Here’s an odd thing. There seems, browsing along my shelves, to be no really standard symbolic metalinguistic shorthand used in elementary books for assigning a truth-value to a wff (say, in the propositional calculus). You would have expected there to be some.
In the first edition of my Introduction to Formal Logic, I borrowed the symbol ‘‘
to abbreviate ‘has the value … [on some given valuation]’ and wrote the likes of e.g.
If and then .
But on reflection this was pretty silly, given that the symbol ‘‘ is already overloaded (not in my book, but elsewhere — like on math.stackexchange! — where, for a start, some use it for the conditional, some use it in place of a turnstile, and some get in a tangle by using it ambiguously for both!). It seems wiser not to add to possible confusion, especially when readers might well simultaneously get to see the double arrow being used in one of these different ways.
A bit of notation that is used, not at all consistently but often enough, is square double-brackets, so ‘‘ is used for ‘the value of …’, and we write the likes of ‘‘. But this seems to me a bit cluttered for elementary purposes — I’m after readability, rather than portability to more sophisticated contexts. And it misses the dynamism(??) of some type of arrow.
So for the upcoming second edition, I’m tentatively minded to use the \mapsto symbol for value-assignment, and write instead
If and then .
(I suppose a colon could be another possibility, but I’d rather have something more distinctive. And the likes of ‘T()’ isn’t so pretty/easy to read in bulk and is conventionally part of an augmented object language.)
Any objection to the revised arrow? Am I missing some sufficiently established (or even just nicer) alternative??