PHIL Exam 2
Main Operator
- Has as its range the largest component and components in a compound statement - There can be only one main operator in a compound statement. Compare: 1. Main operator is (~): ~[(P v ~ Q) · (R⊃S)] 2. Main operator is (·): (P v ~Q) · (R⊃S) 3. Main operator is (v): P v [~ Q v (R⊃S)] 4. Main operator is (⊃): [(P v ~Q) · R]⊃S
5 Logical Operators
1. Negation 2. Conjunction 3. Disjunction 4. Conditional 5. Biconditional
Propositional Logic
1. Provides definitions with: AND, OR, IF, ONLY IF 2. Basic elements are statements (propositions) rather than classes of objects
Biconditional: ≡ (triple bar)
A compound statement consisting of two conditionals - one indicated by "IF"; the other by phrase "ONLY IF" - The triple bar symbol (≡) is used to translate biconditional statements. EX: If you eat your spinach, then you can get your ice cream, and you get ice cream only if you eat your spinach - I ≡ S = You can get your ice cream if and only if you eat your spinach - I ≡ S
Conjunction: (·) dot
A compound statement with 2 distinct statements (conjuncts) connected by the dot symbol (·) (·) translates: AND, IT, BUT, STILL, MOREOVER, WHILE, HOWEVER, ALSO, ALTHOUGH, YET, NEVERTHELESS, WHEREAS. EX: Honesty is the best policy, and lying is for scoundrels. = H · L Honesty is the best policy; moreover, lying is for scoundrels. = H · L Frank and Ernest teach music. = F · E
Disjunction: (v) wedge
A compound statement with 2 distinct statements (disjunct) connected by the wedge symbol (v). (v) translates: OR, UNLESS, OTHERWISE, EITHER... OR EX: Paris is the city of lights or Big Ben is in London = P v B She is either a Pisces or a Scorpio = P v S You have to stay home unless you clean your room = H v R You can't go the party unless you clean your room = ~P v R
Simple Statement
Basic component of propositional logic. We use upper-case letters to symbolize a simple statement. EX: Hamlet is a tragedy. We symbolize as "H"
Complex Transitions
If it's before noon, then I'm drinking coffee Translate: B⊃D It is not summertime Translate: ~S Combine 2 statements into 1 statement: It's not summertime and if it's before noon then I'm drinking coffee = ~S·(B ⊃ B) It is not true that it's not summertime or if it's before noon then I'm drinking coffee: -[~S v (B⊃D)]
Translate the compound sentence
If you feel great, then you look great (conditional) = F⊃L Pizza contains all the best food groups, and only if, you get it with anchovies (bi-conditional) is... P≡A
Choose best translation
It is not true that Titanic is the highest grossing film of all time. Let T = Titanic is the highest grossing film of all time. = ~T
Translate the propositional logic
Joyce has visited Hawaii but Judy has not been there and Eddie has not been there. Answer: J·(~D·-E)
Translations and the Main Operator
Locating the main operator helps to translate sentences and place parentheses accurately: EX: 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) If Julian does not become a lifeguard at the YMA or get the job driving school buses then he will keep working t the bookstore in San Fran and clean houses part time. Answer: - (L v D)⊃(B·C)
Compound Statement
One simple statement as a component EX: Hamlet is a tragedy but The Tempest is a comedy
Negation: (~) tilde
Statements in which the word NOT and the phrase IT IS NOT THE CASE THAT are used to deny the statement that follows them. EX: Today is Monday = M Today is not Monday = ~ M Gold is selling at $1000 an ounce = G It is not the case that gold is selling at $1000 an ounce = ~ G
Logical Operators
Symbols used in translations or ordinary language statements; we use 5 operators.
Conditional: (⊃) horseshoe
The word "if" precedes the antecedent of a conditional statement. The horseshoe symbol (⊃) is used to translate a conditional statement. EX: If you wash the car, then you can go to the movies. = W ⊃ M You can go to the movies, if you wash the car = W ⊃ M - In ordinary language, many words/phrases indicate conditional statements including: 1. Every time P, then Q. 2. Each time P, then Q. 3. Any time P, then Q. 4. Given that P, then Q. 5. Provided that P, then Q. 6. P implies Q. CONDITIONAL STATEMENTS - Distinguish IF from ONLY IF 1. If precedes the antecedent of a conditional statement. 2. Only if precedes the consequent of a conditional statement. EX: If you live in Ohio (antecedent), then you live in the United States (consequent) = O ⊃ U You live in Ohio only if you live in the United States = O ⊃ U You live in Ohio if you live in the United States = O ⊃ U
Biconditional
Truth table for biconditional shows that for any combo of truth values for p,q, "p≡q" will have these truth values: p q and p≡q 1. T,T = T 2. T,F = F 3. F,T = F 4. F,F = T A biconditional is true when both p,q have the same truth value.
Conditional
Truth table for conditional shows that for any combo of truth values for p,q, "p⊃q" will have these truth values: p q and p ⊃ q = 1. T, T = T 2. T,F = F 3. F, T = T 4. F, F = T A conditional will be fake only when the antecedent is true and the consequent is false (line 2). EX: If you drive north on 33, then you will get to Columbus. According to truth table for p ⊃ q, N ⊃ C is false only when the antecedent is true but the consequent is false. In other words, it's false only when you drive north on 33 but you will not get to Columbus. But N ⊃ C is true when the antecedent and the consequent are both false. Weird? Not really because can say N ⊃ C is true when you didn't drive north on 33 (antecedent false) and you didn't get to Columbus (consequent false).
Conjunction
Truth table for conjunction shows that for any combo of truth values for p,q, "p·q" will have these truth values: A conjunction will be true only when each conjunction is true (line 1). p q and p · q = 1. T, T = T 2. T, F = F 3. F, T = F 4. F, F = F EX: Today is Monday (p) and it is raining outside (q)
Disjunction
Truth table for disjunction shows that for any combo of truth values for p,q, "p v q" will have these truth values: p q and p v q = 1. T, T= T 2. T, F = T 3. F,T = T 4. F, F = F A disjunction will be false only when each disjunct is false (line 9). EX: Today is Monday (p) or it is raining outside (q)
Truth Functions
Truth value: true or false Truth function: a truth value of a compound proposition is determined by (i.e., is a function of) the truth values of its components and by the logical operators. Statement variable: A statement variable p,q,r,s... can stand for only statement, simple, or complex.
Complex Statements
Well-formed formula (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). Rule 2: The tilde (~) goes in front of the statement it is meant to negate. Rule 3: The tilde (~) cannot, by itself, go between two statements. Rule 4: Parentheses, brackets, and braces are required to eliminate ambiguity in a complex statement. EX: Determine if it's a WFF a. [S v (Q·~P)] ⊃~R b. (R·~S v P) = R and not S or P c. (P~) · [~S v (R⊃Q)] Answer: a. WFF b. Not WFF: violates Rule 4 (ambiguous ambiguity) c. Not WFF: violates rule 2 (tilde not in front of a statement)
Truth Tables
show every possible truth value for compound proposition. provide truth-functional definitions of logical operators.
Negation
the truth table for negation shows that for every statement (p, ~p) will have the opposite truth value. p, ~p = T F, F T A Albany is the capital of New York (T) ~A It is not the case that Albany is the capital of New York (F)
