Variant Glossary
Definition
In the semantics of first order logic, two interpretations are variants of each other if they agree on everything except some names. For a specific name m, two interpretations are m-variants of each other if they assign the same value to every symbol except (possibly) the name m.
Comments
Notation: where I is an interpretation, m is a name and α is an object in the domain of I, we say that I[m←α] is the m-variant J such that J(m) = α.
The point of considering m-variants is that whereas on any particular interpretation m is just a name, referring to a specific object, over the whole set of m-variants it varies across the entire domain. This makes it possible to define the truth conditions for quantifiers in a smooth way that fits our definition of a formula.
Example
Let I be an interpretation with domain D = {Alice, Bob, Charlie}. Suppose I(a) = Alice, I(b) = Bob and I(F) = {Alice, Charlie}. Let J = I[a←Bob] and let K = J[b←Charlie]. Then:
| I(a) = Alice | J(a) = Bob | K(a) = Bob |
| I(b) = Bob | J(b) = Bob | K(b) = Charlie |
| I(F) = {Alice, Charlie} | J(F) = {Alice, Charlie} | K(F) = {Alice, Charlie} |
So Fa is true for I but false for J and K, while a=b is false for I and K but true for J.