I think you hit on the definition of formal heresy in your question ... "because they thought the Catholic Church taught it".  If they truly believe that and are inculpably ignorant, their accepting something because (they think) the Church taught it speaks to having the correct formal motive of faith, and thus the term "formal" in "formal heresy".  Now, the term has been warped and distorted the past couple centuries to meaning that they're "sincere".  While sincerity is necessary (i.e. you have to sincerely believe the Church taught it and not due to some ulterior motive), it's not sufficient.  Protestants, for instance, even the most sincere, are all formal heretics by definition since they do not believe what they believe due to the authority of the Church, i.e. with the correct supernatural formal motive of faith.

Now, that's for most NORMAL heresies.

Problem with Trinitarian / Christological heresies is that if you don't have true belief in the Holy Trinity, you're not simply a heretic, but an infidel, since you do not actually believe in the Holy Trinity, and there can be no supernatural faith without explicit (true) believe in the Holy Trinity.

So I would hold that Arians were infidels.  Aside from Trinitarian and Christological issues, however, I would hold that if someone does believe the Church taught something then they would be definition be only material heretics.

I think you hit on the definition of formal heresy in your question ... "because they thought the Catholic Church taught it".  If they truly believe that and are inculpably ignorant, their accepting something because (they think) the Church taught it speaks to having the correct formal motive of faith, and thus the term "formal" in "formal heresy".  Now, the term has been warped and distorted the past couple centuries to meaning that they're "sincere".  While sincerity is necessary (i.e. you have to sincerely believe the Church taught it and not due to some ulterior motive), it's not sufficient.  Protestants, for instance, even the most sincere, are all formal heretics by definition since they do not believe what they believe due to the authority of the Church, i.e. with the correct supernatural formal motive of faith.

Now, that's for most NORMAL heresies.

Problem with Trinitarian / Christological heresies is that if you don't have true belief in the Holy Trinity, you're not simply a heretic, but an infidel, since you do not actually believe in the Holy Trinity, and there can be no supernatural faith without explicit (true) believe in the Holy Trinity.

So I would hold that Arians were infidels.  Aside from Trinitarian and Christological issues, however, I would hold that if someone does believe the Church taught something then they would be definition be only material heretics.

Can someone be a material infidel? Like the example of the Arians?

The formal aspect of the theological virtue of Faith is God revealing.  The material aspect is the doctrine itself.  The formal aspect of the sin of heresy is pertinacity in the will against what he knows the Church teaches.  The material aspect is error in the intellect regarding the doctrine the Church teaches.  For the Catholic, this cleanly applies.  However, for the Protestant, it does not cleanly apply.  Why?  Because even though a Protestant knows that the Church teaches a doctrine that he rejects, he does not know that the Church is the the proximate rule of Faith.  Therefore, you cannot say that Protestants are formal heretics by definition if you mean within the context of the sin of heresy.

False.  You're talking about the formal vs. material aspect of DOCTRINE, not of the act of faith.  With regard to the act of faith, there's the what of belief and the why of belief, the former being the material aspect and the later being the formal aspect, the formal motive of faith.

For two to three hundreds years, the same malicious forces of subjectivism that were steadily undermining EENS dogma subtly warped the notion of formal heresy as effectively requiring thinking along the lines of "I know that the Church's teaching is the rule of faith, and I know that the Church teaches X, and I reject X anyway."  That's completely false.

You also keep blustering about the "SIN" of heresy, except the degree to which an individual is culpable and has committed a sin is a matter of the internal forum.  That is where culpability comes in.

Since a Protestant does not have the correct formal motive of faith, the Protestant is by definition a formal heretic, regardless of whether or not he's committed a sin (knowable only to God anyway).  In theory, though practically impossible, but hypothetically you could have a Protestant who read the Bible and somehow came up with a belief that matches Catholic teaching 100%, but because he did not have the supernatural formal motive of faith, he's still a formal heretic.

Protestants are always formal heretics.

Catholics can be formal heretics or material heretics.

Infidelity is always formal, since you cannot formally have the faith while being in error regarding those whings which must be believed by necessity of means for supernatural faith.

The formal aspect of the theological virtue of Faith is God revealing.  The material aspect is the doctrine itself.  The formal aspect of the sin of heresy is pertinacity in the will against what he knows the Church teaches.  The material aspect is error in the intellect regarding the doctrine the Church teaches.  For the Catholic, this cleanly applies.  However, for the Protestant, it does not cleanly apply.  Why?  Because even though a Protestant knows that the Church teaches a doctrine that he rejects, he does not know that the Church is the the proximate rule of Faith.  Therefore, you cannot say that Protestants are formal heretics by definition if you mean within the context of the sin of heresy.

This is totally backwards.  The protestant doesn't know/care that the Church is the proximate rule of Faith because they ALSO reject the Church's authority.  They are formal heretics, without a doubt.

This is totally backwards.  The protestant doesn't know/care that the Church is the proximate rule of Faith because they ALSO reject the Church's authority.  They are formal heretics, without a doubt.

Right, and this comes turning everything to being about sincerity, where a subjective sincerity can supply for an objective lack of truth.

Now, sincerity can enter in to a point, such as when you "sincerely" THINK that the Church taught something but were wrong.  That's the nature of material heresy, where you believe the wrong thing but for the right reasons.  St. Augustine says that the litmus test for identifying this type of heretic is their readiness to accept correction.  When you explain to them that, "oh, that's not what the Church teaches", they'll immediately respond with, "oh, wow, sorry ... did not know that" and then immediately change their minds.  Now, this does not apply to scenarios where you claim that I have some amazingly powerful and undisputable arguments, based on various reasoning and syllogisms, that something is heretical, but where you don't agree.  If there are reasonable numbers of Catholics who profess the True Faith who seem to agree with you, then it's likely not been proposed by the Church's authority with sufficient clarity in order for anyone of good faith to be required to accept it as Church teaching.  Take papal infallibilty before Vatican I, where you'd correctly have argued that it's revealed dogma, but someone who didn't agree with you would not be considered a formal heretic until the Church officially declared it.

That's why it's said that if you deny one dogma, you deny them all, since if you impugn, even implicitly, the teaching authority of the Church, then even the things that you do not (materially) get wrong, you still don't believe them for the right reasons, but because YOU decided they were right, having made yourself (or some false sect) your false rule of faith.

But, if you're a Prot and don't even accept the True Rule of Faith, you simply cannot have the right motive of faith, and you're in the same position going in as that guy above.

You can be lacking the formal motive of faith either by having rejected it positively and having sinned against it, or simply by lacking it.  Just because you're sincere does not put you in possession of the correct formal motive of faith.  You won't be punished for that, as St. Thomas teaches, but that doesn't mean it's somehow supplied as in some Pelagian sense.

Let's say you are baptized a Catholic, but your parents die and you're raised by some atheists.  Once you reach the age of reason and you do not affirm the infused supernatural virtue of faith by real acts of faith, the supernatural virtue fades away, as if by an atrophy, even if you did not directly sin against it.  It's not like this guy could live 50 years as an atheist and as long as he didn't actively sin against the virtue of faith, that somehow means he has it.

There are some things so essential to supernatural faith that you cannot have them "implicitly", and the only difference between someone who lost the virtue by having actively sinned against it or someone who just never had it in the first place is their culpability in terms of actual sin.

But CK here is hung up on the "sin of heresy", meaning an actual sin against the faith, as being an essential component of formal heresy.  No, the formal motive of faith can simply be lacking even without having committed an actual sin against the virtue of faith ... or it can be reject by actual sin, but that speaks only to their degree of culpability, not to whether they actually have or do not have the formal motive of faith.

I think this idea was developed precisely in order to undermine EENS, where we say that after the first couple generations of Prots, subsequent "cradle Prots" could likely be merely-material heretics, and therefore somehow within the Church and therefore somehow saved.  So that's why more and more over time, the notion of formal heresy requiring an actual sin of knowingly rejecting the faith grew more with each succeeding generations.

That reminds me of the meaningless-formula statement on EENS that only those cannot be saved who KNOW that the Catholic Church is the True Church founded by Christ and refuse to join it anyway.  So, whom does this definition exclude, a handful of atheists who say, "Yep, that's the True Church, but I refuse to join it anyway."  Anyone else can be said to be somehow implicitly in the Church simply for not having almost-explicitly rejected it?  Ridiculous.  You can simply NOT be a member of the Church merely for NOT happening to be in it, and you're not somehow saved by default unless you actively forfeit your salvation.  That's Pelagianism in a nutshell, and it's made a dramatic comeback the past few hundred years.

You aren’t accounting for PINO’s. Many Protestants are just born into it and really don’t know what they believe. The same can be said of CINO’s. You have to know something in order to reject it.

  ... No, we just finished explainng that.  You don't have to REJECT something in order to not have it.  I if I have $100 and throw it away, I have rejected the $100.  If I don't have $100, I cannot reject it.  But in both cases I end up without having the $100, the end result being the same, whether I actively rejected it or not.  This notion that they have the supernatural virtue of faith simply because they haven't actively reject the faith simply doesn't follow.  Protestants are formal heretics by definition, since they do not have the proper superntural formal motive of faith.  What you're promoting here is nothing short of Pelagianism.

"It makes a lot of sense that the Vatican II Popes materially and illegitimately hold the position of the Papacy but not formally and legitimately hold the office of the Papacy."

Is that basically the position of +Guerard des Lauriers and his Cassiciacuм Thesis (aka 'the Thesis')?