Is the “free offer” of the gospel really “sincere” if Jesus only died for some men and not all? If there is no atonement available for them, the offer seems insincere.This is a frequent objection, particularly from Amyraldians and Arminians. If you think that the gospel is "Jesus died for you," then this objection makes a lot of sense. If we're supposed to tell people indiscriminately that Christ died for them, but he didn't, that doesn't seem very sincere.
Scriptures, however, don't present the gospel that way. In Scripture, the gospel is expressed in terms of repenting of your sins and believing on (i.e. trusting in) Jesus Christ for salvation. If you trust in Christ and repent of your sins, God will have mercy on you.
There is a world of difference between those two messages. One message makes an unconditional assertion regarding what Christ has done. The other message makes a conditional assertion about what God will do.
Yet, even among those who will grant to us that the gospel is not, "Jesus died for you," some people still don't like the idea of salvation being offered to those for whom God has not made any provision. Indeed, our Amyraldian and Arminian friends sometimes urge on us the idea that such a conditional offer is not "sincere" unless God has made preparations for those people.
The mere absence of enough provision for everyone to be saved, however, doesn't explain this objection. Suppose a company offers to "anyone who is willing to come down here and listen to us explain the benefits of our new tractor," an incentive of "$5, just for coming down and listening to the talk." No one would consider it "insincere" if the company doesn't actually have $5 times the number of people who will hear the offer, so long as they have $5 times the number of people that they think will accept the offer.
So, as long as the provision is sufficient for those who will "accept" the offer, we don't view the offer as insincere. Since, under the Calvinist framework, God has made provision for all who will come to Christ, the offer of the gospel should also be considered to be sincere by this standard.
The intuition behind the objection that remains, however, is that an "offer" doesn't seem sincere, if you have no intention of giving the offered thing to the person to whom you are offering it. For example, when a child offers to share an ice cream cone, it sometimes happens that this is simply an imitation of a parent's offer to share the parent's cone. If the parent were to try to accept the child's offer, the child might greedily refuse to allow the parent to have a bite. So, the child has only offered to share the cone because the child thought the offer would be refused. Such an offer is insincere.
Of course, by this time we are now dealing with the kind of objection that an Amyraldian, or someone like Ponter, cannot consistently make. After all, the problem with the child's offer is not that he doesn't have a cone to share, but that he does not intend to give up the cone. The Amyraldian admits that God does not intend to save the non-elect. Therefore, whether or not a provision is made seems utterly moot.
Nevertheless, for those who insist that God must intend to save, we may still legitimately question the weight of this objection. Isn't it enough that God intends to save everyone who "accepts" the "offer"? The idea that God must intend to save all those whom he knows will refuse seems absurd when expressed that way. Thus, we may conclude that while such an objection may have some limited intuitive appeal, it does not hold up to intellectual scrutiny.