The Negative Inference Fallacy

The argument of the high Calvinist on limited atonement in John 10:15 is thus:
Those for whom the Son died, the Holy Spirit regenerates and grants saving faith. Christ died for the sheep. We know there are sheep and goats. By not stating "Christ died for the sheep and goats" we know that Christ only died for the sheep. Therefore, He did not die for the goats. Therefore, the sins of the goats were not expiated for.

This argument is based on a faulty premise (the sentence in pink), and results in a false conclusion (red). Their mistake is that a disjunctive syllogism (either P or Q, P, therefore Not Q) is only valid for the exclusive case (either P or Q, but not both).  For example, the following exclusive case is valid: he's either dead or alive. The inclusive case however, is not valid: it's either raining or it's warm outside, it's warm, therefore it cannot be raining.
This presumption of the exclusive case may look strong at first, but a closer examination shows this assumption doesn't hold up. Of course the hyper-calvinist will answer this charge by saying that Q is the case NOT P- Christ dies for the elect, therefore His work is not for the not-elect.
But this is a false dichotomy, it's to assume that election is the mirror opposite to reprobation. Reprobation and election are not in the same class, they are not symmetric; scriptures speak of the reprobate being given a fair trial and of condemning themselves, "You lazy and wicked servant, from your own words will you be judged etc" while the elect, being justified, have their case dismissed before God. The categories remain distinct.  So unfortunately, given we know God elects all we can conclude is that He elects. P isn't non Q, it's just P.

If that were not enough there is a second problem with the statement in the modus ponens: "All whom Christ expiates sins for are saved, the goats end up in hell, therefore the goats didn't have their sins expiated on the cross" in that it commits the fallacy of denying the antecedent (A implies B and B is false, therefore A is false). An example of this fallacy might help to show the problem with it: all good math students can do calculus, Sam can't do calculus, therefore Sam is a bad math student.  The faulty premise of the High Calvinist is that expiation from the cross automatically saves regardless of faith, but the Bible teaches us that to get to Heaven we must have both an expiation of sins and faith, (A implies {B AND C}) and unless both conditions are met a person goes to hell.  So even if we know a reprobate is in hell we can only conclude that they failed to have both {faith AND expiation}. We cannot conclude that because they go to hell they had no expiation.

What the hyper-calvinist needs is a negative statement that Christ did not atone for the non-elect.  It would look something like: my death serves no purpose to the non-elect. But in fact the scriptures never give a negative on the nature of expiation. We see an affirmation that Christ dies for all, and we certainly clearly see His death is for the elect, but we are never given the denial that would prove the hypers assertion.  In the final analysis of this verse we know that Christ dies for the sheep, therefore the sheep will be saved.  P, therefore P.