Hypothetical Syllogism

Hypothetical Syllogism – is a syllogism that has a hypothetical proposition as one of its premises.

3 Kinds of Hypothetical Syllogism:

a.       Conditional Syllogism

b.      Disjunctive Syllogism

c.       Conjunctive Syllogism

A.      Conditional Syllogism –  is a syllogism whose major premise is a conditional proposition. The major premise is composed of two parts: antecedent (ante = before) and consequent (sequi = follow). The antecedent is the component which states the condition while the consequent is the result which follows from the antecedent.

Examples:

1.       If you are worthy (antecedent), then you can have my blessing (consequent).

                                       But you are not worthy.

                                      Therefore, you cannot have my blessing.

2.       If the student is responsible enough (ante.), he can pass this subject (cons.).

                                       But he can pass this subject.

                                      Therefore, he is responsible enough.

3.       If the cloud is dark (ante.) , then it will rain (cons.).

But the cloud is dark.

Then, it will rain.

4.       If the blue litmus paper turns red (ante), then the chemical is acid (cons).

But the chemical is not acid.

Then the blue litmus paper will not turn red.

5.       If the tools are here (ante.), then we can start planting (cons.).

If they arrived early, then the tools are here.

Therefore, if they arrived early, then we can start planting.

6.       If August is your birthday (ante.), then you might be a Virgo (cons.).

But if we are not compatible, then you are not a Virgo.

Therefore, if we are not compatible, then August is not your birthday.

Rules in Conditional Syllogism

1.       To affirm the antecedent is to affirm the consequent, but to deny the antecedent does not mean denial of the consequent. Example 1 above is a violation of this rule. There are many ways in which you can have my blessing. It doesn’t mean that because you are not worthy, then you cannot have my blessing.

Other examples of violation of this rule:

               

                        He will attend if she is the presentor.

                But she is not the presentor.

                Therefore, he will not attend.

                If the operation is not successful, then he will die.

                But the operation is successful.

                Therefore, he will not die.

A violation of this rule is called fallacy of denying the antecedent.

2.       To deny the consequent is to deny the antecedent, but to affirm the consequent does not mean affirmation of the antecedent. Example 2 above is a violation of this rule. It doesn’t mean that because he can pass the subject that he is already a responsible student.

Other examples of violation of this rule:

               

                If the book is thick, then it contains a lot of ideas.

                But this book contains a lot of ideas.

                Therefore, it is thick.

                If soldiers are brave, then they will not leave their companion behind.

                But they will not leave their companion behind.

                Therefore, they are brave.

A violation of this rule is called fallacy of affirming the consequent.

Two Valid Conditional Syllogisms:

a.       Modus Ponens – (ponens = affirm)

-          a conditional syllogism in which the antecedent is affirmed in the minor premise and the consequent is affirmed in the conclusion. Example 3 above is a modus ponens.

Other examples:

               

                If she is interested, then she will give me her number.

                But it turns out she is interested.

                Thus, she gave me her number.

                Only when people learn to understand each other can there be genuine peace.

                But people have learned to understand each other.

                Therefore, there can be genuine peace.

                I will vote for him if he is really sincere.

                But he is sincere.

                Hence, I will vote for him.

b.      Modus Tollens – (tollens = deny)

-          a conditional syllogism in which the consequent is denied in the minor premise and the antecedent is denied in the conclusion. Example 4 above is a modus tollens.

Other examples:

                If she is interested, then she will give me her number.

                But she will not give me her number.

                Thus, she is not interested.

                Only when people learn to understand each other can there be genuine peace.

                But there is no genuine peace.

                Therefore, people have not learned to understand each other.

                I will vote for him if he is really sincere.

                But I will not vote for him.

                Hence, he is not sincere.

B.      Disjunctive Syllogism – it is a hypothetical syllogism in which the major premise is a disjunctive proposition.

Examples:

               

1.       Either he is a criminal or he is a non-criminal.

But he is a criminal.

Therefore, he is not a non-criminal.

2.       Either the flag is white or it is red.

But the flag is not red.

Therefore, it is white.

3.       She might be in the library or she is reading book.

But she is not in the library.

Therefore, she is reading book.

4.       Either they will lose or make a compromise.

But they will not make a compromise.

Therefore, there is no other option but for them to lose.

                Two Kinds of Disjunctive Syllogism

a.       Strict Disjunctive – when one, and only one, is true among the disjuncts (parts of disjunctive syllogism).

Rule: If one disjuct is affirmed, then the other must be denied, and if one is denied, then the other must be affirmed. Examples 1 and 2 above are disjunctive syllogism in a strict sense.

Other examples:

                The soul is either immortal or it is mortal.

                But the soul is immortal.

                Therefore, it cannot be mortal.

                The students can be a leftist or non-leftist.

                But these students are non-leftists.

                Therefore, these students are not leftists.

b.      Broad Disjunctive – at least one disjunct is true but both disjuncts can be true.

Rule: If one is affirmed, it does not mean that the other must be denied, since it can also be affirmed. But if one is denied, then, automatically, one is affirmed since at least one of the disjuncts is true. Examples 3 and 4 are of this type.

Other examples:

                The teacher is either in the classroom or he is computing grades.

                But the teacher is computing grades.

                Therefore, he is not in the classroom.

-          invalid

The teacher is either in the classroom or he is computing grades.

                But the teacher is not computing grades.

Therefore, he is in the classroom

-          valid

In this example, the teacher can be both in the classroom and is computing grades. Hence, we cannot say that since the teacher is in the classroom, he is not computing grades.

c.       Conjunctive Syllogism – it is a syllogism whose major premise is a conjunctive proposition.

Examples of a conjunctive proposition:

                One cannot be wealthy and poor at the same time.

You cannot serve both God and money.

                You cannot be both in Cotabato and Manila at the same time.

Examples of a conjunctive syllogism:

                One cannot be wealthy and poor at the same time.

                But you are wealthy.

                Therefore, you are not poor.

You cannot serve both God and money.

                But you love money.

                Therefore, you cannot serve God.

                You cannot be both in Cotabato and Manila at the same time.

                But you are in Manila.

                Therefore, you are not in Cotabato.

Rule: In a conjunctive proposition, only one of the components can be true, but both can be false. Hence if one is affirmed, it necessarily entails that one must be denied. However, if one is denied, it does not necessarily entail that one must be affirmed, for both of them can be denied without contradiction.

                Examples of violation of this rule:

                               

                One cannot be wealthy and poor at the same time.

                But you are not wealthy.

                Therefore, you are poor.

                                - invalid

It doesn’t mean that because you are not wealthy that you are already poor.

You cannot serve both God and money.

                But you don’t serve money.

                                Therefore, you can serve God.

                                                - invalid

                It doesn’t mean that because you don’t serve money you can serve God.

                You cannot be both in Cotabato and Manila at the same time.

                But you are not in Manila.

                                Therefore, you are in Cotabato.

                                                - invalid

It doesn’t mean that since you are not in Manila, then we can conclude that you are in Cotabato.

March 5, 2014

PHILO 110 – LOGIC

Assignment

General Instruction: Put all your answers on the spaces provided in this sheet. No erasure is allowed. Pass the accomplished seat work on Wednesday, March 5 during the class.

Name:                                                                                                                 

A.      Give the CONVERSE of the following propositions. If the proposition has no converse, write No Converse.

1.       All citizens in this country are Filipinos.

                                                                                                                                                                        .

2.       Some politicians are lawyers.

                                                                                                                                                                        .

3.       No indigents are illiterate.

                                                                                                                                                                        .

4.       There are lights in this room which are not functioning.

                                                                                                                                                                        .

5.       The president was in Malaysia.

                                                                                                                                                                        .

B.      Give the OBVERSE of the following propositions.

1.       All Supreme Court judges are intelligent lawyers.

                                                                                                                                                                        .

2.       There is no minority leader who belongs to the Administration’s party-list.

                                                                                                                                                                        .

3.       Some congressmen were involved in the PDAF scandal.

                                                                                                                                                                        .

4.       Some MILF members are not willing to give up their arms.

                                                                                                                                                                        .

5.       UN officials are people who urge China to respect international law.

                                                                                                                                                                        .

C.      Identify the truth values of the following propositions. Base your answer on the given proposition preceding them.

1.       All teachers are professionals. – True

a.       Some non-professionals are teachers. -                               

b.      No non-professionals are teachers. -                     

c.       All professionals are teachers. -                                

d.      Some non-teachers are professionals. -                               

e.      No professionals are teachers. -                               

2.       No boxers are non-fighters. – True

a.       All boxers are fighters. -               

b.      All fighters are boxers. -               

c.       Some non-fighters are boxers. -                              

d.      Some non-boxers are fighters. -                              

e.      No non-boxers are non-fighters. -                    

3.       Some teachers are doctors. – True

a.       All teachers are doctors. -           

b.      No non-teachers are doctors. -                 

c.       No doctors are teachers. -                  

d.      Some doctors are not teachers. -             

e.      Some teachers are not non-doctors. -                   

4.       Some businessmen are not arrogant. – True

a.       Some businessmen are non-arrogant. -                                

b.      No non-arrogant are businessmen. -                     

c.       Some non-arrogant are businessmen. -                                

d.      Some non-arrogant are not non-businessmen. -                              

e.      All arrogant are businessmen. -                                 

Immediate Inference - Eduction

is a type of immediate inference in which we deduce the truth of other propositions with a different structure from a given proposition.

Rule:  Particular terms cannot be turned into universal. What is true to some may not be true to all.

For example:

                All men are mortals.

From this proposition, I can deduce an apparently equivalent proposition: Some mortals are men, but NOT all mortal are men.

Remember that we do not give additional knowledge when we deduce propositions from a given proposition when we use eduction. In fact, all other new propositions deduced from a given proposition are not something new. In meaning, they are the same with the original proposition.

Kinds of Eduction:

1.       Conversion – it is done by simply interchanging the subject and the predicate. We call the original proposition convertend, and the new proposition converse.

Examples:

Convertend                                                                        Converse

All men are mortals.                                                       Some mortals are men.

Some politicians are liars.                                             Some liars are not politicians.

No stones are flowers                                                   No flowers are stones.

Some animals are not mammals.                              no converse

2.       Obversion – it is done by changing the quality of the proposition (from affirmative to negative, from negative to affirmative) and replacing the predicate with its contradictory. We call the original proposition obvertend, and the new proposition obverse.

Examples:

Obvertend                                                                          Obverse

All men are mortals.                                                       No men are non-mortals.

Some politicians are liars.                                             Some politicians are not non-liars.

No stones are flowers.                                                  All stones are non-flowers.

Some animals are not mammals.                              Some animals are non-mammals.

3.       Contraposition 1 – It is done by doing obversion, and then conversion. We call the original proposition contraposend  and the new proposition contraposit 1.

Examples:

Contraposend                                                                   Contraposit 1

All men are mortals.                                                       No non-men are mortals.

Some politicians are liars.                                             no contraposend 1

No stones are flowers.                                                  Some non-flowers are stones.

Some animals are not mammals.                              Some non-mammals are animals.

4.       Contraposition 2 – it is done by doing obversion, then conversion, then obversion again. We call the original proposition contraposend and the new proposition contraposit 2.

Examples:

Contraposend                                                                   Contraposit 2

All men are mortals.                                                       All non-men are non-mortals.

Some politicians are liars.                                             no contraposend 2

No stones are flowers.                                                  Some non-flowers are not non-stones.

Some animals are not mammals.                              Some non-mammals are not non-animals.

Immediate Inference - Oppositional Inference

A.      Oppositional Inference – it is a type of immediate inference in which we deduce the truth value of another proposition (can be A, E, I, O) from a given truth value of a proposition of the same structure.

Example:

                                “All men are mortals” is true.

                                Therefore, “No men are mortals” is _______.

                                The answer is false.

Types of Oppositional Inference:

1.       Contrary – it is the opposition between A and E propositions.

                        Rule: If one is true, the other is false. If one is false, the other is unknown.

Example:

        If one is true:

                        (A) All men are mortals. – True                  (E) No men are angels. – True

                        (E) No men are mortals. – False                 (A) All men are angels. – False

        If one is false:

                        (A) All men are angels. – False                   (E) No men are mortals. – False

                        (E) No men are angels. – Unknown         (A) All men are angels. – Unknown

Explanation: If one is false, we cannot immediately infer that the other is true because it might be the case that if one is false the other might also be false. Hence, the better answer is unknown.

2.       Sub-contrary – it is the opposition between I and O propositions.

                        Rule: If one is true, the other is Unknown. If one is false, the other is true.

Example:

        If one is true:

                        (I) Some animals are mammals. – True                  

                        (O) Some animals are not mammals. – Unknown

                        (O) Some animals are not mammals. – True

                        (I) Some animals are mammals. – Unknown

Explanation: If one is true, we cannot immediately infer that the other is false since it is possible that the other could also be true; hence, Unknown.

        If one is true:

                        (I) Some animals are flowers. – False                     

                        (O) Some animals are not flowers. – True

                        (O) Some mammals are not animals. – False

                        (I) Some mammals are animals. – True

3.       Sub-altern – it is the opposition between A & I propositions and E & O propositions.

Rule:      If the universal proposition is true, the particular proposition is true. If the universal proposition is false, the particular proposition is Unknown.

                If the particular proposition is false, the universal proposition is false. If the particular proposition is true, the universal proposition is Unknown.

Example:

        If the universal is true:

                        (A) All men are mortals. – True                  (E) No men are angels. – True

                        (I) Some men are mortals. – True             (O) Some men are not angels. – True

If the universal is false:

                (A) All men are angels. – False                   (E) No men are mortals. – False

                (I) Some men are angels – Unknown      (O) Some men are not mortals. Unknown

Explanation: If the universal is false, we cannot immediately infer that the particular is also false since it is possible that the particular can be true.

If the particular is true:

                (I) Some animals are mammals. – True

                (A) All animals are mammals. – Unknown

                (O) Some animals are not mammals. – True

                (E) No animals are not mammals. – Unknown

Explanation: If the particular is false, it is possible that the universal can be true or can be false; hence, Unknown.

If the particular is false:

                (I) Some animals are mammals. – False

                (A) All animals are mammals. – False

                (O) Some animals are not mammals. – False

                (E) No animals are not mammals. – False

4.       Contradictory – the opposition between A & O propositions and E & I propositions.

Rule: If one is true, the other is false. If one is false, the other is true.

Examples:

 (A) All men are mortals. – True | False

(O) Some men are not mortals. – False | True

(O) Some men are not mortals. – False | True

(A) All men are mortals. – True | False

(E) No men are angels. – True | False

(I) Some men are angels. – False | True

(I) Some men are angels. – False | True

(E) No men are angels. – True | False