@Mark I like your gloss here. Faith and hope are definitely related. I use the Scriptural definition of faith, which I could try to state as (roughly) belief in things that are true, without having to be shown compelling evidence that they are true beforehand. For example, doing something because it is good, only later to find out that the claims are indeed true, I say would qualify as hope. Hope is also sometimes termed an expectation. In modern parlance, a person could hope to disprove a true claim, but that hope would be vain, it would not be faith.
@Mark In an experiment with a null hypothesis, perhaps the faith is not so much in that a given explanation is true, but rather in the prospect of learning something new and interesting. The faith of many scientists has been gratified by their unexpected discoveries. One does not have to be a good guesser to do science, one only has to put in the effort to learn and weigh the outcomes candidly.
@Mark Scripturally speaking we might be splitting hairs. In many places where the Scriptures use a word rendered by KJV translators as "hope", the underlying connotation aligns more with the English word "expectation". Faith is more expectant than the vanilla modern English version of "hope", which seems more like wishful thinking. But even a desire to believe, even a modernist flavor of "hope" if invested can lead to the dividends of a growing expectation, and legitimate faith, which results in the delicious fruit of knowledge and a growing, living tree that produces more knowledge.
@Mark Belief does appear to be a necessary ingredient to faith, which is required in both of the above Scriptural promises. It doesn't grow all at once though. "exercise a particle of faith, yea, even if ye can no more than desire to believe, let this desire work in you, even until ye believe in a manner that ye can give place for a portion of my words." Alma 32:27 One of the best ways that occurs to my mind to try to measure belief or faith is therefore the degree of investment or amount of space in a person's life he devotes to pursuing and living the doctrine, in agreement with John 7:17.
@RussellMcMahon both ways are mentioned to illustrate the duality. Falsifiability would only matter for false claims, verifiability only matters for true claims. There does not need to be a test for proving that a false claim is true, since that would lead to a contradiction.
@ScottRowe Love of truth urges us to go on with endeavors that will yield fruit under the right conditions. Of course we encounter the halting problem: Those with hope in the truth will eventually find it, whereas those whose pursuit is of falsehood will never find it in all eternity. This is love of truth, that we endure until we find it, and love it when we do. The proof of an unfalsifiable but true statement is fairly made when the experiments to demonstrate its truth have succeeded, and the route of falsification never comes up with anything to the contrary. Thus all things are testable.
@ScottRowe Proof is always personal because the convincing of the individual cannot occur outside of the individual's conscience or awareness. Of course this does not mean that truth is relative; it is not. There is no way to obtain a testimony of truth from afar. In order to believe and come to know what is true, we must be imminently involved and seeking to learn for ourselves. That is what all of the great experimenters and discoverers have learned. The only way to learn is to learn for ourselves. No one else's learning will do it for us. The only way to know is to know for ourselves.
"You don't see Q, therefore P is false." This assumes that the proposition P → Q is true, but this is the proposition under test. That is not a test of P → Q. Have you not heard of Bayes Theorem or experiential learning, or proving causation, or experimental design in general? It is true that P → Q does not imply that Q → P, but that is not even part of the test for the truthfulness of proposition P → Q! It is entirely irrelevant. The above does not depend on any such argument. Simply, if you were to attempt to disprove that P → Q, you do not do so by failing to satisfy condition P.
@NotThatGuy P must be true in order to falsify P → Q. You are testing the wrong proposition. P → Q is analogized to the claim that "if you ask in faith, you will receive an answer from God". P → Q is the truth claim you are attempting to falsify. This cannot be done without supplying P. You provide P, and as you said, if Q does not follow, then P → Q would be false. If you fail to provide P (by not following the experimental protocol), you cannot disprove P → Q.
@NotThatGuy The proposition is that you (and anyone) can come to know the Actual God and that He exists (Q) by satisfying the prerequisites He has given (P). Together, this proposition is P → Q. P and P → Q are not the same test. They are not even the same proposition. P is not even merely a proposition. It is a condition. I have a degree in this field and have taught professionally. If you were doing this as an assignment I would fail you on this.
@NotThatGuy If Q is false, then P is false or P → Q is false. This is true. However, it is not the same test because you failed to disambiguate whether P is false or P → Q is false. Since you did not satisfy P, you can say nothing about P → Q. This is very simple. You don't assume P, you must provide P so that you have a chance at falsifying P → Q (or of failing to falsify it by observing Q).
@NotThatGuy Again, P is not what is under test. I have stated this many times. This really isn't rocket science, it's just the most basic and ubiquitous element of logical proofs used everywhere throughout the world. If P then Q. Is that statement true? In order to find out, you must supply P. Otherwise it remains unfalsified, not because it is unfalsifiable, but because of a simple failure to conduct the experiment at all. It is telling the teacher that the answer to the hypothesis in the assignment is "no" "because I didn't even do the assignment!"
@NotThatGuy I clearly demonstrated to you that you had chosen to conduct the wrong test (not the test advanced in the question nor addressed in this answer), and you had decided not even to entertain the experiment, and then you claimed that the question was faulty, introducing circular reasoning, even though it really is just vanilla Modus Ponens, the first homework assignment we give in propositional logic classes. You can do with that what you will.