 However, for reasons sketched in  §2.4,  type theory is hardly satisfactory. Accordingly, many type-free versions of property theory have been developed over the years and no consensus on what the right strategy is appears to be in sight. Of course, without type-theoretical constraints, given \((\lambda\)-conv) and classical logic (CL), paradoxes such as Russell’s immediately follow (to see this, consider this instance of \((\lambda\textrm{-conv}): [\lambda x {\sim}x(x)]([\lambda x {\sim}x(x)]) \leftrightarrow{\sim}[\lambda x {\sim}x(x)]([\lambda x {\sim}x(x)])\)). In formal systems where abstract singular terms or predicates may (but need not) denote properties (Swoyer 1998), formal counterparts of (complex) predicates like “being a property that does not exemplify itself” (formally, “[\(\lambda x {\sim}x(x)\)]”) could exist in the object language without denoting properties; from this perspective, Russell’s paradox merely shows that such predicates do not stand for properties (similarly, according to Schnieder 2017, it shows that it is not possible that a certain property exists). But we would like to have general criteria to decide when a predicate stands for a property and when it does not. Moreover, one may wonder what gives these predicates any significance at all if they do not stand for properties. There are then motivations for building type-free property theories in which all predicates stand for properties. We can distinguish two main strands of them: those that weaken CL and those that circumscribe \((\lambda\)-conv) (some of the proposals to be mentioned below are formulated in relation to set theory, but can be easily translated into proposals for property theory).