Inference

Inference – mental operation where the mind passes from the

truth of one to the truth of another, logically connected with,
and necessarily following from the first

Types of Inference
1. On the basis of the number of premises: Direct (immediate)
and indirect (mediate);
2. On the basis of the extension of the conclusion: deductive
and inductive;
3. On the basis of the principle involved: a) traditional or
Aristotelian (class inclusion and class exclusion), and b)
relational inference.
Inference
Types of Direct Inference

1) Equipollence – Logical Opposition

2) Eduction or Equivalence – Conversion, Obversion,
Contraposition
3) Relational Inference
4) Inference involving possibility and existence
Equipollence (Oppositional Inference)
The Laws of Inference Governing Logical Opposition
1) The Law of Contradiction – Contradictory propositions (A-O,
E-I) cannot be both true or both false.
2) The Law of Contrariety – Contrary propositions (A-E) cannot
be both true, but they may be both false.
3) The Law of Subalternation – (A-I, E-O)The truth of the
universals carries, involves, implies or includes, the truth of
the particulars, but not vice-versa; The falsity of the
particulars means or involves the falsity of the universals,
but not vice-versa.
4) The Law of Subcontrariety – Subcontrary propositions (I-O)
cannot false, but they may be both true.
Equipollence (Oppositional Inference)
Application of the Laws of Logical Opposition
1) Contradiction (A-O, E-I)
If A is true then O is false;
If O is true then A is false;
If E is true then I is false;
If I is true the E is false.

If A is false the O is true;

If O is false then A is true;
If E is false then I is true;
If I is false then E is true.
2) Contrariety (A-E)
If A is true then E is false;
If E is true then A is false;
If A is false then E is Doubtful/Undetermined;
If E is false then A is Doubtful/Undetermined
Equipollence (Oppositional Inference)
Application of the Laws of Logical Opposition
3) Subalternation (A-I, E-O)
If A is true then I is true;
If E is true then O is true;
If I is true then A is Doubtful/Undetermined;
If O is true then E is Doubtful/Undetermined

If I is false then A is false;

If O is false then E is false;
If A is false then I is Doubtful/Undetermined;
If E is false then O is Doubtful/Undetermined.
4) Subcontrariety (I-O)
If I is false then O is true;
If O is false then I is true;
If I is true then O is Doubtful/Undetermined;
If O is true the I is Doubtful/Undetermined.
Eductions and Relational Inference
Logical Equivalence or Eductions – two propositions involved are
logically equivalent and always have the same meaning and truth
value.

In Logical Equivalence, the mind passes immediately from one

proposition (or premise), to another proposition (conclusion) having
the same meaning and truth value as the first; and vice-versa.

Types:
1) Conversion
2) Obversion
3) Contraposition
Eductions and Relational Inference
Types:
1) Conversion – immediate, eductive, inference which derives a
proposition from another by interchanging the positions of the
subject term and the predicate term, but without changing the
meaning and truth value of the original proposition.
2) Obversion – the mental process of passing from an affirmative
proposition to another having the same meaning as the first
but expressed in the double negative; or the mental process of
passing from a double negative proposition to its equivalent in
the affirmative
3) Contraposition – involves the mental process of obverting,
converting and finally obverting again.
Eductions and Relational Inference
Types:
1) Conversion
Simple E and I propositions
Accidental A propositions
O propositions cannot be converted
2) Obversion applicable to A, E, I and O propositions
Steps:
a.) retain the subject term
b.) negate the verb
c.) contradict the predicate term
3) Contraposition valid when applied to A and O propositions
Mental Exercise:
Give the correct converse of each of the following convertends:
1) No man is a monkey.
2) Some adults are delinquents.
3) Some students are activists.
4) All squares are quadrilaterals of equal sides.
5) All squares are rectangles of equal sides.
6) No woman is an angel.
7) No bat is a bird.
8) All whales are mammals.
9) No dolphin is a fish.
10) Some members are professionals.
Mental Exercise:
Give the correct obverse of each of the following obvertend:
1) No man is without honor.
2) With God’s infinite mercy, no sin is unpardonable.
3) Some laborers were dismissed without previous clearance from
the Department of Labor.
4) All the allegations contained in the complaint have been proved.
5) Some of the refugees are non-Vietnamese.
6) Some members of the OPEC are non-Arabic countries.
7) No test tube baby is naturally conceived offspring.
8) Some of the flying objects have up to now been unidentified.
9) Some prices of prime commodities are uncontrollable.
10) All natural rights are inalienable.
Mental Exercise:
Give the correct contrapositive of each of the following:
1) All whales are marine mammals.
2) Some convicts are not recidivists.
Direct Relational Inference
Characteristics:

 Its premise and its conclusion are not logically equivalent and do
not use the same terms;
 Based mostly on a definition or law of science or mathematics
arrived at and established by generalization or induction (not
class inclusion and/or exclusion);
 This is mostly taken up in modern logic;
 The controlling or guiding principle is the matter or content of the
propositions contained in the argument;
 But still governed by the supreme law of deductive inference –
that the conclusion drawn should be strictly implied in the
premises and should necessarily follow from them.
 Direct Relational Inference
Illustration:
Eductive Inference Relational Inference
No Jew is an Arab. X is to 1 as 10 is to 5
Therefore, no Arab is a Jew. Therefore, x is 2.

All human beings have basic rights. X3 = 1000

Therefore, women have basic rights. Therefore, x = 10.

Some electrical appliances are generators of heat. X is to electricity as the difference in water
Therefore, some generators of heat are electrical appliances. level is to the flow of water.

Therefore, x is the difference in electrical

potential (or the electromotive force).
Inference Involving Possibility and Existence
In point of existence (taking reality as the point of reference):

1. Certainty is stronger than Possibility

 Certainty = actuality, existence, truth, reality

2. Impossibility is stronger than Non-actuality or Non-existence

 Impossibility covers past, present, future
 Non-actuality/Non-existence covers only the past and the present
Inference Involving Possibility and Existence
Laws Governing Inference Involving Possibility and Existence

1. De esse ad posse valet illatio. It is legitimate inference to pass from

actuality or existence to possibility.
2. De posse ad esse non valet illatio. It is not valid to argue from mere
possibility to actuality or existence.
3. De non-posse ad non-esse valet illatio. It is valid inference to pass
from impossibility to non-actuality or non-existence.
4. De non esse ad non posse non valet illatio. It is not valid to infer
impossibility from non-actuality.
Mental Exercise:
In the light of the rules presented, test each of the following inference
for validity:
1. We never won the championship game.
Therefore, we can never win the championship game.
2. The plane must have touched down at Hongkong Airport yesterday
afternoon.
Therefore, the plane actually touched down at Hongkong Airport yesterday
afternoon.
3. Man probably descended from the ape.
Therefore, man actually descended from the ape.
4. There is strong evidence that there are living things in the outer space.
Therefore, there are living things (such as microorganisms) in outer space.
5. Perpetual motion machines are surely possible.
Therefore, there are actually perpetual motion machines.
Mental Exercise:
Explain on the basis of the golden rule of inference (the conclusion should necessarily
flow from the premise) why the following types of reasoning are not valid:

1. Pedro loves Maria. (premise)

Therefore, Maria loves Pedro. (conclusion)
2. Samar is bigger than Bohol.
Therefore, Samar has a bigger population than Bohol.
3. Many lawyers are liars.
Therefore, most lawyers are liars.
4. It is probable that there is life in Mars.
Therefore, there is life in Mars.
5. No price control has ever lowered the prices.
Therefore, no price control can ever lower down the prices.
6. The object is not white.
Therefore, it is black.
The Categorical Syllogism
A syllogism in general is an argument wherein, from two propositions having
a common term, and one of which is a universal, we derive another logically
connected with, and necessarily following from the two.
As a type of mediate inference, a syllogism may be defined as the mental
process in which we compare two ideas and with the aid of a third idea, conclude
whether they agree or disagree.
Composition:
 3 propositions:
 2 premises
 The 1st premise is called the major premise
 The 2nd is called the minor premise
 The conclusion is introduced by therefore (Lt. ergo)
 1 conclusion
 All propositions are categorical propositions
 Being a form of mediate inference, the syllogism contains three (3) terms: (1) the
major term (P), (2) the minor term (S), and (3) the middle term (M)
The Form of a Syllogism
The form of a categorical syllogism consist of:
Figure, which denotes the way the middle term (M) is arranged in the syllogism; and
Mood, which refers to the types of propositions (A, E, I, O) in which the major
premise, the minor premise and the conclusion, respectively appear in the syllogism.
The different figures of a syllogism:

M–P P–M M–P P–M

S–M S–M M–S M–S
S–P S–P S–P S–P
(1) (2) (3) (4)
The Form of a Syllogism
Examples:
1. All metals are elements.
Mercury is a metal.
Therefore, mercury is an element.

2. No shark is a mammal.
All whales are mammals.
Therefore, No whales are sharks.

3. All members are students.

Some members are employees.
Therefore, some employees are students.
Valid Forms of a Syllogism
First Figure Second Figure Third Figure Fourth Figure

AAA – 1 EAE – 2 AII – 3 AEE – 4

EAE – 1 AEE – 2 EIO – 3 EIO – 4
AII – 1 EIO – 2 IAI – 3 IAI – 4
EIO – 1 AOO – 2 OAO – 3
Rules of the Categorical Syllogism
1. There should be three and only three terms in the syllogism to used
in the same meaning throughout;
2. Neither the major term, nor the minor term may be made universal
in the conclusion if it is merely a particular in the premises;
3. The middle term should be used at least once as universal in the
premises;
4. The middle term may not appear in the conclusion;
5. No conclusion follows from two negative premises;
6. No conclusion follows from two particular premises;
7. If one of the premises is particular, the conclusion should also be
particular;
8. If one of the premises is negative, the conclusion should also be
negative.
Syllogistic Fallacies:
1. The fallacy of four terms, (also called “four-legged syllogism,”
quaternitas terminorum, or quaternion) occurs whenever a term is
used in two different meanings – in violation of the first rule of the
syllogism. Example:
All stars are heavenly bodies.
Alma is a star.
Therefore, Alma is a heavenly body.
2. Fallacy of illicit major is an error in syllogistic reasoning wherein the
major term which is only a particular term in the premises, becomes
a universal term in the conclusion. Example:
All dogs are quadrupeds.
Cats are not dogs.
Therefore, cats are not quadrupeds.
Syllogistic Fallacies:
3. Fallacy of illicit minor occurs whenever the minor term is used as
particular or undistributed in the premise but is made universal or
distributed in the conclusion. Example:
All the literates are voters.
All voters are citizens.
Therefore, all citizens are literates.
4. Fallacy of undistributed middle. This fallacy is committed whenever
the middle term is used twice as an undistributed or particular term
in the premises. Example:
All diamonds are precious things.
Gold is a precious thing.
Therefore, gold is a diamond.
 This syllogism violates rule number 3.
Mental Exercise:
1. Only laborers dismissed without just cause are entitled to
reinstatement.
He was dismissed without just cause.
Therefore, he is entitled to reinstatement.

2. No living things are generated by purely inanimate matter.

Microorganisms are living things.
Therefore, microorganisms are not generated by purely
inanimate matter.

3. Any violation of a basic human right is against international law.

Plane highjacking is a violation of a basic human right.
Therefore, plane highjacking is against international law.