interpretations. In particular, there are other types of interpretations that are used in the study of non-classical logic (such as intuitionistic logic), and in the study of modal logic. Interpretations used to study non-classical logic include topological models, Boolean-valued models, and Kripke models. Modal logic is also studied using Kripke models. Intended interpretations Many formal languages are associated with a particular interpretation that is used to motivate them. For example, the first-order signature for set theory includes only one binary relation, ∈, which is intended to represent set membership, and the domain of discourse in a first-order theory of the natural numbers is intended to be the set of natural numbers. The intended interpretation is called the standard model (a term introduced by Abraham Robinson in 1960). In the context of Peano arithmetic, it consists of the natural numbers with their ordinary arithmetical operations. All models that are isomorphic to the one just given are also called standard; these models all satisfy the Peano axioms. There are also non-standard models of the (first-order version of the) Peano axioms, which contain elements not correlated with any natural number. While the intended interpretation can have no explicit indication in the strictly formal syntactical rules, it naturally affects the choice of the formation and transformation rules of the syntactical system. For example, primitive signs must permit expression of the concepts to be modeled; sentential formulas are chosen so that their counterparts in the intended interpretation are meaningful declarative sentences; primitive sentences need to come out as true sentences in the interpretation; rules of inference must be such that, if the sentence is directly derivable from a sentence , then turns out to be a true sentence, with meaning implication, as usual. These requirements ensure that all provable sentences also come out to be true. Most formal systems have many more models than they were intended to have (the existence of non-standard models is an example). When we speak about 'models' in empirical sciences, we mean, if we want reality to be a model of our science, to speak about an intended model. A model in the empirical sciences is an intended factually-true descriptive interpretation (or in other contexts: a non-intended arbitrary interpretation used to clarify such an intended factually-true descriptive interpretation.) All models are interpretations that have the same domain of discourse as the intended one, but other assignments for non-logical constants. Example