Logic

Logical Properties of sentences: Consistency

A true sentence

Statements Intuition

process of making generalizations on insight

Example Write the inverse for each of the following statements. Determine whether the inverse is true or false. 1. If a person using a stolen passport, he is breaking the law. 2. If a line is perpendicular to a segment at its midpoint, it is the perpendicular bisector of the segment. 3. Dead men tell no tales.

. * the inverse of a given conditional statement is formed by negating both the hypothesis and conclusion of the conditional statement. 1. hypothesis = " a person travels abroad with a stolen passport"; conclusion= he is breaking the law. negation= a person is not traveling abroad with a stolen passport. inverse= if a person is not traveling abroad with a stolen passport, then he is not breaking the law. The inverse is false, since there are more ways to break the law than by identity theft. 2. hypothesis= contains 2 conditions: a. the line is perpendicular to the segment b. the line intersects the segment at the midpoint. Negation= of statement a and statement b is not statement a or not statement b. Thus, it is the line is not perpendicular to the segment or it doesn't intersect the segment at the mid point." Conclusion negation= the line is not the perpendicular bisector of a segment. Inverse= if a line is not perpendicular to the segment or does not intersect the segment at the midpoint, then the line is not the perpendicular bisector o the segment. This inverse is true because if either of the conditions holds ( the line is not perpendicular; the line does not intersect at the midpoint), then the line cannot be a perpendicular bisector. 3 Hypothesis= the man is dead conclusion= the man tells no tales. Inverse= if a man is not dead, then he tell tales. It is false because many witnesses to crimes are still alive but they have never told their stories to the police.

Logic

...

Basic Principles, Laws, and Theorems

1. The Law of the Excluded Middle= Any statement is either true or false. 2. The Law of Contradiction= a statement cannot be both true and false. 3. The converse of a true statement is not necessarily true. 4. The converse of a definition is always true. 5. For a theorem to be true, it must be true for all cases. 6. A statement is false if one false instance of the statement exists. 7. The inverse of a true statement is not necessarily true. 8. Contrapositive of a true statement= true Contrapositive of a false statement= false 9. If the converse of a true statement is true, then the inverse is true. Likewise, if the converse is false, the inverse is false. 10. Logically equivalent= Statements that are either both true or both false. 11. If a given statement and its converse are both true, then the conditions in the hypothesis of the statement are both necessary and sufficient for the conclusion of the statement. 12. If a given statement is true but its converse is false, then the conditions are sufficient but not necessary for the conclusion of the statement. 13. If a given statement and its converse are both false, then the conditions are neither sufficient nor necessary for the statement's conclusion.

Logical Truth

A sentence is logically true if it is impossible for it to be false. That is the denial of the sentence is inconsistent. Example: Either Mars is a planet or Mars is not a planet.

Logical Properties of sentences: Inconsistency

A sentence that is impossible to be true. Example At least one odd number is not odd

Logical Indeterminacy (Contingency)

A sentence that is logically indeterminate if it is nether logically true nor logically false. Example: Einstein was a physicist and Pauling was a chemist.

Given three statements, P, Q, and R, suppose it is known that R is true. Which one of the following must be true? A. (P upside down V, Q) right arrow R. B. R right arrow ( P V Q) C. ( P right arrow Q) upside down V, R. D. ( R right arrow Q) V P.

A. Conditional statement= X right angle Y is true in all instances except when X is true and Y is false. Let X= P upside down V Q Y= R. Since R is true and we do not know the true value of " P upside down V, Q, we have either "true right arrow true or false right arrow true. In either case, ( upside down Q) right arrow R must be true.

Given any two statements P and Q, where Q is a false statement, which one of the following must be false? A. Not P and Q B. Not (P and Q) C. P or not Q D. P implies Q.

A. Given that Q is a false statement, the statement "not P and Q" must be false. Any compound statement with the conjunction operator ( which is "and") is false unless both component part are true.

Statements Validity

An argument is valid if the truth of the premises means that the conclusions must also be true.

Deductive Reasoning Syllogism

An arrangement of statements that would allow you to deduce the third one from the preceding two. broken down into three parts: 1. Major Premise: general statement concerning a whole group. 2. Minor Premise: specific statement which indicates that a certain individual is a member of that group. 3. Deduction: a statement to the effect that the general statement which applies to the group also applies to the individual.

Statements Conjunction

And statements " a and b" denoted a upside down v, b. Example: P= Ann is in Durango, Colorado. AND ( Upside down V) Q= Paul is in Denver, Colorado. (p upside down v, Q) Therefore, Ann is in Durango, Colorado and Paul is in Denver, Colorado.

Sentence

Any expression that can be labeled either true or false. Example: It is raining where I am standing. My name is George.

Which of the following is an example of a biconditional statement? A. A triangle has an angle less than 90 degrees If it is acute. B. A rectangle is a square if and only if it has four equal sides. C. A circle has both a radius and a diameter. D. A cube has either six edges or it has more than six edges.

B. Biconditional statement= if it can be written I the form "P if and only if Q." P= A rectangle is a square Q= it has four equal sides. Then the statement "A rectangle is a square if and only if it has four equal sides" is biconditional. * a biconditional statement may also appear in the form "if P then Q, and if Q then P."

What is the contrapositive of the statement "If Johnny is in Las Vegas, then Marie will be angry"? A. If Johnny is not in Las Vegas, then Marie will be angry. B. If Marie is not angry, then Johnny is not in Las Vegas. C. If Johnny is not in Las Vegas, then Marie will not be angry. D. If Marie is angry, the Johnny is in Las Vegas.

B. The contrapositive of "if P then Q" is "if not Q then not P." P= "Johnny is in Las Vegas" Q= Marie will be angry By substitution, for the statement "if Johnny is in Las Vegas, then Marie will be angry" The contrapositive statement is "If Marie is not angry,. then Johnny is not in Las Vegas."

What is the inverse of the statement "If it is snowing, then people stay indoors"? A. If people board a plane, then it is not snowing. B. If it is not snowing, then people do not board a plane. . C. If people do not board a plane, then it is not snowing. D. If it is not snowing, then people board a plane.

B. The inverse of "If P then Q" is "if not P then not Q." Le P represent the statement " it is sowing" and let Q represent statement " People board a plane." By substitution, for the statement " If it is sowing, then people board a plane," the inverse statement is "if it is not snowing, then people do not board a plane.

Drill Questions If P and Q represent statements, which one of the following is equivalent to "Not P and Not Q?" A. Not P or not Q B. Not P or Q C. Not (P or Q) D. Not (P and Q)

C. One of De Morgan's laws for sentences is ~ (X V Y) double arrow ~ X upside down Y. Substituting P for X and Q for Y, we have ~(P V Q) double arrow ~ P upside down V, ~Q. This statement is read as follows" "Not (P or Q) is equivalent to "Not P and not Q."

For which one of the following statements is the converse true? A. If you live in Toledo, then you live in Ohio. B. If your pet is a cat, then it has a tail. C. If a number is negative, then its square is positive. D. If a geometric figure has three sides, then it is a triangle.

D.

Let R and S represent statements. Consider the following: I. If R then S II. Not R and S III. If S then R. Which of the above statements Is (are) equivalent to the statement " R is a necessary condition for S"? A. Only I B. I and II C. II and III D. Only III

D. If R is a necessary condition for S, then by definition S implies R. Also, the statement that "S implies R" is equivalent to the statement "if S then R," which represents item III. Neither of items I or II is equivalent to the given statement.

What is the negation for the statement " Nationality is important or Immigration matters"? A. Nationality is important and immigration does not matter. B. Nationality is not important or immigration does not matter. C. Nationality is important or immigration does not matter. D. Nationality is not important and immigration does not matter.

D. The negation of (P V Q) is ~ (P V Q). By one of De Morgan's Laws, the statement ~ (P V Q) is equivalent to ~ P upside down V, ~ Q. P= "Nationality is important Q= Immigration matters. Then the negation for "Nationality is important or immigration matters" is "Not (nationality is important or immigration matters). The latter statement is equivalent to "nationality is not important and immigration does not matter."

Syllogism( deduced argument) Example A. Major Premise: All planes have wings. B. Minor Premise: Qantas Airways is a plane company. C. Deduction: Qantas Airways has wings.

Deductive reasoning= the technique of employing syllogism to arrive at a conclusion. If a major premise that is true is followed by an appropriate minor premise that is true, a conclusion can be deduced that must be true and the reasoning is valid.

Logical Equivalence Versus Meaning the Same!

Equivalence is different as meaning the same Same meaning = double negative ~ ~ X(not-not) and X itself.

Truth Tables and Basic Logical Operations. The Truth Table

For a sentence X is the exhaustive list of possible logical values of X.

Logical Equivalence Versus Meaning the Same Theorem 1- Double Negation Equals Identity

For any sentence X ~ ~ X double Arrow X

Fundamental Properties of Operations Theorem 3- Properties of Disjunction Operation

For any sentences X, Y, Z, the following properties hold: 1. Commutavity: X V Y double arrow Y V X 2. Associativity: X V (Y VZ) double arrow (X V Y) V Z

Fundamental Properties of Operations Theorem 7 Proof by Contradiction

For any sentences X, Y, the following holds: X right arrow Y double arrow ~ Y right arrow ~ X

Fundamental Properties of Operations Theorem 5- De Morgan's Laws for Sentences

For any sentences X, Y, the following laws hold: 1. ~ (X upside down V, Y) double arrow (~ X) V (~Y). 2. ~ (X V Y) double arrow (~X) upside down v (~ Y).

Fundamental Properties of Operations Theorem 6 Two Logical Identities

For any sentences X, Y, the sentences X and (X upside down V, Y) V (X upside down V ~ Y) are logically equivalent. (X upside down V, Y) V (X upside down V, ~ Y) double arrow X

Fundamental Properties of Operations Theorem 2- Properties of Conjunction Operation

For any sentences, X,Y, Z, the following properties hold: 1. Commutativity: X upside down V Y double arrow Y upside down X 2. Associativity: X upside down V (Y upside down V, Z ) double arrow (X upside down V Y) upside down V, Z

The truth table Negation AND/ OR

For sentences X and Y, the disjunction "X AND/OR Y,"= X v Y= denotes that the sentence that is true if either or both X and Y are true. V= the disjunction operator and is a binary operation, transforming the par of sentences X, Y into a unique image sentence X VY. Example: X sentences= Jane is traveling the world. Y= Marvin is traveling the United States X VY= Jane is traveling the world AND/OR Marvin is traveling the United States.

Fundamental Properties of Operations Theorem 4- Distributive Laws

For sentences X, Y, Z, the following laws hold 1. X V (Y upside down v, Z) double arrow (X V Y) upside down V ( X V Z). 2. X upside down v (Y V Z) double arrow (X upside down v, Y) V (X upside down V, Z).

Statements Negation A and B

I am both traveling and happy A= I am traveling B= I am happy Negation of A and B I am not traveling or I am not happy

The Truth Table Negation IFF and AND

IFF= if and only if. AND sentences = is represented by an upside down V, which is the conjunction coordinator and is a binary operation, transforming a pair of sentences into a unique image sentence. For sentences X and Y, the conjunction " X AND Y" represented by X upside V Y= sentence is true and IFF both X an Y are true. Example: For X= Jane is traveling to Canada. Y= All Canadians love hockey. X AND Y= Jane is traveling to Canada AND all Canadians love hockey.

The Truth Table Negation

If X is a sentence, the ~X represents the negation, the opposite, or the contradiction of X. ~ = the negation operation on sentences. Example: X= "Jane is traveling the world" ~X= "Jane is not traveling the world. Negation Operation= called unary, transforming a sentence into a unique image sentence.

Conjunction

If a and b are statements, then a statement of the form " a and b" is a conjunction of a and b, denoted by an upside down v and b.

Biconditional Statement

If and only if statements Example: A triangle is isosceles if and only if the triangle has two congruent (equal) sides

Implication

If/then statement Example: If the Northern lights is visible, then the sky is not overcast.

Necessary And Sufficient Conditions

Let P and Q represent statements. If P, then Q is a conditional statement in which P is a sufficient condition for Q, and similarly Q is a necessary condition for P.

Negation Summary

Not statement

Statements Disjunction

Or statement denoted by a V b Example: P= Ann is in Durango, Colorado. Q= Paul is in Denver, Colorado. OR (V) Therefore, Ann is in Durango, Colorado or Paul is in Denver, Colorado.

Necessary and Sufficient Conditions Biconditional statements

P if and only if Q, P is necessary and sufficient condition for Q and vice versa.

the truth table Negation IF-THEN

Sentences X and Y= X right arrow Y (implication). IF X THEN Y. X right arrow Y is false IFF X is true and Y is false; otherwise it is true. The right arrow symbol= implication operator. It is a binary operation, transforming the pair sentences X and Y into a unique image sentence X right arrow Y.

Biconditional statement Example Rick gets paid if and only if he works

Sufficient and necessary condition= Rick gets paid and his working

Necessary and sufficient conditions Example: If it rains, then Jane will go to Antelope Canyon.

Sufficient condition= if it rains necessary condition= Jane will travel to Antelope Canyon

Implication

The compound statement "if a, then b," denoted by a right arrow b, is called a conditional statement or implication

Statements Negation A or B

The negation of a statement q is the statement " not q" denoted by ~ q. Example: A or B You are either traveling or staying home. Consider the statement "You are either traveling or staying home." For this statement to be false, you can't be traveling and you can't stay home. In other words, the opposite is to be not traveling and not staying at home. Or if we rewrite it in terms of the original statement we get "You are not traveling and not staying at home." If we let A be the statement "You are traveling" and B be the statement "You are staying at home", then the negation of "A or B" becomes "Not A and Not B."

Negation

The negation of a statement q is the statement not q, denoted by -q.

Syllogism (deduced argument) Example A. Major Premise: All people who live abroad have at least a working visa. B. Improper Minor Premise: Jane has at least a working visa. C. Illogical Deduction: Jane lives abroad.

This is an example of a major premise that is true, but, is followed by a inappropriate minor premise that is also true, but, the conclusion cannot be deduced.

Statements Negation If a, then B

Transforms into A and Not B. If I am rich, then I am happy. A= I am in Iceland. B= I am traveling. If A, then B becomes A and Not B I am in Iceland and I am not traveling.

Logical Equivalent of Sentences

Two sentences are logically equivalent if and only if it is impossible for one of the sentences to be true while the other sentence is false. If it is impossible for the two sentences to have different truth values. Example: Chicago is in Illinois, and Pittsburgh is in Pennsylvania is logically equivalent to "Pittsburgh is in Pennsylvania and Chicago is in Illinois.

Proof of Part 1 of Theorem 5

We can prove ~ (X upside down V, Y) double arrow (~X) V (~Y) by developing a truth table over all possible combinations of X and Y and observing that all vales assumed by the sentences are the same.

Logical Equivalence

X and Y sentences= X double arrow Y is true IFF X and Y have the same truth value; otherwise, it is false. The double arrow = logical equivalence, IFF. It is a binary operation. Example: X= Jane is traveling to Canada Y= Canadians love hockey. X double arrow Y= Jane travels to Canada IFF Canadians love hockey.

Statements Converse

a right arrow b = b right arrow a Conditional: "If the Northern lights came out, then green streaks were seen." Converse: "If green streaks were seen, then the Northern Lights came out. interchange the hypothesis and conclusion of the original statement.

Statement Contrapositive

a right arrow b = b right arrow ~ a. Switching the hypothesis and conclusion of a conditional statement and negating both. For example, the contrapositive of "If it is raining then the plane is cleaned" is "If the plane is not wet then it is not raining.

Statements Inverse

a right arrow b= ~ a right arrow ~b Negating both the hypothesis and conclusion of a conditional statement. For example, the inverse of "If it is raining then the plane will get cleaned" is "If it is not raining then the plane won't get cleaned.

Logical falsity

a sentence is false if it is impossible for it to be true. That is, the sentence is inconsistent. Ex. Mars is a planet and Mars is not a planet.

Statement

a sentence that is either true or false, but, not both.

Statements

a sentence that is either true or false, but, not both.

The Truth Table Logical Value

of a sentence X is true (or T) if X is true, and false (or F) if X is false.