Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > df-sb | GIF version |
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 when 𝑦 is properly substituted for 𝑥 in the
wff
𝜑." We can also use [𝑦 / 𝑥]𝜑 in place of the "free for"
side condition used in traditional predicate calculus; see, for example,
stdpc4 1755.
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, "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 1820, sbcom2 1967 and sbid2v 1976). Note that our definition is valid even when 𝑥 and 𝑦 are replaced with the same variable, as sbid 1754 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 1971 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 1974. When 𝑥 and 𝑦 are distinct, we can express proper substitution with the simpler expressions of sb5 1867 and sb6 1866. In classical logic, another possible definition is (𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑) 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 𝜑. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
df-sb | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | 1, 2, 3 | wsb 1742 | . 2 wff [𝑦 / 𝑥]𝜑 |
5 | 2, 3 | weq 1483 | . . . 4 wff 𝑥 = 𝑦 |
6 | 5, 1 | wi 4 | . . 3 wff (𝑥 = 𝑦 → 𝜑) |
7 | 5, 1 | wa 103 | . . . 4 wff (𝑥 = 𝑦 ∧ 𝜑) |
8 | 7, 2 | wex 1472 | . . 3 wff ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) |
9 | 6, 8 | wa 103 | . 2 wff ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
10 | 4, 9 | wb 104 | 1 wff ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
Colors of variables: wff set class |
This definition is referenced by: sbimi 1744 sb1 1746 sb2 1747 sbequ1 1748 sbequ2 1749 drsb1 1779 spsbim 1823 sbequ8 1827 sbidm 1831 sb6 1866 hbsbv 1921 nfsbv 1927 |
Copyright terms: Public domain | W3C validator |