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Introduction to Propositional Logic Propositional logic: – Provides precise definitions for sentences containing and, or, if, and only if. – Captures much more of ordinary language than is possible with categorical logic. – Its basic elements are statements rather than classes. – Unlike A, E, I, O, contains an unlimited number of complex statements. Logical Operators and Translations We learned that the validity of a deductive argument is purely a function of its form. Therefore, evaluating an argument begins by isolating its form. Unfortunately, ordinary language often obscures the argument’s form. Thus, logic offers techniques for simplifying the form. In chapter 5 and 6, we reduced syllogisms to standard form, using letters to represent the terms in the argument. In propositional logic, the fundamental elements are no longer terms, but whole statements. Statements are represented by letters, and these letters are combined by means of logical operators, or connectives. To understand the symbolic representation used in propositional logic, we must distinguish simple statements from compound statements. A simple statement is a statement that does not contain any other statement as a component. Simple Statements: Harrisburg is the capital of Pennsylvania. (H) Wednesday is hump day. (W) Grilled hamburgers taste delicious. (G) Detective novels make great movies. (D) Compound statements: It is not the case that drinking hot coffee reduces sweating. (It is not the case that D.) Hamlet is a tragedy and Kung Fu Panda is a comedy. (H and K) Either we reduce carbon emissions or global warming will get worse. (Either C or G) If the IRS processed my return, then I should get my refund this week. (If I, then R) You will graduate if and only if you meet all university requirements. (G if and only if U) Connective Name Logical Function Translates ~ v Tilde Dot Wedge Horseshoe Triple bar negation conjunction disjunction conditional equivalence The previous examples, then, translate as: It is not the case that D H and K Either C or G If I then R G if and only if U ~D HK CvG IR GU not, it is not the case that and, also, moreover or, unless if…then…, only if if and only if The Negation: All of the following statements are negations. Barak Obama is not a Republican. ~B It is not the case that home foreclosures have peaked. ~H It is false that gold is selling at $1000 per ounce. ~G All of these statements are negations: ~B ~ (G H) ~ [(A F) v (C G)] Note that the ~ is the main operator in each of the above statements. The main operator is the one that governs the largest number of components in the statement. The Conjunction: The dot symbol is used to translate the following conjunctions: and, but, still, moreover, while, however, also, etc. All of the following statements are conjunctions: Ford builds trucks, and Chrysler builds minivans. Ford builds trucks, but Chrysler builds minivans. Ford builds trucks; however, Chrysler builds minivans. Frank and Ernest teach music. All of these statements are conjunctions: K ~L (E v K) (G v H) [(R T) v (S T)] [(W X) v (Y Z)] FC FC FC FE The Disjunction: The wedge is used to translate: or, unless, otherwise, and “either…or.” All of the following statements are disjunctions: You can have steak or chicken. She is either a Pisces or a Scorpio. Boeing assembles planes unless Lockheed constructs rockets. Unless Boeing assembles planes, Lockheed constructs rockets. SvC PvS BvL BvL All of these statements are disjunctions: ~C v ~D (F H) v (~K ~L) [(R T) (S T)] v [(W X) (Y Z)] The Conditional: All of the following statements are conditionals. If Guinness brews beer, then Jameson distills whiskey. GJ Guinness brews beer if Jameson distills whiskey. JG Guinness brews beer only if Jameson distills whiskey. GJ Schweppes makes tonic provided that Beefeater makes gin. B S Schweppes makes tonic on the condition that Beefeater makes gin BS Schweppes's making tonic implies that Beefeater makes gin. S B All of these statements are conditionals: H ~J (A v C) (D E) [K v (S ~T)] [~F v (M O)] More on the conditional: see page 292. All of the following English expressions are symbolized as p q q if p if p then q p only if q q so long as p q provided that p q assuming that p q on the condition that p Do not confuse the antecedent and the consequent. “If” precedes the antecedent, while “only if” precedes the consequent. The antecedent represents the sufficient condition while the consequent represents the necessary condition. Note these equivalences: p q = ~q ~p p q = ~p v q The Biconditional: All of the following statements are biconditionals You get ice cream if and only if you eat your spinach. SI If you eat your spinach, then you get ice cream, and you get ice cream only if you eat your spinach. (S I) (I S) Kodak's introducing a new film is a sufficient and necessary condition for Fuji's doing so. KF All of these statements are biconditionals: M ~T (B v D) (A C) [K v (F C)] [~L (G v H)] Note that A B is equivalent to (A B) (B A), and also to (A B) v (~A ~B) Section 7.B: Complex Statements Well-formed formulas (WFFs): Compound statement forms that are grammatically correct. Rule 1: The dot, wedge, horseshoe, and triple bar symbols must go between two statements (either simple or compound). WFFs: (P v Q) ~R (S · P) v (Q · S) Not WFFs: ·P P PQv PQ≡ Rule 2: The tilde (~) goes in front of the statement it is meant to negate. WFFs: ~ (P v Q) ~R Not WFFs: P~ (S · P) v ~(~Q · S) (PvQ)~ ~(S·P)~ Rule 3: The tilde (~) cannot, by itself, go between two statements. WFF: P v ~Q Not a WFF: P~Q Rule 4: Parentheses, brackets, and braces are required in order to eliminate ambiguity in a complex statement. WFFs: P v (Q · R) Not a WFF: PvQ(·R) ~[(P v Q) · (~ R S )] The Main Operator: Three important factors: – Has as its range the largest component or components in a compound statement. – Either one of the four operators that go between statements or else the negation operator. – There can be only one main operator in a compound statement. Compare: Main operator is (~): ~[(P v ~Q) · (R S)] Main operator is (· ): Main operator is (v): Main operator is (): (P v ~Q) · (R S) P v [~Q v (R S)] [(P v ~Q) · R ] S Translations and the Main Operator: Locating the main operator helps to translate sentences and place parentheses accurately: Either Tracy or Becky owns a DVD player, but Sophie owns one for sure. (TvB)· S Not both Suzuki and Honda are Japanese-owned companies. ~(S ·H) Which is equivalent to: ~S v ~H