The propositional logic is the oldest and simplest forms of logic . Using a primitive representation of language, it allows us to represent and manipulate assertions about the world around us. It is also called the Propositional Calculus .
Summary
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- 1 Fundamentals
- 2 Propositions
- 1 Logically equivalent propositions
- 2 Simple propositions
- 3 Composite propositions
- 3 Logical operators
- 1 Conjunction
- 2 Weak disjunction
- 3 Strong disjunction
- 4 Denial
- 4 Notation
- 5 Referential publications
- 6 Sources
Fundamentals
Propositional logic allows reasoning, through a mechanism that first evaluates simple sentences and then complex sentences, formed by using propositional connectives, for example Y ( AND ), O ( OR ). This mechanism determines the veracity of a complex sentence, analyzing the truth values assigned to the simple sentences that comprise it.
Propositions
The proposition is a fundamental element of mathematical Logic that works only with propositions or Propositional Logic or Propositions. Therefore, the first thing is to recognize which sentences or phrases constitute propositions and which do not.
Logically equivalent propositions
Two propositions are said to be logically equivalent if they both have the same truth values for all combinations of values of the simple propositions that compose them. That is, in each of the interpretations of both, the truth values of both propositions are equal.
Simple propositions
A proposition or statement is a sentence that can be true or false but not both at the same time. In other words, the main property of a proposition is that it takes one of possible truth values, either they are true or they are false.
Composite propositions
A compound proposition is one that is not just composed of a simple proposition. Examples:
- p: Two and two are four; four and two are six
- q: 4 <6 and 6 <8
- r There is good sun, a sea of foam, fine sand and Pilar wants to go out to wear her feather hat.
- s: There is no final in the history of the National Baseball Series as contested as this year’s 2002.
- t: The final of the 2002 Soccer World Cup was won by Brazil and did not disappoint his fans.
- u: The number 8 is even or odd.
Logical operators
They are those that allow to join simple propositions to form compound propositions.
Conjunction
The conjunction operator is used to connect two propositions that must be met in order for a true result to be obtained. In other words, the compound proposition that contains a conjunction is affirming that the simplest propositions that compose it are fulfilled. It is represented using the symbol ∧. It is also known as logical multiplication, operator “and”, operator “and”. Other symbols it is represented with are “.” and “∩”.
Weak disjunction
With the weak disjunction operator a true result is obtained when any of the propositions is true. It is generally indicated by the symbol ∨. Also known as logical sum, operator “or”, operator “or”. Other symbols also used to represent it are “+ ∪” and “”. As is known proposition with these other names: inclusive disjunction , weak disjunction . “Copulativa disjunction” and “” no rigorous disjunction ” [1]
Strong disjunction
Using the strong disjunction operator there is a certain output exactly if one of the operands is true and the other is false. If the two operands have the same truth value, the output is false. It is called exclusive disjunction, rigorous disjunction , and divisive disjunction . It is represented by a v with a. in the ‘interior’, also with the uppercase Greek letter delta. Strong disjunction logically implies weak disjunction. [2]
(p ← / → p) → (poq), symbolically
Negation
The function of the negation operator is to negate another proposition. This means that if any proposition is true and the negation operator is applied, its complement or negation (false) will be obtained. This operator is indicated by the following symbols: ‘, ¬, -, ∼.
Notation
To represent the propositions, the lower case letters of the alphabet will be used, generally beginning with the letters p, q, r … z. Sometimes it is necessary to represent more propositions. For these cases you can use any other letters of the alphabet, but always in lower case.
