Formula'-- any horizontal string of symbols 'Formula'-- any horizontal string of symbols 'Left-hand grouper' -- a '(', '[', or '{' 'Left-hand grouper' -- a '(', '[', or '{' 'Matching right-hand grouper’-- the mirror image of a left-hand grouper 'Matching right-hand grouper’-- the mirror image of a left-hand grouper 'Binary Connective' -- any connective other than a tilde 'Binary Connective' -- any connective other than a tilde

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A WFF is either: (a) a statement letter (a) a statement letter (b) a tilde followed by a WFF, (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 (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. Note:Every compound WFF (those not covered under (a)) is a substitution instance of a statement form.

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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. 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.

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Identifying WFF’s Download the Handout on Well-Formed Formulas and discuss the examples with your classmates via the bulletin board. Download the Handout on Well-Formed Formulas and discuss the examples with your classmates via the bulletin board.Handout 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. 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.http://www.poweroflogic.com

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

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5 Logical Operators (Connectives) NameEnglish Symbol Negationnot tilde (~) Conjunctionand dot ( ) Disjunctionor wedge ( ▼ ) Conditionalif, then arrow ( ) Biconditionalif & only if double arrow ( )

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Our 5 logical operators produce statement forms that are truth-functional Negation~p Negation~p Conjunctionp q Conjunctionp q Disjunctionp ▼ q Disjunctionp ▼ q Conditionalp q Conditionalp q Biconditionalp q Biconditionalp q

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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. The logical operators in a statement form are constants.

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Conjunction A conjunction is composed of two component statements called conjuncts A conjunction is composed of two component statements called conjuncts o The component statements may be either simple or compound A conjunction is true only when both of the conjuncts are true A conjunction is true only when both of the conjuncts are true Conjunction is commutative and associative Conjunction is commutative and associative

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Disjunction A disjunction is composed of two component statement called disjuncts A disjunction is composed of two component statement called disjuncts A disjunction is true whenever either or both of the disjuncts is true A disjunction is true whenever either or both of the disjuncts is true Disjunction is commutative and associative Disjunction is commutative and associative

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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. 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.

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Negation A negation is composed of a tilde and a constituent element A negation is composed of a tilde and a constituent element A negation is true when the constituent element is false A negation is true when the constituent element is false Remember: Negation is a logical operation. ALWAYS represent negation with a tilde Remember: Negation is a logical operation. ALWAYS represent negation with a tilde

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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 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 A conditional is true if either the antecedent is false or the consequent true

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Biconditional A biconditional is composed of two elements A biconditional is composed of two elements A biconditional is true when the elements agree in truth value (both true or both false) A biconditional is true when the elements agree in truth value (both true or both false)

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

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Dominant Operators (Main Connectives) The connective which determines the statement form of a compound statement is called the dominant operator (or main connective) 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) The dominant operator is the connective with the greatest scope (the fewest groupers around it)

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

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Putting Words Into Symbols Statements are either simple (represented by a statement letter) or compound. Statements are either simple (represented by a statement letter) or compound. A compound statement is any statement containing at least one connective 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. In our language a Capital letter stands for an entire simple statement. A dictionary is used to indicate which letters stand for which statements.

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When Symbolizing an English Sentence, Identify the Dominant Operator First, and Group AWAY from it. Paraphrasing Inward Paraphrasing Inward Identify the statement forms of the component sentence(s) and repeat Identify the statement forms of the component sentence(s) and repeat

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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). 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. The sentence is a conditional, so begin by identifying the antecedent and consequent of it. Underline the antecedent and italicize the consequent. Underline the antecedent and italicize the consequent.

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You get: You get: If Jones wins the nomination or Dexter leaves the party, then Williams is the sure winner. 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 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 The antecedent is a disjunction, so show that (Jones wins the nomination ▼ Dexter leaves the party) Williams is the sure winner (Jones wins the nomination ▼ Dexter leaves the party) Williams is the sure winner Finally, replace statements with statement letters Finally, replace statements with statement letters (J ▼ D) W and you are done! (J ▼ D) W and you are done!

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Practice some on your own Download the Handout on Symbolization Exercises and work the problems. Download the Handout on Symbolization Exercises and work the problems. Handout Handout

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Key Ideas Logical Syntax Logical Syntax WFF’s WFF’s Substitution Instance Substitution Instance Dominant Operator Dominant Operator

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CSNB143 – Discrete Structure Topic 4 – Logic. Learning Outcomes Students should be able to define statement. Students should be able to identify connectives.

CSNB143 – Discrete Structure Topic 4 – Logic. Learning Outcomes Students should be able to define statement. Students should be able to identify connectives.

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