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|Description: Define the not-free
predicate for wffs. This is read "𝑥 is not free
in 𝜑". Not-free means that the
value of 𝑥 cannot affect the
value of 𝜑, e.g., any occurrence of 𝑥 in
bound by a "for all" or something that expands to one (such as
exists"). In particular, substitution for a variable not free in a
does not affect its value (sbf 2199). An example of where this is used is
stdpc5 2137. See nf5 2216 for an alternate definition which
quantifiers on the same variable.
Not-free is a commonly used constraint, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the not-free notion within the logic itself rather than as a metalogical side condition.
To be precise, our definition really means "effectively not free", because it is slightly less restrictive than the usual textbook definition for not-free (which only considers syntactic freedom). For example, 𝑥 is effectively not free in the formula 𝑥 = 𝑥 (see nfequid 1970), even though 𝑥 would be considered free in the usual textbook definition, because the value of 𝑥 in the formula 𝑥 = 𝑥 cannot affect the truth of that formula (and thus substitutions will not change the result).
This definition of not-free tightly ties to the quantifier ∀𝑥. At this state (no axioms restricting quantifiers yet) 'non-free' appears quite arbitrary. Its intended semantics expresses single-valuedness (constness) across a parameter, but is only evolved as much as later axioms assign properties to quantifiers. It seems the definition here is best suited in situations, where axioms are only partially in effect. In particular, this definition more easily carries over to other logic models with weaker axiomization.
The reverse implication of the definiens (the right hand side of the biconditional) always holds, see 19.2 1934.
This predicate only applies to wffs. See df-nfc 2912 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 24-Sep-2016.) Convert to definition. (Revised by BJ, 6-May-2019.)
|df-nf||⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))|
|1||wph||. . 3 wff 𝜑|
|2||vx||. . 3 setvar 𝑥|
|3||1, 2||wnf 1746||. 2 wff Ⅎ𝑥𝜑|
|4||1, 2||wex 1742||. . 3 wff ∃𝑥𝜑|
|5||1, 2||wal 1505||. . 3 wff ∀𝑥𝜑|
|6||4, 5||wi 4||. 2 wff (∃𝑥𝜑 → ∀𝑥𝜑)|
|7||3, 6||wb 198||1 wff (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))|
|Colors of variables: wff setvar class|
|This definition is referenced by: nf2 1748 nfi 1751 nfri 1752 nfd 1753 nfrd 1754 nfbiit 1813 nfbidv 1881 nfnf1 2091 nf5r 2122 nfbidf 2156 nf5 2216 nf6 2217 nfnf 2266 sbnf2 2291 bj-nfimt 33511 bj-ssbft 33532 bj-nnfnfTEMP 33805 bj-nfnnfTEMP 33811|
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