Can someone explain the linguistic philosophical distinction between 'believe in' and 'believe that'?
(HH Price came up with the initial idea of the two entities)
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Not entirely sure what you mean by "linguistic philosophical distinction" nor am I completely clear on the origins, though a few websites do seem to validate sourcing it to H.H. Price.
I am going to assume that "faith-in" and "faith-that" are used synonymously to the pair you mention. The difference in philosophy of religion and in discussions about faith is that
believe that is used to mark the use of belief with respect to propositional truths.
believe in is used to mark trust in a person, institution, or society.
The value of making this distinction is that a belief-that P ties things up in a large amount of literature on propositional truths and their justification and then entangles things in questions of whether you can believe false propositions and then what it means to both "know" something is false and that you "believe" it to be true.
Conversely, belief-in-Q or faith-in is to take Q as a veridical source in some way for the self.
Using the common example of God. Belief-that-God-exists and Belief-in-God differ in that the former is "belief" whatever that will turn out to be with respect to the truth of (or potentially the falsity of) G. Belief in God conversely does not entail anything about the proposition G. Instead, it's an implication about the relationship the subject has to G and how that impacts their world.
The same could be said about an organization. I believe-in the US government is not identical to I believe-that the US government exists, etc. The one indicates whether you think they accomplish something; the other whether they exist (or are truthful, etc.)
Here's a few articles that use the distinction with respect to faith:
Robert Audi, “Faith, Belief, and Rationality,” in Philosophical Perspectives 5 (1991): 213-39.
William Alston, “Belief, Acceptance, and Religious Faith,” in Faith, Freedom, and Rationality: Philosophy of Religion Today, ed. Jeff Jordan and Daniel Howard-Snyder (Lanham, MD: Rowman & Littlefield, 1996), 3-29.
Benjamin W. McCraw, “Faith and Trust,” International Journal for Philosophy of Religion 77, no. 2 (April 2015): 141-58.
Trent Dougherty, “Zagzebski, Authority, and Faith,” European Journal for Philosophy of Religion 6, no. 4 (December 22, 2014): 47-59, at 58.