On Aquinas's view, formal falseness consists in the intention to say what is false; a material falseness consists in the saying of something that is in fact false. One could, for instance, intend to say something true but what one says is actually false. That is materially false, and it is not sufficient for lying. You can lie without any material falseness at all: if you intend to lie but what you say happens to be true, you still can be said in some sense to have lied, even though you did not even say something false. As Aquinas says, your action has falseness essentially but truth accidentally.
Now there are three potential pitfalls one may face when it comes to proper interpretation of this claim, one of which is obvious and therefore easy to avoid and the other two of which are more subtle and easy to fall into. The first subtle one is that the falseness of which Aquinas speaks is not a mere mismatch between words and things but a falseness of action by which words and things are related. It is the falsity of the manifesting or enunciating that makes the enunciation formally false. We are talking about acts of reason, and in particular, they are reason's particular acts of manifesting the true and the false. It is entirely because of this that lying, which by its nature involves falseness in the very act of manifestation of the true and the false, is in opposition to the virtue of truthfulness, which (of course) involves truth in the rational act of manifestation of the true and the false. It is also what is involved in the distinction between material falseness and formal falseness: in material falseness, that which is enunciated is false, while in formal falseness, the enunciating itself exhibits falseness: it is an intention to say what is false in the sense that it is a will for false speaking, a will to be false in speaking -- the falseness is built into the intention itself even if it fails actually to issue in anything that is false. A little more on this below.
The second subtle pitfall is to confuse formal falsehood with the intention to deceive. What makes this one tricky is not Aquinas's actual discussion so much as the fact that Aquinas very clearly distinguishes these two things that we sometimes conflate. The intention of speaking the false is what lying is, but the intention to deceive has to do with a natural effect of being false in speaking. The latter is not formal falseness but effective falseness, i.e., falseness in the effect of one's speaking. Smith fails to keep this properly in mind, I think, when she says:
Thus I believe Aquinas would not approve of misleading Nazis who were attempting to kill Jews even with true speech. I believe his principles mean someone answering the door of his neighbor's house (who is harboring Jews) and responding to a Nazi demanding that he "Turn over to me any Jews in your house" would lie if he said "There are no Jews in my house." Such is true speech, but said with the will of saying something false [voluntas falsum enuntiandi], which Aquinas finds to be the essence of a lie.
Now, whether or not Aquinas would approve of this, we cannot conclude this from his discussion of lying because Smith is clearly wrong about what kind of falseness is involved here. While one could say what the person in the scenario says with voluntas falsum enuntiandi, nothing about the scenario Smith describes requires it; everything in the scenario is consistent with a will to speak truly. One could want the Nazi to be deceived, i.e., one could be aiming at effective falseness. (Strictly speaking, nothing in the scenario Smith gives requires this, either, since one could very well be answering in shock, or befuddlement, or honest confusion, or even, not expecting the Nazi to be deceived at all, simply to delay or give him pause.) But this is not the same as willing to speak falsely, and Aquinas is very clear about the distinction. Even if one assumes that Aquinas thinks that trying to deceive people always involves formal falseness, it does not follow from anything he actually says about lying. And it leads to further confusion later, when Smith runs into a puzzle over why Aquinas explicitly allows circumstances in which we can hide our purpose and meaning, one that her interpretation has made insoluble.
The other pitfall, which is easy to avoid, because Aquinas himself says that it is formal falseness that makes something have to do with lying, is to think that this distinction between formal falseness and material falseness is a distinction between formal lies and material lies. Later moral theologians use the terminology; it is not Aquinas's own distinction, which is between formal falseness, which is in itself lying, and material falseness, which is not in itself lying at all. Smith, for what reason I do not know, jumps right into this pit, by summarizing Aquinas's view as saying that there are two kinds of lies, material and formal. The result is a much more extreme position than I think the text can reasonably be stretched to suggest.
Smith goes on to say that Aquinas thinks that lying is wrong because it violates the purpose of enunciative speech, which in a sense is true; but it is more completely accurate to say that Aquinas thinks lying is wrong because false enunciative speech violates the purpose of reason from which it springs. Enunciative speech, remember, is not a thing the rational act extends to; it is an act of reason and is subject to the ends of reason itself. And since reason, with truth as a fundamental end, has truth as its end in having signs at all, false enunciative speech is reason's use of signs in a manner inconsistent with truth as the end of reason. Words have their value as signs of acts of mind, and thus exist to express the mind: false enunciative speech is their use by a mind to express itself in a way inconsistent with actual expression of itself. It is a use of what naturally expresses the mind in order to misexpress it; this is an injustice in broad sense of the term and a disorder of reason itself. One can definitely call this a violation of the purpose of enunciative speech, but the 'purpose of enunciative speech' doesn't just jump out of nowhere, but arises from the fact that enunciative speech is an act of reason. This is perhaps easy to overlook if you focus too narrowly on ST 2-2.110.3; but all of Aquinas's discussion of lying presupposes his discussion of the virtue of truthfulness; he is discussing lying because it is opposed to truthfulness. And truthfulness is, broadly speaking, the virtue whereby the mind expresses itself to another mind in a way consistent with itself.
It's clear that Smith runs into trouble in this regard, because what she immediately goes on to puzzle about is where the purpose of enunciative speech comes from. And when she discusses it, she discusses it not in terms of a mind expressing itself, which are the terms in which Aquinas discusses it, but in terms of "how language operates." Now, language does not operate at all except insofar as it is used by reason; language does not "operate to convey the truth," but reason can express itself truly by language. Several of Smith's examples of false enunciations fail completely because of her sloppiness on this point. Cordialities, for instance, are obviously often forms of figurative speech, and reason can express itself truly by figurative speech. This is true as well of fiction and scholarly speculations. Other examples she gives are cases where reason has a particular aim in saying something -- encouragement, consolation, and so forth -- that concerns the effects of saying it. However, as should be clear from our discussion above, not a single one of these is relevant: they merely identify the obvious fact (one presupposed, in fact, by Aquinas's account of lying) that you can deliberately aim at other things in addition to the end of reason's expressing itself at all.
It is extremely odd. Most of Smith's argument against Aquinas makes the bizarre assumption that Aquinas doesn't understand that language can be used to do things like encourage and console and entertain. The sheer ignorance of language that would entail is just stunning, and it is not consistent with (for instance) Aquinas's commentary work, and it should have given Smith more pause than it apparently did. And it is a strong sign that her interpretation of Aquinas is not wholly adequate to Aquinas's actual position.
Much of Smith's own view of the matter I find somewhat difficult to understand. It's a variation of the 'right to know' account of lying but a somewhat unusual one, and I don't know yet what to make of most of her claims, nor do I understand all of her arguments. So I'll leave them alone for now, particularly since I still intend at some point to post on some of the philosophical problems faced in general by 'right to know' accounts of lying. I'll also mull her account over a bit more to see if time will remedy my perplexities over it, and if so, I might post on it in particular.