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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.
While the above compound sentence is itself a statement, because it is true, the two parts, "Ganymede is a moon of Jupiter" and "Ganymede is a moon of Saturn", are themselves statements, because the first is true and the second is false. However, it is sometimes used to name something abstract that two different statements with the same meaning are both said to "express".
In this usage, the English sentence, "It is raining", and the French sentence "Il pleut", would be considered to express the same proposition; similarly, the two English sentences, "Callisto orbits Jupiter" and "Jupiter is orbitted by Callisto" would also be considered to express the same proposition.
However, the nature or existence of propositions as abstract meanings is still a matter of philosophical controversy, and for the purposes of this article, the phrases "statement" and "proposition" are used interchangeably.
, is that branch of logic that studies ways of combining or altering statements or propositions to form more complicated statements or propositions.
then...", "because", and "necessarily", are all operators.
A logical operator is said to be on the truth or falsity of the statements from which they are constructed.
In English, words such as "and", "or", "not", "if ...(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.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.
Since most of their original works -- if indeed, many writings were even produced -- are lost, we cannot make many definite claims about exactly who first made investigations into what areas of propositional logic, but we do know from the writings of Sextus Empiricus that Diodorus Cronus and his pupil Philo had engaged in a protracted debate about whether the truth of a conditional statement depends entirely on it not being the case that its antecedent (if-clause) is true while its consequent (then-clause) is false, or whether it requires some sort of stronger connection between the antecedent and consequent -- a debate that continues to have relevance for modern discussion of conditionals.