The Hypothetical Syllogism

Hypothetical Syllogism is a syllogism that has a hypothetical proposition as one of its premise

Kinds of Hypothetical Syllogism:

1. Conditional Syllogism (“If…, then…”)

2. Disjunctive Syllogism (“Either…, or…”)

3. Conjunctive Syllogism (“Not both…, and…”)

Relationship of an Antecedent and its consequent

Note: a. An antecedent is false when only one premise is false, as well as when both premises are false.

b. Where the sequence is invalid, there is, strictly speaking, no sequence, antecedent, or consequent at all. (When the sequence is invalid, the apparent premises and conclusion are not related to one another.)

Relationship of an Antecedent to its Consequent

1. If the antecedent is true and sequence valid, the consequent is true. (A particularized statement of the principle of contradiction.)

2. If the antecedent is true and sequence invalid, the consequent is doubtful.

Every dog is an animal;

but no cat is a dog;

therefore, no cat is an animal.

3. If the antecedent is false and sequence valid, the consequent is doubtful.

Every dog is an animal;

but every cat is a dog;

therefore, every cat is an animal.

4. If the antecedent is false and sequence invalid, the consequent is doubtful.

Every cat is a monkey;

but no cat is a dog;

therefore, no dog is an monkey.

Relationship of a Consequent to its Antecedent

1. If the consequent is false and the sequence valid, the antecedent is false. (Only truth can flow from truth, every antecedent from which a false statement can flow must itself be false.)

2. If the consequent is false and the sequence invalid, the antecedent is doubtful. (When the sequence is invalid, anything can come after anything, since the consequent and the antecedent are not related to one another at all.)

Every cat is a dog;

but no cat is a terrier;

therefore, no terrier is a dog.

3. If the consequent is true and sequence valid, the antecedent is doubtful.

Squares have three sides;

but triangles are squares;

therefore, triangles have three sides.

4. If the consequent is true and the sequence invalid, the antecedent is doubtful. (If the antecedent of a true consequent is doubtful even when the sequence is valid, it is also doubtful when the sequence is invalid.)

The Basic Laws which serve as basis of Valid Inference

1. If the antecedent is true and the sequence valid, the consequent is true.

2. If the consequent is false and the sequence valid, the antecedent is false.

CONDITIONAL SYLLOGISM

A Conditional Syllogism is one whose major premise is a conditional proposition.

2 Types of Conditional Syllogism:

1. Mixed Conditional (the minor premise is a categorical proposition)

2. Purely Conditional (both of whose premises are conditional propositions)

Conditional Propositions is a compound proposition of which one member (the “then” clause) asserts something as true on the condition that the other member (the “if” clause) is true.

“If it is raining, the roof is wet.”

- The “if” clause or its equivalent is called the antecedent.

- The “then” clause or its equivalent is called the consequent.

Rules of the Mixed Conditional Syllogism:

1. If the antecedent is true and the sequence valid, the consequent is true.

Procedure:

1. Posit the antecedent in the minor premise and posit the consequent in the conclusion.

2. If the consequent is false and the sequence valid, the antecedent is false.

2. Sublate the consequent in the minor premise and sublate the antecedent in the conclusion.

Example of a Valid Form Conditional Syllogism:

Major Premise

“If your have acute appendicitis, you are very sick.”

Conclusion

Posit the Consequent

“Therefore you are very sick.”

Sublate the Antecedent

“Therefore you do not have acute appendicitis”

Minor Premise

Posit the Antecedent

“But you have acute appendicitis.”

Sublate the Consequent

“But you are not sick.”

Example of a Invalid Form Conditional Syllogism:

Major Premise

“If your have acute appendicitis, you are very sick.”

Conclusion

Posit the Antecedent

“Therefore you have acute appendicitis.”

Sublate the Consequent

“Therefore you are not very sick.”

Minor Premise

Posit the Consequent

“But you are very sick.”

Sublate the Antecedent

“But you do not have acute appendicitis.”

Purely Conditional Syllogism

The Purely Conditional Syllogism, which has conditional propositions for both its premises, has exactly the same forms and the same rules as the mixed conditional syllogism except that the condition expressed in the minor premise must be retained in the conclusion.

If A is a B, then C is a D;

but if X is a Y, then A is a B;

therefore, if X is a Y, then C is a D.

Exercise: Indicate the form, or procedure, illustrated by each of the following, and state whether the example is valid or invalid.

1. If the dentist is not skillful, he will cause his patient much pain;

but the dentist is skillful;

therefore he will not cause his patient much pain.

2. If this book possesses literary merit, it will be widely read;

but it will surely be a best seller;

therefore it must possess literary merit.

3. “If you have bad eyes, you will never make the team.”

“But my eyes are all right;

therefore you must admit that I will make the team.”

4. If materialism is true, you would expect an intimate connection between the condition of a man’s brain and his powers of thinking;

but there is such connection;

therefore materialism must be true.

5. If that bill passes, rents will rise;

but the bill will not pass;

therefore rents will not rise.

DISJUNCTIVE SYLLOGISM

A Disjunctive Syllogism is one whose major premise is a disjunctive proposition, whose minor premise sublates (or posits) one or more members of the major premise, and whose conclusion posits (or sublates) the other member or members.

A Disjunctive Syllogism is one that presents various alternatives and asserts that an indeterminate one of them is true. It consists of two or more members joined by the conjunctions “either … or…”. It is sometimes called an alternative proposition.

2 Kinds of Disjunctive Syllogism:

1. Strict Disjunctive (only one member is true and the others are false. If all the members except one are false, the remaining member must be true; and if one is true, the remaining members must be false).

2. Broad Disjunctive (at least one member is true but more than one may be true).

Rules for Disjunctive Syllogism:

1. If the minor premise posits one or more members of the major premise, the conclusion must sublate each of the other members.

It is either raining or not raining;

but it is raining;

therefore it is not not raining.

It is either raining or not raining;

but it is not raining;

therefore it is not raining.

2. If the minor premise sublates one or more of the members of the major premise, the conclusion posits the remaining members, one of which must be true. If more than one member remains, the conclusion must be a disjunctive in the strict sense.

It is either raining or not raining;

but it is not raining;

therefore it is not raining.

It is either raining or not raining;

but it is not not raining;

therefore it is raining.

Broad Disjunctive

In a Broad Disjunctive Syllogism, the major premise is a disjunctive proposition in a broad or improper sense. There is only one valid procedure: to sublate one (or more – but not all) of the members in the minor and posit the remaining member (or members) in the conclusion.

It is either A, or B, or C, or D – at least one of them;

but it is either A nor B;

therefore it is either C or D – at least one of them.

Exercise: If possible, complete the following syllogism. Are the major premises disjunctive propositions in the strict sense or in the broad sense?

1. He is either not speaking or lying;

but he is not speaking;

therefore he is …..

2. He is either not speaking or lying;

but he is lying;

therefore he is …..

3. John failed to pass such and such an exam, and is therefore either lazy or lacking in talent;

but John is lacking in talent;

therefore John is …..

4. John is either lazy or lacking in talent;

but John is not lacking in talent;

therefore John is …..

5. Either the man who drafted the Constitution of the United States were animated by the desire to protect their property and privileges, or they were trying to create a just government based on the ethical standards of right. Historical research has shown that the members did indeed wish to protect their property and privileges. And so it is certain that they …..

Criticize the following. Some are valid, others are not. Examine the disjunctive propositions to see if they include all possible alternatives. Are they disjunctive propositions in the strict or in the broad sense?

1. The order in the world owes its origin to mere chance or to an intelligent designer;

but the order of the world cannot be due to mere chance;

therefore it must be due to an intelligent designer.

2. He either violated the law, or else he was arrested unjustly;

but he did violate the law;

therefore he was not arrested unjustly.

3. Jesus Christ is either God or the world’s greatest deceiver;

but it is impossible to admit that He is the world’s greatest deceiver;

therefore we are compelled to admit that He is God.

CONJUNCTIVE SYLLOGISM

A Conjunctive Syllogism is one whose major premise is a conjunctive proposition, whose minor premise posits one or more members of the major premise, and whose conclusion sublates the other member of the major premise.

A Conjunctive Syllogism is one that denies the simultaneous possibility of two alternatives.

“A thing cannot both be and not be in the same respect”

Rules for Conjunctive Syllogism:

1. Posit one member in the major premise and sublate the other in the conclusion.

He cannot be in Manila and Cebu at the same time;

but he is now in Manila;

therefore he cannot now be in Cebu.

Exercise: 1. You cannot be married and be single too;

but he is married;

therefore he cannot be single.

2. A diplomat, it is sometimes said, is either not honest or not successful;

but John is a diplomat who is not successful;

therefore it looks as though John is at least honest.

3. It is impossible to study properly and at the same time to listen to the radio;

but he is listening to the radio;

therefore he cannot be studying properly.