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The idea of a "leap of faith" is unfairly defined by Wikipedia (https://en.wikipedia.org/wiki/Leap_of_faith) as:
In philosophy, a leap of faith is the act of believing in or accepting something not on the basis of reason. The phrase is commonly associated with Danish philosopher Søren Kierkegaard.
What an almighty put-down - and entirely simplistic! The entry continues by saying this:
As an idiom, leap of faith can refer to the act of believing something that is unprovable. The term can also refer to a risky thing a person does in hopes of a positive outcome.
I have a bit more sympathy for this second, idiomatic usage. The point for me is that it all depends on what we regard "having faith in something" means for us.
To me, the definition given by Dictionary.com conveys a better sense of what "leap of faith" means:
A belief or trust in something intangible or incapable of being proved. For example, "It required a leap of faith to pursue this unusual step of transplanting an animals' heart into a human patient."
The Wikipedia article goes on to say that Kierkegaard never said "leap of faith" as such. It seems to be one of those ideas that has emerged through a process of discussion and literary exploration, while the broad concept nevertheless remains associated with Kierkegaard.
An excellent quote often associated with a leap of faith is this one attributed to J.R.Rim:
“Wings are like dreams. Before each flight, a bird takes a small jump, a leap of faith, believing that its wings will work. That jump can only be made with rock solid feet.”
2. Faith, Trust and Belief
In line with my comments on Paul H's article on Faith, the notions of Trust and Faith all have to do with the degrees and nature of personal Belief - which in itself really has to do with the nature of Knowledge, epistemology, and how we personally come to know things and have any confidence in them.. Trust is quite a weak, informal form of personal belief, with Faith being the most profound strongest sort of belief requiring considerable commitment. Al belief involves some notion of evidence, either for or against which in turn involves reason.
Trust is informal confidence and belief that things will behave as expected, will act similarly to past experiences and are unlikely to surprise us with unexpected poor performance. Trust is typically contingent on circumstances and does not represent a hard and fast guarantee - it is what is likely to happen, what is expected to happen. Trust is more about what is likely and may be quite malleable, particularly when the world changes, our trust has to change with it.
To use Mark's "bus driver" example: we may have trust that the bus driver will deliver us to our desired destination in good order and in reasonable time. That doesn't mean that this will happen - the driver could have a heart attack, the bus may be involved in an accident, there may be unexpected delays in traffic and so on - any number of ways that what we had trusted in when we took the bus doesn't happen in the way we had expected. When we trust something, it doesn't mean that it is necessarily the case - instead, trust is always contingent upon how circumstances pan out this time around.
Beliefs are personal claims about the world, people and ideas. We think that claims could either be true or false - but in fact our claims are generally partial statements and incomplete. This means claims may contain true and false elements and this leads to degrees of truth for these claims.
Faith is a completely different kettle of fish. Although faith is a form of belief, the nature of evidence is very different. Faith in anything at all is about making an initial commitment without incontrovertible evidence that the something you have faith in is actually true. Faith requires this leap into faith, a step into what is at the time of commitment, the unknown. If what you have faith in is actually true, then strong evidence will only become apparent later. Just because you once lept into faith about something, this still means you could leap out of it later - evidence still plays a role but is often more about maintaining a belief, rather than initially establishing it.
People sometimes say that faith is all about believing without reason - well, that would lead to a very weak form of faith indeed and would I think be akin to a superstitious form of trust - easily gained and equally easily lost. As someone with a background in Mathematical Logic and so on, I equate the notion of faith as something that is initially held as being axiomatically true - withno logical precursor. Some people will accept this statement as being foundationally true - whereas others with a different way of thinking may adopt other statements as being foundationally true - this is the nature of personal belief and faith. Logically, such axioms are taken as being unconditionally (tautologically) true by that person. Later evidence, if fairly understood without bias or prejudice, may either confirm, deny or have nothing relevant to say about the actual truth (or otherwise) of those faith beliefs.
The notion of a leap of faith (or as I prefer, a leap into faith c.f. leap into the unknown) can be illustrated by reusing Mark's bus driver analogy - if instead of our usual bus, another bus came along to our standard bus stop, driven by a bus driver we didn't recognise, with a bus number we hadn't seen before, we will then have to wonder if we should trust taking this bus to complete our journey. We then ask the driver and they confirm we could safely use this bus. If we get on, we are nonetheless taking a risk, making an assumption in accepting what we have been told by someone we don't recognise - this is a leap into trusting what we have been told.