Let, and be non-empty and disjoint sets. Let " " be a simple order. Krantz et al. (1971) argued the quadruple is a polynomial conjoint system if and only if the following axioms hold.
- WEAK ORDER.
- SINGLE CANCELLATION. The relation " " satisfies single cancellation upon A whenever if and only if holds for all and . Single cancellation upon P and U is similarly defined.
- DOUBLE CANCELLATION. The relation " " upon satisfies double cancellation if and only if for all and, and therefore is true for all . The condition holds similarly upon and .
- JOINT SINGLE CANCELLATION. The relation " " upon satisfies joint single cancellation such that if and only if is true for all and . Joint independence is similarly defined for and .
- DISTRIBUTIVE CANCELLATION. Distributive cancellation holds upon if and only if, and implies is true for all and .
- DUAL DISTRIBUTIVE CANCELLATION. Dual distributive cancellation holds upon if and only if
, and implies is true for all and .
- SOLVABILITY. The relation " " upon is solvable if and only if for all and, there exists and such that .
- ARCHIMEDEAN CONDITION.
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Famous quotes containing the word axioms:
“The axioms of physics translate the laws of ethics. Thus, the whole is greater than its part; reaction is equal to action; the smallest weight may be made to lift the greatest, the difference of weight being compensated by time; and many the like propositions, which have an ethical as well as physical sense. These propositions have a much more extensive and universal sense when applied to human life, than when confined to technical use.”
—Ralph Waldo Emerson (18031882)
“I tell you the solemn truth that the doctrine of the Trinity is not so difficult to accept for a working proposition as any one of the axioms of physics.”
—Henry Brooks Adams (18381918)