Computability and Logic

SECOND-ORDER LOGIC 281
(The foregoing gives the standard notion of interpretation and truth for secondorder
logic. In the literature nonstandard notions, euphemistically called ‘general’,
are sometimes considered, where an interpretation has separate domains of individuals
and of relations and functions. These will not be considered here.)
22.2 Example (The definition of identity). The Leibniz definition of identity in Example
22.1 is unnecessarily complicated, since the following simpler Whitehead–Russell definition
will do:
We don’t need a biconditional on the right!
c = d ↔∀X(Xc→ Xd)
Proof: ∼Pc∨ Pd or Pc→ Pd is true in an interpretation just in case the set P
denotes either fails to contain the individual c denotes or contains the one d denotes.
Hence a set R satisfies Xc→ Xd just in case it either fails to contain the individual c
denotes or contains the one d denotes. Hence ∀X(Xc→ Xd) is true just in case every
set either fails to contain the individual c denotes or contains the one d denotes. If c
and d denote the same individual, this must be so for every set, while if c and d do not
denote the same individual, then it will fail to be so for the set whose one and only
element is the individual c denotes. Thus ∀X(Xc→ Xd) is true just in case c and d
denote the same individual, which is to say, if and only if c = d is true. (Intuitively,
the Whitehead–Russell definition is valid because among the properties of a is the
property of being identical with a; hence if the individual b is to have all the properties
of a, it must in particular have the property of being identical with a.)
22.3 Example (The ‘axiom’ of enumerability). Let Enum be the sentence
∃z∃u∀X((Xz& ∀x(Xx → Xu(x))) →∀xXx).
Then Enum is true in an interpretation if and only if its domain is enumerable.
Proof: First suppose Enum is true in an interpretation M. This means there exists
an individual a in |M| and a one-place function f on |M| that satisfy
∀X((Xz& ∀x(Xx → Xu(x))) →∀xXx).
Thus, if we add a constant 0 and let it denote a, and a one-place function symbol ′
and let it denote f , then
∀X((X0 & ∀x(Xx → Xx ′ )) →∀xXx)
is true. This means every subset A of |M| satisfies
(X0 & ∀x(Xx → Xx ′ )) →∀xXx.
In particular this is so for the enumerable subset A of |M| whose elements are all and
only a, f (a), f ( f (a)), f ( f ( f (a))), and so on. Thus if we add a one-place predicate
N and let it denote A, then
(N0 & ∀x(Nx → Nx ′ )) →∀xNx