First thing first, we have to solve a textual dilemma. The issues pertains to the disputed reading as to which verb for ‘believe’ we should accept. It comes down to whether one accepts the aorist pisteusete (attested by Sinaiticus2 A C D L N W f1 f13 33) or the present pisteuete (attested by P66 Sinaiticus* B 0250). The material difference is based on the presence or absence of the letter sigma in the verb to indicate either an aorist subjunctive or present subjunctive verb. The actual difference is alleged to come down to whether one adopts an aorist subjunctive ‘you might come to believe’ (pisteusete) or a present subjunctive ‘you might continue to believe’ (pisteuete). The former would be consistent with trying to bring people to faith (i.e. evangelism) and the later would be consistent with reinforcing belief (i.e. discipleship). The external evidence is evenly weighted though perhaps marginally favors the aorist pisteusete. But what breaks the deadlock is the internal evidence because the conjunction hina (‘in order that’) in the Fourth Gospel is normally followed by an aorist subjunctive and the aorist is more likely here given the previous appearance of pisteusete in 19:35 which uses the same verbal form for believing in a dramatic purpose clause (though again with some textual variants but the aorist is even better attested there).
The Johannine Purpose Statement (John 20:31) Part 1 – Text Critical Issues