It is important to think of theological beliefs as a system rather than a series of isolated doctrines, otherwise we may hold positions that logically contradict in which case one ought to rectify such a contradiction. This is important as, assuming one takes a realist position as to the nature of truth, two propositions that are logically contradictory cannot both be true as it would violate the law of non-contradiction-one of the most basic laws of thought. If it is the case, as I intend to (briefly) demonstrate, that the soteriological position of Arminianism and the Penal Substitutionary theory of Atonement are logically incompatible, then it behooves Christians who hold to both theories to drop one (or both) or modify one (or both) of the theories in order to find a more consistent theological model. Now, a brief prefatory comment before beginning to lay out the doctrines; that the two are inconsistent by no either is incorrect, or that either is correct (cards on the table, I personally adhere to neither), merely that an intellectually honest Christian cannot adhere to both. Now then, prior to any attempt to demonstrate incompatibilities, it is necessary to lay out, in brief, what the respective doctrines teach.
The Penal Substitutionary theory of Atonement is a result of the Reformers expanding Anselm’s Satisfaction Theory to correct what they saw as inadequacies. Anselm’s Satisfaction Theory teaches (in brief) that Man’s rebellion offended the divine honor of God and Christ’s sacrifice afforded a surplus of honor as His life of perfect obedience to the Father afforded an unquantifiably large sum of honor. In contrast to this concentration of divine honor, Calvin and his Reformer brethren moved the emphasis to divine Justice believing that, while Anselm’s theory was good it had an insufficient view of sin. In this theory, Christ’s sacrifice is taking the guilt of the sinners’ upon Himself and bearing their punishment and taking upon Himself the Just Wrath of God towards the sins of the redeemed releasing them from the weight of their inequities.
Arminian soteriology (at least the parts relevant to this discussion) holds that, by His Prevenient Grace, God counter-acts the detrimental affects of the Fall allowing all people the possibility of responding to the gospel. Every person receives a “call” to salvation from God, it is up to their free-will to decide whether or not they would respond positively or negatively. While God, in His foreknowledge and atemporality knows who will respond how, He does not cause any to reject the gospel; the offer is truly open to all and the choice is made by each individual.
So, Whats the big deal? Well Penal Substitutionary Atonement necessitates a “closed” atonement- the sins of each individual sinner are placed upon Christ. This means that the redeemed, and only the redeemed, are covered by the Atonement of Christ On the other hand, in order for the offer to be open to all in a meaningful way, Arminianism necessitates an “open” atonement in which anyone can partake in it. Now, as is apparent by now, the contradiction lies in the one necessitating an open atonement whereas the other necessitates a closed one. It is possible to reconcile the two by changing Penal Substitutionary view to be Christ’s substitution for the Church (corporate Atonement) or for the penalty due to Sin in general rather than sins in particular; however, as is typically articulated Penal Substitutionary Atonement and Arminianism are incompatible.