"Everyone knows each other", said someone on BBC Radio 4 this morning, speaking about some tight-knit community. And instantly I saw that this was the key to a definitive argument against the logic of the opponents of singular they. I wonder if I can make you see how awesomely beautiful the insight is.
The -s suffix on the present-tense verb knows tells us that the subject is morphosyntactically singular. That is, it counts as singular for purposes of subject-verb agreement. But each other, famously, requires a semantically plural subject. That is why They know each other is grammatical and *He knows each other is not. From this and nothing else it follows that semantic plurality and morphosyntactic singularity are compatible in English. No prescriptivist has suggested that there is something grammatically wrong with Everyone knows each other. But because of that, the logical objection to singular they just collapses. Everyone knows themselves has no grammatically relevant property that isn't already instantiated by Everyone knows each other.
I recently discussed the ravings of the apparently half-mad David Gelernter's description of the consequences of singular they (which he falsely believes was a malign invention of feminists some time "since the 1970s"): he said that grammar simply "collapsed in a heap after agreement between subject and pronoun was declared to be optional." He means antecedent and pronoun. But there is no failure of agreement.
Syntactically, when everyone is a subject it demands singular agreement on the verb of its clause; semantically, when it is an antecedent it denotes a collectivity of human beings and thus allows plural anaphoric elements. Everyone, including Gelernter (surely), would agree that its meaning is fully compatible with plural predicates, so that Everyone knew each other is grammatical. What happens in Everyone knows each other is that you see a plural-requiring predicate and a singular verb in the same clause. And Everyone knows themselves simply illustrates the same point with a reflexive pronoun instead of a reciprocal.
Avoid singular they if you want to; nobody is making you use it. But don't ever think that it is new (it goes back to early English centuries ago), or that it is illogical (there is no logical conflict between being syntactically singular and semantically plural), or that it is ungrammatical (it is used by the finest writers who ever used English, writers who uncontroversially knew what they were doing).
By the way, Arnold Zwicky points out to me that there are verbs like disperse that can only have collectivity-denoting subjects, and this yields the possibility of cases where the anaphoric pronoun for everyone must be they, like Everyone dispersed when they heard the shots. He, like he or she, would be simply unimaginable here.
This tells us that the logical case against singular-anteceded they is really in tatters. Gelernter's reasoning has collapsed in a heap, one might say. If Everyone dispersed when they heard the shots is fine, why should we assume there would have to be anything logically or grammatically wrong with Everyone gets out of the way when they hear gunshots ? If the evidence was clear that people never said such things, we grammarians would have to accept it, of course. But instead the evidence is that everyone says them all the time. So we have a logically impeccable construction that expert users of the language regularly employ and experienced listeners unhesitatingly accept. I wonder what more one would need to take something to be grammatical.