There has been some discussion around the internet of the modern form of Catholic integralism, in part due to Kevin Vallier's recent All the Kingdoms of the World. In general it has tended to be framed in terms of 'liberalism vs. integralism', which leaves out people like myself who are neither*, but even in those terms has tended to be slippery -- integralists tend to defend not integralism but their own particular integralist preferences, while liberal critics of integralism, when they aren't spouting gibberish, erratically slide among different versions (and imaginations of what implications different versions might have) in ways that often result in their criticisms being simply incoherent.
Trying to give any kind of adequate definition of integralism (or liberalism, for that matter) is quite difficult. I'm inclined to think it needs to be characterized typologically, by identifying a pure form of it, in terms of the following set of principles:
(1) Human nature is naturally fulfilled only in a civil society, which is a society complete in itself for achieving its intrinsic ends (the old name for this is societas perfecta, and the usual description of these intrinsic ends of civil society is peace in the sense of the tranquillity of just order, although sometimes people prefer to describe it metonymically as civil friendship).
(2) Civil society forms an instrumental part of itself to coordinate and organize itself by, for the achievement of its intrinsic ends; this instrumental part is what we call the state; as instrumental to the ends of civil society, it can only be used in a manner consistent with the ends of civil society.
(3) Christians by their baptism are not merely members of civil society but have a 'dual citizenship' in another, very different, society, the Church, which is also a societas perfecta, with its end usually summarized as salvation of souls; it is organized by the hierarchy, which is very different from the state but is its analogue in the very different society of the Church.
(4) The ends of civil society and the ends of the Church (and thus the ends of the state and of the hierarchy which serve these societies respectively) are not equal; the ends of the Church, being directly received from God, have preeminence over those of civil society, without squashing or eliminating them.
(5) Just interaction between these societies requires both their distinction, because their ends are distinct, and their integration, in which the intrinsic ends of civil society are explicitly recognized as having, beyond their value in themselves, an ancillary function with respect to the intrinsic ends of the Church.
(6) The performance of this ancillary function in particular requires cooperation between state and hierarchy, but since the ends of the two are not equal, the hierarchy is the superior partner in the cooperation; this is not a general superiority, but a superiority with respect to matters directly involved in this ancillary function.
(7) In its role as superior partner in this ancillary function, the hierarchy may at least authorize the state to perform activities beneficial to the Church, where these are not inconsistent with the ends of the civil society, and may also under certain conditions prevent or veto state action that is not beneficial to the Church. (Contrary to what Vallier always assumes, different integralists disagree on the question of whether and to what extent and under what conditions the hierarchy may require the state to engage in certain activities.)
(8) In return, the Church supports the civil society in ways that are appropriate to the ends of the Church; which may under certain conditions involve the state authorizing the hierarchy to perform certain state functions or preventing the hierarchy from directly harming the ends of civil society. (Different integralists also differ about whether and to what extent and under what conditions the state can require the hierarchy as such to do things.)
In formulating these principles, I have had to make some terminological choices that are far from universal. For instance, it is not always clear when either integralists or liberals use the term 'state' whether they use it to mean what I here call 'the civil society' or what I here call 'the state'. And as I previously noted, this is a typological rather than essential definition; different particular versions of what is called 'integralism' may recede from this (e.g., by muddling or only loosely or in a limited way exemplifying some of these) or go beyond it (e.g., by adding other substantive principles that are essential to that version). It's also possible that one could improve the statement of one of these principles. But I would argue that this is superior to a lot of the characterizations of integralism one finds. First, because it clarifies the connection of the position to Catholic theology; there is an excellent argument that something along the lines of (1)-(4) are required by Catholic theological principles, and while (5)-(8) are more controversial, in this form it's not difficult to explain why integralists, at least, might think they are also required, or at least highly preferable to alternatives. At the same time, the more controversial status of (5)-(8) clarifies the ways in which one might have a Catholic political theology that is not integralist. Second, because it is much less tendentious than most descriptions of integralism, rigging it neither in favor of or against the integralist. Third, it avoids the common 'chunkiness' which tends to plauge the discussions of both liberals and integralists; there's a bad habit, arising from how things are handled in certain liberal societies, of treating Church (ambiguous between Church and hierarchy in the above terms) and State (ambiguous between civil society and state) as if they interacted only in big-block, large-scale-policy ways, whereas in fact they can interact in all sorts of ways at all sorts of levels.
There are some complications that are not completely accounted for in the above characterization. The first is that every civil society has a civil religion, something that serves at least for ceremonial and solidary purposes. It's the flags-and-parades-and-funerals-and-marriages-and-prayers portion of society.This is sometimes an established religion (i.e., a religion that is part of the state, like a national church), but it might take all sorts of other forms; the United States has a traditional civil religion that is deliberately generic, and is celebrated by Thanksgiving and Memorial Day and Old Glory and "The Star-Spangled Banner" and 'In God We Trust', and serves both to unite citizens and to make it easier for the state to operate in a fairly religious society while being relatively neutral with respect to the major religions of the population. Exactly how to understand and treat this is an extremely difficult issue for both liberals and integralists, and ends up throwing all sorts of spanners into otherwise good arguments. In criticizing integralists, liberals sometimes forget that liberal societies have civil religions and therefore are already doing things that they sometimes criticize integralists for wanting actual Church authorization for. On the other side, integralists have a tricky line to walk and sometimes don't properly distinguish between having an explicitly Catholic civil religion and the Church itself; confusing them treats the Catholic (i.e, Universal, i.e., Not Merely National) Church as if it were a national church. At the same time, it's obvious that the civil religion would have to be a major interaction point between state and hierarchy in an integralist regime (as indeed it already de facto is in any liberal regime with a large Catholic population).
The second major complication, which is less serious, is that in reality the Church has not only a hierarchy but a state -- Vatican City State -- which is an officially Catholic state whose entire purpose is to be ruled by the Holy See for the benefit of the Church. It is neither liberal nor integralist. It is in fact anomalous in a large number of ways, in part due to the fact that it is not a natural state but a juridical state (i.e., a state formed entirely by international law) which has a juridical continuity with a now-defunct natural state (the States of the Church or Papal States) that was also ruled entirely by the Holy See. The Church has another non-territorial agency with state structure due to its being a stump of a previous natural state, the Sovereign Military Order of Malta, which as a religious order is related to the hierarchy in a completely different way. These juridical states serve as another point of interaction, one that exists for liberal and integralist states alike, and are often forgotten by both liberals and integralists in making general claims about political frameworks.
The third complication is that, regardless of the regime, Catholic citizens can't help but act as Catholic and as citizens in any civil society to which they belong, and the two parts of this 'dual citizenship' inevitably interact in complicated ways. This intersectionality at the very level of citizenship itself is one reason why focusing only on state vs. hierarchy can be a mistake; much of the interaction between the two, in any society whatsoever, is downstream from the answers that Catholic citizens already give to questions of how to make practical sense of their 'dual citizenship'. It is also one reason why I think Vallier's preferred characterization of integralism in terms of a "natural common good" and "supernatural common good" has some problems. (Similar problems arise with attempts to put the matter in terms of "natural end" and "supernatural end", but in fact more serious, because all things have a supernatural end, they just aren't oriented to it in the same way.) Certainly a civil society has a common good and the Church has a common good; every complete society has a complete common good, a constituting and preserving good all members of that society share in common in order to be in society at all. But common goods, which form communities, are also not exclusive, and in the case of Catholic citizens, they already overlap; Catholic citizens have to try to uphold both. A liberal society with a significant Catholic population already has to make some accommodation of the common good of the Church, or it is actively unjust toward its Catholic population, deliberately imposing unnecessary hardship on them for being Catholic. At the same time, the distinction of citizenship and baptism will even in an integralist regime require some accommodation of non-baptized citizens, just on the grounds of what the ends of civil society itself require. These accommodations between civil society and Church are built in; you can't have a just society without them. This mutual accommodation in a reasonably functional and just society is distinct from the much stronger cooperations covered by (5)-(8), but its necessary existence often muddies the waters when people get into details.**
However, if these additional complications are kept in mind, the eight principles above do a good job of capturing the essential integralist idea, and the fundamental points. Let's take a simple example of one particular way this might work, depending on how the cooperation between state and hierarchy in general works. Under canon law, organizations calling themselves 'Catholic' in the sense used in talking about the Catholic Church must have the permission of the relevant bishop or ordinary; this is usually enforced quite lightly, but if the bishop explicitly denies the right of an organization to call itself Catholic, it is a violation of canon law and can subject the organization and potentially its members to canonical penalties. In the United States, the state does not recognize the hierarchy as having any authority to determine who uses the term 'Catholic'. But it would be entirely possible to have a society in which the state does recognize this authority, therefore treating 'Catholic' as something analogous to a trademark, but with the state deferring to the hierarchy's standards about whether an entity is using the term with authorization rather than (as with trademarks) applying its own standards. We already know that the hierarchy in Catholic theology and practice has the authority to determine whether an organization is Catholic. And this fits the integralist pattern in (7), and does so even if the hierarchy only takes its judgment to apply to Catholic matters (which, trivially, it does, since the claim is not that the hierarchy owns the word 'Catholic' but that its authority applies to whether it can be used when used specifically to represent an organization as Catholic in the sense of the Catholic Church; non-Catholics using the term in another sense would be a separate matter).
If we use the eight-point characterization above, however, it is clear that many arguments against integralism are not particularly effective. For instance, Vallier has what he calls the Justice Argument, which goes roughly like this:
(i) The Church may direct the state to enforce canon law. [Integralist assumption]
(ii) This enforcement must conform to two norms of justice: (a) coercion into the faith is unjust and (b) coercion to keep the faith is just. [Integralist assumption]
(iii) The dividing line between the application of the two norms can only be baptism. [Integralist assumption]
The above three are taken to characterize integralism; the remaining part of the argument is the actual argument against.
(iv) To serve as a dividing line between the application of two norms, baptism must function as a normative transformer, transforming religious coercion from unjust to just.
(v) Baptism cannot act as a normative transformer with respect to these two norms.
(vi) Therefore, integralism is false. [(i)-(v) by reductio]
In light of the above characterization, however, we can see that this argument, and any argument much like it, does not actually identify anything that is essential to integralism itself. (i) is not strictly required by the basic principles of integralism; what is required is that the hierarchy can authorize the state to enforce (at least to some extent and in some way) at least some aspects of canon law. (This latter follows from (7) above plus some very uncontroversial assumptions about the role of canon law in the activities of the hierarchy.) (iia) follows from principles of Catholic theology. (iib), however, does not; but it also does not strictly follow from any of the eight principles above. It might perhaps do so if one interpreted 'coercion' not as a force but as 'enforcement'; but it doesn't seem that it could have that meaning in (iia), and equivocation on the term would cause problems for (iv). (7) does allow for the possibility of the hierarchy authorizing coercive action, although it doesn't allow for the authorizing of just any kind of coercive action (e.g., they would have to be actions that the state could do in a manner consistent with its own ends). There have been people who have argued, I think, that the state can't give any differential coercive treatment of baptized and non-baptized. This is not particularly obvious (states, including liberal states, exercise at least some kinds of differential coercive treatment given some kinds of differences between the people to whom the treatment is applied, e.g., with minors vs. non-minors, adults of draftable age vs. children and non-draftable adults, citizens vs. residents, citizens and residents vs. non-resident non-citizens, felons vs. non-felons, people in their right mind vs. people not in their right mind), but even if it is assumed, this doesn't actually affect any of the eight principles; it just means that the hierarchy-state interaction will involve other things besides coercive action. Since baptism is the difference between being fully 'in the faith' (i.e., in the Church) and 'not in the faith' in Catholic theology, (iii) can be accepted if (ii) is.
But it is also Catholic doctrine that baptism is itself a normative transformer -- it changes your deontic and juridical status in the Church, and puts you in the legal jurisdiction overseen by the hierarchy on behalf of the Church (canon law, as Vallier usually puts it, although in the strict sense canon law is only one of the kinds of laws that operate in the legal jurisdiction of the Church). This is a very explicit part of Catholic doctrine. Thus any Catholic integralist would have to reject (v) as a general claim, even if they accepted (ii), because baptism's function as a normative transformer in the Church is built in. For instance, within the Church there are kinds of coercive action that can be exercised on the baptized that are not exercised on the unbaptized -- the hierarchy can under certain conditions excommunicate and impose penances on the baptized but not the unbaptized and is on the other side required to protect certain rights pertaining to the baptized but not the unbaptized in Church proceedings. The reason for this is that the baptized are in the jurisdiction of the Church and the unbaptized are not. Introducing the state does nothing to change this; all it does is raise the question whether the normative transformation introduced by baptism in the context of the Church can be recognized by the state. If it can, as integralists claim, then it is in fact capable of being a normative transformer in the context of civil society, as well, even if only by some indirect mechanism. Thus the only question is whether (at least in certain circumstances) the hierarchy can delegate certain aspects of its own coercive authority to the state, or can work cooperatively with the state in a way that uses the coercive authority of each.*** (7) & (8) allow for this at least sometimes, and thus this is just part of what integralists are saying. Therefore there are only two ways that (v) can be true: either Catholic theology is already assumed to be false, or Catholic integralism is already assumed to be false. The former already implies the latter, and therefore, either way, the argument begs the question against the integralist.
------
* I am a pluralist; in fundamental political matters it is pointless to optimize ('best'), so one must settle for satisficing ('good enough'), and as it happens there are quite a few very different political frameworks that meet the criterion of 'good enough'. Some possible liberal frameworks are good enough; the ones actually in place now are not, but you could have ones that are. Some possible integralist frameworks are also good enough, as are probably quite a few other political frameworks that do not fall clearly into either the liberal or the integralist camp. Roughly, all the 'good enough' frameworks share an essential principle, which is negatively describable as repudiation of totalitarian temptations and positively describable as reasonable plurality of societies; that is to say, they all (not necessarily in the same way) recognize that human persons are necessarily members of more than one society with ends fundamental to living fully as a person, and that no such society can be allowed to destroy or pervert or disregard all the others. (This is a harder condition actually to meet than it might seem; as I've noted elsewhere, many modern political views tend to converge toward having a totalitarian structure, and this is true of some things that would often be called liberal and some things that are proposed as integralist, despite the fact that this always introduces an incoherence into both.) 'Good enough' regimes would likewise share an additional feature: explicit guarantees of the rights of the Church, both as distinct human society and as a society with an essential mission (summarized in the Great Commission, and essentially having to do with its doctrinal and sacramental life) with which the state cannot justly interfere. I think (like Rosmini, who I suppose would count as a Catholic liberal) that the explicit guarantees are necessary for 'good enough' because the consistent experience of Catholic populations across a wide variety of societies is that freedoms, privileges, and rights Catholics aren't explicitly guaranteed tend to get chipped away over time, as states try to qualify, minimize, trivialize, reinterpret, or interfere with them in the name of various state projects.
** This also raises the complicated question of exactly how far apart liberalism and integralism actually are. They both by their very structure oppose totalitarian statism, so they have that in common. They are also committed as a matter of principle to some of the same means for opposing it, like some kind of due process and formal procedures. On the other side, there have definitely been attempts to blend anti-clericalism and liberalism, and in fact, almost all liberal societies have had anti-clericalist phases or cycles of varying vehemence. Likewise, there could be integralist societies that were definitely very far in custom and practice from any actual liberal society. But how far different will a well-run and consistent liberal society with an active population of Catholic citizens be from a well-run and consistent integralist society with a similar Catholic population? The liberal society would not definitively recognize its ends as being ancillary to the ends of the Church; but well-run and consistent liberal societies recognize their ends as being part of and subordinate to a larger moral order, and if the population is very Catholic, their conception of that larger moral order may well be very Catholic-tinged. The liberal state would not recognize the hierarchy as a superior partner; but if the citizens are actively Catholic, it could very well be that the hierarchy would have a significant indirect influence on the operation of the state. (This has occasionally happened, for instance, in cities like New York or Chicago or Boston in the United States, where the local bishop backed by the Catholic population has sometimes had a significant say in city policy. Liberal societies don't have any way to prevent this if the liberal state is to remain responsive to popular movements and protests.) There might not be a formalized cooperation structure; but there could very well be a lot of informal cooperation structures, and there will certainly have to be accommodations in both directions in order to have a functioning society at all. On the other side, an integralist society meeting all eight principles could have a lot of practices and customs that we usually associate with liberal societies; for instance, the hierarchy might not actually ask for very much beyond the ordinary mutual accommodation that even liberal societies have. How far apart are the maximally Catholic version of liberalism and the minimum version of Catholic integralism, really? This is relevant to one of Vallier's arguments against integralism, the Transition Argument, which argues that there is no moral way to reach integralism; this primarily seems to require that there be a large gap between liberalism and integralism, but as liberalism and integralism are not diametrical opposites, share at least some moral goals, and can both only be just with accommodations, it's unclear that this is so.
*** It's not relevant to the immediate point but it's worth noting that there is no dispute that the hierarchy can call upon the state to exercise the state's coercive authority under conditions appropriate to civil society (e.g., even in a liberal regime, the pastor of a parish can call the police and charge you with trespassing if you are asked to leave church grounds and don't, and churches can call on the state to enforce contracts that they have made, or report vandalism, or make any other call upon state authority that any other organization recognized by the state can do).