All philosophical objections [that I’ve heard] to Actual Atonement (better known as Limited Atonement) are mistakes in logic. Perhaps the most common is the objection that a limited view of the atonement makes the universal offer of the gospel insincere.
First, we might say that if the Bible teaches on the one hand that God only intends to eternally redeem the elect, and on the other hand that we should offer salvation to all, we should conclude that God’s offer must be genuine even if our pre-conceived philosophical understanding makes the legitimacy of such an offer a genuine mystery.
Second, this objection misunderstands the nature of the offer. The universal offer of salvation is always contingent. The offer is not intended to benefit everyone, only those who repent and believe. Thus, the nature of the offer itself astronomically limits the scope of its intended benefactors by virtue of its built-in conditionality. The offer, therefore, is just as genuine as the offer “Whosoever meets the requirements for enrollment to SBTS, as well as the requirements for discounts on tuition, will be able to receive such benefits.” The offer is intended for, and voiced to, all seminary students indiscriminately, but the benefit is only intended for a select group. This contingency does not ruin the genuine nature of the offer.
Many of the other objections leveled against an actual view of the atonement are really objections against Calvinism as a whole—that it contradicts the concept of a loving God, that it is unfair, that it prohibits people who sincerely desire to be saved from actually being saved. These objections impose philosophical definitions of love, justice, and grace that are foreign to the Bible. They also misunderstand the nature of responsible Calvinism.