Argument based on 1 Corinthians 13
P1. There is a time T when the perfect will come and spiritual gifts such as tongues, prophecy and knowledge will cease. Before T, these spiritual gifts are available. After T, they cease (From 1 Cor 13:1-2,8-10)
P2. There will be a time T when we will see face to face. (From 1 Cor 13:12)
P3. The time T when we will see face to face is the same time T when the perfect will come. (From 1 Cor 13:9-12)
P4. The time T when we will see face to face is the same time T when we will have full access to the mysteries of the kingdom of God. (From 1 Cor 13:1-2,9-12)
P5. We don't have full access to the mysteries of the kingdom of God now. (From 1 Cor 13:9 and common sense)
C1. Therefore, the time T when we will see face to face has not yet passed (i.e. now <= T). (From P5 and P4)
C2. Therefore, the time T when the perfect comes has not yet passed (i.e. now <= T). (From C1 and P3)
C3. Therefore, spiritual gifts such as tongues, prophecy and knowledge are still available (i.e. they haven't ceased yet, because now <= T). (From C2 and P1)
C4. Therefore, Continuationism is true. (From C3 and the definition of Continuationism)
Defense of the premises
Premise 1. There is a time T when the perfect will come and spiritual gifts such as tongues, prophecy and knowledge will cease. Before T, these spiritual gifts are available. After T, they cease (From 1 Cor 13:1-2,8-10)
This one is fairly uncontroversial, in my opinion. Here are the verses:
If I speak in the tongues of men and of angels, but have not love, I am a noisy gong or a clanging cymbal. 2 And if I have prophetic powers, and understand all mysteries and all knowledge,
[...]
8 Love never ends. As for prophecies, they will pass away; as for tongues, they will cease; as for knowledge, it will pass away. 9 For we know in part and we prophesy in part, 10 but when the perfect comes, the partial will pass away.
It is very clear that these spiritual gifts are regarded by Paul as partial/imperfect, and that they will become obsolete and therefore cease once the perfect comes. In other words, if we call T the time when the perfect comes, before T the gifts are available and after T they become obsolete and get replaced by the perfect.
Premise 2. There will be a time T when we will see face to face. (From 1 Cor 13:12)
This comes directly from verse 12:
For now we see in a mirror dimly, but then face to face. Now I know in part; then I shall know fully, even as I have been fully known.
Explicitly stated, trivially true.
Premise 3. The time T when we will see face to face is the same time T when the perfect will come. (From 1 Cor 13:9-12)
9 For we know in part and we prophesy in part, 10 but when the perfect comes, the partial will pass away. 11 When I was a child, I spoke like a child, I thought like a child, I reasoned like a child. When I became a man, I gave up childish ways. 12 For now we see in a mirror dimly, but then face to face. Now I know in part; then I shall know fully, even as I have been fully known.
It's very clear that Paul is talking about and comparing now vs. then (in the future). Now we have these partial tools (spiritual gifts), but then the perfect will come, and we will see face to face, and we will know fully, and the spiritual gifts will become obsolete and cease. Therefore, it becomes clear that the expressions "see face to face" and "the perfect" are pointing to a same event in the future.
Premise 4. The time T when we will see face to face is the same time T when we will have full access to the mysteries of the kingdom of God. (From 1 Cor 13:1-2,9-12)
Basically the same defense as before. But also, a brief commentary on the expression "mysteries of the kingdom of God". Verse 12 reads:
12 For now we see in a mirror dimly, but then face to face. Now I know in part; then I shall know fully, even as I have been fully known.
Even though it doesn't appear in the chapter, I use the expression "mysteries of the kingdom of God" because the passage very clearly talks about attaining full knowledge, just like we have been fully known (by God). This divine knowledge will be attained when the perfect comes. And it will be complete, unambiguous, not subject to controversies or different interpretations.
Premise 5. We don't have full access to the mysteries of the kingdom of God now. (From 1 Cor 13:9 and common sense)
Perhaps the most controversial premise of the argument. Verse 9 says:
9 For we know in part and we prophesy in part,
Paul acknowledged that, at least at the time verse 9 was spoken, they had partial, imperfect knowledge of God and His mysteries. So premise 5 was definitely true at their time. The question is: What about now (August 2021)? I would argue that we don't have perfect knowledge of God and His mysteries now either. And it's very easy to see why: If we already had perfect knowledge now, then this perfect knowledge should be complete, unambiguous, uncontroversial. If it is not complete, then it is partial, and if it is ambiguous, then it is subject to misinterpretations, misunderstanding and can lead to controversies and erroneous conclusions. But that's exactly the kind of knowledge we have right now: we don't know everything, we haven't figured out yet everything about the laws of physics, we don't know everything about the spirit realm, there are tons of mysteries that will probably be revealed in Heaven which we aren't currently aware of, our understanding of the Scriptures is very imperfect and full of ambiguities, controversies, etc.
Responding to objections in the comments
Objection. It is very revealing (and to be honest, unsurprising) that I cannot find the name of Jesus Christ anywhere in this 'argument'. If the Spirit were truly present in all of this, we would hear of Christ (and not the Spirit) because : Howbeit when he, the Spirit of truth, is come, he will guide you into all truth: for he shall not speak of himself; He shall glorify me: for he shall receive of mine, and shall shew it unto you..
Answer. This is only a problem if it invalidates one or more of the premises. If it doesn't, then I don't see the point of this objection. On the other hand, one could argue that the phrase "see face to face" in verse 12 implies seeing Jesus Christ face to face in Heaven, after the resurrection of the death. See the answers to the related question When shall we see “face to face”? 1 Corinthians 13:12 that quote many similar passages that explicitly mention the person of Jesus.
Objection. Deductive reasoning won't lead to a conclusion based on the weight of Biblical evidence however, which is often what we do want.
Answer. When you talk about the weight of the evidence, that sounds pretty much like abductive reasoning. But different reasoning paradigms are not incompatible with each other. In fact, quite the contrary, you can use abductive arguments, inductive arguments or even other deductive arguments (in short, any kind of argument) to defend the premises of a deductive argument. For example, see this answer that attempts to deduce the doctrine of the Trinity. Another example is William Lane Craig's defense of the premises of the Kalam cosmological argument.
Relevant follow-up question posted on BH.SE by @GratefulDisciple: Can 1 Cor 13:8-10 be used as to prove that radical Continuationism is necessarily true? Stay tuned :-)
Edit: link to my answer to said question.