I have seen both versions of the word, falsy and falsey.
It can mean "something that is equivalent to false" in computer science, such as "The only two falsy values in the Ruby Language are false and nil".
What is the correct usage of this word?
I've always seen falsy and truthy. Falsey is a perfectly acceptable alternative and gives me just as many search results. The word is unfortunately too new to provide good sources. The ECMAScript Language Specification uses “⟦ToBoolean⟧” to refer to the interpretation of of non-Boolean values as Booleans, but makes no use of truthy or falsy.
As tchrist points out, Perl is a notable exception because it lacks a Boolean type, so falsy values are just called false, in the adjectival sense.
The only such word currently attested by the OED is the plural noun falsies, whose sense is given as:
It has its own Wikipedia entry, one which I must advise you is unlikely to be work-friendly in sensu stricto.
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