Antilogism and the Validity of Categorical Syllogisms

Antilogism is another method to test the validity of categorical syllogisms. This test of validity is a type of indirect proof in which the conclusion of the syllogism to be tested is replaced by its contradictory. The antilogism of a valid syllogism must meet the three requirements, namely:

1. There must be two universal propositions and one particular proposition, or two equations and one inequation.
2. The two universal propositions (two equations) must have a common term between them which is once negative and once affirmative.
3. The other two terms must appear unchanged in the particular proposition (inequation).

Let us consider the example below.

Antilogism and the Validity of Categorical Syllogisms

Antilogism is another method to test the validity of categorical syllogisms. This test of validity is a type of indirect proof in which the conclusion of the syllogism to be tested is replaced by its contradictory. The antilogism of a valid syllogism must meet the three requirements, namely:

1. There must be two universal propositions and one particular proposition, or two equations and one inequation.
2. The two universal propositions (two equations) must have a common term between them which is once negative and once affirmative.
3. The other two terms must appear unchanged in the particular proposition (inequation).

Let us consider the example below.

Example 1:

All men are mortal.
All Filipinos are men.
So, all Filipinos are mortal.

How do we determine the validity of the syllogism above using the antilogism method?

First, let us symbolize the syllogism in the algebraic notation. Let M stand for men and F for Filipinos. The algebraic notation of the above syllogism is as follows:

Next, let us construct its antilogism by replacing the conclusion with its contradictory. The contradictory of a proposition in algebraic form is easily formulated by changing an inequality (particular) to an equality (universal), or an equality (universal) to an inequality (particular). Thus, the antilogism of the example above is:

Now, let us check to see if the antilogism meets the three requirements  mentioned above. As we can see:

1. There are three equations, namely: propositions (that is, premises) 1 and 2, and 1 inequation (that is, the conclusion).
2. There is a common term between the equations (universal propositions), which is once negative and once affirmative, namely:
3. The other two terms are unchanged in the inequation (conclusion), namely:

Hence, the above syllogism is valid because it meets the three requirements for antilogism of valid syllogisms.

Let us consider another example.

Example 2:

All professionals are former amateurs.
But some former amateurs are wealthy persons.
Therefore, some wealthy persons are professionals.

Let us check whether the syllogism is valid or invalid.

1. The first requirement is met: the first premise and the conclusion are equalities, that is, universal propositions.
2. The second requirement is also met: there is a common term between the equations (universal propositions) which is one negative and the other affirmative, namely:

3. But the third requirement is not met: the other two terms in the equations (that is, universal propositions), namely

are changed in the inequation (that is, particular proposition), namely: F and W.

Hence, the above syllogism is invalid because it does not meet the three requirements for the antilogism of a valid syllogism.

Practice Test

Determine the validity of the arguments or syllogisms below using the antilogism method.

Example 1:

All criminals are guilty of a felony.
But some politicians are guilty of a felony.
Therefore, some politicians are criminals.

Example 2:

Some drivers are traffic law violators.
Some government employees are drivers.
Therefore, some government employees are traffic law violators.

Example 3:

Nurses are sweet lovers.
But Kit is a nurse.
Therefore, Kit is a sweet lover.

Note: Please email  learnphilosophy@philonotes.com for the answers.