Liberal Debutante

Soooo this time ‘round I will be working with making some of
the questions I’ve faced in the past a little more precise and how to get some
actual answers.

First consider an informal example. We could expand the
language by adding a three-lace sentential connective symbol #, called the
majority symbol. We allow now as a wff expression (#αβγ) when ever α, β and γ
are wffs. In other words, we add a sixth formula building operation to our list:

Є#(α,β,γ) = (#αβγ).

Then we must give the interpretation of this symbol. That
is, we must say how to compute v|((#αβγ)), given the values v|(α), v|(β) and v|(γ).
We choose to define

V|((#αβγ)) is to agree with the majority of v|(α), v|(β),
v|(γ).

We claim thatthis extension has gained us nothing, in the
following precise sense: For any wff I the extended language, there is a
tautologically equivalent wff in the original language. (On the other hand, the
wff in the original language may be much longer than the wff in the extended
language.) We will prove this (in a more general situation) below; here we just
not that it relies on the fact that (#αβγ) is tautologically equivalent to

(α^β) V (α^γ) V (β^γ). 

(We note parenthetically that our insistence that v|((#αβγ))
be calculable from (v|(α), v|(β), v|(γ)) plays a definite role here. In
everyday speech, there are unary operators like “it is possible that” or “I
believe that.” We can apply one of these operators to a sentence, producing a
new sentence whose truth or falsity cannot be determined solely on the basis of
the truth or falsity of the original one.)

In generalizing the foregoing example, the formal language
will be more of a hindrance than a help. We can restate everything using only
functions. Say that a k-place Boolean
function is a function from {F,T}^k
into {F,T}. (A Boolean function is then anything which k-place Boolean function for some k. We stretch this slightly by permitting F and T themselves to be
0-place Boolean functions.) Some Boolean functions are defined by the equations
(where X Є {T,F}) 

I^n(X1,…Xn) = Xi,

N(F) = T, N(T) = F,

K(T, T) = T, K(F, X) = K(X, F) = F,

A(F, F) = F, A(T, X) = A(X, T) = T,

C(T, F) = F, C(F, X) = C(X, T) = T,

E(X, X) = T, E(T, F) = E(F, T) = F. 

From a wff α we can extract a Boolean function. For example,
if α is the wff A1^A2, then we can make a table (below). The 2^2 lines of the
table correspond to the 2^2 truth assignments for {A1, A2}. For each of the 2^2
pairs X, we set Bα(X) equal to the truth value α receives when its sentence symbols
are given the values indicated by X.

In general, suppose that α is a wff whose sentence symbols
are at most A1,…,An. We define an n-place
Boolean function Bα^n (just Bα if n
seems unnecessary), the Boolean function realized by α, by 

Bα^n(X1,…,Xn)= the truth value given to α when

 A1,…,An
are given the values X1,…,Xn.

Or in other words Bα^n(X1,…,Xn) = |v(α), where v is the
truth assignment for {A1,…,An} for which v(Ai) = Xi. Thus Bα^n comes from
looking at |v(α) as a function of v, with α fixed.

…There, I’d do some proofs, but its 2:40 am and I have class
in the morning, so you’ll just have to trust me, because I probably won’t do
proofs tomorrow. I will however probably do 0-ary Connectives and Ternary
Connectives. 

Okay fine one quickly:
Let α and β be wffs whose sentence symbols are among A1,…,An.
Then 

a) α
|= β for all X {F,T}^n, Bα (X) (less than or equal to) Bβ(X)

b) α
|= =| iff Bα = Bβ

c) |=
α if Bα is the constant function with value T.

Proof α |= β iff for all 2^n truth assignments v for A1,…,An.
Whenever |v(α) = T, then also |v(β) = T. (this is true even if the sentence
symbols in α and β do not include all of A1,…,An) Thus

Α |= β  iff for all 2^n
assignments v,  |v(α) = T => v|(β) = T

 Iff for all
2^n n-tuples X,  Bα^n(X) = T => Bα^n(X)
= T

 Iff for all
2^n n-tuples X,  Bα^n(X) (less than
or equal to) Bβ^n(X),

Where F < T.

In addition to identifying tautologically equivalent wffs,
we have freed ourselves from the formal language. We are now at liberty to
consider any Boolean function, whether it is realized or not.