Material heresy is an idea which is contrary to an infallible teaching of the Magisterium. Whenever we refer to material heresy, we are talking about ideas, not persons.
Formal heresy is a sin committed by a person, the sin of believing or asserting or teaching material heresy. If the error in question does not rise to the level of heresy, which is a type of severe doctrinal error, then the person is not guilty of formal heresy.
A person who believes, asserts, or teaches material heresy, without knowing that it is contrary to definitive Church teaching, is not a formal heretic. Formal heresy requires a knowing deliberate choice. And such a person is not properly called a heretic at all, since they have not committed that sin. There is no such thing as a “material heretic”.
Conversely, it is also not correct to say that someone has “taught formal heresy.” The idea being taught may be material heresy, and the act of teaching material heresy, knowingly and deliberately, is the sin of formal heresy. But the ideas themselves are not formal heresy.
Material heresy is an idea. Formal heresy is a sin. An idea cannot be formal heresy, and a sin cannot be material heresy.
Formal heresy always includes material heresy, by definition. The sin of believing, asserting, or teaching material heresy is formal heresy. If a person is teaching material heresy, that person is only a heretic if he realizes that the idea is contrary to definitive Church teaching.
Heretics are always formal heretics. Heretical ideas are always material heresy.
Many times, especially on the internet, I have noticed Catholics teaching material heresy, along with the claim that the heretical idea is either a correct understanding of Magisterium teaching or at least a sound theological opinion. Such persons are not, as far as I can tell, guilty of formal heresy. But they are guilty of teaching material heresy. Such persons may be guilty of grave sin, for teaching on important matters of faith, morals, and salvation, without first having learned what the Church teaches. They harm souls through grave negligence.
In a few cases, I have found Catholics teaching material heresy, after correctly stating the infallible teaching of the Church on the subject. I suppose that such a person could be so foolish and confused as to think that the material heresy they are teaching, which directly contradicts their own correct explanation of infallible magisterial teaching, is somehow tenable as an interpretation. But it is also certainly possible that some such persons are guilty of formal heresy.
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