Unrestricted functional logic Joop Leo August 17, 2016 Abstract An extremely simple logic is developed in which we may quantify un- restrictedly over everything whatsoever. Distinctive features of the logic are that equality is its sole predicate, and that the things talked about may have an input–output functionality. Because things are themselves potent, interpretations can be defined in a straightforward way without giving rise to the much-discussed problems with unrestricted quantifica- tion. Satisfiability of formulas obviously depends on what there is. We show that some natural assumptions about what there is give the logic at least the same expressive power as higher-order predicate logic. Furthermore, for theories with first-order restricted quantification we have compactness, completeness, and a variant of the L¨ owenheim-Skolem property. 1 Introduction In [Leo16] I argued that standard logic misleads us in the following respects: (i) it assumes that arguments of a relation always come in a linear order, and (ii) it implicitly suggests that relations are universals. Furthermore, I mentioned as a weakness that it cannot say anything about strict symmetry. 1 In the same paper, I presented a new logic that does not have these problems. A distinguishing feature of the logic is that it has no predicates, except equality. Its structures are collections of entities that may have input–output functionality. The logic required an alternative mathematical foundation with no axiom of extensionality and no axiom of foundation. With this alternative foundation, the logic was shown to have the same expressive power as first-order predicate logic with set theory. I called the logic coordinate-free logic. In one respect coordinate-free logic is very similar to predicate logic: quantifi- cation is always understood as being restricted. In predicate logic, the domains of the structures are sets or classes, and in coordinate-free logic the structures are collections of entities. However, I now think that demanding quantification to be restricted is a bad choice for two reasons. First, not all intended models have restricted domains. 1 A relation R is strictly symmetric if for all x, y the complex Rxy is identical to the complex Ryx. 1