Description: Define proper
substitution. Remark 9.1 in [Megill] p. 447
(p. 15 of the
preprint). For our notation, we use to mean "the wff
that results from the proper substitution of for in the wff
." That is, properly replaces . For example,
is the same as , as
shown in elsb42179. We
can also use in place of the "free for" side condition
used in traditional predicate calculus; see, for example, stdpc42094.

Our notation was introduced in Haskell B. Curry's Foundations of
Mathematical Logic (1977), p. 316 and is frequently used in textbooks
of
lambda calculus and combinatory logic. This notation improves the common
but ambiguous notation, "() is the wff that
results when
is properly substituted for in (x)." For
example, if the
original (x) is , then () is , from
which we obtain that (x) is . So what exactly does
(x) mean? Curry's
notation solves this problem.

In most books, proper substitution has a somewhat complicated recursive
definition with multiple cases based on the occurrences of free and bound
variables in the wff. Instead, we use a single formula that is exactly
equivalent and gives us a direct definition. We later prove that our
definition has the properties we expect of proper substitution (see
theorems sbequ2117, sbcom22189 and sbid2v2201).

Note that our definition is valid even when and are replaced
with the same variable, as sbid1996 shows. We achieve this by having
free in the first conjunct and bound in the second. We can also achieve
this by using a dummy variable, as the alternate definition dfsb72199 shows
(which some logicians may prefer because it doesn't mix free and bound
variables). Another version that mixes free and bound variables is
dfsb32115. When and are distinct, we can express proper
substitution with the simpler expressions of sb52174
and sb62173.

There are no restrictions on any of the variables, including what
variables may occur in wff . (Contributed by NM,
10-May-1993.)