From: Jörg Knappen <jknappen_at_web.de>

Date: Fri, 19 Jul 2013 11:54:29 +0200 (CEST)

**Gesendet:** Freitag, 19. Juli 2013 um 09:43 Uhr

**Von:** "Stephan Stiller" <stephan.stiller@gmail.com>

**An:** "Unicode Public" <unicode@unicode.org>

**Betreff:** Re: symbols/codepoints for necessity and possibility in modal logic
Received on Fri Jul 19 2013 - 04:57:15 CDT

Date: Fri, 19 Jul 2013 11:54:29 +0200 (CEST)

I think,

U+25C7 WHITE DIAMOND

is the best choice, followed by

U+27E1 WHITE CONCAVE-SIDED DIAMOND • never (modal operator)

The latter has a more fancy shape and might not be the one the reader expects. As a plus, it comes also with versions having right and left ticks, needed in some extensions of modal logic. I couldn't locate WHITE DIAMOND WITH LEFTWARDS TICK in UNicode.

(U+2662 WHITE DIAMOND SUIT would also look OK, but I think this is symbol abuse. Can be used as a fallback when the font of choice has this one, but none of the two above.)

For the properties of mathematical symbols, see also

http://www.unicode.org/reports/tr25/

---but I have to admit that the report does not answer the specific question posed here.

Maybe this mapping table is more useful (but harder to read):

http://www.w3.org/Math/characters/unicode.xml

--Jörg Knappen

P.S. I'd consider U+22C4 DIAMOND OPERATOR as wrong because it is used as a binary operator which has a very different

spacing than the unary modal operator needed here.

"wrong" is strong wording and goes beyond what I suggested or implied, but it's not clear to a user of Unicode that it's the right fit either. There are a couple of indicators factoring in:What is wrong with using DIAMOND OPERATOR?

- The charts mention modal logic in conjunction with ◻ (U+25FB) and ⟠ (U+27E0) but not with ⋄ (U+22C4).
- The glyph in the code charts is tiny (and that of Cambria Math is tiny as well). Typographically you see various things (a lozenge, fallback to letter-M) in esp older books, but it feels like it's meant to be an orthogonal diamond of perhaps slightly less area than the box but descending a little above and below the box, which is somewhat taller than x-height. The book by {Blackburn, de Rijke, Venema} has glyphs that look right. This is more than a guess: it makes sense if they have similar visual weight, as they are – literally – defined to be duals of one another; but whether you can make them geometrically congruent symbols of equal area I haven't tested (this might have the diamond ascend too far).
- The vague notion of "operator" (a word with different meanings in math, from
*logical relation*to*[non-logical/non-relational] mapping of type A×A→A or perhaps A×A→B*to*(linear) map (between say vector spaces) in linear algebra*) in this context (in the code charts) seems to refer to something like my middle meaning, which is likely to use a smaller symbol around x-height in placement and dimensions. - The glyph of ⬦ (U+2B26) seems to have a more appropriate name, but in the charts I like ◇ U+25C7. The differently sized square-like symbols are hard to semantically tell apart in/from the charts anyway.
- These symbols are the first two visually distinct ones you define in modal logic, so they're well-known and standardized in meaning for anyone who had had contact with the field. It's surprising they're not explicitly named in the charts. (There's stuff like the outdated horseshoe for logical implication popping up in the relevant books, but that is a leftover or outdated logic notation in general.) So for box and diamond it's quite reasonable to be expecting a standard math font to provide them just right out of the box; for whatever commonly used box-like symbols in math there are, one would assume that there are corresponding codepoints; otherwise you'd have to choose a different font.

Stephan

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