Logical Formulations
Equality is always defined such that things that are equal have all and only the same properties. Some people define equality as congruence. Often equality is just defined as identity.
A stronger sense of equality is obtained if some form of Leibniz's law is added as an axiom; the assertion of this axiom rules out "bare particulars"—things that have all and only the same properties but are not equal to each other—which are possible in some logical formalisms. The axiom states that two things are equal if they have all and only the same properties. Formally:
- Given any x and y, x = y if, given any predicate P, P(x) if and only if P(y).
In this law, the connective "if and only if" can be weakened to "if"; the modified law is equivalent to the original.
Instead of considering Leibniz's law as an axiom, it can also be taken as the definition of equality. The property of being an equivalence relation, as well as the properties given below, can then be proved: they become theorems. If a=b, then a can replace b and b can replace a.
Read more about this topic: Equality (mathematics)
Famous quotes containing the word logical:
“It was at that moment, just after Krug had fallen through the bottom of a confused dream and sat up on the straw with a gasp—and just before his reality, his remembered hideous misfortune could pounce upon him—it was then that I felt a pang of pity for Adam and slid towards him along an inclined beam of pale light—causing instantaneous madness, but at least saving him from the senseless agony of his logical fate.”
—Vladimir Nabokov (1899–1977)