CHAPTER 7. EDUCTIONS

7. EDUCTIONS
a) Conversion and Obversion and other Immediate inferences.
b) Laws of Thought as applied to propositions.

a) Conversion and Obversion and other Immediate inferences.

A proposition that falls in the category of traditional classification, i.e. that is either universal affirmative, or universal negative or particular affirmative or particular negative, has seven more types of relations or ways to express the same subject and predicate terms. These relationships are called Eduction relations.

The concept of Immediate inferences or Eduction relations like obversion conversion etc. is based exactly on this. Let us see how this works:

Here, we are writing the original term relation in a proposition as S==P. To show that we are using the term that is opposite to the original one we are drawing a line above the term. So, when the negative of subject term is used, we write S. Similarly, when we are using the term that is negative of predicate term we write P.

by using the mathematical combination rule, we can get total eight combinations where subject term and predicate term appears once in a statement and each one is either affirmative or negative. This means, if we have s-p as original, it is one of the eight combinations. Rest seven are its relations. This can be written as follows:

S – P – – – P – S

S – ~P – – – P – ~S

~S – P – – – ~P – S

~S – ~P – – – ~P – ~S

The table below can explain these relations & names of each relation at a glance.

S==P	ORIGIONAL	P==S	Converse
S== P	Obverse	P==S	Obverted Converse
S==P	Partial Inverse	P==S	Partial Contrapositive
S==P	Full Inverse	P==S	Full Contrapositive

To understand how this is done, we must see how to check validity of proposition used in any relation of above types by taking example of each type of proposition and converting it in all the above relationships.

The conversion method and understanding of the meaning of the converted statements itself can explain why in some cases no conversion is possible.

Remember, for accepting any type as an equivalent expression of any type of proposition, it must follow the basic Logic rules.

1. It must clear the distribution test
2. It must not distort the original meaning.

Let us take A proposition;

e.g. let us say “All study is a useful thing”

We write it as 'S a P'

Let us see Eduction relations of this.

Here, we need to check for all the four proposition type options for each relation.

Original : All study is a useful thing S a P

Obverse: = S e P

All study is a non-useful thing. A

No study is a non-useful thing. E

Some study is a non-useful thing. I

Some study is not a non-useful thing. O

Converse: P i S

All useful thing is a study. A

No useful thing is a study. E

Some useful thing is a study. I

Some useful thing is not a study. O

Obverted Converse: P o S

All useful thing is non-study. A

No useful thing is non-study. E

Some useful thing is non-study. I

Some useful thing is not non-study. O

Partial Inverse: S o P

All non-study is a useful thing. A

No non-study is a useful thing. E

Some non-study is a useful thing. I

Some non-study is not a useful thing. O

Full Inverse: S i P

All non-study is a non-useful thing. A

No non-study is a non-useful thing. E

Some non-study is non-useful thing. I

Some non-study is not non-useful thing. O

Contra-positive (partial): P e S

All non-useful thing is a study. A

No non-useful thing is a study. E

Some non-useful thing is a study. I

Some non-useful thing is not a study. O

Contra-positive (full): P a S

All non-useful thing is a non-study. A

No non-useful thing is a non-study. E

Some non-useful thing is non-study. I

Some non-useful thing is not non-study. O

Let us take E proposition;

e.g. let us say “No study is a useless thing”

We write it as 'S e P'

Let us see Eduction relations of this.

Here, we need to check for all the four proposition type options for each relation.

Original : No study is a useless thing S e P

Obverse: = S a P

All study is a non-useless thing. A

No study is a non-useless thing. E

Some study is a non-useless thing. I

Some study is not a non-useless thing. O

Converse: P e S

All useless thing is a study. A

No useless thing is a study. E

Some useless thing is a study. I

Some useless thing is not a study. O

Obverted Converse: P a S

All useless thing is non-study. A

No useless thing is non-study. E

Some useless thing is non-study. I

Some useless thing is not non-study. O

Partial Inverse: S i P

All non-study is a useless thing. A

No non-study is a useless thing. E

Some non-study is a useless thing. I

Some non-study is not a useless thing. O

Full Inverse: S o P

All non-study is a non- useless thing. A

No non-study is a non-useless thing. E

Some non-study is non- useless thing. I

Some non-study is not non-useless thing. O

Contra-positive (partial): P i S

All non- useless thing is a study. A

No non- useless thing is a study. E

Some non- useless thing is a study. I

Some non- useless thing is not a study. O

Contra-positive (full): P o S

All non-useless thing is a non-study. A

No non-useless thing is a non-study. E

Some non-useless thing is non-study. I

Some non-useless thing is not non-study. O

Let us take I proposition;

e.g. let us say “Some study is a useful thing”

We write it as 'S i P'

Let us see Eduction relations of this.

Here, we need to check for all the four proposition type options for each relation.

Original : Some study is a useful thing S i P

Obverse: = S o P

All study is a non-useful thing. A

No study is a non-useful thing. E

Some study is a non-useful thing. I

Some study is not a non-useful thing. O

Converse: P i S

All useful thing is a study. A

No useful thing is a study. E

Some useful thing is a study. I

Some useful thing is not a study. O

Obverted Converse: P o S

All useful thing is non-study. A

No useful thing is non-study. E

Some useful thing is non-study. I

Some useful thing is not non-study. O

Partial Inverse: S x P

All non-study is a useful thing. A

No non-study is a useful thing. E

Some non-study is a useful thing. I

Some non-study is not a useful thing. O

Full Inverse: S x P

All non-study is a non-useful thing. A

No non-study is a non-useful thing. E

Some non-study is non-useful thing. I

Some non-study is not non-useful thing. O

Contra-positive (partial): P x S

All non-useful thing is a study. A

No non-useful thing is a study. E

Some non-useful thing is a study. I

Some non-useful thing is not a study. O

Contra-positive (full): P x S

All non-useful thing is a non-study. A

No non-useful thing is a non-study. E

Some non-useful thing is non-study. I

Some non-useful thing is not non-study. O

Let us take O proposition;

e.g. let us say “Some study is a not useless thing”

We write it as 'S o P'

Let us see Eduction relations of this.

Here, we need to check for all the four proposition type options for each relation.

Original : Some study is not a useless thing S o P

Obverse: = S i P

All study is a non-useless thing. A

No study is a non-useless thing. E

Some study is a non-useless thing. I

Some study is not a non-useless thing. O

Converse: P x S

All useless thing is a study. A

No useless thing is a study. E

Some useless thing is a study. I

Some useless thing is not a study. O

Obverted Converse: P x S

All useless thing is non-study. A

No useless thing is non-study. E

Some useless thing is non-study. I

Some useless thing is not non-study. O

Partial Inverse: S x P

All non-study is a useless thing. A

No non-study is a useless thing. E

Some non-study is a useless thing. I

Some non-study is not a useless thing. O

Full Inverse: S x P

All non-study is a non- useless thing. A

No non-study is a non-useless thing. E

Some non-study is non- useless thing. I

Some non-study is not non-useless thing. O

Contra-positive (partial): P i S

All non- useless thing is a study. A

No non- useless thing is a study. E

Some non- useless thing is a study. I

Some non- useless thing is not a study. O

Contra-positive (full): P o S

All non-useless thing is a non-study. A

No non-useless thing is a non-study. E

Some non-useless thing is non-study. I

Some non-useless thing is not non-study. O

Let us see EDUCTION at a glance in brief:

Original	Obverse	Partial Inverse	Full Inverse	Converse	Obverted Converse	Partial Contra-positive	Full Contra-positive
S - P	S - P	S - P	S - P	P - S	P - S	P - S	P - S
S a P	S e P	S o P	S i P	P i S	P o S	P e S	P a S
S e P	S a P	S i P	S o P	P e S	P a S	P i S	P o S
S i P	S o P	S x P	S x P	P i S	P o S	P x S	P x S
S o P	S i P	S x P	S x P	P x S	P x S	P i S	P o S

In detail:

Relation	Changed	Type	Original
Original = S-P			All S is P	No S is P	Some S is P	Some S is not P
			A	E	I	O
Obverse	All S is non P	A		All S is non P
S-P	No S is non P	E	No S is non P
	Some S is non P	I				Some S is non P
	Some S is not non P	O			Some S is not non P
Converse	All P is S	A				X
P-S	No P is S	E		No P is S		X
	Some P is S	I	Some P is S		Some P is S	X
	Some P is not S	O				X
Obv Converse	All S is non P	A		All S is non P		X
P-S	No S is non P	E				X
	Some S is non P	I				X
	Some S is not non P	O	Some S is not non P		Some S is not non P	X
Part Inverse	All non S is P	A			X	X
S-P	No non S is P	E			X	X
	Some non S is P	I		Some non S is P	X	X
	Some non S is not P	O	Some non S is not P		X	X
Full Inverse	All non S is non P	A			X	X
S-P	No non S is non P	E			X	X
	Some non S is non P	I	Some non S is non P		X	X
	Some non S is not non P	O		Some non S is not non P	X	X
Part Contra +ve	All non-P is S	A			X
P-S	No non P is S	E	No non P is S		X
	Some non P is S	I		Some non P is S	X	Some non P is S
	Some non P is not S	O			X
Full Contra +ve	All non P is non S	A	All non P is non S		X
P-S	No non P is non S	E			X
	Some non P is non S	I			X
	Some non P is not non S	O		Some non P is not non S	X	Some non P is not non S

b) Laws of Thought as applied to propositions.

In 18th, 19th, & early 20th Century, scholars who followed the Aristotelian and Medieval tradition in logic, spoke of the “laws of thought” as the basis of all logic.

The usual list of logical laws includes three axioms:

The law of identity,

The law of non-contradiction, and

The law of excluded middle.

The thinking in logic must have a solid base and these three laws provide this base. They are the foundation of logical thinking.

The law of identity could be summarized as the patently unremarkable but seemingly inescapable notion that things must be, of course, identical with themselves. Expressed symbolically: “A is A,” where A is an individual, a species, or a genus. Although Aristotle never explicitly enunciates this law, he does observe, in the Metaphysics, that “the fact that a thing is itself is [the only] answer to all such questions as why the man is man, or the musician musical.”

This suggests that he does accept, unsurprisingly, the perfectly obvious idea that things are themselves. If, however, identical things must possess identical attributes, this opens the door to various logical maneuvers.

One can, for example, substitute equivalent terms for one another and, even more portentously, one can arrive at some conception of analogy and induction. Aristotle writes, “all water is said to be . . . the same as all water . . . because of a certain likeness.” If water is water, then by the law of identity, anything we discover to be water must possess the same water-properties.

Aristotle provides several formulations of the law of non-contradiction, the idea that logically correct propositions cannot affirm and deny the same thing:

“It is impossible for anyone to believe the same thing to be and not be.”

“The same attribute cannot at the same time belong and not belong to the same subject in the same respect.” “The most indisputable of all beliefs is that contradictory statements are not at the same time true.” Symbolically, the law of non-contradiction is sometimes represented as “not (A and not A).”

The law of excluded middle can be summarized as the idea that every proposition must be either true or false, not both and not neither. In Aristotle’s words, “It is necessary for the affirmation or the negation to be true or false.” Symbolically, we can represent the law of excluded middle as an exclusive disjunction: “A is true or A is false,” where only one alternative holds. Because every proposition must be true or false, it does not follow, of course, that we can know if a particular proposition is true or false.

Despite challenges to these so-called laws, Aristotelians inevitably claim that such counterarguments have unresolved ambiguity equivocation, on a conflation of what we know with what is actually the case, on a false or static account of identity, or on some other failure to fully grasp the implications of what one is saying.

In short, we can say that our thinking naturally follows some thumb rules that are listed as the three main laws. They are called as laws of thought. These are, law of identity, law of non-contradiction, and law of excluded middle.

Let us see these laws in a simple way:

1. LAW OF IDENTITY: This law says that something is what it is. In short, we can say, “A is A”. That means, to prove or state the existence of something that already is, we need not have any other proof. The presence of anything is self-proven. This is where we say, “If I am, then I am. Or, I am existing, therefore I am existing. Or, I am myself.” Another common way of expressing law of identity is, “Sun is Sun”, “Moon is Moon”, “Tree is Tree” and so on.
2. LAW OF NON-CONTRADICTION: This law is also written as and called as Law of Contradiction by some people. This states a simple thing, a thing cannot be true and false at the same time at the same place. If someone is saying so, he is telling a lie. If a thing is existing, then it cannot be absent from the same place at the same time when and where it is claimed to exist. This means, two contradictory statements cannot be true together. For example, if I say, “I have Logic book in my hand” I cannot say at the same time, in the same place, “I do not have Logic in my hand.”
3. LAW OF EXCLUDED MIDDLE: This law states that there is no third option between a statement and its contradiction. This means, when we give two contradictory options for anything, there is no third way possible. This law is useful especially when we have to categorically state some options about something. Use of this law removes all ambiguity & vagueness of expression. For example, when I say, “Either I believe in what you say or I do not.” there ios no third way. The person to whom I am talking cannot say that I believe in him and bot believe.at the same time, he cannot talk of any third possibility.

This is how we describe the laws of thought. We must remember that these are the foundation of logical thinking and all of us have been using them in our thinking much before we learned that they are the basis of thinking. They form the basic foundation of any logical activity. Experiments may show that even animals and insects use these laws in their thinking when they think and choose to do anything.