Fideism in 1 Corinthians 15

No chapter argues more significantly or more voluminously for the truth of the resurrection than 1 Corinthians 15. Interestingly, however, Paul is neither asserting nor proving that truth to unbelievers. Rather, he speaks to Christians (albeit weak and even carnal ones) who are part of the church at Corinth. So his argument actually works through a different means and in a different direction than how it is usually preached. Normally the passage is used apologetically--that is, it is given as if to say the historical veracity of the resurrection is empirically undeniable, having been witnessed by those identified in verses 5-9. But that direction is not where Paul takes the argument at all.
The heart of his argument for the resurrection is in the syllogism he presents in verses 13-19, to which he then adds verses 20ff to make obvious the impact of the syllogism and to clarify that the relationship between the resurrection of Christ and everyone else's resurrection is efficient rather than simply implicit. To save space, take DR to represent the proposition "that there is a resurrection of the dead," CR to be "that Christ has risen from the dead," PV to be "that the apostle's preaching is vain," FV to be "that the Corinthians' (believers') faith is vain," SS to be "that the believers are still in their sins," SP to be "that those who sleep (have physically died) in Christ are destroyed," and MM to be "that believers are the most miserable (pitiable) of people." Also, take "that we have hope in Christ in this life only" as equivalent to the denial of DR ("that there is a resurrection of the dead." Paul's syllogism then should be fairly obvious, beginning in verse 13 and ending in verse 19. (For those not familiar with the symbols, "~" means "not," "-->" means "implies," "^" means "and," and ". :" means "therefore.")
Verse 13 provides the first premise (in this case the minor premise) and is supported by (or simply restated in) verse 16. Verse 14 provides the beginning of the second premise (in this case the major premise) completed and supported in verses 15, 17, and 18. Finally, verse 19 provides the conclusion, which is, as it turns out, a reduction to absurdity (a reductio ad absurdum argument.) So the syllogism starts out looking like this:
~DR --> ~CR
~CR --> PV ^ FV ^ SP ^ SS
. : ~DR --> MM
The lack of syllogistic structure above is resolved with the simple observation that Paul refers to believers as the most miserable (pitiable) of people in verse 19 in order to abbreviate his (hypothetical) claim that the apostle's preaching is vain (in verse 15), that the believers' faith is vain (in verse 17), that the dead in Christ are simply rotting in the grave (in verse 18) and that the Corinthians are still in their sins (in verse 17.) (To be a bit more precise, SS is actually an implication of FV in Paul's argument, but that fact does not change the value of this structure.) With that equivalence, the structure is straight up.
~DR --> ~CR
~CR --> MM
. : ~DR --> MM
Here's the point, especially clarifying verse 19: Paul is not saying that if there is no resurrection, Christian life is an abysmal disappointment. Rather, Paul expects his readers to see MM as an absurdity. Since the structure of his argument is undeniable and he has emphasized (if not proven) the truth of the implicit relationship between each subject and predicate in the premises, then the absurdity of the predicate of the conclusion can lead to only one conclusion, the denial of the subject of the first (minor) premise, that the dead rise not. Like this:
~DR --> MM
~MM
. : DR
In other words, Paul assumes (for good reason) the Corinthians will respond to verse 19 by asserting that they certainly are not the most pitiable of all people. That is, he believes they will assert that the preaching they have heard is not vain, that their faith is not vain, that they have been delivered from their sins, and that their dead loved ones are not just decaying in the grave. And he has argued such that if they make that assertion, they will have to acknowledge that the dead do indeed rise. That acknowledgment then at least allows for the possibility that Christ also rose. Of course, the possibility of Christ rising is not at all the final disposition of the argument for Paul, which is why he asserts in the next verse (20) not only that Christ has risen from the dead, but that Christ's resurrection not only makes the resurrection of the dead necessarily possible (the contrapositive of the original first premise in verse 13,) but more importantly that Christ's resurrection of the dead paves the way for the resurrection of the dead more generally.
Now back to the weight of the reductio ad absurdum argument in verses 13-19: Paul does not begin with the fact of the resurrection then argue that the Corinthians ought to be more holy as a result of the fact for which he has provided evidence. Rather, he argues that their commitment to the faith is already real (although poorly lived, as the rest of the epistle attests) and that therefore the resurrection is undeniable. It is a more fideist than evidentialist argument. Of course, it could hardly be otherwise, written not as an apology to unbelievers but an appeal to believers.
The point here is simply that if believers' lives have been changed by the gospel then the objective content of the gospel message, including the literal nature of the resurrection, is thereby validated. As always, miracles (even the greatest miracle: the resurrection) do not produce faith; rather, in this case epistemologically, faith (and its subsequent transforming power in the lives of the Corinthians) justifies the miracle--a fideist position indeed!

An explanatory note and a followup question:
An explanatory note: Fideism here is not just that salvation is by faith alone, nor does it include the idea that faith is purely subjective, without regard to objective content. It is simply the advocation of initial, volitional faith as the foundation within which Christianity is preached both to the lost and saved. That commitment of faith is independent of sufficient evidence or any other prior cause and therefore an expression of the free will (a gracious gift from God to humans.) The lost ought to make that commitment-choice. The saved have made that commitment-choice.
A followup question: But is it not possible that Paul made his evidential apologetic (based on the witnesses) first, then argues from that fact to the conclusion that without the resurrection they would be the most pitiable of men? The answer is no, because he already introduced the belief they have in reality in verse 2. In fact, he premises all his claims with the necessary relationship between their faith (unless you believed in vain,) their salvation, and their standing in what he preached to them. First they believe in the content of his message, which includes the objective truth of the resurrection. Everything else follows.