Hypothetical Syllogism

Hypothetical Syllogismy A syllogism which contains hypothetical propositions as

premises.y There are 3 types of hypothetical syllogism:

Conditional 2. Disjunctive 3. Conjunctive1.

Conditional Syllogismy Contains a conditional proposition as a major premise y The minor premise and the conclusion are both categorical y The conditional proposition which is the major premise

contains the two parts, the antecedent and the consequent

Hypothetical Syllogismy The antecedent, the part introduced by the If is the condition y The consequent, the part introduced by the then is the result of

the effect

y If Peter is a surgeon, then, he is a doctor.

But Peter is a surgeon Ergo, he is a doctory If a person is a Catholic, then he is a Christian

My friend is not a Christian Therefore, he is not Catholic

Hypothetical Syllogismy The antecedent is symbolized by p and the consequent by q y The symbol of conditional is the horse shoe/ ellipse y The symbol of a curl (tilde) ~ before the antecedent or the

consequent indicate that it is negativey If Peter is a surgeon, then, he is a doctor. p

q p q

But Peter is a surgeon Ergo, he is a doctor

Hypothetical Syllogismy If a person is a Catholic, then he is a Christian

p -q -p

q

My friend is not a Christian Therefore, he is not Catholic The Laws of the Conditional:1. 2. 3. 4.

If the antecedent is true, the consequent is also true If the consequent is false, the antecedent is also false If the antecedent is false, the consequent is doubtful If the consequent is true the antecedent is doubtful

Deriving a Valid Conclusiony Positing (Modus Ponens) when the minor accepts the

antecedent and the conclusion accepts the consequenty If the man is Catholic, then he is Christian.

p

He is Catholic He is a Christian

q p q

Deriving a Valid Conclusiony Denying (Modus Tolens) when the minor premise rejects

the consequent and the conclusion rejects the antecedenty If the man is a catholic then he is a Christian

p

He is not Christian He is not a Catholic

q ~q ~p

Disjunctive Syllogismy Consists of a disjunctive proposition as the major premise

and categorical propositions as minor premise and conclusiony If one part is denied, the other part must be accepted y If one part is accepted, the other part may be denied or

accepted (depending on the type of disjunction)

Disjunctive Syllogismy The senate either approves or rejects the bill

The senate approves the bill therefore, the senate did no reject the bill pvq p ~q

Disjunctive Syllogismy TwoTypes of Disjunction y 1. Proper (exclusive/ strict) if the parts of the disjunction

are mutually exclusive or contradictoriesy 2. Improper (inclusive) if the parts are not mutually

exclusive. If one is false then the other must be true but if one is true, the other maybe true or false

y Proper (exclusive)

The teacher is either present or absent. p v q But the teacher is present P He is not absent ~q The teacher is either present or absent. He is not present He is absent pvq ~p q

y Improper (inclusive)

The student either studies math or english. But he is not studying math He is studying english The student either studies math or english He is studying math He is not studying english

y Valid Process of Disjunctive Syllogism

Positing when the minor premise accepts one part and the conclusion denies the other part 2. Denying when the minor premise denies one part and the conclusion accepts the other part1.

Conjunctivey Consists of conjunctive proposition as the major premise and

categorical propositions as minor premise and conclusiony The suspect cannot be in Aparri and Jolo at one time

But report says he is in Aparri Therefore, he cannot be in Jolo

Validityconsequent in the conclusion) p q ~p q P ~p q ~qy Validity of Conditional 1. Modus Ponens (affirm the antecedent in the minor, affirm the

2. Modus Tolens (deny the consequent in the minor, deny the antecedent in the conclusion) p ~q ~p q q ~q ~p p

Validityy Validity of Disjunction

Symbolic Logicy ...by the aid of symbolism, we can make transition in

reasoning almost mechanically be the eye, which otherwise would call into play the higher faculties of the brain. AlfredNorthwhitehead

y The use of symbols enables us to analyze statements and

arguments quickly and efficientlyy Modern logics approach does not focus on syllogisms but

upon logical connectives that are fundamental in deductive arguments

Basic Conceptsy Propositional Logic the logic of compound statements. It

uses variables and special symbols for operations. The logic of compound statements which relies solely on three key concepts: TRUTH VALUE, LOGICAL OPERATORS and VARIABLESy Simple Statement one that does not contain any other

statement as a component

Ex: Nature is changing drastically Hydrogen is a gas Shakespeare wrote, Hamlet.

Basic Conceptsy Compound Statement one that contains one simple or

atomic statement as a componentEx: It is not the case that Jose Rizal wrote Hamlet If acorns are planted in the spring then oak trees will grow in the fall. Either Moriarty committed the murder or Stapleton lied. John is a student or Mary is a teacher. Gloria is an entrepreneur and Lalo is an employee.

Basic Conceptsy Connective any words attached to one or more statements in

order to create a new statement y They join simple statements into compound statements * in Symbolic Logic, they are called, TRUTH FUNCTIONAL CONNECTIVE (v,

.

,

, )

a statement connective wherein the truth or falsity of such statement created by its use would be depending solely on the truth or falsity of the statement(s) to which it is attachedIt is not the case that all lawyers are liars Either the class will have an open forum or forever hold their silence If Napoleon wins the war and France will regain power then they could rule over Europe

Basic Conceptsy Truth Function a truth function that takes one or more

truth-values as its input and returns a single truth-value as its output (Conjunction, Disjunction, Implication, Equivalence)y Truth value refers to the falsity or truth of each statement y Logical Variables the letters which stand for any logical unit

(Q, P, R, S,T...etc)

Compound Statements and their ConnectivesFunction Conjunction Disjunction Conditional Biconditional Negation ~ Operator Name Ampersand Wedge Horseshoe Triple Bar Tilde English Translation and or If...,then If and only If not

.v

y (Pedro passed the final exam) and (the course.)

p.c (Either Pedro passed the final exam) or (he passed the course). pvq (If Pedro passes the final exam) then (he will pass the course). p c (Pedro will pass the course) if and only if (he passed the final exam). p c Pedro passed the course but not the final exam. p . ~c

Types of Compound StatementsConjunction this is a compound statement whose component statements are called conjuncts. -The connective is a centered dot (.) or the ampersand (&) which means and -other conjunction words used instead of and are: but, yet, however, although, whereas, nevertheless, a comma, and a semicolon1.Maria and Celia are gamblers. The president is happy but tired. The students went to the event yet disappointed someone. Pepito loves to eat crabs; though he is a vegetarian.

2. Disjunction a compound statement whose component parts are called disjuncts- The statement is connected by a vee or wedge which means

or Either the teacher passes the students or they fail. Romeo either marries Juliet or he kills himself. Either one is present or one is absent.

y 3. Material Implication / Conditional a statement that expresses

if...then relationship between its component statementsy Other connectives

if P, Q P implies Q P entails Q P only if Q P thus Q P therefore Q

P hence Q P if Q P since Q P because of Q P for Q P when Q

y 4. Material Equivalence / Biconditional two conditional

statements which have been reversed but involving the same propositions, conjoined together.y Expresses an if and only if relationship between each compound

statements Ex:The plants are healthy if and only if my mother is home. The students are behaved if and only if the teacher is present. Some animals are in danger or extinction if and only if human beings do not take care of the environment.

y 5. Negation reversal of a statement

Ex: I am not worried about your grades in logic. Mrs. Pirtola does not like working with plastic in her artwork. It is impossible for a sensible woman to like a conceited man.

Truth Table for Compound Statementsy 1. ConjunctionP T T F F Q T F T F P.Q T F F F

y P . Q is true only if each component is true y Future lawyers were the ones who started the riot last night

and many people got hurt

Truth Table for Compound Statementsy 2. DisjunctionP T T F F Q T F T F PvQ T T T F

y The only case in which a disjunction is false is when both its

disjuncts are false y Either it was the group s decision to create trouble or it was their leaders decision alone.

Truth Table for Compound Statementsy 3. Material Implication / ConditionalP T T F F Q T F T F P Q T F T T

y The only instance when the conditional is false is when the

antecedent is false and the consequent is true y If Philip eats a piece of stone, then his teeth might break.

Truth Table for Compound Statementsy 4. Material Equivalence / Bi conditionalP T T F F Q T F T F P Q T F F T

y Friday is the happiest day of the week if and only if I am able

to sleep for 8 hours. y Goodbyes are not sorrowful if and only it is for the best interest of the parties involved.

Truth Table for Compound Statementsy 5. NegationP T F ~P F T

y Immanuel is not a teacher. y The villagers are not contented. y No fisherman wished to sail that fateful night. y Neither Julian nor Martha, opted for a good but meaningless

life.

ExerciseIf A, B and C are TRUE and X,Y and Z are false, which of the following statements are true? Apply the truth table.1.2. 3. 4. 5. 6.

B Y (A.C) X ~(A C) v (X.Z) (X.A) v (B.Z) (ZvX) (CvB) ~[(AvB) . (CvX)] (C ~Y)

Tautologies,Contradictories, Contingent

Tautologies,Contradictories, Contingenty A compound statement is tautologous

if it is true

regardless of the truth value of its componentsy All bachelors are either male or not male.M T F ~M F T Mv~M T T

y All crows are either black or not black.

Tautologies,Contradictories, Contingenty A compound statement is self-contradictory if it is false

regardless of the truth value of its components.y I am a liar and that is not true.M T F ~M F T M.~M F F

y Never say never.

Tautologies,Contradictories, Contingenty A compound statement is contingent if its truth value

varies depending on the truth value of the components.M T T F F I T F T F M I T F T T

y If Marcos was a great public official, then Imelda was an

honest official.

Formal Proofs of Validityy 1. Disjunctive Syllogism (DS) an argument that consists of a

disjunctive premise, a premise that denies one of the disjuncts and a conclusion that affirms the other disjuncty pvq

~p q Thisj will apologize or Cornelius will be angry Thisj will not apologize Cornelius will be angry

Formal Proofs of Validityy 2. Pure Hypothetical Syllogism (HS) a syllogism whose

propositions are conditional statements.y p

q q r p r

If Louise is a leader, then France is a good adviser If France is a good adviser, then Jaimee is jobless If Louise is a leader, then Jaimee is jobless

Formal Proofs of Validityy 3. Modus Ponens (MP) affirming the antecedent

p q p q

If there is storm tomorrow, then flood is likely to occur. There is storm tomorrow. Therefore flood is likely to occur.

Formal Proofs of Validityy 4. Modus Tollens (MT) denying a consequent

p q ~q ~p

If there is storm tomorrow, then flood is likely to occur. There is no storm tomorrow. Therefore flood is not likely to occur.

Formal Proofs of Validityy Constructive Dilemma (CD) the first premise consists of two

conjoined conditional statements and, the second premise asserts the truth of one of the two antecedents. The conclusion which follows logically via 2 modus ponens steps asserts the truth value of at least one of the consequents. q . (r s)

y p

pvr qvs If this punch contains gin, then Elisa will like it, and if it contains vodka, then Carmen will like it. This punch contains either gin or vodka Therefore, Elisa will like it or Carmen will like it.

Formal Proofs of Validityy 6. Destructive Dilemma the second premise asserts the falsity of at

least one of the consequents. The conclusion which follows via 2 modus tollens asserts the falsity of atleast one of the antecedents.

y p

q . (r s) ~q v ~s ~p v ~r

If this punch contains gin, then Elisa will like it, and if it contains vodka, then Carmen will like it. Either Elisa will not like it or Carmen will not it. Therefore, either this punch does not contain gin or it does not contain vodka

Formal Proofs of Validityy 7. Absorption (Abs) p

q p (p.q)

IF the price of oil increases, then the sales of cars will decrease. Therefore if the price of oil increases, the price of oil increases and the sales of cars decreases.

Formal Proofs of Validityy 8. Simplification (Simp) p . q

p June is practicing in the gym while classmates have gone home. June is practicing in the gym.y 9. Addition (Add)- p

pvq Today is Thursday. Ergo, today is Thursday or else I will be wearing my favorite blouse.

Formal Proofs of Validityy 10. Conjunction p

q p.Q Cornelia is beautiful. Cornelia is smart. Cornelia is beautiful and smart.