# A Class-Room Introduction to Logic

## May 4, 2009

### Unit-V: Nature of Proposition

Filed under: Logic,Proposition — Dr Desh Raj Sirswal @ 8:53 am

Proposition:

Propositions are the material of our reasoning. A proposition links nouns, pronouns and phrases to other words in a sentence. The word or phrase that the proposition introduces is called the object of the proposition.A proposition is a judgment expressed in a language and a judgment is a mental act in which two or more than two ideas are combined together.

Judgments have two types:

1. Affirmative- Indians are laborious.

2. Negative- Indians are not dull.

A proposition usually indicates the temporal, spatial or logical relationship of its object to the rest of the sentence as in the following examples:

The book is on the table.

The book is beneath the table.

The book is leaning against the table.

The book is beside the table.

She held the book over the table.

She read the book during class.

Components of Proposition:

Every proposition has three components called as term. Any word or word phrase, which is used in a proposition as a subject or predicate, is called as term.:

1. Subject- About whom something asserts or denies.
2. Predicate-What assert or deny.
3. Copula- Conjunct both subject and predicate terms. Copula will be negative or affirmative.

For example:

Sonia is a good orator.

S      C                  P

Difference with Sentences:

1. Propositions are different from sentences. Sentences have many kinds like questions, exclamations etc. But none of these can be asserted and denied. Truth and Falsity apply always to proposition, but not apply to questions or commands or exclamations.
2. Propositions must be differentiated from sentences by means of what they are asserted. Two different sentences consisting the same proposition.
3. Sentences are parts of some language, but propositions are not tied to any given language i.e. “It is raining”, “Barsat ho rahi hai”, both consist the same content.
4. A sentence called as proposition when it’s both term (subject and predicate) are nouns i.e. Ram is a man. “Flower is beautiful” is not a proposition because its predicate is adjective.
5. Proposition is always in present tense. But sentences are expressed in all tenses.
6. Proposition explains quantity and quality but sentence does not explain it.
7. All propositions are sentences but not all sentences are propositions.

Types of Proposition

According to the relation of terms proposition has three types:

Categorical Proposition: It is a type of proposition which has no condition for their assertion. – Roshan is a student.

Conditional or Hypothetical Proposition: A type of compound proposition, it is false only when the antecedent is true and the consequent is false.- If Ram will pass, then he will get a bicycle.

Disjunctive Proposition: A type of compound proposition; if true, at least one of the component of propositions must be true.-Ram is honest or clever.

Categorical Proposition:

A categorical proposition is simply a statement about the relationship between categories. It states whether one category or categorical term is fully contained with another, is partially contained within another or is completely separate.

A dog is an animal.

Some dogs are friendly.

No dog is a cat.

Propositions may have quality: either affirmative or negative.

They may also have quantity: such as ‘a’, ‘some’, ‘most’ or ‘all’. The ‘all’ quantity is also described as being universal and other quantities particular.

Aristotelian Four-fold Classification of Categorical Propositions:

Aristotle classified categorical proposition in four, based on Quality and Quantity distribution:

Universal Affirmative – All S is P. – A Type Proposition– All men are mortal.

Universal Negative     – No S is P. – E Type Proposition–  No men are immortal.

Particular Affirmative – Some S is P. –  I Type Proposition– Some men are wise.

Particular Negative    – Some S in not P. –O Type Proposition– Some men are not wise.

### Distribution of Terms

Both subject and predicate of a proposition are called as term. A term is a word or group of words which is either a subject or a predicate of a proposition.

A term is said to be distributed if it refers to all the members of a class. In the other words, a term is distributed when it includes or excludes all the members of a class. If a term includes or excludes only some members of a class, then it is undistributed.

In a categorical syllogism the distribution of terms depends on the quantifier:

A Type:      In “All A are B”-propositions the subject (A) is distributed.

E Type:      In “No A are B”-propositions both the subject (A) and the predicate (B) are distributed.

I Type :      In “Some A are B”-propositions neither the subject nor the predicate are distributed.

O Type :     In “Some A are not B”-propositions the predicate is distributed.

 Form Type Quality Quantity Distribution of X Distribution of Y All S is P A Affirmative Universal Distributed Undistributed No S is P E Negative Universal Distributed Distributed Some S is P I Affirmative Particular Undistributed Undistributed Some S is not P O Negative Particular Undistributed Distributed

Copi and Cohen state two rules about distribution of terms in valid syllogisms:

• 1. The middle term must be distributed  at least in one premise.
• 2. If the major term or the minor term is distributed in the conclusion, then it must be distributed    in the premises.

Venn and Boolean Expression of Categorical Proposition:

The modern interpretation of categorical logic also permits a more convenient way of assessing the truth-conditions of categorical propositions, by drawing Venn diagrams, topological representations of the logical relationships among the classes designated by categorical terms. The basic idea is fairly straightforward:

All S is P. – Universal Affirmative –A Type Proposition

Or S non-P = 0

No S is P. – Universal Negative- E Type Proposition

Or S P = 0

Some S is P.-Particular Affirmative-I Type Proposition

Or S P ≠ 0

Some S is not P.-Particular Negative-O Type Proposition

Or S non-P ≠0

Denotation and Connotation of Terms

Denotation denotes the objects, connotation connotes the characteristics. Denotation of a term refers to  the objects or things which possess the quality. Connotation refers to the set of characteristics essentially possessed by every object denoted by the term. For example, Man, Gita, Mohan, Kamal etc. Man means that possess morality and rationality.

Man= Denotation      Morality and rationality = Connotation