For Chrysippus see already chi 568 (and chi 567).
[1] What follows depends, almost verbatim, on Diogenes Laertius 7.186-7 (= FDS 1205), where some examples of fallacies are given. No definition of this term is attributed by ancient sources to the Stoics directly. Sextus Empiricus says (in Pyrrhoniae Hypotyposes 2.229 = FDS 1200) that for those who venerate dialectic 'a fallacy is a persuasive and tricky argument that makes a false or similar to false conclusion to be admitted'. Diog.Laert. 7.43, quoting the summary of Diocles, divides the fallacies according to the level in which they are produced. Thus there are fallacies in the level of language and in the level of concepts, which correspond with the levels of dialectic: dialectic about language (fwnh/), and dialectic about meaning (lekta/). In both levels there exist ambiguities which could become fallacies. Thus arguments that change their truth value (FDS 1025) are caused by the self-structure of the proposition, while the fallacy of the carriage that goes through the mouth is caused by a purely linguistic ambiguity.
[2] Diog.Laert. had put this proposition negatively: 'what is not in the city, is not in the house either'.
[3] This argument is known as "Nobody", because of the episode of Odysseus and the Cyclops in Homer, Odyssey 9.366-414. It is said that it was invented by Diodorus of Megara (Meg. 10). It occurs in a lacunar form in Diog.Laert. 7.82: the opposition between 'here' and Rhodes, instead of Athens and Megara. According to Proclus (In Parm. 889, 8-13), Chrysippus could have used it to criticize the platonic theory of Forms, because entities which for the Stoics are real, i.e. individual bodies, do not produce paradoxical conclusions, but qualities –- with which Chrysippus associated Ideas -- do. Brunschwig 1988 emphasizes that this argument is a 'test of reality': just the bodies pass it with success. At a linguistic level it shows the need to establish a difference between o)/noma (proper name) and proshgori/a (appellative, common name), because this last type is the linguistic counterbalance of qualities (and Platonic Forms), and for this reason it produces a paradoxical conclusion. This distinct behaviour allowed that the Stoics could resist the tendency imposed by Alexandrian grammarians to gather both categories into a simple one, considering proshgori/a as a subordinate species (cf. Dionysius Thrax, Téchne grammatiké §11).
[4] The argument of the carriage that goes through the mouth is related to the notion of a)sw/maton 'incorporeal'. According to Clement of Alexandria (Stromata 8.9,26,5,97,3-7 = FDS 763), who transmits the same argument changing the noun from a(/maca 'carriage' to oi)ki/a 'house', the theory behind the argument points to the fact that the thing that goes through the mouth is not the house, but the linguistic/grammatical case in which the material-house has its counterbalance. The case, because of its linguistic character, is an incorporeal.
[5] Aulus Gelius, Attic Nights 16.2.11, wrote about a way of avoiding this type of fallacy, which seems to have been very famous in ancient times, answering the question 'What you have lost, is it something you had or something you had not?', with a formula like 'What I had, I had if I have not lost it'.