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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
." We
can also use in
place of the "free for"
side condition used in traditional predicate calculus; see, for example,
stdpc4 1702.
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 1765, sbcom2 1908 and sbid2v 1917). Note that our definition is valid even when and are replaced with the same variable, as sbid 1701 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 1912 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another alternate definition which uses a dummy variable is dfsb7a 1915. When and are distinct, we can express proper substitution with the simpler expressions of sb5 1812 and sb6 1811. In classical logic, another possible definition is but we do not have an intuitionistic proof that this is equivalent. There are no restrictions on any of the variables, including what variables may occur in wff . (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
df-sb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 | |
2 | vx | . . 3 | |
3 | vy | . . 3 | |
4 | 1, 2, 3 | wsb 1689 | . 2 |
5 | 2, 3 | weq 1435 | . . . 4 |
6 | 5, 1 | wi 4 | . . 3 |
7 | 5, 1 | wa 102 | . . . 4 |
8 | 7, 2 | wex 1424 | . . 3 |
9 | 6, 8 | wa 102 | . 2 |
10 | 4, 9 | wb 103 | 1 |
Colors of variables: wff set class |
This definition is referenced by: sbimi 1691 sb1 1693 sb2 1694 sbequ1 1695 sbequ2 1696 drsb1 1724 spsbim 1768 sbequ8 1772 sbidm 1776 sb6 1811 hbsbv 1862 |
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