PROPOSITIONAL LOGIC

Propositional logic, also known as sentential logic, is the branch of logic that studies ways of joining and/or modifying entire propositions, statements or sentences to form more complicated propositions, statements or sentences, as well as the logical relationships and properties that are derived from these methods of combining or altering statements.

In propositional logic, the simplest statements are considered as indivisible units, and hence, propositional logic does not study those logical properties and relations that depend upon parts of statements that are not themselves statements on their own, such as the subject and predicate of a statement.

The most thoroughly researched branch of propositional logic is classical truth-functional propositional logic, which studies logical operators and connectives that are used to produce complex statements whose truth-value depends entirely on the truth-values of the simpler statements making them up, and in which it is assumed that every statement is either true or false and not both.

Joining two simpler propositions with the word "and" is one common way of combining statements. When two statements are joined together with "and", the complex statement formed by them is true if and only if both the component statements are true. Because of this, an argument of the following form is logically valid:

Paris is the capital of France and Paris has a population of over two million.
Therefore, Paris has a population of over two million.

Propositional logic largely involves studying logical connectives such as the words "and" and "or" and the rules determining the truth-values of the propositions they are used to join, as well as what these rules mean for the validity of arguments, and such logical relationships between statements as being consistent or inconsistent with one another.

Propositional logic also studies way of modifying statements, such as the addition of the word "not" that is used to change an affirmative statement into a negative statement. Here, the fundamental logical principle involved is that if a given affirmative statement is true, the negation of that statement is false, and if a given affirmative statement is false, the negation of that statement is true.

Pelé is the king of soccer. [True]
p

NEGATION: It´s not the case that Pelé is the king of soccer.
~p: False

It is not the case that Steve Wonder is not a singer.
~(~p)
~(F)
T
Conclusion: Steve Wonder is a singer.

Propositional logic can be thought of as primarily the study of logical operators. A logical operator is any word or phrase used either to modify one statement to make a different statement, or join multiple statements together to form a more complicated statement. In English, words such as "and", "or", "not", "if ... then...", "because", and "necessarily", are all operators.

A logical operator is said to be truth-functional if the truth-values (the truth or falsity, etc.) of the statements it is used to construct always depend entirely on the truth or falsity of the statements from which they are constructed.

The English words "and", "or" and "not" are (at least arguably) truth-functional, because a compound statement joined together with the word "and" is true if both the statements so joined are true, and false if either or both are false, a compound statement joined together with the word "or" is true if at least one of the joined statements is true, and false if both joined statements are false (Thus, the logical statement ``It is raining or the sun is shining '' means it is raining, or the sun is shining or it is raining and the sun is shining.
), and the negation of a statement is true if and only if the statement negated is false.

More must be said about the meaning or semantics of the logical operators '&', 'v', '→', '↔', and '¬'.

These are used in place of the English words, 'and', 'or', 'if... then...', 'if and only if', and 'not', respectively.

However, the correspondence is really only rough, because the operators of PL are considered to be entirely truth-functional, whereas their English counterparts are not always used truth-functionally.
Consider, for example, the following statements:
(1) If Bob Dole is president of the United States in 2004, then the president of the United States in 2004 is a member of the Republican party.

(2) If Al Gore is president of the United States in 2004, then the president of the United States in 2004 is a member of the Republican party.

For those familiar with American politics, it is tempting to regard the English sentence (1) as true, but to regard (2) as false, since Dole is a Republican but Gore is not. But notice that in both cases, the simple statement in the "if" part of the "if... then..." statement is false, and the simple statement in the "then" part of the statement is true. This shows that the English operator "if... then..." is not fully truth-functional. However, all the operators of language PL are entirely truth-functional, so the sign '→', though similar in many ways to the English "if... then..." is not in all ways the same.