1) Hume's "dilemma of determinism": the problem that our actions are either causally determined or random. In either case, we are not responsible for them.
2) Hume's distinction between 'is' and 'ought'. 
3) Hume's distinction between demonstrative and probable reasoning. (see below) 
The third distinction is between two different areas of human study:
- All the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of fact. Of the first kind are the sciences of Geometry, Algebra, and Arithmetic ... [which are] discoverable by the mere operation of thought ... Matters of fact, which are the second object of human reason, are not ascertained in the same manner; nor is our evidence of their truth, however great, of a like nature with the foregoing.
Hume's fork is often stated in such a way that statements are divided up into two types:
- Statements about ideas - these are analytic, necessary statements that are knowable a priori.
- Statements about the world - these are synthetic, contingent, and knowable a posteriori.
In modern terminology, members of the first group are known as analytic propositions and members of the latter as synthetic propositions. This terminology comes from Kant (Introduction to Critique of Pure Reason, Section IV).
Into the first class fall statements such as "2 + 2 = 4", "all bachelors are unmarried", and truths of mathematics and logic. Into the second class fall statements like "the sun rises in the morning", "the Earth has precisely one moon", and "water freezes at 32 degrees Fahrenheit".
Hume wants to prove that certainty does not exist in science.
First, Hume notes that statements of the second type can never be entirely certain, due to the fallibility of our senses, the possibility of deception (see e.g. the modern brain in a vat theory) and other arguments made by philosophical skeptics. It is always logically possible that any given statement about the world is false. (Note that statements like "either the Earth has precisely one moon, or not" are really truths of logic, and say nothing about the world).
Second, Hume claims that our belief in cause-and-effect relationships between events is not grounded on reason, but rather arises merely by habit or custom. Suppose one states: "Whenever someone on earth lets go of a rock, it falls." While we can grant that in every instance thus far when a rock was dropped on Earth it went down, this does not give us any reason to think that in the future, rocks will fall when in the same circumstances. It would give us reason only if we add as a premise that the future will resemble the past. But that isn't something that we can know based on past experience -- all past experience could tell us is that in the past, the future has resembled the past.
Third, Hume notes that relations of ideas can be used only to prove other relations of ideas, and mean nothing outside of the context of how they relate to each other, and therefore tell us nothing about the world. Take the statement "An equilateral triangle has three sides of equal length." While some earlier philosophers (most notably Plato and Descartes) held that logical statements such as these contained the most formal reality, since they are always true and unchanging, Hume held that, while true, they contain no formal reality, because the truth of the statements rests on the definitions of the words involved, and not on actual things in the world, since there is no such thing as a true triangle or exact equality of length in the world. So for this reason, relations of ideas cannot be used to prove matters of fact.
The results claimed by Hume as consequences of his fork are drastic. According to him, relations of ideas can be proved with certainty (by using other relations of ideas), however, they don't really mean anything about the world. Since they don't mean anything about the world, relations of ideas cannot be used to prove matters of fact. Because of this, matters of fact have no certainty and therefore cannot be used to prove anything. Only certain things can be used to prove other things for certain, but only things about the world can be used to prove other things about the world. But since we can't cross the fork, nothing is both certain and about the world, only one or the other, and so it is impossible to prove something about the world with certainty.
If accepted, Hume's Fork makes it pointless to try to prove the existence of God (for example) as a matter of fact. If God is not literally made up of physical matter, and does not have an observable effect on the world, making a statement about God is not a matter of fact. Therefore, a statement about God must be a relation of ideas. In this case if we prove the statement "God exists," it doesn't really tell us anything about the world; it is just playing with words. It is easy to see how Hume's Fork voids the causal argument and the ontological argument for the existence of a non-observable (non-material) God. However, this does not mean that the validity of Hume's Fork would imply that God definitely does not exist, only that it would imply that the existence of God cannot be proven as a matter of fact without worldly evidence.
Hume famously rejected the idea of any meaningful statement that did not fall into this schema, saying:
- If we take in our hand any volume; of divinity or school metaphysics, for instance; let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames: for it can contain nothing but sophistry and illusion.
This same idea has perdured in relatively contemporary schools of philosophy, such as logical positivism. This general attitude is strikingly present in Alfred Ayer's logical empiricist work Language, Truth and Logic.
There are two notable objections to Hume's fork. Kant, for example, famously defended the idea of synthetic a priori propositions. W.V.O. Quine also argued against the well-defined distinction between analytic and synthetic propositions required by Hume's fork in his landmark paper Two Dogmas of Empiricism.
- ↑ Blackburn, Simon. 'Hume's Fork.' Oxford Dictionary of Philosophy. OUP 1996. ISBN 0-19-283134-8. Note: Blackburn does not mention any of the other possible referents of "Hume's Fork."
- ↑ Nicholas Bunnin and Jiyuan Yu. 'Hume's Fork.' Blackwell's Dictionary of Western Philosophy. Blackwells, 2004.
- ↑ ibid.