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The substitution property of equality, or Leibniz's Law (though the latter term is usually reserved for philosophical contexts), generally states that, if two things are equal, then any property of one, must be a property of the other.
If a theory has a predicate that satisfies the Law of Identity and Substitution property, it is common to say that it "has equality," or is "a theory with equality." [ 27 ] The use of "equality" here is a misnomer in that an arbitrary binary predicate that satisfies those properties may not be true equality, and there is no property or list of ...
In logic and mathematics, statements and are said to be logically equivalent if they have the same truth value in every model. [1] The logical equivalence of and is sometimes expressed as , ::, , or , depending on the notation being used.
An implication of model theory is that the properties defining a relation can be proved independent of each other (and hence necessary parts of the definition) if and only if, for each property, examples can be found of relations not satisfying the given property while satisfying all the other properties. Hence the three defining properties of ...
The most common convention, known as first-order logic with equality, includes the equality symbol as a primitive logical symbol which is always interpreted as the real equality relation between members of the domain of discourse, such that the "two" given members are the same member. This approach also adds certain axioms about equality to the ...
For example, ↑ 0 is the identity substitution, leaving a term unchanged. A finite list of terms M 1.M 2...M n abbreviates the substitution M 1.M 2...M n.(n+1).(n+2)... leaving all variables greater than n unchanged. The application of a substitution s to a term M is written M[s]. The composition of two substitutions s 1 and s 2 is written s 1 ...
And, substitution allows one to derive restrictions on the possible values, or show what conditions the statement holds under. For example, taking the statement x + 1 = 0, if x is substituted with 1, this implies 1 + 1 = 2 = 0, which is false, which implies that if x + 1 = 0 then x cannot be 1.
The converse of this axiom follows from the substitution property of equality. 2) Axiom Schema of Specification (or Separation or Restricted Comprehension ): If z is a set and ϕ {\displaystyle \phi } is any property which may be satisfied by all, some, or no elements of z , then there exists a subset y of z containing just those elements x in ...