Equals: In logic: Term for identity.
The term 'equals' (or 'is the same as', 'is identical with') is a logical term and is used to convey that two terms refer to the same thing.
There are a number of problems with 'equals' and with identity in logic, mostly relating to substitution and to necessity, that are best discussed after stating a version of the rules for '=' that are often used in formal logic:
For every x, x=x.
For every x, every y, if x=y then y=x.
For every x, every y, every z, if x=y and y=z then x=z.
From (Fx) and (x=y) it follows that (Fy).
The first three rules or axioms lay down basic properties of identity, namely that it is reflexive (everything is equal to itself), symmetric (if x equals y then y equals x) and transitive (if x equals y and y equals z then x equals z).
The fourth rule relates identity and substitution in the sense that it lays down that if x equals y then y can be substituted for x.
It is especially with this rule that a number of problems are associated, of which I name five.
First, as stated the rule is much like it is stated for first-order logic, but the problem is that the letter F must be schematic and stand in for any arbitrary predicate - which is a notion which is not part of first-order logic (that contains no quantifiers for predicates) yet something like it is necessary to assure that equal terms can be substituted for each other.
Second, there is the problem that in some contexts it seems intuitively that equals may not always be substitutable in truths with truths as result. Two such contexts are intensional contexts and modal contexts. Thus, to illustrate the first, while the King may know he drinks water, the King may not know that water is the same as H20, and thus the King may not know that he drinks H20, even though he does. And to illustrate the second, if the number of planets of the earth is 9, and 9 is necessarily greater than 7, the number of planets of the earth may not be necessarily greater than 7.
Third, there is a fairly intuitive definition of equality that goes back to Leibniz:
- Two things are equal iff the things have all their properties in common
The problems with this are that already its formulation involves quantification of properties and that usually the properties involved are somehow restricted and e.g. do not involve properties of terms but only properties of things, and then usually also only properties of certain kinds, in certain domains.
Fourth, there is the problem that is connected to the previous problem, what it is that identities or equalities are about: Terms, things, the relations between terms and things? Thus, Leibniz's definition of identity seems to be about things, whereas at least many instances of identities seem to involve, rather, that two different terms, such as '2+2' and '2.2', represent the same thing (viz. the number 4 represented by '4').
Fifth, there is the problem that the rules and axioms for identity as they are normally used make it hard or impossible to distinguish necessary and merely contingent identities, and seem to make all identities necessary identities.