Change “Relation” to “Logical operator”: U+2263 (STRICTLY EQUIVALENT TO)

Ivan Panchenko ivanpan3 at
Sat Jul 23 18:07:32 CDT 2022

The character U+2263 (≣ STRICTLY EQUIVALENT TO) is found under the subhead
“Relations”. I think it would be more appropriate to put it under “Logical
operator” (for comparison: U+2227) because it stands for a connective in
modal logic: 𝑝 is strictly equivalent to 𝑞 if 𝑝 necessarily implies 𝑞
and 𝑞 necessarily implies 𝑝. Source: Fitch (1952, p. 77).

One might object that “is strictly equivalent to” (as opposed to
“necessarily if and only if”) is used in metalanguage for a relation
between logical formulas (→ use–mention distinction). However, this is not
what the symbol “≣” itself actually means, it is just that an alternative
to saying “if and only if” is to say “is equivalent to” and mention (rather
than use) the linked logical formulas. Likewise, one might read “𝑝 → 𝑞”
either as “if 𝑝 then 𝑞” or as “𝑝 (materially) implies 𝑞”. This does not
change the fact that “≣” and “→” are symbols of the logical OBJECT language.

(As a side note, usage of the triple bar “≡” and of “identity” in
mathematics is convoluted: In ordinary language, two distinct things might
be said to be “equal” when they are equal in a certain respect (e.g.,
“sexual equality”). In mathematics, “equals” (=) is simply used in the
sense of strict identity rather than for equivalence relations or
congruence relations in general, though convention has it that the equals
sign is more often read as “equals” or “is equal to” than “is identical
to”, and “(solving an) equation” is used while “identity” occurs in
“identity function” and what is expressed by a statement of equality can be
called an identity (e.g., “Euler’s identity”). As described so far, there
is no actual difference between ‘is equal to’ and ‘is identical to’ at all,
however, it seems that because we are only justified in proclaiming that an
identity holds if the statement is generally valid, this usage of
“identity” got CORRUPTED into saying things like “This equation is an
identity” (meaning that the equation holds for all values) and “is
identically equal to” (≡); you can even find a few Google hits for
“identically less”/“identically greater”. 🥴 Besides, “≡” is used for
equivalence relations and for the logical equivalence connective.

When John Conway (in “On Numbers and Games”) used “≡” for identity
(expressing that two objects are one and the same object) and “=” for
equality in a weaker sense than described above for mathematics, he might
have been influenced by the fact that “≡” is sometimes read as “is
identical(ly equal) to”, even though this so-called “identity” is something
different from Conway’s identity altogether. Donald Knuth used the symbols
the other way round, which I like better.)
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