'Else what shall they do who are baptised for the dead? If the dead are not raised at all, why then are they baptised for them?'
This first argument was possibly based on a custom at that time of baptising the dead by proxy (or possibly the mortally ill who could be described as 'dead' and in no condition to be baptised?). It would seem that it had become a custom in Corinth for people to be baptised on behalf of Christians who had died unbaptised or were so close to death that they could not be baptised. So 'for the dead' means being baptised for Christians who had died before they could be baptised, or were about to die and could not be baptised, so that there might be no loss by their not being baptised. The use of the article with 'dead' indicates from its use elsewhere that it is Christian dead who are in mind. When used of the general dead it does not generally have the article.
One point that he might be making here might be that once someone has died their spirit has passed on. If, as some of the Corinthians stated, their body is now finished with, what on earth point is there in someone going through a bodily ceremony for them? The baptism would be declaring these dead to be Christians, of what point would that be on bodies which had been cast off?
Others have referred it to being baptised on behalf of relatives who had died before the Gospel had reached Corinth, in the hope that it might be effective for them as those who had had no opportunity to receive Christ, or even on behalf of friends and relatives, in a general hope for those who had died unsaved. But Paul would hardly have accepted such ideas without protest. He has made quite clear earlier that it was the word of the cross that saved, not baptism (1 Corinthians 1:17-18).
Thus the first is the most likely to be the case, otherwise it would have smacked of that very thing that in chapter 1 he had rejected, that baptism was necessary to salvation, and could bring about salvation, in contrast with his view that it was the word of the cross that saved. He would hardly have quietly accepted that.
For in the case of the baptism on behalf of Christians who had died or were mortally ill Paul could see it as declaring to the world that the person had died (or was dying) trusting in Christ, saved by the word of the cross, and so could be seen as a physical and outward manifestation by proxy that he belonged to Christ. Thus he could go along with it. But what, says Paul, would be the point of that if the dead are not raised? It is the body which is being indicated to be Christ's, not the spirit.
This practise is not witnessed to anywhere else in the New Testament and is found in none of the earliest Christian literature, even though baptism had by then gained a deeper significance. It must therefore be assumed that it was a local practise. But all the Corinthians were seemingly involved in it. Indeed these whom he was disputing with clearly had a 'high' view of baptism (1 Corinthians 1:13; 1 Corinthians 10:2), so he points out that their practise is contrary to what they teach.
Other interpretations include the suggestion that ‘baptised' here refers to the baptism of suffering which Jesus faced and which would face at least some of the Apostles (Luke 12:50; Mark 10:38-39). Thus by this Paul would be saying, ‘Why should those who have suffered overwhelmingly in order to bring to Christ those who have now died in Christ, have done so if there is no resurrection?' This would fit well the following verses, where the same thought would then be applied to Paul personally. Or alternately that the meaning is ‘baptised in readiness for being dead ones'. In other words why are Christians themselves baptised at all if they are not to rise from the dead? For Paul saw baptism as a depiction of that rising from the dead (Romans 6:4).