DEFINING IN TYPE LOGIC ALL LOGICAL OPERATIONS IN TERMS OF
APPLICATION, ABSTRACTION, AND IDENTITY
1. We introduce 1 as a name of a tautology.
For this, we need to find a tautology expressed only with λ, ( ( )), and =.
Let α 2 VARt.
λα.α denotes the identity function ID = {<1,1>,<0,0>}.
Since this function is self-identical, (λα.α = λα.α) is a tautology.
We define:
1 := (λα.α = λα.α)
Three facts:
-λα.1 denotes the constant function on 1: C1 = {<1,1>,<0,1>}
-λα.α = 1 is equivalent to λα.α
This is easy to see: λα.α = 1 (1) is equivalent to (1 = 1) and hence to 1
λα.α = 1 (0) is equivalent to (0 = 1) and hence to 0
Thus λα.α = 1 denotes {<1,1>,<0,0>}
-Let x 2 VARa.
λx.1 denotes the characteristic function of Da, hence, set-theoretically, it denotes Da.
2. We introduce 0 as a name of a contradiction.
For instance, λα.α denotes the identity function, λα.1 denotes the constant function on 1.
These functions are not the same function. Expressing that they are the same is a
contradiction.
We define:
0 := (λα.α = λα.1)
Obviously, λα.0 denotes the constant function on 0, C0 = {<1,0>,<0,0>}.
3. Fact: λα.α = 0 denotes NEG = {<1,0>,<0,1>}
This is easy to see: λα.α = 0 (1) is equivalent to (1 = 0) and hence to 0
λα.α = 0 (0) is equivalent to (0 = 0) and hence to 1
Thus λα.α = 0 denotes {<1,0>,<0,1>}.
We define:
 := λα.α = 0
4. Let x 2 VARa, φ 2 EXPt.
λx.φ denotes the set of all a-entities that have φ (such that φ is true).
We saw above that λx.1 denotes the set of all a-entities, Da.
Hence, (λx.φ = λx.1) is true iff the set of all a-entities coincides with the set of all aentities that have φ, i.e. iff all a-entities have φ.
We define:
xφ := (λx.φ = λx.1)
5. Tarski's definition of conjunction.
Let f 2 VAR<t,t>, α, β 2 VARt
^ := λαλβ.f[ (f(α) = f(β)) = β ]
Correctness proof:
We need to prove,for φ,ψ 2 EXPt, that f[ (f(φ) = f(ψ)) = ψ ] is equivalent to (φ ^ ψ) (on
the standard semantics of the latter).
We prove this by the following equivalences:
1. v f[ (f(φ) = f(ψ)) = ψ ] bM,g = 1
2. for every k 2 D<t,t>: v (f(φ) = f(ψ)) = ψ ] bM,fk = 1
1 and 2 are equivalent by the semantics of .
3. v
v
v
v
(f(φ) = f(ψ)) = ψ
(f(φ) = f(ψ)) = ψ
(f(φ) = f(ψ)) = ψ
(f(φ) = f(ψ)) = ψ
bM,gfID = 1 and
bM,gfNEG = 1 and
bM,gfC1 = 1 and
bM,gfC0 = 1
2 and 3 are equivalent, because D<t,t> = {ID, C1, C2, NEG}.
4. v
v
v
v
(λα.α(φ) = λα.α(ψ)) = ψ bM,g = 1 and
(λα.α(φ) = λα.α(ψ)) = ψ bM,g = 1 and
(λα.1(φ) = λα.1(ψ)) = ψ bM,g
= 1 and
(λα.0(φ) = λα.0(ψ)) = ψ bM,g
=1
3 and 4 are equivalent by the coinsiderations in the above points 1-4:
λα.α denotes ID, λα.α denotes NEG, λα.1 denotes C1, λα.0 denotes C0.
5. v
v
v
v
(φ = ψ) = ψ bM,g
= 1 and
(φ = ψ) = ψ bM,g = 1 and
(1 = 1) = ψ bM,g
= 1 and
(0 = 0) = ψ bM,g
=1
4 and 5 are equivalent by λ-conversion.
6. v (φ = ψ) = ψ bM,g = 1 and v ψ bM,g = 1
5 and 6 are equivalent by the following observations:
a) (φ = ψ) and (:φ = :ψ) are equivalent, hence also
(φ = ψ) = ψ and (φ = ψ) = ψ are equivalent (by extensionality). Thus the second
clause in 5 is equivalent to the first, and can be dropped.
b) (1 = 1) is equivalent to 1; (0 = 0) is also equivalent to 1. Hence (by extensionality),
(1 = 1) = ψ and (0 = 0) = ψ are equivalent to (1 = ψ), and hence to (ψ = 1). But, by λconversion, (ψ = 1) is equivalent to λα.α = 1(ψ), and we saw above that λα.α = 1 has the
same denotation as λα.α, hence (by extensionality) λα.α = 1(ψ) is equivalent to λα.α(ψ),
and hence (by λ-conversion) to ψ. Hence, both the third and fourth clause in (5) are
equivalent to v ψ bM,g = 1.
Combining (a) and (b), we see that indeed 5 and 6 are equivalent.
7. v φ bM,g = 1 and v ψ bM,g = 1
6 and 7 are equivalent by the fact that (φ = ψ) = ψ and φ are equivalent. The latter fact is
easily shown in a truth table:
φ
ψ
(φ = ψ)
(φ = ψ) = ψ
1
1
0
0
1
0
1
0
1
0
0
1
1
1
0
0
Thus, we have shown that 1 and 7 are equivalent, which, of course, indeed means that the
expression which defines ^ picks out the correct two-place truth function for conjunction.
As usual, with  and ^ defined, we can define all other truth functions, and with xφ
defined, for any x and any φ, we can define xφ for any x and any φ. This proves that we
need only application, abstraction, and identity to define all other logical operations.
It should be clear that identity plays a crucial role in these considerations: the power of
the theory lies in the combined power of abstraction and identity, and identity itself
cannot be defined with just application and abstraction.