The Sproulian view of the peccability of Christ ends in either in an abstraction of the human nature from the second Person or else it attributes human personhood to the Son. Either way the denial of the impeccability of Christ implicitly, yet unwittingly, denies Chalcedon. (At the 21 minute mark I interact with Sproul, though I don’t get into modality in the Sunday school class.)
It’s really as simple as modus tollens.
1. If it is possible that Jesus could sin, then it is possible that God could sin.
2. It is false that it is possible that God could sin.
3. Therefore, it is false that it is possible that Jesus could sin.
Given the validity of the form of the argument, which premise (1 or 2) is disputed by those who’d deny Christ’s impeccability? It’s hard to say given that the focus is typically on the possible sin of Christ’s humanity, and not on the possible sin of Christ in his humanity. Notwithstanding, in order to deny impeccability one must affirm that it’s possible for the Son to sin. Otherwise the debate is misunderstood.
Possible world semantics are also useful here. Consider, is there a possible world in which the incarnate Son of God sins? (The answer to the question is kind of built into the definition of God, but I won’t get ahead of myself.)
Modality considerations:
We would do well to distinguish (a) narrow or strict logical possibility from (b) broad logical possibility or metaphysical possibility. One might say that “God sins” is logically possible in a strict sense because the proposition does not immediately entail a logical contradiction. But that would not imply that it is broadly logically possible, metaphysically speaking, for God to sin.

An analogy might be useful here. A state of matter cannot be solid and not solid at the same time and in the same way. To affirm the contrary would entail logical impossibility in a strict sense, as it would violate the law of non-contradiction in an immediate inferential sense. (It’s critical to grasp at this point that one needn’t know what solid, gaseous, liquid and plasma states entail for it to be known that such a phase of matter (a form that is both solid and not solid…) is a strict (or narrow) logical impossibility. The logical contradiction in view is formal and according to the law of non-contradiction (aside from any semantic considerations). It merely pertains to: something cannot be x and ~x…
However, it would not immediately entail a logical contradiction for a phase to be simultaneously solid and gaseous; yet how is such a state of being relevantly possible? Well, it’s not. It can’t be actualized. We might say that such a form of matter is not strictly (or narrowly logically) impossible, but that’s merely because no formal law of logic is immediately violated by the term solid-gas. What’s lacking in the immediate or strictly logical inference of the possibility of a solid-gas is the meaning, or qualitative differences, of two distinct truths about forms of matter. Yet once we know the semantic implications of solid and gaseous states, then we may infer from additional premises that no solid can be simultaneously gaseous. Accordingly, we may then further deduce that a phase that is both solid and gaseous is more broadly logically (or metaphysically) impossible. Furthermore, a solid-gas is just as relevantly impossible as a solid that is not a solid!
Back to impeccability. Like a solid-gas, a God-man who can sin is a contradiction in terms. Such contemplations are broadly illogical due to the nature of things.
2 ways one might go:
Without grasping the relevant implications of divinity as it relates to the doctrine of Christ, one might assert the metaphysical possibility of Jesus sinning. Furthermore, it’s not immediately inferable that it’s logically impossible for all possible humans, including Jesus, to sin. Yet if one grasps Chalcedon and incorporates God’s nature into the deduction, one may more modestly concede the latter option, that it is narrowly logically possible for Jesus, a human being, to sin. Whereas the former option lacks the use of relevant information about God’s nature, the latter, although more sophisticated, would have little or nothing to do with the doctrine of Christ’s impeccability, which is a metaphysical, broadly logical consideration. (Moreover, I’ve never seen such a subtle distinction of modality articulated as the basis for one’s denial of the doctrine of the impeccability of Christ, which is not to say that some haven’t had such reflections without having the semantic categories to articulate such a position.)
Those who hold to a doctrine of peccability either are confusing modalities or else they’re latent Nestorians:
Christians who affirm a doctrine of peccability typically do so without any self-conscious reference to a modality maneuver. Notwithstanding, to assert peccability as true doctrine entails a misunderstanding of temptation that in turn undermines the two natures in one subsistence. It’s not as though they affirm only strict logical possibility over possible actuality. Rather, in affirming peccability, they affirm the actual (metaphysical, broadly logically) possibility of an unfaithful Christ (and consequently affirm strict logical possibility too). In doing so, they abstract the human nature from the divine person, which falls to the same type error as positing a solid gas. In confusion, they might additionally attribute distinct personhood to the human being, Christ (Nestorianism).
Further reflection:
Christians embrace the incarnation of the divine Son as a union of two distinct natures in one hypostasis. Yet given a doctrine of peccability, is it further supposed that the human nature could possibly have sinned apart from the Person having sinned? In other words, by sinning would the Second Person (God) have committed sin only in his humanity but not personally? It’s hard to tell whether people like Sproul think that the whole person of Christ could possibly sin in his humanity. After all, Sproul’s position entails an unorthodox abstraction that “Satan was not trying to get God to sin. He was trying to get the human nature of Christ to sin, so that he would not be qualified to be the Savior.”
Wrapping up:
Given the meaning or ontological import of Jesus is Son, we may safely maintain it is metaphysically or broadly logically impossible for Jesus to sin in any actualizeable (feasible) world, which is the only relevant scope of possibility in this regard. Since God cannot possibly actualize a world in which the Son sins, in what Christological sense might Christ possibly sin? Given God’s nature, an implication of Chalcedon is Jesus was indeed impeccable.
There are other missteps Sproul makes. I’ll briefly touch on a few.
“But if Christ’s divine nature prevented him from sinning, in what sense did he obey the law of God as the second Adam?”
False dichotomy: When God prevents us from sinning in the face of temptation, are we not truly obeying? Accordingly, operative grace does not undermine either obedience or true temptation.
Moreover, God’s free knowledge of the divine decree presupposes the causal divine determinism of ordinary providence. Consequently, Sproul’s question smacks of Incompatibilism for God cannot but ultimately and causally determine the incarnate Son’s willful intentions through the intentional ordering of states of affairs, about which God pre-interprets the particulars consistent with a Reformed understanding of concurrence.
“I may be wrong, but I think it is wrong to believe that Christ’s divine nature made it impossible for his human nature to sin. If that were the case, the temptation, the tests, and his assuming of the responsibility of the first Adam would have all been charades. This position protects the integrity of the authenticity of the human nature because it was the human nature that carried out the mission of the second Adam on our behalf. It was the human nature uniquely anointed beyond measure by the Holy Spirit.”
What is it to be “uniquely anointed beyond measure by the Holy Spirit” other than to attribute something additional to the Second Adam that was not granted to our first father by the Holy Spirit? Moreover, how might Sproul capitalize on the Spirit’s anointing in a way that distinguishes it in any relevant sense from the ordinary empowering of the human will that might have come to Christ’s humanity from the Son’s ubiquitous divine nature, which is shared with the Father and the Spirit? How many divine beings are there after all? Moreover, the incarnation entails a perichoresis in the sense that the omnipresent divine nature of Christ penetrates his human nature, as it does ours yet to a lesser degree, though always without a transfer of properties. The penetration is also one directional and never from the human nature to the divine nature.
Lastly, regarding the human nature and Christ’s mission, was it the human nature that kept itself from sinking under the infinite wrath of God? Moreover, did the human nature alone give worth and efficacy to the sufferings of Christ? No to both. A human person could not have possibly redeemed! Accordingly, Sproul is not only wrong for abstracting the humanity of Christ from Christ, he’s also mistaken in thinking that the divine nature of the Son contributes nothing to our salvation. (See my post on strict vs. pactum justice.)
We are saved by a divine Person, not by an abstracted impersonal nature or even a human person. Accordingly, Sproul simply is incorrect that “the human nature carried out the [redemptive] mission.” Rather, it was requisite that a person carry out the mission, and that the person be God incarnate, as Sproul’s confessional Standards rightly teach:
Westminster Larger Catechism:
Q. 38. Why was it requisite that the Mediator should be God?
A. It was requisite that the Mediator should be God, that he might sustain and keep the human nature from sinking under the infinite wrath of God, and the power of death; give worth and efficacy to his sufferings, obedience and intercession; and to satisfy God’s justice, procure his favor, purchase a peculiar people, give his Spirit to them…
(As with the false doctrine of Christ’s peccability, so it is with Molinism. As I argue here, Molinism posits true narrow-sense possibilities that cannot be actualized even though there are an “infinite number” of these “logical” possibilities. And here, I made a passing remark about impeccability in a post primarily pertaining to Dabney’s unhappy employment of Middle Knowledge. That passing remark was a seed thought to the current post.)
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