CHAPTER 4. MODERN CLASSIFICATION OF PROPOSITIONS

4. MODERN CLASSIFICATION OF PROPOSITIONS
a) Aim of Modern classification, kinds of Simple and Compound propositions

b) Basic Truth Tables for Compound propositions.

Traditional logic deals with limited types of propositions. So, it was difficult to deal with many types of propositions. This is the reason why Modern Logic or formal logic came into existence. It follows and expands on Logic given by Aristotle.

This logic simplifies the way in which we reason. It also makes difference between form and content of propositions and arguments. This logic has introduced mathematical formal methods in logic and with the help of these methods, we can test the valid relationships between terms and propositions in no time.

Let us see the modern classification and its details:

a) Aim of Modern classification,.

Modern logic aims at re-organizing the logical concepts and expanding the boundaries of logical thinking. While doing so, we look at the statements used in logic with a different perspective.

This is the reason why we classify them a bit differently here on the basis of terms, verbs and connectives used in them. This way to classify the propositions makes it easy to understand the relationship between parts of the propositions in an argument as here we make them have objective and mathematical appearance.

Modern classification tries to simplify our thinking and also organize it more effectively so that more types of reasoning can be included in the classification.

Kinds of Simple and Compound propositions & basic Truth Tables

In modern logic, simple proposition is defined as one with only one verb in it. Such a proposition has no connective in it. The simple proposition have no connective. They have only one verb and do not indicate any complicated meaning.

The Simple propositions are classified into two types,

a) subject-less propositions, b) subject-predicate propositions,

The subject-predicate propositions are further classified into

i) relational propositions and ii) class membership proposition.

Let us see the simple proposition types in details:

a) Subject-less propositions, are propositions that have only predicate and no subject. These are symbolized by using single alphabet that stands for predicate.

b) Subject-predicate propositions, are the propositions that have a subject, a predicate and a verb. The subject-predicate propositions are further classified into two types. Relational and class-membership. Let us see these types:

i) Relational propositions are the propositions that show some type of relationship between the term of subject and that of predicate. This means in this type, both the subject and predicate are singular terms.

ii) Class membership proposition shows that the subject term belon gs to the class indicated by predicate. So, here, predicate term is general.

Modern logic also defines a compound proposition that has one or more components connected using one or more connectives.

The compound propositions have at least one connective used in them. They have one or more component that connectives join meaningfully.

When we express these propositions in an objective way, we can explicitly state whether the given compound proposition is true or not on the basis of truth or falsity of the components it connects and the type of connective used.

In modern logic the connecting words, commonly called as connectives, are classified into two types, viz. Monadic and Diadic.

Monadic connective is a connective that works on only one proposition.

The class of monadic connectives has only one connective in it.

This is negation.

This means in modern logic, negative proposition is no more with different quality.

It is a compound proposition.

A negation is expressed by words like 'no, never, not' etc.

While symbolizing a negation, we use the symbol ' ~ ' that is called curl or tilde.

A negation is true when the component to which it is attached is false.

Diadic connectives are connectives that work on two propositions. We have four diadic connectives. They are; conjunction, disjunction, implication and equivalence.

Conjunction is expressed by words like 'and, but'.

While symbolizing this, we use the symbol ' . ' called a dot.

A proposition with conjunction is true only when both its components are true.

Dis-junction is expressed by words like 'either, or.'

While symbolizing this, we use the symbol ' v ' called a vedge.

A proposition with disjunction is false only when both its components are false.

Implication is expressed by words like 'If...then, unless...'

While symbolizing this, we use the symbol ' ' called a horse-shoe.

A proposition with implication is false only when its antecedent, i.e. the first component is true and the consequent, i.e. the second component is false.

Equivalence is expressed by words like 'if and only if... then.'

While symbolizing this, we use the symbol ' ' called a dot.

A proposition with conjunction is true only when both its components are true.

Let us see this classification at a glance:

Proposition

Sentence that asserts

|

| |

Simple Compound (with connective)

No connective one or more components

| |

| | | |

Subject-less Subject-predicate Monadic Diadic

No subject | one component two component

| | …........................|

| | Negation 1 = Conjunction = .

Relational Class-membership =No, Not 2 = Dis junction = V

= ~ 3 = Implication =

4 = Equivalence=

b) Basic Truth Tables for Compound propositions

We saw the connectives and their symbols. Now let us see how the propositions are symbolized in modern classification.

Compound propositions are symbolized in modern classification by taking a capital alphabet for the first letter of the predicate of first component simple statement, and a capital alphabet for the first letter of the predicate of the second component simple statement.

Between these two alphabets, we put the symbol for the connective that is connecting these two components.

This means, if we have a proposition,

'If Logic is easy, then many will learn it.'

we take 'E' for 'logic is easy' and ' L' for 'many will learn it'.

The connective here is implication. The symbol for it is, .

We write this in between E and L. This reads as 'E L'

This is how we can symbolize any given proposition in modern logic.

So, if we take standard alphabets P for first component and Q for second, we can express all compound proposition types as follows:

Negation: ~P

Conjunction: P . Q

Dis-junction: P v Q

Implication: P Q

Equivalence: P Q

The method we use to check the validity of their relations is called the method of constructing truth tables. While doing this, we check the possibilities of truth and falsity in both the components.

We arrange these possibilities here using the 2ⁿ method of calculating the possibilities. Here 2 stands for the two truth value options, viz. True and false. The alphabet 'n' stands for number of variables present in the compound proposition.

If a proposition has only one variable, that means only one simple proposition, even if it is repeated, then we have 2¹ = 2 possibilities of truth value combinations.

If a proposition has two different simple statements as components, then we have 2² = 4 possibilities of truth value combinations.

If a proposition has three different simple statements as components, then we have 2³ = 8 possibilities of truth value combinations.

If a proposition has four different simple statements as components, then we have 2⁴ = 16 possibilities of truth value combinations.

Of course, for learning the basic truth-functional tables, we need to see only the first two options, i.e. the statements with 2 and 4 combination options.

When we have a single component as in ~P, we write the truth table as:

P ~P

T F

F T

When we have two components as in P . Q, P v Q, P Q, P Q, we make the truth tables by using the terms of validity of each connective as follows:

Let us write possibilities for all proposition types together for easy understanding.

P Q P . Q P v Q P Q P Q

T T T T T T T T T T T T T T

T F T F F T T F T F F T F F

F T F F T F T T F T T F F T

F F F F F F F F F T F F T F

On the basis of the above table, we can pick up the table for any relavent proposition type to be symbolized and form a truth table for it.

While doing this, follow the following steps:

Write the first part of 'P Q' and the truth values under it

then write the proposition type as per the connective.

Like,

Negation: ~P

Conjunction: P . Q

Dis-junction: P v Q

Implication: P Q

Equivalence: P Q

Then form the relevant truth table for it.

Suppose we have a proposition like, 'Law is useful and Religion is peaceful”

We symbolize it as 'U . P' Then we form a truth table for it as:

U P U . P

T T T T T

T F T F F

F T F F T

F F F F F

Suppose we have a proposition like, 'Law is useful or Religion is peaceful”

We symbolize it as 'U v P' Then we form a truth table for it as:

U P U v P

T T T T T

T F T T F

F T F T T

F F F F F

Suppose we have a proposition like, 'If Law is useful then Religion is peaceful”

We symbolize it as 'U P' Then we form a truth table for it as:

U P U P

T T T T T

T F T F F

F T F T T

F F F T F

Suppose we have proposition, 'If & only If Law is useful then Religion is peaceful”

We symbolize it as 'U P' Then we form a truth table for it as:

U P U P

T T T T T

T F T F F

F T F F T

F F F T F