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|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 when
is properly substituted for in the wff
can also use in
place of the "free for"
side condition used in traditional predicate calculus; see, for example,
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 1875, sbcom2 1990 and sbid2v 1998).
Note that our definition is valid even when and are replaced with the same variable, as sbid 1828 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 1995 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 1872. When and are distinct, we can express proper substitution with the simpler expressions of sb5 1915 and sb6 1914.
There are no restrictions on any of the variables, including what variables may occur in wff .
|1||wph||. . 3|
|2||vx||. . 3|
|3||vy||. . . 4|
|4||3||cv 1585||. . 3|
|5||1, 2, 4||wsbc 1814||. 2|
|6||2||cv 1585||. . . . 5|
|7||6, 4||wceq 1586||. . . 4|
|8||7, 1||wi 3||. . 3|
|9||7, 1||wa 337||. . . 4|
|10||9, 2||wex 1615||. . 3|
|11||8, 10||wa 337||. 2|
|12||5, 11||wb 219||1|
|Colors of variables: wff set class|
|This definition is referenced by: sbimi 1817 drsb1 1819 sb1 1820 sb2 1821 sbequ1 1822 sbequ2 1823 sbn 1877 sb6 1914 subsym1 14951|