How do Christians define knowledge and faith?

There isn't a single Christian perspective on what it means to know or believe a fact (or indeed whether those should be treated as distinct ideas). Because Christianity has been around for two thousand years, and of course sees itself in continuity with the Jewish tradition going back at least a thousand years more, Christian beliefs have been expressed in all sorts of philosophical frameworks.

That said, Michael Polanyi's critical realist epistemology has my vote. ;-) His book Personal Knowledge is well worth reading. He was a Jewish convert to Catholicism, but his epistemology doesn't grow out of any particular belief system.

Justified True Belief

I do not accept the JTB standard because, so far as I can determine, JTB does not meet its own criteria:

• Justified: even philosophers who agree on JTB disagree on what constitutes a "justified" belief, suggesting to me that dogmatic views on the "J" criteria, and what it means in actual practice, are not justified. As noted in the link cited by the OP, the extreme end of the "J" spectrum leads to the conclusion that we can know nothing...therefore including in its definition the view that we cannot know that justification is necessary for knowledge. This is self-defeating.
• True: How do we know that the JTB standard is correct? On JTB's criteria, we would know that this standard is correct because it is a justified true belief...but this is arguing in a circle. With no criteria given for testing this proposition, we are left with an axiom that cannot be falsified. One might accept JTB because it is functionally superior to some other standard, and this might make JTB useful...but it would not make it true. Additionally, as discussed under "justification", if we cannot know if anything is true, we cannot know that JTB is true.
• Belief: no concrete objection here; some people do appear to genuinely believe in the JTB standard.

If JTB lacks a claim to justification or truth, accepting JTB as a definition of knowledge is itself not a justified true belief.

For what it's worth, I do believe that the Gettier problems pose a significant obstacle to the validity (or at least the usefulness) of JTB.

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Christian definitions

Faith:

Now faith is the substance of things hoped for, the evidence of things not seen (Hebrews 11:1)

Even so faith, if it hath not works, is dead, being alone. (James 2:17)

"Hope" in the New Testament does not describe sentiments such as "I hope it doesn't rain tomorrow", but rather refers to a motivation based upon an expected result. The "hope" of eternal life is not wishful thinking, but being motivated by the promise of eternal life.

"Faith" then, is having trust/confidence that provides motivation enough to act.

Knowledge:

a) an awareness that comes as a gift from God, through the Holy Spirit:

And I will give them an heart to know me, that I am the Lord: and they shall be my people, and I will be their God: for they shall return unto me with their whole heart. (Jeremiah 24:7)

And their eyes were opened, and they knew him (Luke 24:31)

Hereby know we that we dwell in him, and he in us, because he hath given us of his Spirit. (1 John 4:13)

b) the confidence & certainty that comes from experience:

If any man will do his will, he shall know of the doctrine, whether it be of God, or whether I speak of myself. (John 7:17)

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Conclusion

do (some) Christians claim to "know" that God exists?

Yes, in fact John was rather emphatic on the matter. He did not use the JTB standard in his testimony in 1 John.

Distinction between faith & knowledge

See my thoughts in this post

Is faith a special kind of belief?

In the scriptural sense, useful faith is faith in things that are true. One may well exercise faith in something that isn't true, but this will not produce the power discussed in Hebrews 11.

How do Christians define knowledge and faith?

A simple definition would be almost too simplistic here. Knowledge requires reasons. But even reason must be define to understand the nuances of knowledge.

Faith is the belief in the truth of something that does not require any evidence and may not be provable by any empirical or rational means. Reason is the faculty of the mind through which we can logically come to rational conclusions and being understood becomes knowledge.

But basic definitions do not explain the whole picture.

Here is how the Catholic Encyclopedia explains the essentials of knowledge.

Knowledge

Knowledge, being a primitive fact of consciousness, cannot, strictly speaking, be defined; but the direct and spontaneous consciousness of knowing may be made clearer by pointing out its essential and distinctive characteristics. It will be useful first to consider briefly the current uses of the verb "to know". To say that I know a certain man may mean simply that I have met him, and recognize him when I meet him again. This implies the permanence of a mental image enabling me to discern this man from all others. Sometimes, also, more than the mere familiarity with external features is implied. To know a man may mean to know his character, his inner and deeper qualities, and hence to expect him to act in a certain way under certain circumstances. The man who asserts that he knows an occurrence to be a fact means that he is so certain of it as to have no doubt concerning its reality. A pupil knows his lesson when he has mastered it and is able to recite it, and this, as the case may be, requires either mere retention in memory, or also, in addition to this retention, the intellectual work of understanding. A science is known when its principles, methods, and conclusions are understood, and the various facts and laws referring to it co-ordinated and explained. These various meanings may be reduced to two classes, one referring chiefly to sense-knowledge and to the recognition of particular experiences, the other referring chiefly to the understanding of general laws and principles. This distinction is expressed in many languages by the use of two different verbs—by gnônai and eidénai, in Greek; by cognoscere and scire, in Latin, and by their derivatives in the Romance languages; in German by kennen and wissen.

Essentials of knowledge

(1) Knowledge is essentially the consciousness of an object, i.e. of any thing, fact, or principle belonging to the physical, mental, or metaphysical order, that may in any manner be reached by cognitive faculties. An event, a material substance, a man, a geometrical theorem, a mental process, the immortality of the soul, the existence and nature of God, may be so many objects of knowledge. Thus knowledge implies the antithesis of a knowing subject and a known object. It always possesses an objective character and any process that may be conceived as merely subjective is not a cognitive process. Any attempt to reduce the object to a purely subjective experience could result only in destroying the fact itself of knowledge, which implies the object, or not-self, as clearly as it does the subject, or self.

(2) Knowledge supposes a judgment, explicit or implicit. Apprehension, that is, the mental conception of a simple present object, is generally numbered among the cognitive processes, yet, of itself, it is not in the strict sense knowledge, but only its starting-point. Properly speaking, we know only when we compare, identify, discriminate, connect; and these processes, equivalent to judgments, are found implicitly even in ordinary sense-perception. A few judgments are reached immediately, but by far the greater number require patient investigation. The mind is not merely passive in knowing, not a mirror or sensitized plate, in which objects picture themselves; it is also active in looking for conditions and causes, and in building up science out of the materials which it receives from experience. Thus observation and thought are two essential factors in knowledge.

(3) Truth and certitude are conditions of knowledge. A man may mistake error for truth and give his unreserved assent to a false statement. He may then be under the irresistible illusion that he knows, and subjectively the process is the same as that of knowledge; but an essential condition is lacking, namely, conformity of thought with reality, so that there we have only the appearance of knowledge. On the other hand, as long as any serious doubt remains in his mind, a man cannot say that he knows. "I think so" is far from meaning "I know it is so"; knowledge is not mere opinion or probable assent. The distinction between knowledge and belief is more difficult to draw, owing chiefly to the vague meaning of the latter term. Sometimes belief refers to assent without certitude, and denotes the attitude of the mind especially in regard to matters that are not governed by strict and uniform laws like those of the physical world, but depend on many complex factors and circumstances, as happens in human affairs. I know that water will freeze when it reaches a certain temperature; I believe that a man is fit for a certain office, or that the reforms endorsed by one political party will be more beneficial than those advocated by another. Sometimes, also, both belief and knowledge imply certitude, and denote states of mental assurance of the truth. But in belief the evidence is more obscure and indistinct than in knowledge, either because the grounds on which the assent rests are not so clear, or because the evidence is not personal, but based on the testimony of witnesses, or again because, in addition to the objective evidence which draws the assent, there are subjective conditions that predispose to it. Belief seems to depend on a great many influences, emotions, interests, surroundings, etc., besides the convincing reasons for which assent is given to truth. Faith is based on the testimony of someone else - God or man according as we speak of Divine or of human faith. If the authority on which it rests has all the required guarantees, faith gives the certitude of the fact, the knowledge that it is true; but, of itself, it does not give the intrinsic evidence why it is so. Faith is based on the testimony of someone else, God or man according as we speak of Divine or of human faith.

Kinds of knowledge

(1) It is impossible that all the knowledge a man has acquired should be at once present in consciousness. The greater part, in fact all of it with the exception of the few thoughts actually present in the mind, is stored up in the form of latent dispositions which enable the mind to recall it when wanted. Hence we may distinguish actual from habitual knowledge. The latter extends to whatever is preserved in memory and is capable of being recalled at will. This capacity of being recalled may require several experiences; a science is not always known after it has been mastered once, for even then it may be forgotten. By habitual knowledge is meant knowledge in readiness to come back to consciousness, and it is clear that it may have different degrees of perfection.

(2) The distinction between knowledge as recognition and knowledge as understanding has already been noted. In the same connection may be mentioned the distinction between particular knowledge, or knowledge of facts and individuals, and general knowledge, or knowledge of laws and classes. The former deals with the concrete, the latter with the abstract.

(3) According to the process by which it is acquired, knowledge is intuitive and immediate or discursive and mediate. The former comes from the direct sense perception, or the direct mental intuition of the truth of a proposition, based as it were on its own merits. The latter consists in the recognition of the truth of a proposition by seeing its connection with another already known to be true. The self-evident proposition is of such a nature as to be immediately clear to the mind. No one who understands the terms can fail to know that two and two are four, or that the whole is greater than any one of its parts. But most human knowledge is acquired progressively. Inductive knowledge starts from self-evident facts, and rises to laws and causes. Deductive knowledge proceeds from general self-evident propositions in order to discover their particular application. In both cases the process may be long, difficult, and complex. One may have to be satisfied with negative conception and analogical evidence, and, as a result, knowledge will be less clear, less certain, and more liable to error.

Faith, unlike knowledge, can not be reached exclusively reached by cognitive faculties. We may take God’s word as being Divine Truth simply as such. That is faith...

How is faith defined?

The meaning of the word

(Pistis, fides). In the Old Testament, the Hebrew means essentially steadfastness, cf. Exodus 17:12, where it is used to describe the strengthening of Moses' hands; hence it comes to mean faithfulness, whether of God towards man (Deuteronomy 32:4) or of man towards God (Psalm 118:30). As signifying man's attitude towards God it means trustfulness or fiducia. It would, however, be illogical to conclude that the word cannot, and does not, mean belief or faith in the Old Testament for it is clear that we cannot put trust in a person's promises without previously assenting to or believing in that person's claim to such confidence. Hence even if it could be proved that the Hebrew does not in itself contain the notion of belief, it must necessarily presuppose it. But that the word does itself contain the notion of belief is clear from the use of the radical, which in the causative conjugation, or Hiph'il, means "to believe", e.g. Genesis 15:6, and Deuteronomy 1:32, in which latter passage the two meanings — viz. of believing and of trusting — are combined. That the noun itself often means faith or belief, is clear from Habakkuk 2:4, where the context demands it. The witness of the Septuagint is decisive; they render the verb by pisteuo, and the noun by pistis; and here again the two factors, faith and trust, are connoted by the same term. But that even in classical Greek pisteuo was used to signify believe, is clear from Euripides (Helene, 710), logois d'emoisi pisteuson tade, and that pistis could mean "belief" is shown by the same dramatist's theon d'ouketi pistis arage (Medea, 414; cf. Hipp., 1007). In the New Testament the meanings "to believe" and "belief", for pisteon and pistis, come to the fore; in Christ's speech, pistis frequently means "trust", but also "belief" (cf. Matthew 8:10). In Acts it is used objectively of the tenets of the Christians, but is often to be rendered "belief" (cf. 17:31; 20:21; 26:8). In Romans 14:23, it has the meaning of "conscience" — "all that is not of faith is sin" — but the Apostle repeatedly uses it in the sense of "belief" (cf. Romans 4 and Galatians 3). How necessary it is to point this out will be evident to all who are familiar with modern theological literature; thus, when a writer in the "Hibbert Journal", Oct., 1907, says, "From one end of the Scripture to the other, faith is trust and only trust", it is hard to see how he would explain 1 Corinthians 13:13, and Hebrews 11:1. The truth is that many theological writers of the present day are given to very loose thinking, and in nothing is this so evident as in their treatment of faith. In the article just referred to we read: "Trust in God is faith, faith is belief, belief may mean creed, but creed is not equivalent to trust in God." A similar vagueness was especially noticeable in the "Do we believe?" controversy—one correspondent says—"We unbelievers, if we have lost faith, cling more closely to hope and — the greatest of these — charity" ("Do we believe?", p. 180, ed. W. L. Courtney, 1905).

I would say that knowledge is a subset of faith, since all knowledge depends on faith. Religion has nothing to do with it. This is provable:

1. No knowledge is more certain than mathematical knowledge.
2. Godel's second incompleteness theorem proves no consistent mathematical system can prove its own consistency. Consistency means the mathematical system will never contradict itself.
3. Therefore, since we cannot prove the most certain form of knowledge does not contradict itself, neither can we prove the consistency of any other kind of knowledge.

Mathematicians, and every other knowledge worker, must take it on faith they have a consistent body of knowledge.

The two words are generally synonymous (as seen in a thesaurus). Different philosopher's have tried to draw a distinction between the two in order to claim some sort of qualitative distinction between their beliefs and teachings vs others - a form of elitism. The earliest example I can think of would be Plato who, if I recall correctly, maintained that the difference is that knowledge concerns the Ideas/Forms whereas belief/opinion (even true, well reasoned belief) concerns in some way the unreal physical, material realm. Mathematics holds a place in between these two for him.

While Plato was at one point the height of education, his philosophy is now a neat historical footnote. His and Aristotle's impact was huge and long-lasting to be sure - especially on the Church - but for the average individual they probably at best know a few references like that Socrates was considered wisest man alive for he claimed to know nothing. And while these philosophers surely coined a number of terms or gave words a distinct meaning within the course of their philosophy, it does not mean that people in generally adopted those definitions. Nor should we seek to replace the common meaning of words with each philosopher's distinct spin - better to stick to the common usage unless the context calls for something different.

The scriptures were written in common (koine) Greek and so one way to research the biblical meaning of these terms would be to look them up in a Greek Concordance. For example, if we look up faith/belief (πίστις) in Strong's Greek Concordance, we find the following:

assurance, belief, believe, faith, fidelity.

From peitho; persuasion, i.e. Credence; moral conviction (of religious truth, or the truthfulness of God or a religious teacher), especially reliance upon Christ for salvation; abstractly, constancy in such profession; by extension, the system of religious (Gospel) truth itself -- assurance, belief, believe, faith, fidelity.

see GREEK peitho

So then, biblical "belief" or "faith" carries with it a sense of being persuaded, a sense of conviction, of assurance, of fidelity.

We can also look up the Greek word for knowledge (γνῶσις):

knowledge, science.

From ginosko; knowing (the act), i.e. (by implication) knowledge -- knowledge, science.

see GREEK ginosko

Now, whenever you say you "know" something - and especially when you choose to use that word in contrast to "believing" something - you are in essence simply asserting a strong conviction that what you hold to be true is in fact so. But this returns us to the same qualities of biblical belief/faith. There is thus no meaningful distinction between knowledge and belief - save that some choose try to use one or the other as a form of elitism without being able to provide an objective distinction (unless you think Plato's realm of Ideas is an objective basis for distinction)

Knowledge exists in (among other locations) the human brain. It is the intimate awareness of some reality.

Faith is confidence of what you do not see. When I don't see the sun (because it's night time, or I am indoors) I still am confident it is there because my mind has knowledge.

One theologian said true faith is knowing. If you know then you'll be confident even when you aren't looking at it. If you don't know, you won't be confident.

If you have knowledge, you have the ability to have faith. It's important to venture outside of what you know and into the implications of what you know, as well as into more knowledge Asking questions of wise and knowledgable people is important. But just because you have faith doesn't mean you have knowledge. Faith based on lies is dangerous.

Faithlessness is turning our backs on knowledge. This is the thing that continues to destroy generation after generation of humanity. We turn our backs on what our parents know. Our parents did not live perfectly, but they have knowledge. Failing to honor the knowledge they do have ensures we do not enjoy a better life than they did.

Jesus allowed his disciples to see he was resurrected. That was His idea, not theirs. He rebuked them, pre resurrection, and rebuked Thomas post resurrection for being so slow to believe, not for requiring knowledge. All of Thomas's friends were telling him they saw Jesus alive, yet he didn't say something like, "wow it's so hard to believe. I hope I get to see Him too.". He said that he would absolutely refuse to believe unless he saw Jesus himself. He was more concerned with verifying it for himself than he was with whether or not the resurrection was real. Thomas did not allow fascination with reality to lead him.

Faith is confident of what is real even when the person is not looking directly at it. I'd add that faith is interested in what's real. Science is a discovery process that requires faith that there's something worth being discovered.

In the Middle Ages, Christian society graduated out of believing in cows, boobs, rain and all the things we see and started recognizing there are things we don't see. They started having faith. Since then we've discovered gravity, sound waves, bacteria, electromagnetic waves, DNA and all kinds of other things that aren't visible to the human eye. We started recognizing that there's more than what we see. And that faith has led to an extraordinary amount of knowledge.

These are some logical constructs.

Personally, I'm inclined to accept the JTB definition of knowledge; I didn't find the Gettier problems convincing, and would say that justification, generally, is a sound and valid logical deduction, even one that may be subconscious, eg seeing something with your own two eyes. There may be exceptions, but I'm happy to call them just that, exceptions, although I can't think of any off the top of my head.

Faith, on the other hand, I would simply call belief, one which may or may not have justification, or be in something true, but which is not known. Faith might be justified, but untrue, or vice versa. Thus you might have blind faith, or faith with at least some justification (there is a popular apologetics ministry called Reasonable Faith).

Thus, you could have faith that you know something. For example, I have faith that I know God exists.