@Nihilist Virus
You seem to try and exploit the difficulties inherent in classification and definition. You say you know what a Toad is, but do you really?
Toad is a popular classification of certain frogs, esspecially of Bufonidae, but there is no such group in scientific nomenclature. One group's toad is another's frog and people differ on whether size or leathery skin or shorter legs or what have you, is required. It is polythetic, but such classifications are nested in one another and often fuzzy borders, like classifying monotremes as mammals as another example.
There are popular concepts of Holy and Sin, there are technical ones. These overlap but aren't the same. Often the technical ones differ markedly. It is like animal classification, for before Linnaeus, Aristotle classified Cetaceans separate from mammals and most segmented groups together. Now with cladistics, we rewrite our taxonomy once again. Why do you expect such precision from Religion when the most basic of the sciences cannot even deliver it, even more so with popular uses of terms?
Ludwig Wittgenstein adressed this in his language theory, which simplistically is often put as "meaning is use". A definition does not alter meaning: it is deductively derived therefrom and creates a new abstract concept of the 'definition of X'. This is like your toad. You know what a toad is, but I doubt you would be able to define it in such a precise way, and when seeing a toad, you likely would not reach for your definitional armamentarium to decide whether an amphibian was one or not. Likewise, with Holy and Sin, and while addressing concepts couched in those terms, you either have to apply their inherent meaning in the context of the sentence or culturally, or rope in external definition, but neither would allow the sort you are insisting upon before addressing an attempted syllogism. It is specious to expect it to, and being a former Christian and I assume culturally western, you understand the concept even if its reificatory definition is not necessarily evident. The path you trod would otherwise eradicate all meaning presented in language that can be ambigiously understood or indeterminate and therefore leave all discussion moot outside of highly abstract systems like Mathematics.
Seeing though that you are seeking a syllogistic answer, I would point you to Aristotle, the father thereof. You would address this in the manner of Apodeixis, as he would call it. We Christians know Sin exists, we know Christ forgave it and that the Crucifixion was integral to Atonement. The operative method of Atonement is immaterial to know it works, just like I can drive a car without understanding the internal combustion engine.
The Material cause is Jesus and His suffering; the Formal cause is the Crucifixion; the Efficient cause is our Sin and Christ's sinlessness and the Final cause is Atonement. If we cannot agree each of these, then it cannot be placed in a syllogistic framework. For to say A thus B, etc. assumes that agreement is present already on the validity of A for instance, so that the logical interdependance of the concepts or not, can be investigated. This courtesy you seem not to have extended to us, so to argue the concept until you accept it and then syllogistically present it to you, is beyond the scope of the reasonable.
My 2 cents though: I know that the Crucifixion is the operable event in history that brings about Atonement. I know many theories of Atonement, such as Poenal Substitution, Moral Influence, Scapegoat, Christus Victor, etc. Many of them aren't mutually exclusive. They may all be right, they may all be wrong. This does not change the fact of Atonement on the Cross; that I cannot properly describe its minutiae; for this is again the dichotomy between the real use of something and the abstraction thereof. To syllogistically present it, is no different from explaining how Gravity makes a stone fall. I know it does, I can give clever ways to describe it and support it, but I cannot definitively and exhaustively prove it; it is assertoric after all; without axiomatic appeals or ignoring the basic teleological question of why it does so.
