In formal logic, an interpretation is a way of assigning meaning to the symbols and formulas of a logical system. An interpretation provides a set of objects, relations, and functions that correspond to the symbols used in the logical system. The interpretation specifies how the symbols should be understood and how they should be related to each other.

For example, in first-order logic, an interpretation of a formula assigns a set of objects to the variables in the formula and specifies how the predicates and functions in the formula should be applied to those objects. The interpretation determines whether the formula is true or false in the context of the assigned objects and relations.

It is important to note that a system of formal logic can have multiple interpretations that are equally valid. This is because the symbols in a logical system are abstract and do not have inherent meaning outside of the interpretation assigned to them. Different interpretations may assign different meanings to the symbols, but as long as the interpretation is consistent and follows the rules of the logical system, it is considered valid.

For example, the formula "∀xP(x) → Q(x)" can be interpreted in multiple ways depending on the interpretation assigned to the predicates "P" and "Q". In one interpretation, "P" might be interpreted as "is a prime number" and "Q" might be interpreted as "is an odd number". In another interpretation, "P" might be interpreted as "is a mammal" and "Q" might be interpreted as "is a vertebrate". Both interpretations are equally valid as long as they are consistent and follow the rules of the logical system.

However, a counterexample can invalidate an interpretation by demonstrating that it leads to a contradiction or a falsehood. For example, if the interpretation of "P" as "is a prime number" and "Q" as "is an odd number" leads to the formula being false for a specific value of "x", then this interpretation is not valid. The counterexample shows that there is at least one object that does not satisfy the formula under the given interpretation, and thus the interpretation is not consistent with the logical system.

In summary, an interpretation is a way of assigning meaning to the symbols and formulas of a logical system, and a system of formal logic can have multiple but equally valid interpretations. However, a counterexample can invalidate an interpretation by demonstrating that it leads to a contradiction or a falsehood.