1. According to traditional logic the two types of universal proposition (affirmative 'Every S is P' and negative 'no S is P') are contrary: they cannot both be true. Hence, their two subcontrary propositions 'some S is P' and 'some S is not P', which are by definition the negations of the 'contrary' propositions, cannot both be *false*.
2. But if each of the subcontrary propositions has 'existential import', i.e. implies the existence of the subject, S, it follows that the subject term cannot be empty, i.e. the subject exists.
3. All or most traditional logicians recognised that there were empty terms (e.g. 'goat-stag', 'chimera')
4. This flaw is pretty obvious and no such obvious flaw in a system evades scrutiny over a long period (in this case, over 20 centuries).
Clearly, one of these propositions must be false. Parsons argues that it is (2). Traditional logicians, he says, did not think that the O proposition 'some S is not P' has existential import. It does not imply the existence of any S's. It simply is the strict negation of 'every S is P'. Thus there is no obvious flaw in the system.
I reply as follows. First, traditional logic developed in Latin, and the Latin form of the O proposition from Boethius was 'quoddam S est not P', i.e. 'a certain S is not P'. The word 'quoddam' (neuter, 'quidam' masculine) is strongly existential. It means not just 'someone' or 'something', but indicates that the speaker knows the identity of the object in question, but is not divulging it. 'Someone in particular', 'a certain someone' &c.
Given that traditional logicians did think the subcontraries have 'existential import', how do we resolve the apparent contradiction? Answer: traditional logicians had a different sense of 'empty' than modern logicians. They thought that 'some S is not P' is existential in the sense that it implies something is an S that is not P. Thus there are no empty terms, in this sense. But they thought that some things (chimeras, goat-stags) do not exist in reality. Thus 1-4 are only inconsistent if we read 'empty' in (3) in the same sense as it is used in (2). If we read it in (3) as meaning 'has a subject that does not exist in reality', and in (2) as 'has a subject', there is no contradiction.
One of the best bits of evidence they thought this is that scholastic logicians tried to construct ontological arguments. If 'every perfect being is perfect' implies 'some being is perfect', and if 'some being is perfect' is existential, then the existence of God is a logical truth. So why did no traditional logician prove the existence of a perfect being in this way? Because 'some being is perfect' is not existential. It does imply 'some perfect being exists in reality'. (I owe this thought to Morris and Laserowitz, 1948).
Traditional logic … ignores the question of existence or non-existence, because it inquires only how things are thought.… Consequently there is implication between Every sea serpent is bearded and Some sea serpent is bearded, in spite of the fact that there are no sea serpents whatever. And its rule comes to this: if you think every sea serpent is bearded, you must think some are. (Wade)
Thus Parson's argument is anachronistic. Given the Brentano-Venn interpretation of particular proposition (i.e. that some A is B logically implies Some AB exists), Parson's argument is sound. But that interpretation was one of the most keenly fought battles in logic, begun in the 1850's, and not conceded by the majority until the 1950's – a real Hundred Years' War! If we do not accept that interpretation – and the evidence is that few traditional logicians accepted it – then it is perfectly consistent to hold (1) that there are empty subject terms, in the sense that the subject does not exist in reality (2) that there are no empty subject terms, because they all denote a 'something' (3) that subcontraries imply the existence of a 'something', though not necessarily an existing something.
I am neutral about whether the traditional interpretation of non-existential character of particular propositions is correct. But I would end by noting that
Not all characters in War and Peace were real historical characters
clearly implies
Some characters in War and Peace were not real historical characters
despite the fact that names like 'Natasha Rostov' are empty. Indeed, it is the emptiness (in one sense) of such names, that makes the implication valid.