From this view of things, then, comes the axiom that if you visit to discover the author of any bad action, seek first to discover the person to whom the perpetration of that bad action could be in any way advantageous.

It is an algebraic axiom, which makes us proceed from a known to an unknown quantity, and not from an unknown to a known; but sit down, sir, I beg of you.''

``I have been nearly mad; and you know the axiom, non bis in idem.

It is an axiom of criminal law, and, consequently, you understand its full application.''

In short, doctor although I know you to be the most conscientious man in the world, and although I place the utmost reliance in you, I want, notwithstanding my conviction, to believe this axiom, errare humanum est.''

But the latter went on without pity:''`Seek whom the crime will profit,'says an axiom of jurisprudence.''

In his address to the British Association, in 1858, he speaks (page li) of "the axiom of the continuous operation of creative power, or of the ordained becoming of living things."

It has been maintained by several authors that it is as easy to believe in the creation of a million beings as of one; but Maupertuis' philosophical axiom "of least action" leads the mind more willingly to admit the smaller number; and certainly we ought not to believe that innumerable beings within each great class have been created with plain, but deceptive, marks of descent from a single parent.

"It has long been an axiom of mine that the little things are infinitely the most important.

In propositional logic a resolution proof of a clause formula_1 from a set of clauses C is a directed acyclic graph (DAG): the input nodes are axiom inferences (without premises) whose conclusions are elements of C, the resolvent nodes are resolution inferences, and the proof has a node with conclusion formula_1.

The "axiom of choice" in mathematical set theory is used to show that even the great deserts of Asia can be included in the "set" Indian Ocean through the logic of dialectical opposition.

He hinted that the court's role in maintaining such public confidence was imputed in the common law, holding that "is ... an axiom of the common law that justice should not only be done, but should manifestly and undoubtedly be seen to be done".

If one also assumes the Axiom of Choice, then all sets can be enumerated so that it coincides up to relabeling with the most general form of enumerations.

If one does "not" assume the axiom of choice or one of its variants, "S" need not have any well-ordering.

Even if one does assume the axiom of choice, "S" need not have any natural well-ordering.

If one works in Zermelo-Fraenkel set theory without the axiom of choice, one may want to impose the additional restriction that an enumeration must also be injective (without repetition) since in this theory, the existence of a surjection from "I" onto "S" need not imply the existence of an injection from "S" into "I".

Most important, Krippendorff allies himself with L. Wittgenstein’s definition of meaning as use, culminating in the axiom that "Humans do not see and act on the physical qualities of things, but on what they mean to them".

Note that this axiom still does not uniquely characterize the hyperbolic plane up to isometry; there is an extra constant, the curvature "K"0, which must be specified.

Then, if formula_77 is a set and formula_116 is any functional relation, the 'axiom of replacement' assures that formula_144 is a set in ZFC.

The Axiom of Infinity of ZFC tells us that there is a set "A" which contains formula_208 and contains formula_209 for each formula_210.

In NFU, it is not obvious that this approach can be used, since the successor operation formula_209 is unstratified and so the set "N" as defined above cannot be shown to exist in NFU (it is interesting to note that it is consistent for the set of finite von Neumann ordinals to exist in NFU, but this strengthens the theory, as the existence of this set implies the Axiom of Counting (for which see below or the New Foundations article)).

By the axiom of foundation, the restriction of the membership relation to the transitive closure of "A" is a well-founded relation.

An axiom of extensionality for this simulated set theory follows from E's extensionality.

From its well-foundedness follows an axiom of foundation.

There remains the question of what comprehension axiom E may have. Consider any collection of set pictures formula_250 (collection of set pictures whose fields are made up entirely of singletons).

Under the Axiom of Cantorian Sets described in the New Foundations article, the strongly cantorian part of the set of isomorphism classes of set pictures with the E relation as membership becomes a (proper class) model of ZFC (in which there are "n"-Mahlo cardinals for each "n"; this extension of NFU is strictly stronger than ZFC).

When a Sahlqvist formula is used as an axiom in a normal modal logic, the logic is guaranteed to be complete with respect to the elementary class of frames the axiom defines.

If the associativity axiom of a semigroup is dropped, the result is a magma, which is nothing more than a set "M" equipped with a binary operation "M" × "M" → "M".