Chapter 7

Main Operator

-Either it is one of the four operators that go between statements or the negation operator its typically the last thing you do example P v [~Q v (R ⊃ S)] the v is the main operator

Truth Value

-Every statement in our language of propositional logic is either true or false. - We call true and false the two truth values. -Each statement has exactly one truth value.

Ordinary language examples of Conditional

-Every time P, then Q. -Given that P, then Q. -Each time P, then Q. -Provided that P, then Q. -Any time P, then Q. -P implies Q

Propositional Logic

-Its basic elements are statements rather than classes -Unlike categorical logic that is restricted to A,E,I, and O propositions, it contains unlimited number of complex statements

Facts about main operator

-There can only be one main operator in a compound statement -Locating the main operator helps to translate sentences

Step 1 of Order of Operations: ~ (P · Q) v Q

1.) Determine the truth values under the dot in this example P Q |~ (P · Q) v Q T T.............T T F.............F F T.............F F F.............F

Precise recursive definition of WFF

1.) Every atomic sentence is a WFF. 2.) If p is a WFF, then ~p is a WFF. 3.) If p and q are WFF, then (a) (p ·q) (b) (p v q) (c) (p ⊃ q) (d) (p ≡ q) ...(a) - (d) are WFF. 4.) Nothing else is a WFF. * 1 -> base clause 2 & 3 -> recursive clause 4 -> exclusion clause

Determining Validity through Truth Tables

1.) Find the truth values for the premises and the conclusion 2.) Then find a line (row) where the two premises are true and the conclusion is false. If this exists the argument is *invalid*, and if it does not exist than the argument is *valid*

Step 2 of Order of Operations: ~ (P · Q) v Q

2. Determine the truth values under the tilde P Q ~ (P · Q) v Q T T....F..... T T F....T......F F T....T......F F F....T......F the opposite values of the dot

Step 3 of Order of Operations: ~ (P · Q) v Q

3. Finally, determine truth values under the wedge P Q ~ (P · Q) v Q T T....F..... T.......T T F....T......F.......T F T....T......F.......T F F....T......F.......T the result of both the tilde and the dot gives the values for the wedge

Truth Function

A compound proposition whose truth value is completely determined by the truth values of its components (atomic sentences) ex. A ⊃ B you can determine the truth function if you know the truth values of A and B

Self-contradiction

A logical falsehood: it is necessarily false. ex. a statement form p · ~p will always be false

Tautology

A logical truth of propositional logic: it is necessarily true. ex. a statement the form p v ~p will always be true

Statement Form

A pattern of statement variables and logical operators ex. ~(p v q)

Non-contingent statement

A statement such that the truth values in the main operator column do not depend on the truth values of the component parts -Tautology -Self-contradiction

Compound Statement

A statement that has at least one simple statement as a component ex. Hamlet is a tragedy and Kung Fu Panda is a comedy this would be called H and K

Simple Statement

Also called atomic sentences are ones that do not have any other statement as a component ex. Hamlet is a tragedy this would be called H ex. Wednesday is hump day this would be called W they are called by the first letter of whatever the main subject is

Conditional statements: antecedent and consequent

If *antecedent* then *consequent* the antecedent always goes first in conditional statements

Conditional Statements: Distinguishing if and only if

If is before the antecedent and Only if is before the conditional ex. If the light is on (antecedent), then the neighbors are home (consequent) L ⊃ N ex. Only if the light is on will the neighbors be home N ⊃ L

Negation (Truth Table)

If p is true than ~p is false If p is false than ~p is true ex. A Albany is the capital of New York. (T) ~A It is not the case that Albany is the capital of New York. (F)

How to know the amount of lines in a truth table?

L = 2 ^ n L = number of lines in truth table n = number of different simple propositions in the statement. ex. 1 ~ (*P* · Q) v *Q* L = 2^2 = 4 4 lines in truth table ex. 2 ~ (*P* · *Q*) v *R* L = 2^3 = 8 8 lines in truth table

When will a conjunction be true?

Only when each conjunct is true

When is a conditional false?

Only when the antecedent (the first atomic sentence) is true and the consequent is false ex. (antecedent) If you drive South on 195, (consequent) then you will get to Canada antecedent = true consequent = false P ⊃ Q is false!

Arranging the Truth Values (bigger example)

P Q R | ~ (P · Q) v R T T T | T T F | T F T | T F F | F T T | F T F | F F T | F F F | and the second line here is two true, and two false alternating

Conjunction (Truth Table)

P Q | P · Q T - T .....T T - F .....F F - T .....F F - F .....F

Biconditional (Truth Table)

P Q | P ≡ Q T T | T T F | F F T | F F F | T

Conditional (Truth Table)

P Q | P ⊃ Q T T | T T F | F F T | T F F | T

Arranging the Truth Values

P Q ~ (P · Q) v Q T T T F F T F F first line will always be half true and half false, and the last line will always alternate true and false

Disjunction (Truth Table)

P Q| P v Q T - T .....T T - F .....T F - T .....T F - F .....F

Rule 4 of WFF

Parentheses, brackets, and braces are required in order to eliminate ambiguity in a complex statement WFF: P v (Q ·R) ~[(P v Q) ·(~R ⊃ S)] NOT WFF: Pv Q ( ·R)

Rule 1 of WFF

Rule 1: The dot, wedge, horseshoe, and triple bar symbols must go between two statements (either simple or compound) examples of NON WFFs · P ⊃ P PQ ≡

Logical operators

Special symbols that can be used as part of ordinary language statement translations

Well-formed formulas (WFFs)

Statement forms that are grammatically correct

Contingent Statement

Statements that are neither necessarily true nor necessarily false (they are sometimes true, sometimes false).

Conjunction

Symbol is a . A compound statement that has two distinct statements (called conjuncts) connected by the dot symbol (.) words that indicate conjunction: and, also, moreover ex. Honesty is the best policy, *and* lying is for scoundrels H . L

The Order of Operations Rule

The main operator will be the last step!

Disjunction

The symbol is v A compound statement with two distinct statements (called disjuncts) connected by the wedge (v) words that are associated with disjunction: or, unless ex. She is either a Pisces or a Scorpio P v S

Negation

The symbol is ~ the word "not" and "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

Biconditional

The symbol is ≡ -A compound statement consisting of two conditionals: if and only if -The triple bar symbol (≡) is used to translate a biconditional statement ex. You get ice cream if, and only if, you eat your spinach I ≡ S

Conditional

The symbol is ⊃ In ordinary language the word "if" typically precedes the antecedent of a conditional words that are associated with conditional: if ... then, only if ex. If you wash the car, then you can go to the movies W ⊃ M

Rule 3 of WFFs

The tilde (~) cannot, by itself, go between two statements. WFF: Pv ~Q NOT WFF: P~Q

Rule 2 of WFF

The tilde (~) goes in front of the statement it is meant to negate. examples of NON WFFs P~ (Pv Q)~ ~(C·P)~

Inconsistent Statements

Two (or more) statements that do NOT have even one line on their respective truth tables where the truth value in the main operator column is T for both statements at the same time.

Consistent Statement

Two (or more) statements that have at least one line where the truth value in the main operators column is T for each statement. so true for both statements at the same time (only need one line that this happens)

When is a biconditional true?

When both p and q have the same truth values

When will a disjunction be false?

When each disjunct is false

Logically Equivalent

When two truth-functional statements have identical final columns in their truth tables

Inclusive Disjunction

a compound statement in which both disjuncts can be true at the same time ex. Today is Monday or it is raining outside.

Exclusive Disjunction

a compound statement in which both disjuncts cannot be true at the same time ex. Today is Monday or today is Wednesday (M v W) · ~(M · W) *do not need to know how to do it*

Statement Variable

a lowercase letter (p,q,r,s,...) that serves as a placeholder for any statement (simple or complex)

Truth Tables

show every possible truth-value distribution for propositions, and, in particular, compound propositions the more atomic sentences you have the bigger the truth table