CHAPTER 3 PROPOSITIONS

3. PROPOSITIONS

a) Traditional classification of proposition into Categorical and Conditional

b) four- fold classification.
c) Reduction of sentences to their logical forms.
d) Distribution of terms in A, E, I, O propositions.

Propositions are sentences used in logic. These are of various types, as we can express a matter of fact in any way. To convey any meaning, we may just use a subject less statement like 'it rains.' or we may use just a subject predicate relational statement like 'India is larger than Japan in land area.' we may also use subject predicate class membership statement like 'some subjects are easy,' or 'no subjects are easy to study for exam.' These propositions are either simple, i.e. have only one subject and predicate, or compound, i.e. having at least two subjects and two predicates and connecting words that join the two or more simple propositions in a link to convey some meaningful relationship between them.

In traditional logic, given by Aristotle, only one type of propositions were treated useful in logical arguments. They were subject predicate class membership type of statements. Aristotle had classified these subject predicate class membership propositions that he used to call propositions; into two types on the basis of their attributes. Attributes are characteristics that are basis of any piece of expression.

These attributes were, quality and quantity.

Quality states assertion or denial of information indicated in piece of expression.

Quantity states number of individuals indicated by subject term in a proposition.

Each attribute has two sub attributes.

So, quality is of two types, affirmative and negative.

Also, quantity is of two types they are, singular and general.

General propositions are further classified in two types, universal and particular.

Let us see this classification of traditional propositions at a glance:

Each proposition has to have at least one quality & at least one quantity. So, we have six types of traditional propositions. These proposition types are as follows; singular affirmative, singular negative, universal affirmative, universal negative, particular affirmative, particular negative.

a) Traditional classification of proposition into Categorical and Conditional

in traditional logic given by Aristotle, we study Only the simple subject predicate propositions that indicate class membership. Here, we have two types of relationships in the subject and predicate. Depending on these relations, the propositions are classified into categorical and conditional.

If the affirmation or denial in a proposition depends on any condition, then the proposition is conditional. If it is not dependent on any condition, then it is categorical.

According to some other definition, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the subject term) are included in another (the predicate term). Categorical propositions can be simple or compound, but they are necessarily class membership type propositions. In simple categorical proposition, we have one subject and one predicate, in a compound, we have two subjects & two predicates.

A conditional proposition is a proposition that has only one affirmation in which one part of the proposition depends on the other part.

Some others say that Conditional propositions are the compound propositions that contain at least two subjects and two predicates, where we have two simple statements such that the truth of one depends on the truth of another.

In either case, we must admit that conditional proposition is the proposition that talks of some condition under which one part of the proposition is true.

So, when we go further to convert the traditional propositions into propositional form, we may be able to say that the universal propositions are conditional in nature and particular are categorical in nature.

b) Four- fold classification.

We have seen above that propositions have a quality and quantity. The quantity of a proposition depends on its subject. It is this quantity that decides whether the proposition is going to be just categorical or conditional.

When subject of proposition represents single individual, it is singular proposition. When the subject of a proposition represents a group of individuals, it is a general proposition. General propositions are further classified into Universal and Particular.

When the subject tells something about the whole group represented by it, the proposition is known to be universal. When the subject tells something about some members of the group represented by it, the proposition is known to be particular.

The propositions are also classified using another criteria of quality and this makes them affirmative or negative.

So, both singular and general propositions are either affirmative or negative.

Aristotle in his classification used to count singular proipositions in the category of universal propositions. Only difference they had with normal universal propositions was that these did not have their particular counterparts.

As a result, we have four types of general propositions, on the basis of both the quality as well as quantity.

This is how the four types of general propositions are; A, E, I, & O.

The quantity and quality of these are as follows:

Type	S	P
A	Universal	Affirmative
E	Universal	Negative
I	Particular	Affirmative
O	Particular	Negative

We can express any relationship of subject and predicate into these four ways.

The relationship between propositions having same subject and predicate but having different quality and / or quantity is called as the relation of opposition of propositions. Relation of opposition between these propositions is as follows:

When two universal propositions differ in quality, they are called CONTRARY. When two particular propositions differ in quality, they are SUB-CONTRARY. When two propositions of same quality, differ in quantity, they are SUB-ALTERN. When propositions differ both in quality and quantity they are CONTRADICTORY.

This relationship is also shown in a square of opposition.

The square of opposition of proposition shows two universals on the top and two particulars at the bottom. On the left side of the square, we have affirmative type of proposition and on the right side we have negative type of proposition. So, TOP is universal, BOTTOM is particular, LEFT is affirmative, RIGHT is negative. Since all 4 sides of square are connected, each side shows one quality & one quantity.

Each proposition is either true or false. So, if a given proposition is true, the proposition having same subject and predicate and that differs in quality or quantity or both may be true or false or uncertain. This relationship is called as the relation of truth value between various propositions having opposition relation.

The relation of truth values between these opposite propositions is as follows:

If a universal proposition is true, its contrary is false, its sub-altern is true and its contradictory is false.

If a universal proposition is false, its contrary is uncertain, its sub-altern is uncertain and its contradictory is true.

If a particular proposition is true, its sub-contrary is uncertain, its sub-altern is uncertain and its contradictory is false.

If a particular proposition is false, its sub-contrary is true, its sub-altern is false and its contradictory is true.

The opposition relation of proposition is the foundation of all traditional logic, syllogism, and various types of mediate and immediate arguments used in traditional logic. The immediate inferences having relationship based on opposition of proposition, is also called the relationship of EDUCTION where we can convert one type of proposition into another maintaining its meaning and validity.

But to do all this, it is necessary to understand the relationship of subject and predicate in a given proposition in a perfect way. This can be done easily by taking help of mathematical logic and set theory.

The expression of proposition in such a format by using set theory to indicate the relationship between subject and predicate is called venn diagram.

Venn Diagrams:

Venn diagram (also known as a set diagram or logic diagram) is a diagram that shows all possible logical relations between a finite collection of different sets. Venn diagrams were conceived around 1880 by John Venn.

The general propositions represent the relation of two groups indicated by the subject and predicate, and so, they can be represented symbolically using the venn diagram method used in mathematics.

To do this, we use two intersecting circles. The circle on the left represents the subject, and the one on the right, represents the predicate.

To represent "A" proposition, we shade the part of the circle of subject, that is outside that of predicate. This shows that set of subject outside predicate is empty.

ATo represent "E" proposition, we shade the part of the circle of subject, that is inside predicate. This shows that the set of subject inside the predicate is empty.

E

To represent "I" proposition, we put a cross in part of the circle of subject, that is inside that of predicate. This shows that set of subject inside predicate is not empty.

I

To represent "O" proposition, we put a cross in part of subject circle, outside that of predicate. This shows that the set of subject outside the predicate is not empty.

O

When these propositions are symbolized, we change them in a specific format so that we can show the class membership of the subject and the predicate terms.

c) Reduction of sentences to their logical forms.

We have four standard forms of categorical propositions A, E, I and O. They have following structure & form.

A = 'All S is P'

E = 'No S is P’

I = 'Some S is P'

O = 'Some S is not P'

The method of symbolizing propositions using their quality and quantity is known as the method of Quantification and the symbols used to indicate the quantity of the subject are known as quantifiers.

Let us see how this is done:

Singular propositions:

Affirmative:

“Ramu is a boy” is symbolized as Br

Negative:

“Sita is not a boy” is symbolized as ~Bs

General propositions:

These are of four kinds as we have seen earlier. They are symbolized as follows:

"A" proposition:

Subject-less:

“Everything perishes” will be written as “Given any x, x is Perishable.”

This is symbolized as follows:

Given any x = (x), x is perishable = (Px),

so whole proposition is, (x)(Px)

Subject-predicate:

“All crows are birds” will be written as

“Given any x, if x is a crow, then x is a bird”

This is symbolized as follows:

Given any x = (x), If x is a crow = Cx, then = , , x is a bird = Bx

So, the whole proposition is,

(x)(Cx Bx)

"E" proposition:

Subject-less:

“Nothing is Permanent” will be written as Given any x, x is not Permanent.

This is symbolized as follows

Given any x = (x), x is not permanent = (~Px),

so whole proposition is, (x)(~Px)

Subject-predicate:

“No crows are red” will be written as

“Given any x, if x is a crow, then x is not red”

This is symbolized as follows:

Given any x = (x), If x is a crow = Cx, then = , , x is not red = ~Rx

So, the whole proposition is,

(x)(Cx ~Rx)

"I" proposition:

Subject-less:

“Lions exist” will be written as There is an x, such that, x is a Lion.

This is symbolized as follows:

There is an x such that = (x), x is a lion = Lx

So, the whole proposition is, (x)(Lx)

Subject-predicate:

“Some roses are red” will be written as

“There is an x such that, x is a rose, and x is red”

Here, since both words begin with R, we take R for subject & D for predicate,

This is symbolized as follows:

There is an x such that, = (x), x is a rose = Rx, then = ., x is red = Dx

So, the whole proposition is,

(x)(Rx . Dx)

"O"proposition:

Subject-less:

Ghosts do not exist is be written as There is an x, such that, x is not a Ghost.

This is symbolized as follows:

There is an x such that = (x), x is not a ghost = ~Gx

So, the whole proposition is, (x)(~Gx)

Subject-predicate:

“Some buses are not red” will be written as

“There is an x such that, if x is a bus, then x is not red”

This is symbolized as follows:

There is an x such that = (x), x is a bus = Cx, and = . , x is not red = ~Rx

So, the whole proposition is,

(x)(Bx . ~Rx)

When we symbolize, the first letter of the subject term is taken as a capital letter and small x is written after it to indicate the singular variable that is quantified in the beginning. This is how we symbolize general propositions in traditional classification

Thus, we know that the logical structure of any categorical proposition exhibits the following four items Quantifier (Subject term) copula (predicate term) in order.

Here the first item is the 'quantifier' or more precisely the words expressing the quantity of the proposition. This is attached to the subject term.

The second item in any regular logical proposition is the subject term.

The third item is copula, placed in between the subject and predicate term. The quality of the proposition is expressed in and through the copula. .

The fourth item is predicate term, that expresses something about the subject. This comes after the copula in a proposition that has a regular order.Thus, a categorical proposition which is in standard form must exhibit explicitly the subject, the predicate, the copula, its quality and quantity. Let us call a categorical proposition regular if it is in its standard form, otherwise it is called irregular.

In our ordinary language most categorical propositions are irregular in nature. Even irregular categorical propositions can be put in their regular form. It should be noted that while reducing an irregular categorical proposition into its standard form, we should pay enough attention to the meaning of the proposition so that the reduced proposition is equivalent in meaning to its irregular counter-part.

Before describing the method of reduction of irregular propositions into their regular forms, it is good to understand the reasons for irregularity. The irregularity of any categorical proposition may be due to one or more of these following factors.

1. Copula is not clear or it is mixed with verb which forms part of predicate
2. Logical ingredients are not arranged in their proper logical order.
3. Quantity is not expressed by a proper word like 'all', 'no' (or none), 'some' etc.
4. All exclusive, exceptive and interrogative propositions are clearly irregular.
5. Quality is not specified by attaching the sign of negation to the copula.

In light of this, let us describe systematically the method of reduction of an irregular categorical proposition into its standard form (or into a regular proposition).

Let us see with examples the method of reduction.

I. Reduction of categorical propositions whose copula is not stated explicitly.

Let us consider an example of irregular proposition, where copula is not explicit.

"All sincere students deserve success".

This is an irregular proposition. Here, the copula is mixed with main verb.

The method of reducing such irregular sentences into regular ones is as follows:

The subject and the quantifier of the irregular proposition should remain as they are, while the rest of the proposition may be converted to a class forming property (i.e. term) which would be our logical predicate.

In our above example 'All' is the quantifier attached to the subject 'sincere students'. We should not touch the quantifier nor the subject term of the proposition, they should remain where they are.

On the other hand, the rest of the proposition 'deserve success' should be converted into a class forming property 'success deserving'.

This should be our logical predicate. Then we link the subject term with the predicate term with a standard copula.

Thus, "All sincere students deserve success." Irregular proposition.

"All sincere students are success deserving." = A - Proposition.

"All people seek power." Irregular proposition.

"All people are power seekers." A – Proposition.

"Some people drink Coca Cola." Irregular proposition.

"Some people are Coca Cola drinkers." I – proposition

II. Irregular propositions where the usual logical ingredients are all present but are not arranged in their logical order.

Consider the following examples of irregular propositions.

"All is well that ends well" and "Ladies are all affectionate."

In these cases, first we have to locate the subject term and then rearrange the words occurring in the proposition to obtain the regular categorical proposition.

Such reductions are usually quite straight forward.

Thus we reduce the above two examples as given below.
"All is well that ends well." Irregular proposition
"All things that end well are things that are well." A - Proposition
"Ladies are all affectionate." Irregular proposition
"All ladies are affectionate." A – Proposition

III. Statements in which quantity is not expressed by proper quantity words.

Some propositions do not contain word like 'All', 'No', 'some' or contain no words to indicate the quantity. We reduce such a type of irregular proposition into its logical form as explained below.

Here we have to consider two sub-cases :

(i) where there is indication of quantity but no proper quantity words like 'All', 'No', on 'Some' are used

(ii) where the irregular proposition contains no word to indicate its quantity.

These errors are of the following types:

(a) Affirmative sentences that begin with words like 'every', 'any', 'each' are to be treated as A-propositions, where such words are to be replaced by the word "all" and rest of the proposition remains as it is or may be modified as necessary. The followings are some of the examples of this type.

"Every man is liable to commit mistakes." Irregular proposition.

"All men are persons who liable to commit mistakes." A – Proposition.

"Each student took part in the competition." Irregular proposition.

"All students are persons who took part in the competition." A – Proposition

"Any one of my students is laborious." Irregular proposition.

"All my students are laborious." A – Proposition.

A negative sentence that begins with a word like 'every', 'any', 'each', or 'all' is to be treated as an O-proposition. Any such proposition may be reduced to its logical form as shown below.

"Every man is not honest". Irregular proposition

"Some men are not honest." O – Proposition

"Any student cannot get first class." Irregular proposition.

"Some students are not persons who can get first class." O – Proposition.

"All is not gold that glitters." Irregular proposition.

"Some things that glitter are not gold." O - Proposition.

(b) Sentences with singular term or definite singular term without the sign of negation are to be treated as A-proposition.

For example, "Ram is mortal.",

"The oldest university of Orissa is in Bhubaneswar." are A-propositions.

Here the predicate is affirmed of the whole of the subject term. On the other hand, sentences with singular term or definite singular term with the sign of negation are to be treated as E-propositions.

For example, "Ram is not a student" and "The tallest student of the class is not a singer" are to be treated as E-propositions. These are cases where the predicate is denied of the whole of the subject term.

IV. Sentences beginning with the words like 'no', 'never', 'none' are to be treated as E-propositions.

The following sentence is an example of this type.

"Never men are perfect." Irregular proposition

"No man is perfect." E – Proposition

V. Affirmative sentences with words, like 'a few', 'certain', 'most', 'many' are to be treated as I-propositions, while negative sentences with these words are to be treated as O-propositions.

Since the word 'few' has a negative sense, an affirmative sentence beginning with the word 'few' is negative in quality. A negative sentence beginning with the word 'few' is affirmative in quality because it involves a double negation that amount to affirmation. The following are examples of above type.

"A few men are present." Irregular proposition.

"Some men are present." I – proposition.

"Certain books are good." Irregular proposition.

"Some books are good." I – proposition.

"Most of the students are laborious." Irregular proposition.

"Some students are laborious." I – proposition.

Here 'most' means less then 'all' and hence it is equivalent to 'some'.

"Many Indians are religious." Irregular proposition.

"Some Indians are religious." I – proposition.

"Certain books are not readable." Irregular proposition

"Some books are not readable." O – Proposition

"Most of the students are not rich." Irregular proposition.

"Some students are not rich." O – Proposition

"Few men are above temptation." Irregular proposition

"Some men are not above temptation." O – Proposition

"Few men are not selfish." Irregular proposition

"Some men are selfish.'

VI. Any statement whose subject is qualified with words like 'only', 'alone', 'none but', or 'no one else but' is called an exclusive proposition.

Here, the term qualified by any such word applies exclusively to the other term.

In such cases the quantity of the proposition is not explicitly stated.

This is the reason why such statements are tricky and they can mislead or indicate a contrary meaning if not reduced to correct form in the right way.

The propositions beginning with words like 'only', 'alone', 'none but' etc are to be reduced to their logical form by the following procedure.

While converting such statements, first interchange the subject and the predicate.

Then replace the words like 'only', 'alone' etc with 'all'.

Now it will become a regular proposition.

For example,

"Only Oriyas are students of this college." Irregular proposition.

"All students of this college are oriyas." A – Proposition.

"The honest alone wins the confidence of people." Irregular Proposition.

"All persons who win the confidence of people are honest." A-proposition.

VII. Propositions in which the predicate is affirmed or denied of the whole subject with some exception is called an exceptive proposition.

An exceptive proposition may be definite or indefinite. If the exception is definitely specified as in case of "All metals except mercury are solid" then the proposition is to be treated as universal and if the exception is indefinite, as in case of "All metals except one is solid", the proposition is to be treated as particular.

"All metals except mercury are solid." is a universal proposition.

It means, "All non-mercury metals are solid."

Now let us consider an example where the exception is indefinite.

For example, "All students of my class except a few are well prepared".

This is to be reduced to an I-proposition as given below.

"All students of my class except a few are well prepared" is Irregular proposition.

"Some students of my class are well prepared." is an I – proposition.

VIII. There are impersonal propositions where the quantity is not specified.

Consider for example, "It is cold", "It is ten O'clock".

In such cases propositions in question are to be reduced to A-proposition because the subject in each of these cases is a definite description.

"It is cold". Irregular proposition

"The whether is cold." A – Proposition.

"It is ten O'clock." Irregular proposition.

"The time is ten O'clock." A – Proposition.

There are some propositions where the quantity is not specified. In such cases we have to examine the context of its use to decide the quantity.

For example, consider following sentences

(1) "Dogs are carnivorous",

(2) "Men are mortal",

(3) "Students are present."

In first two examples, the quantity has to be universal but in the third case, it is particular. Thus, their reductions into logical form are as follows.

"Dogs are carnivorous." Irregular proposition.

"All dogs are carnivorous." A – Proposition.

This is so because we know that "being carnivorous' is true of all dogs.

"Men are mortal." Irregular proposition.

"All men are mortal." A – Proposition

Here 'being mortal' is generally true of men.

But in the proposition "Students are present",

we mean to assert that some students are present".

So the proposition "Men are mortal" is reduced to

"All men are mortal"

But in the example "Students are present",

'being present' is not generally true of all students.

So the proposition "Students are present" is reduced to

"Some dents are present" which is an I-proposition.

Thus the context of use of a proposition determines the nature of the proposition.

IX. Problematic propositions are particular in meaning.

For example "The poor may be happy" should be treated as a particular proposition, because what such a proposition asserts is that it is sometimes true and sometimes false.

Thus, "The poor may be happy" is reduced to "Some poor people are happy", which is an I-proposition.

X. Similarly, there are propositions where the quantity is not specified but their predicates are qualified by the words like 'hardly', 'scarcely', 'seldom'.

Such propositions should be treated as particular negative.

For example, "Businessmen are seldom honest", is an irregular proposition.

It is reduced to "Some businessmen are not honest".

If such a proposition contains the sign of negation that these proposition is to be treated as an I-proposition.

For example, "Businessmen are not seldom honest." is to be reduced to "Some businessmen are honest", which is an I - proposition.

This is so because it involves a double negation which is equivalent to affirmation.

d) Distribution of terms in A, E, I, O propositions.

Distribution of Terms : When we state something about the entire group indicated by the Terms, the Term is distributed. In a universal proposition Subject is Distributed and in a negative proposition Predicate is Distributed.

Quantity of Proposition : It is the quantity of the group of the subject of a proposition. This is of two types. Universal & Particular. The Universal quantity distributes the subject not the particular.

Quality of Proposition : - It is the quality of the Predicate of the proposition. This is affirmative or negative. Affirmative says that subject or its group belongs to the group of predicate. Here the predicate terms is not distributed. Negative quality says that the subject or its group does not belongs to the group of predicate. Here the predicate is distributed.

TABLE explaining the DISTRIBUTION of terms

Type	S	P
A	Universal	Affirmative X
E	Universal	Negative
I	Particular X	Affirmative X
O	Particular X	Negative