|Description: Define proper
substitution. Remark 9.1 in [Megill] p. 447
(p. 15 of the
preprint). For our notation, we use [𝑦 / 𝑥]𝜑 to mean "the wff
that results from the proper substitution of 𝑦 for 𝑥 in the
𝜑." That is, 𝑦
properly replaces 𝑥. For example,
𝑦]𝑧 ∈ 𝑦 is the same as 𝑧 ∈ 𝑥, as shown in elsb4 2433. We
can also use [𝑦 / 𝑥]𝜑 in place of the "free for"
used in traditional predicate calculus; see, for example, stdpc4 2351.
Our notation was introduced in Haskell B. Curry's Foundations of
Mathematical Logic (1977), p. 316 and is frequently used in textbooks
lambda calculus and combinatory logic. This notation improves the common
but ambiguous notation, "𝜑(𝑦) is the wff that results when 𝑦
is properly substituted for 𝑥 in 𝜑(𝑥)." For example, if the
original 𝜑(𝑥) is 𝑥 = 𝑦, then 𝜑(𝑦) is 𝑦 = 𝑦, from
which we obtain that 𝜑(𝑥) is 𝑥 = 𝑥. So what exactly does
𝜑(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 sbequ 2374, sbcom2 2443 and sbid2v 2455).
Note that our definition is valid even when 𝑥 and 𝑦 are
with the same variable, as sbid 2112 shows. We achieve this by having 𝑥
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 2453 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 2372. When 𝑥 and 𝑦 are distinct, we can
substitution with the simpler expressions of sb5 2428
and sb6 2427.
There are no restrictions on any of the variables, including what
variables may occur in wff 𝜑. (Contributed by NM,