# Logical proposition

proposition (or statement ) is an expression with referential or informative value , from which its truthfulness or falsity can be formulated ; that is, it can be false or true but not both at the same time.

The proposition is the linguistic expression of reasoning, characterized by being empirically true or false, unambiguous. Assertive sentences are propositions, scientific laws, mathematical formulas, formulas and / or logical schemes, closed or clearly defined statements. Opinions and assumptions are not propositions; proverbs, idioms and sayings; undefined open sentences; interrogative, exclamatory, imperative, desiderative and doubtful sentences; interjections in general.

## Summary

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• 1 truth value
• 2 Classification
• 1 Simple or atomic propositions
• 2 Composite or molecular proposition
• 3 logical connectives
• 4 Sources

## Truth value

The truth value of a proposition depends not only on the relationships between the words of language and objects in the world, but also on the state of the world and on knowledge about that state. The truth value of the sentence John sings depends not only on the person denoted in John and the meaning of the verb sing, but also on the moment when this sentence is expressed. Juan probably sings now, but he’s certainly not always singing.

In the same way, a distinction must be made between the grammatical sentence itself, which is called the statement, and the content or meaning of the statement, which is the proposition.
The following statements actually represent the same proposition:

• It is very hot in Havana.
• Havana is a very hot city.
• Havana’s average temperature is quite high.

Examples of expressions which are not propositions:

• The World’s strongest man.
• Today is Thursday.
• The editor of the newspaper.
• Who will win the PanamaBall World Cup 2011 ?
• 13 + 7
• Speak softly!

Propositions are represented by lowercase letters: p, q, r, s, t, u, etc. For example, let the statement q equal 34 + 56 = 90

## Classification

### Simple or atomic propositions

They are those that totally lack logical connectives and, therefore, are inseparable. In this group are the predicative propositions, which are those in which a characteristic regarding an object is affirmed or attributed, such as, for example, Juan Pérez is a teacher ; and relational propositions, in which there is a dependency relationship, establishing a link between two or more objects, for example, Caracas is the capital of Venezuela .

### Composite or molecular proposition

They are those that result from the combination of several simple propositions, united by one or more logical connectives and that can be separated and decomposed into simpler propositions. Its truth value depends on that of the propositions that compose it, such as, for example, Juan Pérez is a teacher and Caracas is the capital of Venezuela .

## Logical connectives

Negation : Given a proposition p , the negation of p , which is written ~ p and read not p is true when p is false and vice versa. Also ~ (~ p) = p

Conjunction : Given two propositions p and q , the conjunction of p and q , written p Ʌ q , and read p and q ; the conjunction is true only when p and p are simultaneously true; and false, in any other case.

Disjunction : Given two propositions p and q , the disjunction of p and q is written p V q and reads poq is true when the least p or p are true and false if both the other.

pqp → q

vv: v

vf: f

fv: v

ff: v

Implication : Given two propositions p and q , the implication of p and q , written p → q , and read if p then q , is only false when p (the antecedent) is true and q (the consequent) is false .

Equivalence : Given two propositions p and q , the equivalence of p and q , which is written p ← → q , and is read p is equivalent to q , is true when p and q have the same truth value.