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287 BCE), did recognize a need for the development of a doctrine of "complex" or "hypothetical" propositions, i.e., those involving conjunctions (statements joined by "and"), disjunctions (statements joined by "or") and conditionals (statements joined by "if...then..."), but their investigations into this branch of logic seem to have been very minor.(These notions are defined below.) 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.
A logical operator is said to be on the truth or falsity of the statements from which they are constructed.
In addition to classical truth-functional propositional logic, there are other branches of propositional logic that study logical operators, such as "necessarily", that are not truth-functional.
There are also "non-classical" propositional logics in which such possibilities as (i) a proposition's having a truth-value other than truth or falsity, (ii) a proposition's having an indeterminate truth-value or lacking a truth-value altogether, and sometimes even (iii) a proposition's being both true false, are considered.
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 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, as well as logical properties of propositions, such as being tautologically true, being contingent, and being self-contradictory.