**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
." We
can also use in
place of the "free for"
side condition used in traditional predicate calculus; see, for example,
stdpc4 2073.
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 ." For example, if the
original is , then is , from
which we obtain that is . So what exactly does
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 sbequ 2109, sbcom2 2163 and sbid2v 2173).
Note that our definition is valid even when and are replaced
with the same variable, as sbid 1943 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 dfsb7 2171 shows
(which some logicians may prefer because it doesn't mix free and bound
variables). Another version that mixes free and bound variables is
dfsb3 2105. When and
are distinct, we can express proper
substitution with the simpler expressions of sb5 2149
and sb6 2148.
There are no restrictions on any of the variables, including what
variables may occur in wff . (Contributed by NM, 5-Aug-1993.) |