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(For more information on these alternative forms of propositional logic, consult Section VIII below.) The serious study of logic as an independent discipline began with the work of Aristotle (384-322 BCE).Generally, however, Aristotle's sophisticated writings on logic dealt with the logic of categories and quantifiers such as "all", and "some", which are not treated in propositional logic.For example, both of the following statements are true: Here, the first example is true but the second example is false.
(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.
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.
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, and the negation of a statement is true if and only if the statement negated is false. One example of an operator in English that is not truth-functional is the word "necessarily".
Whether a statement formed using this operator is true or false does not depend entirely on the truth or falsity of the statement to which the operator is applied.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.However, there are other forms of propositional logic in which other truth-values are considered, or in which there is consideration of connectives that are used to produce statements whose truth-values depend not simply on the truth-values of the parts, but additional things such as their necessity, possibility or relatedness to one another.These are, of course, cornerstones of classical propositional logic.There is some evidence that Aristotle, or at least his successor at the Lyceum, Theophrastus (d.is that branch of truth-functional propositional logic that assumes that there are are only two possible truth-values a statement (whether simple or complex) can have: (1) truth, and (2) falsity, and that every statement is either true or false but not both.