we're no longer talking about signalling as it was originally conceived.
The word "signal" dates back to the 14th century. The use of the word as a verb dates back to at least the 17th century. The specific meaning you are trying to use seems to have started in the mid-to-late 20th century. That's the issue. Signaling means what you say it means, but it also has a broader meaning. If I do an action that I wouldn't do were it not for the fact that others observe me doing it, it seems very likely that part of my motivation is signaling. The manager clearly qualifies for this, as she would not be "acting decisively" but for the fact that she is being observed. (I also think that Gresham's Law is the wrong one, or that you need a bit of an explanation to tie it into this behaviour, but that's besides the point; the fact that there is a more precise name for a problem does not make that the only name for the problem.).
Unenforceable pre-commitments are still precommitments. If I promise never to cheat on my spouse again, despite a long history of cheating, I've made a commitment. It's not a very credible commitment, but it still belongs in the set labeled, "commitments." If you define "commitment" to only count "credible commitment," you've essentially created a new word.
As with any debate over definitions, this can get circular rather quickly. My point is this: if you want people to use the word "signal" to mean something very specific, and to abandon the conventional use of the word, you need to provide a viable alternative definition, and you need to explain why it would be more productive to abandon the conventional use of the term. I do not think your definition is viable, because it necessarily involves an arbitrary cost threshold. Even if your definition were viable, I don't see how you've shown that there is a problem with the conventional use of the term. Yes, there are different types of signals that differ in important ways, but I don't see why this warrants completely changing how we use the term, rather than specifying weak vs. strong signals.
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My objections would indeed not apply if a new term were used. You can define a new term however you like; that's the point of making a new term. You can't just declare that a commonly used term has a specific meaning without providing some justification for abandoning its other existing meanings.
If I wanted to argue that the definition of "bachelor" is "an unmarried man," I could do so rather easily, by citing this for example. If I were arguing over what counts as "theft," I could offer an argument as to why a particular act should or should not fit under the general definition. An argument like the OP's could theoretically include evidence (of common usage, of confusion, etc.) or argumentation, but the OP's post does not really seem to do this. It declares, "The definition should be X" and then rejects certain usages as not fitting the definition. If you're using an extremely common word like "signaling," you don't get to arbitrarily redefine it.