Hypothetical Syllogism (Full version)

Hypothetical Hypothe tical Syllogism

Hypothetical Hypothe tical Syllogism y

A syllogism which contains hypothetical hypothetical propositions propositions as premises.

There are 3 types of hypothetical syllogism: 1. Conditional 2. Disjunctive 3. Conjunctive y

Conditional Syllogism y

Con Contains ins a conditi itional proposit sition ion as a majo majorr pre premise mise

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Thee min Th minor premise mise and the conclusio sion are bot both categori goriccal

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The conditional proposition which is the major premise contain ains th thee two parts, arts, the antecedent and the consequent

Hypothetical Hypothe tical Syllogism y

y

y

y

The antecedent, the part introduced by by the ´If µ is the condition condition The consequent, the part introduced by the ´thenµ is the result of  the effect If Peter is a surgeon, then, he is a doctor. But Peter is a surgeon Ergo, he is a doctor If a person is a Catholic, then he is a Christian My friend is not a Christian Therefore, he is not Catholic

Hypothetical Hypothe tical Syllogism y y y

y

The antecedent is symb The symboolize ized by p and and th thee consequ sequeent by q The symbol of conditional is the horse rse shoe/ ellipse  The symbol of a curl (tilde) ~ before the antecedent or the consequ sequeent indica icate that hat it is negat gative ive If Peter is a surgeon, then, he is a doctor. doctor. p  q But Peter is a surgeon p Ergo, he is a doctor q

Hypothetical Hypothe tical Syllogism y

If a person is a Catholic, then he is a Christian My friend is not a Christian Therefore, 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 antece dent is false, the consequent is doubtful If the consequent is true the antecedent is doubtful

pq -q -p

Deriving y

y

a Valid Conclusion

Positing (Modus Ponens) ² when the minor accepts the ante antece cede dent nt and and th thee conc concllusio usionn acce accept ptss th thee cons conseq eque uent nt If the man is Catholic, then he is Christian. He is Catholic He is a Christian

pq p q

Deriving y

y

a Valid Conclusion

Denying (Modus Tolens) ² when the minor premise rejects the consequ sequeent and the conclusio sion rejects cts the ant antecedent If the man is a catholic then he is a Christian He is not Christian He is not a Catholic

pq ~q ~p

Disjunctive y

y

y

Syllogism

Consists of a disjunctive proposition as the major premise and and cate catego gori rica call prop propos osit itio ions ns as mino minorr prem premis isee and and conc conclu lusi sion on If one part is denied, the other part must be accepted If one par t is accepted, the other par t may be denied or acce accept pted ed (dep (depen endi ding ng on th thee type type of disj disjun unct ctio ion) n)

Disjunctive y

Syllogism

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 y

y

y

Syllogism

Two Types ypes of Disju Disjunct nctio ionn 1. Proper (exclusive/ strict) ²if the parts rts of the disjunction are are mutu mutual allly excl exclus usiv ivee or cont contra radi dict ctori ories es 2. Improper (inclusive) ²if the par ts are not mutually exclusive. If one is false then the other must be true but if  one is true rue, the other maybe true rue or false

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Proper (exclusive)

The tea The teach cher er isis eith either er pres presen entt or or abs absen ent. t. 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

Improper (inclusive) The student either studies math or english. But he is not studying math He is studying english y

The student either studies math or english He is studying math He is not studying english

y

alid Process of Disjunctive Syllogism

V

1. Positing ² when the minor minor premise premise accepts accepts one part part and

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

Conjunctive y

y

Consists of conjunctive proposit sition as the major premise and cate catego gori rica call prop propos osit itio ions ns as mino minorr prem premis isee and and conc conclu lusi sion on The suspect cannot be in Aparr rrii and Jolo at one time But report says he is in Aparr rrii Therefo efore, he cannot be in Jolo

Validity alidity of Conditional 1. Modus Ponens (affirm the antecedent in the minor, minor, affirm the consequent in the conclusion) y

V

pq P q

~pq ~p ~q

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

Validity y

alidity of Disjunction

V

Symbolic Logic y

...by the aid of symbolism, we can make transition in reasoning almost mech chaanically be the eye, which otherwise Alfred would call into play the higher faculties of the brain. ² Alfred Northwhitehead

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The use of symbols enables us to analyze statements and argu argume ment ntss quick quicklly and and effi effici cien entl tlyy Moder n logic·s approach does not focus on syllogisms but upon logic gical connectives that are fundamental in deductive arguments

Basic Concepts y

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Propositional Propositional Logic ² the logic of compound statements. statements. It uses variables and special symbols for operations. The logic of compound statements which relies solely on three key concepts: concep ts: TRUTH VALUE, LOGICAL OPERATORS OPERATORS and VARIABLES Simple Statement ² one that does not contain any other sta statemen ment as a component

Ex: Ex: Natu Naturre is changi changing ng dras drasti tica callllyy Hydrogen is a gas Shake Shakespea speare re wrote wrote,, ´Haml ´Hamletµ. etµ.

Basic Concepts y

Compound Statement² one that contains one simple or atomi tomicc sta stateme tement nt as a comp compon onen entt

Ex: It is not the case that Jose Rizal wrote ´Hamletµ If acorns are planted in the spring then oak trees will grow in the fall. Eith Either er Moria oriarty rty commi ommitt tted ed th thee murder der or Sta Stapleto letonn lied lied.. John is a student or Mary is a teacher. Gloria ria is an ent entrepr epreneu eneurr and Lalo is an employee. ee.

Basic Concepts Connective ²any words attached to one or more statements in order to creat eate a new state tatem ment ent Theey join Th join simp simple le sta stateme tement ntss into into comp compou ound nd sta stateme tement ntss * in Symbolic Logic, they are called, TRUTH FUNCTIONAL y

y

CONNECTIVE (v, (v,

.

, , )

a statement ent connecti ectivve where erein th thee truth ruth or falsity of such statement ent created by its use would be depending solely on the trut ruth or falsity of the stateme ement(s) to which it is attach cheed I t is not the case that all all lawy lawyers ers are are liar liarss Either  the class will have an open forum rum or  fore foreve verr hold hold thei theirr silen silence ce If  Napoleon eon wins the the war and  Fra France nce will ill rega egain pow power then they could rul rule over

Europe

Basic Concepts y

y

y

Truth Function ² a tr uth function that takes one or more trut ruth-values as its input and returns rns a single trut ruth-value as its output output (Conjun (Conjunctio ction, n, Disjunct Disjunction, ion, Implica Implication tion,, Equival Equivalence ence)) Trut ruth value ² refers to the falsity or trut ruth of each statement Logical Varia riables ² the letters which stand for any logic gical unit (Q, (Q, P, R, S,T.. S,T...e .etc tc))

Compound

Statements and their Connectives Function

Operator

Name

English Translation

Conjunction

.

Ampersand

and

Disjunction

v

Wedge

or

Conditional



Horseshoe

If...,then

Biconditional



Triple Bar

If and only If  

Negation

~

Tilde

not

y









(Ped (Pedrro pass passed ed the the fina finall exam exam)) and and (the (the cour course se.) .) p.c (Ei (Either ther Pedr edro pass passed ed the the fina finall exam exam)) or (he (he pass passed ed the the cour course se). ). pvq (If (If Pedr edro passe assess the the fina finall exam exam)) then then (he (he will will pass ass the the cour course se). ). pc (Pedro wil will pass the course) rse) if and only if (he passed the the final exa exam). pc Pedr edro passe assedd the the cou course rse but not the the fina finall exam exam.. p . ~c

Types

of Compound Statements

1. Conjunction ² this is a compound statement whose comp compon onen entt stat statem emen ents ts are are call called ed conjuncts.

-The connective is a centered dot (.) or the ampersand (&) whic whichh mean meanss ´and ´andµµ -other conjunction words used instead of ´andµ are: but, yet, however, althou hough, gh, wher hereas, neverthe rthelless, ss, a comma, ma, and and a semicolon Mari Mariaa and and Celi Celiaa are are gamb gamble lers rs.. Thee pres Th presid iden entt is happ happyy but tir tired. ed. Thee stud Th studen ents ts went ent to the the even eventt yet yet disa disapp ppoi oint nted ed some someon onee. Pepi epito lov loves to eat eat crab crabs; s; thou though gh he is a veget egetar ariian. an.

2. Disjunction Disjunction ² a compound compound statement statement whose component component parts are called disjuncts whic h means - The statement is connected by a ´veeµ or wedge which ´orµ 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.

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3. Mate Materi riaal Imp Implic licatio tion / Cond onditio itionnal ² a sta stateme temennt th thaat expr expres essses if  then rela relati tion onsh ship ip betw between een its its comp compon onent ent statem statement entss ... ...

Other Other connec connectiv tives es if P, Q P implies Q P entails Q P only if Q P thus Q P therefore Q y

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

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4. Material Equivalence / Biconditional ² two conditional statements which have been reversed but involving the same propositi propositions, ons, conjoined conjoined together. together. Expr Express esses es an if and only if relat elatio ions nshi hipp betw between een each each comp compou ound nd statements

Ex:Th :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 bein eings do not take care of the env environment ent.

. Negation ² reversal reversal of a statemen statementt Ex: I am not worr worried ied about your grades in logic. Mrs. Pirtola does not like like working working with with plastic in her her artwork. It is impossible for a sensible woman to like a conceited man. y

5

Truth Table y

y y

for Compound Statements

1. Conjunction P

Q

P . Q 

T

T

T

T

F

F

F

T

F

F

F

F

P . Q is true only if each component is true Future lawyers were the ones who started the riot last night and many people people got hurt hur t

Truth Table y

y

y

for Compound Statements

2. Disjunction P

Q

P v Q 

T

T

T

T

F

T

F

T

T

F

F

F

The only case in which a disjunction is false is when both its disjunct disjunctss are false false Either it was the group ¶s decision decision to create trouble or it was their leader·s leader·s decision alone.

Truth Table y

y

y

for Compound Statements

3. Material Implication / Conditional P

Q 

PQ 

T

T

T

T

F

F

F

T

T

F

F

T

The only instance when the conditional is false is when the antecedent is false and the consequent is true If Philip eats a piece of stone, then his teeth might break.

Truth Table y

y

y

for Compound Statements

4. Material Equivalence / Bi conditional P

Q 

PQ

T

T

T

T

F

F

F

T

F

F

F

T

Friday is the happiest happies t day of the week if and only if I am able to sleep for 8 hours. Goodbyes are not sorrowful if and only it is for the best interest of the parties involved.

Truth Table y

y y y y

for Compound Statements

. Negation

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P

~P

T

F

F

T

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

Exercise If A, B and a nd C are TRUE TRUE and an d X, Y and Z are false, which of the following statements are true? tr ue? Apply the truth tr uth table. 1. BY 2.

(A.C)X

3.

~(AC) v (X.Z)

4.

(X.A) v (B.Z)

5.

(ZvX)  (CvB)

6.

~[(AvB) . (CvX)]  (C~Y)

Tautologies,Contradictories, Contingent

Tautologies,Contradictories, Contingent y

y

y

A compound statement is tautologous if it is true regardless of the trut ruth value of its components All bach cheelors are either male or not male. M

~M

Mv~M

T

F

T

F

T

T

All crows are either black or not black ck..

Tautologies,Contradictories, Contingent y

y

y

A compound statement is self-contradictory if it is false regardless of the truth value of its components. I am a liar and that is not true. M

~M

M.~M

T

F

F

F

T

F

Never say never.

Tautologies,Contradictories, Contingent y

y

A compound statement is contingent if its tr uth value varie ries depending ing on the trut ruth value of the compo mponents. M

I

MI

T

T

T

T

F

F

F

T

T

F

F

T

If Marcos was a great public official, then Imelda was an hone honest st offic official ial..

Formal y

Proofs of Validity

1. Disjunctive Syllogism (DS) ² an argument that consists of a disju sjunctiv tive premise, a premise that deni enies one of the disjuncts and a conc onclusio usionn th thaat affir ffirms ms th thee oth ther er disju isjunnct

pvq ~p q y

Thisj will apologiz gize or Cornel rneliius will be angry gry This Th isjj will ill not not apolo pologi gizze Corn Cornel eliu iuss will will be angry ngry

Formal y

y

Proofs of Validity

2. Pure Hypothetica Hypotheticall Syllogism (HS) ² a syllogism whose propositions are conditional statements. pq qr pr

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

Formal y

Proofs of Validity

3. Modus Ponens Ponens (MP) ² affirming the antecedent 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 y

Proofs of Validity

4. Modus Modus Tollens ollens (MT) ² denying denying a consequ consequent ent

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

Formal y

y

Proofs of Validity

Constructive Dilemma (CD) ² the first premise consists of two conjoined conditional statements and, the second pre premise asserts rts the truth of one of the two antecedents. The conclusion which follows logically via 2 modus ponens steps asserts rts the truth value of at least one of th thee cons conseq eque uent ntss. p q.(rs)

pvr qvs If this punch contains gin, then Elisa will like it, and if it contains vodka, then then Carme armen n will will lik like it. it. This This pun punch cont ontain ains eith either er gin gin or vodka dka Therefore, Eli Elisa will like it or Carmen will like it.

Formal y

y

Proofs of Validity

. Destructive Dilemma ² the second premise asserts rts the falsity of at leas east one of the consequ equents. The concl nclusion which follows via 2 modus toll tollen enss asse assert rtss th thee falsi alsity ty of atlea tleast st one one of th thee ante antece cede dent ntss. 6

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 then Carme armen n will will lik like it. 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 contain vodka vodka

Formal y

Proofs of Validity

. Absorptio Absorptionn (Abs) ² p  q

7

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 y

Proofs of Validity

8. Simpl Simplific ificatio ationn (Simp)² p . q p

 June is practicing in the gym while classmates classmates have have gone home.  June is practicing in the gym. 9. Addition Addition (Add)(Add)- p pvq Today is Thurs Thursday day.. Ergo, today is Thursday or else I will be wearing my favorite blouse. y

Formal y

Proofs of Validity

10. Conjunc Conjunction tion ² p q p.Q

Cornelia is beautiful. beautiful. Cornelia is smart. Cornelia is beautiful and smart.