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An axiom, also known as a presupposition, is an assumption in a logical branch or argument from which premises can be fed, implications derived, et cetera. Different sets of axioms being used are called "logical branches". The branch of classical logic, founded around -350 by Aristotle, has the three axioms of:
- The law of identity: A = A, that is, A is identical to itself.
- The law of non-contradiction: A ^ ~A = False, that is, A is mutually exclusive with anti-A.
- The law of the excluded middle: A V ~A, everything is either true or not true with no graduations of validity.
Branches of logic exist which add axioms or remove them. Some axioms, called presuppositions, are actually just premises in disguise, which are typically implicitly stated like an axiom is. For instance, in Alice in Wonderland:
Alice has an implicit premise that she is to get in at all, as the Footman explicitly states. This is still compatible with the laws of classical logic, but it simply adds a premise. The premises of science are explained here, for instance. One might be tempted to call these "axioms", and in a way, they are, but depending on the context, most axioms can be reclassified as presuppositions (although "presupposition" is typically used as a snarl word, as in presuppositionalism.
Branches of logic which remove axioms from classical logic include the various forms of fuzzy logic, which replaces the law of the excluded middle with a continuum, and paraconsistent logic, which rejects the principle of explosion (that anything validly follows from a contradiction) in various ways (for instance by rejecting the law of the excluded middle). Dialetheism, which is arguably a version of paraconsistent logic, humorously enough gets rid of the law of non-contradiction. Intuitionism or constructivism rejects the law of excluded middle as an axiom as well, though the most common versions retain explosion and do not deny that the law of excluded middle is correct (it's just not an axiom).
Axioms are wonderful things in logic, as they define logic outside of presuppositionalism, which has as its axioms that axioms are defined by God and what God defines as axioms are in His holy book. In some branches, axioms are seen as unquestionable, while some branches openly invite the criticism of even its foundations. It makes for a fun time when the foundations of a logical branch is criticized by a member of that branch (almost like a civil war) or when members of different logical branches get into arguments. Axioms are typically hidden, and the other guy is assumed to share the same axioms, leading to awkwardness when they don't. Examples include most debates with fundies.
 Examples of logical branches and their axioms
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- Classical logic (Aristotle, circa 350 BCE)
- Stated above.