Properties as Recursive Relations

• A property P characterises some object x as x is a bearer or a posseser of x. No matter how exactly it can be defined, we will use Px as understanding it as x is P or x has P.
• The properties are not so clear at what they are, because it might be that Px only when S, where S is some kind of a rule (a conditional proposition) for x or Px.
• Since having this we can assume some rules S' can characterise x or Px only due to the fact of x. In other words, Px is true by x or Px.
• This last idea may be represented in the next way:

Px → Px (1)

• But there is no sure whether "→" should be that one definer. To make the formulation (1) a little bit finer, let's rewrite it via relations as follows:

R(Px,Px) (2)

• This time using double P isn't needed, so to make some corrections to (2), there will be:

R(x,x) (3)

• Only what we have to add to (3) is to equal it to the quite demonstrative form:

Px = df R(x,x) (4)

• Such a way allows to reformulate every formula of the type Fμ into formulas H(μ,μ), where F is a given property, H is a certain relation, and μ is an object. Finally, the general formula of it:

(∀μ)(∀F)(∃H).(Fμ = H(μ,μ))