Why is boolean value f (false) defined as a parsing-word while t (true) is not in Factor?
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The answer is indicated on the page for f. Specifically, "[t]he f object is the singleton false value, the only object that is not true." In contrast, t is defined simply as SINGLETON: t where SINGLETON: defines a class and thus a class word which is the sole instance of that class (i.e. itself).
For the purposes of logical comparisons, there is absolutely nothing special about t. Literally any object (other than f) could be used instead. As the docs say, t is just a "canonical" choice.
f, on the other hand, is uniquely special. f couldn't be defined the same way as t, i.e. as a SINGLETON: because then f and its class word would be the same thing, but the class word of f should be a perfectly normal object, and thus behave as true, and only its instance should behave as false.
Ultimately, the f object (as opposed to the f parsing word) is deeply wired into the Factor implementation. t gets some mild special handling by the implementation because the implementation needs to sometimes return t from the C++ implementation code. But it is not defined by the C++ code, while the f object is.
To be clear, f being a parsing word has nothing to do with the behavior you saw with random. The reason f random works is because f is an instance of an immutable-sequence and thus a sequence for which random is defined. t is not. Factor, like many languages, especially dynamically typed languages with "generalized" booleans, fails to distinguish between "no result" and "a valid result that happens to be f". In other words, { } random and { f } random both return f but for different reasons. f random is behaving more like the former as f behaves like an empty sequence.
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