 Dorr and Hawthorne  (2013)  advocate a similar flexibility about the syntax of naturalness claims. They suggest that these claims should be regimented within in a higher-order logic, where for any syntactic categories (“types”) \(\alpha_1,\ldots ,\alpha_n\), there is a type \(\langle\alpha_1,\ldots ,\alpha_n\rangle\), expressions of which can combine with expressions of types \(\alpha_1,\ldots ,\alpha_n\) to form sentences. Using \(e\) for the type of names, an ordinary monadic predicate is of type \(\langle e\rangle\), so a higher-order predicate taking such a predicate as an argument is of type \(\langle\langle e\rangle\rangle\). Sentences (or, more generally, formulas) can be identified with expressions of type \(\langle\rangle\); so binary operators like \(\wedge\) (‘and’) and \(\vee\) (‘or’) are of type \(\langle\langle\rangle,\langle\rangle\rangle\). The proposal is to introduce, for each type \(\alpha\), expressions ‘PerfectlyNatural\(_\alpha\)’ of type \(\langle\alpha\rangle\), ‘MoreNatural\(_\alpha\)’ of type \(\langle\alpha ,\alpha\rangle\), and so on. We can thus formulate questions like ‘Is it the case that MoreNatural \(_{\langle\langle\rangle,\langle\rangle\rangle}(\text{and}, \text{or})\)?’ Standard higher order languages also admit variables and quantifiers of all types, so we can also say things like ‘\(\neg\exists x^{\langle\langle \rangle ,\langle\rangle\rangle}\text{MoreNatural}_{\langle\langle\rangle,\langle\rangle\rangle}(x,\wedge)?\)’—‘No binary operation is more natural than conjunction’.