phil week 5

Example: If "X Y" is true, then which of the following is correct? (a) X must be true. (b) X must be false. (c) X could be true or false.

(c) X could be true or false

statement form

A pattern of statement variables and logical operators, such as ~ ( p v q )

Example A. [ S v (Q · ~ P ) ] ~ R B. ~ [ ( S R ) v (Q · P ) ] C. ( R · ~ S v P ) D. ( P ~ ) · [ ~ S v ( R Q ) ]

A. WFF: [ S v (Q · ~ P ) ] ~ R B. WFF: ~ [ ( S > R ) v (Q · P ) ] C. Not a WFF: ( R · ~ S v P ) D. Not a WFF: ( P ~ ) · [ ~ S v ( R > Q ) ]

Example: If you feel great, then you look great.

F > L ^ ^ Ant. Consquent

Example: Pizza contains all the basic food groups if, and only if, you get it with anchovies

P ≡ A (P > A) (A > P)

Well-Formed Formulas - rule 1

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 Q v > P P Q ≡

Example [ ( A · C ) > ~ B ] v D

The main operator is the (v) sign.

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.

Inclusive disjunction

both disjuncts can be true at the same time. Today is Monday or it is raining outside.

Exclusive disjunction

both disjuncts cannot be true at the same time. Today is Monday or today is Wednesday.

Compound statement:

has at least one simple statement as a component "Hamlet is a tragedy, and The Tempest is a comedy." We can symbolize this as: H and T.

Logical operators:

symbols used in translations of ordinary language statements; we use 5 operators

Simple statement:

the basic component of propositional logic. We use an upper case letter to symbolize a simple statement. "Hamlet is a tragedy." We can symbolize this as: H.

truth value

true or false

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 (propositions) rather than classes of objects. - Enables us to make a long sequence of valid inference from one proposition to another

Conditional: sideways (horseshoe) >

- The word if typically precedes the antecedent of a conditional statement. - The horseshoe symbol (horshoe) is used to translate a conditional statement. - 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 and phrases indicate conditional statements, including: Every time P, then Q. Given that P, then Q. Each time P, then Q. Provided tat P, then Q. Any time P, then Q. P implies Q.

truth tables

- show every possible truth value for compound propositions - provide definitions of logical operators

Five Logical Operators

1. Negation 2. Conjunction 3. Disjunction 4. Conditional 5. Biconditional We use these operators to translate English sentences into symbols.

Biconditional: ≡ (triple bar)

A compound statement consisting of two conditionals—one indicated by the word IF and the other indicated by the phrase ONLY IF. - The triple bar symbol (≡) is used to translate a biconditional statement. - IF you eat your spinach, then you get ice cream, and you get ice cream ONLY IF you eat your spinach. = You get ice cream IF AND ONLY IF you eat your spinach. I ≡ S

Conjunction: · (dot)

A compound statement with two distinct statements (conjuncts) connected by the dot symbol (·). (·) translates and, but, still, moreover, while, however, also, moreover, although, yet, nevertheless, and whereas. - 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 (Frank teaches music and Ernest teaches music.)

Disjunction: v (wedge)

A compound statement with two distinct statements (disjuncts) connected by the wedge symbol (v) . (v) translates or, unless, otherwise, and either ... or. - 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 (She is a Pisces or she is a Scorpio.) - You can't go to the party unless you clean your room. (= Either you clean your room or you can't go to the party.) P v R

statement variable

A statement variable p, q, r, s .... can stand for any statement, simple or complex

Example: I will stop by to visit only if I have finished my homework

Answer: (compound) Let V = I will stop by to visit, and H = I have finished my homework. V only if H. V > H

Conditional Statements

Distinguishing IF from ONLY IF: - IF typically precedes the antecedent of a conditional statement. - ONLY IF precedes the consequent of a conditional statement. - IF the light is on, then the neighbors are home. L > N ^ ^ antecedent consequent - The light is on ONLY IF the neighbors are home. - IF the neighbors are home, then the light is on. N > L - ONLY IF the light is on will the neighbors be home.

Example: If Julian does not become a lifeguard at the YMCA or get the job driving school buses, then he will keep working at the bookstore in San Francisco and clean houses part time. Answer

IF Julian does not become a lifeguard at the YMCA or get the job driving school buses, then he will keep working at the bookstore in San Francisco and clean houses part time. ~ ( L v D ) > ( B · C )

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. ( T v B ) · S NOT both Suzuki and Honda are Japanese-owned companies. ~ ( S · H )

Well-Formed Formulas - rule 2

Rule 2: The tilde (~) goes in front of the statement it is meant to negate. WFFs: ~ ( P v Q ) ~ R ( S · P ) v ~ ( ~ Q · S ) Not WFFs: P ~ ( P v Q ) ~ ~ ( S · P ) ~

Well-Formed Formulas - rule 3

Rule 3: The tilde (~) cannot, by itself, go between two statements. WFF: P v ~ Q Not a WFF: P ~ Q

Well-Formed Formulas - rule 4

Rule 4: Parentheses, brackets, and braces are required in order to eliminate ambiguity in a complex statement. WFFs: P v ( Q · R ) ~ [ ( P v Q ) · ( ~ R > S ) ] Not a WFF: P v Q ( · R )

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. - Today is Monday. M - Today is not Monday. wavey M - Gold is selling at $1000 an ounce. G - It is not the case that gold is selling at $1000 an ounce. wavey G

Example ~ [ ( A · C ) > ( B v D ) ]

The main operator is the (~) sign

Conjunction

The of truth values for p, q (2 × 2 = 4), "p . q" will have the following truth values: A conjunction will be true only when each conjunct is true (line 1). Today is Monday and it is raining outside. ^ ^ p q truth table definition for conjunction shows that for any combination

Disjunction

The truth table definition for (inclusive) disjunction shows that for any combination of truth values for p, q, "p v q" will have the following truth values: A disjunction will be false only when each disjunct is false (line 4). Today is Monday or it is raining outside. ^ ^ p q

Negation

The truth table definition for negation shows that for any statement p, ~ p will have the opposite truth value: A Albany is the capital of New York. (T) ~ A It is not the case that Albany is the capital of New York. (F)

Biconditional

The truth table definition for the biconditional shows that for any combination of truth values for p, q, "p ≡ q" will have the following truth values: A biconditional is true when both p, q have the same truth value (lines 1 and 4).

conditional

The truth table definition for the conditional shows that for any combination of truth values for p, q, "p q" will have the following truth values: A conditional will be false only when the antecedent is true and the consequent is false (line 2). If you drive south on I-15, then you will get to Los Angeles. ^ ^ p (antecedent) q (consequent)

Main Operator

Two important factors: - Has as its range the largest component or components in a compound statement. - There can be only one main operator in a compound statement. Compare: - Main operator is (~): ~ [ ( P v ~ Q ) · ( R > S ) ] - Main operator is (· ): ( P v ~ Q ) · ( R > S ) - Main operator is (v): P v [ ~ Q v ( R > S ) ] - Main operator is ( > ): [ ( P v ~ Q ) · R ] > S

Complex Statements

Well-formed formulas (WFFs): Compound statement forms that are grammatically correct.