If Paul didn’t use logic, then we wouldn’t be certain that logical arguments are proper. If logical arguments weren’t proper, then apologists above all, would be most miserable. But, Paul argues Modus Tollens. So, logical arguments are proper and not only apologists have reason to argue positively for Christ, but all Christians should do the same.
Did you see what I did there?
More importantly, let’s look at what Paul did in 1 Corinthians 15.
The main argument
1 Corinthians 15:1-19 is an argumentative necessity when it comes to apologetics and the Resurrection of Christ. It was Paul’s defense, if you will, of Jesus of Nazareth-raised from the dead. To be more accurate, the passage is a positive case for Jesus’ Resurrection. It is apologetics on the offense.
Paul didn’t merely answer the objection “Jesus was not raised”. He went further. Paul made a positive case for the Resurrection of Christ using evidence, namely eyewitnesses. He destroyed uncertainty about who Jesus is by inviting the reader to look at the evidence. Paul described the Resurrection as an integral part of the Gospel; what has been taught; what has been predicted; and what has been witnessed. It was more than an answer to who asked about his hope.
The first eleven verses do not seem to take the form of a “reply” introduced by identifying a topic
Paul’s positive case for Christ
Jesus died (according to the Scriptures), and he lives again (according to the Scriptures and several eyewitnesses). That is Paul’s offensive apologetic to skeptics. It is the Gospel.
But, Paul wasn’t satisfied simply making the case for the Resurrection. If there is any doubt that Paul was motivated to take apologetics on the offense, it is dispelled in this passage. In his words, Paul engaged in destroying all arguments (2 Cor 10:5) that oppose the Christian Faith. Notice, I did not say that Paul’s target was merely answering the question of a single skeptic. If Paul’s idea of apologetics was mere defensive, he intercepted the ball after one of his linebackers blindsided the QB; he sprinted down the sidelines for a pick-six! Stopped short of the goal line, Paul had two tight ends, a fullback, and a tailback lined just behind him. I’ll let you guess whether he considered himself to be on offense.
Seest thou how excellently he reasons, and proves the resurrection from the fact of Christ’s being raised, having first established the former in many ways? -John Chrysostom
Paul’s logical playbook
Contained in Paul’s playbook is a logical argument known as Modus Tollens. In support of his main argument for Christ’s Resurrection, Paul switched gears in v.12-19.
It’s evident that Paul felt the need, after arguing for Christ’s Resurrection, to make an argument for resurrections in general. He did this to support the main argument that Christ is truly raised. Maybe there were skeptics within the church at Corinth. Maybe there were philosophical and cultural pressures on the new church to disbelieve in life after death.
Yet just as the previous theory embodied two different aspects (gnosticism and overrealized eschatology), so this third approach demands differentiation between the issue of (a) belief in the sovereign power of God to create out of nothing (ἐκ τῶν νεκρῶν) and to raise the σῶμα to a transformed (15:42–44) mode of “somatic” existence; and (b) the notion that the body (σῶμα) cannot be a vehicle for this transformed mode of existence which enters the kingdom of God (15:50–54).40
It is likely that Paul was arguing against the Epicurean worldview that confused the church in Corinth. The idea was that there is no resurrection. At least we can say that the significance of Paul’s argument against the Epicureans is confirmation that a valuable part of apologetics ought to be arguing against opposing world-views.
“Now if Christ is proclaimed as raised from the dead, how can some of you say that there is no resurrection of the dead? But if there is no resurrection of the dead, then not even Christ has been raised. And if Christ has not been raised, then our preaching is in vain and your faith is in vain. We are even found to be misrepresenting God, because we testified about God that he raised Christ, whom he did not raise if it is true that the dead are not raised.
For if the dead are not raised, not even Christ has been raised. And if Christ has not been raised, your faith is futile and you are still in your sins. Then those also who have fallen asleep in Christ have perished. If in Christ we have hope in this life only, we are of all people most to be pitied.” (1 Corinthians 15:12–19, ESV)
Paul’s argumentative tactic
Paul argued backward from resurrections in general (soul, dualism, life after death) to Christs resurrection. Even more interesting is Paul’s assertion that if there were no resurrection per se, then Christ has not been raised. This seems to be an analytical argument rather than ontological. Paul used arguments in the classical sense. If Socrates is a man, and all men are mortal, then Socrates was a mortal. In other words, Paul argued that if men are not resurrected and Christ is a man, then Christ is not resurrected. This is a form of the Modus Tollens logical argument.
Modus Tollens is a Latin phrase describing a logical argument that denies the consequent of a deductive argument: If P, then Q; not-Q; therefore not-P. In other words, Paul’s Modus Tollens argument denied resurrections. The purpose of this form of arguing, at least in this instance, was to show the absurdity of believing in Christ’s Resurrection by denying resurrections.
Why does the reality of Christ’s Resurrection matter?
In v.14 Paul strained in his argument to show the eventual absurdity. If there are no resurrections Christ’s Resurrection did not happen. If Christ has not been raised, then your faith is in vain.
Here are the consequences of truth. If what Paul and the Apostles have been preaching (the resurrection of Jesus) is untrue, then not only is their preaching in vain, faith is empty! The consequences of an empty faith are dire.
It is silly it to believe in an absurdity. Those who hope in fiction are most miserable. How vain is a faith that rests on a dead man?
The consequences of no resurrection are shown to lead to hopelessness and eventual eternal death. Not only is faith in any resurrection futile, humanity remains “in their sins” and those who have passed on are perished.
It’s interesting how Paul framed the consequences.
First, he rightly excluded the possibility of atheism when he concluded that people are still in their sins. By making that claim Paul implied that even though resurrections may not happen it does not follow that God doesn’t exist. In fact, we may conclude that he meant to say that a holy God remains a problem for humanity, especially now that the hope of resurrection has been excluded.
The worst consequences
Furthermore, Paul posed another problem that didn’t cease merely because resurrections have been presupposed to not take place. Life after death remains. It does not necessarily follow that a person ceases to exist even though they may not be resurrected. But, because faith in Christ’s Resurrection is in vain, those who have died have perished.
A quick word study on the word perish reveals its meaning. ἀπόλλυμιa; ἀπώλειαa, ας f; λυμαίνομαιb: to destroy or to cause the destruction of persons, objects, or institutions—‘to ruin, to destroy, destruction.’ 
But, we can use our ability to analyze language here and understand something about what Paul meant when he said those who have fallen have perished. He mustn’t have meant that those who have fallen asleep have merely died. That would be like saying that those who have died have died. That’s what Paul meant when he said folks have fallen asleep. They have died.
To die twice?
What did he mean though, if he didn’t mean to state that those who have died have died? He used the word perish symbolically. Paul used the word perish to describe a destruction or ruin that happens after physical death (falling asleep). People who die, Paul argued, are destroyed or ruined after they die. Paul used the word perish to reintroduce God’s wrath to the readers now miserable circumstance. This is a consequence of no resurrection.
So, Paul concluded that if there is no resurrection per se and if in Christ we only have hope in this life, we are of all people most to be pitied. Our faith is in vain because Christ is still dead. We are still in our sins and a holy God (still) exists. Destruction awaits us and has already ruined those who have gone before.
Paul argued to absurdity to show us that Christ raised is truly our hope. He did this to support his main argument that Christ has been raised! Paul used a version of Modus Tollens to do this. He made a positive case for Christ’s Resurrection by arguing the negative consequences if it were not so.
We ought to use this as an example for our own apologetic because Paul was correct. If there is no resurrection, there is no hope. The truth of the Resurrection matters. Paul argued Modus Tollens to prove it.
 Thiselton, Anthony C.
The First Epistle to the Corinthians: A Commentary on the Greek Text. New International Greek Testament Commentary. Grand Rapids, MI: W.B. Eerdmans.
 Schaff, Philip, ed.
Saint Chrysostom: Homilies on the Epistles of Paul to the Corinthians, vol.12. A Select Library of the Nicene and Post-Nicene Fathers of the Christian Church, First Series. New York: Christian Literature Company.
 Louw, Johannes P., and Eugene Albert Nida
Greek-English Lexicon of the New Testament: Based on Semantic Domains. New York: United Bible Societies.