Principle of sufficient reason
The principle of sufficient reason states that nothing is without a ground or reason why it is. The formulation of the principle is usually attributed to Leibniz, although the idea was conceived and utilized in various philosophers that preceded him, including Anaximander, Parmenides, Archimedes, Thomas Aquinas, Anaximander of Miletus, and Spinoza. Some philosophers have associated the principle of sufficient reason with "ex nihilo nihil fit".
The principle has a variety of expressions, all of which are perhaps best summarized by the following:
- For every entity X, if X exists, then there is a sufficient explanation for why X exists.
- For every event E, if E occurs, then there is a sufficient explanation for why E occurs.
- For every proposition P, if P is true, then there is a sufficient explanation for why P is true.
A sufficient explanation may be understood either in terms of reasons or causes, for like many philosophers of the period, Leibniz did not carefully distinguish between the two. The resulting principle is very different, however, depending on which interpretation is given.
Note that the principle of sufficient reason can't be applied to axioms within a logic construction like a mathematical or a physical theory. Axioms are propositions accepted as having no justification possible within the system. The principle declares that all propositions considered to be true within a system should be deducible from the set axioms at the base of the construction (with some theoretical exceptions: see Gödel's theorem).
Leibniz's view 
In fact Leibniz opposed fatalism and had a more nuanced and characteristic version of the principle, in which the contingent was admitted on the basis of infinitary reasons, to which God had access but humans did not. He explained this while discussing the problem of future contingents:
We have said that the concept of an individual substance (Leibniz also uses the term haecceity) includes once for all everything which can ever happen to it and that in considering this concept one will be able to see everything which can truly be said concerning the individual, just as we are able to see in the nature of a circle all the properties which can be derived from it. But does it not seem that in this way the difference between contingent and necessary truths will be destroyed, that there will be no place for human liberty, and that an absolute fatality will rule as well over all our actions as over all the rest of the events of the world? To this I reply that a distinction must be made between that which is certain and that which is necessary. (§13, Discourse on Metaphysics)
Without this qualification, the principle can be seen as a description of a certain notion of closed system, in which there is no 'outside' to provide unexplained events with causes. It is also in tension with the paradox of Buridan's ass. Leibniz denied that the paradox of Buridan's ass could ever occur, saying:
In consequence of this, the case also of Buridan's ass between two meadows, impelled equally towards both of them, is a fiction that cannot occur in the universe....For the universe cannot be halved by a plane drawn through the middle of the ass, which is cut vertically through its length, so that all is equal and alike on both sides.....Neither the parts of the universe nor the viscera of the animal are alike nor are they evenly placed on both sides of this vertical plane. There will therefore always be many things in the ass and outside the ass, although they be not apparent to us, which will determine him to go on one side rather than the other. And although man is free, and the ass is not, nevertheless for the same reason it must be true that in man likewise the case of a perfect equipoise between two courses is impossible. (Theodicy, pg. 150)
As a Law of Thought 
The principle was one of the four recognised laws of thought, that held a place in European pedagogy of logic and reasoning (and, to some extent, philosophy in general) in the 18th and nineteenth century. It was influential in the thinking of Leo Tolstoy, amongst others, in the elevated form that history could not be accepted as random.
Schopenhauer's Four Forms 
According to Schopenhauer's On the Fourfold Root of the Principle of Sufficient Reason, there are four distinct forms of the principle.
First Form: The Principle of Sufficient Reason of Becoming (principium rationis sufficientis fiendi); appears as the law of causality in the understanding.
Second Form: The Principle of Sufficient Reason of Knowing (principium rationis sufficientis cognoscendi); asserts that if a judgment is to express a piece of knowledge, it must have a sufficient ground or reason, in which case it receives the predicate true.
Third Form: The Principle of Sufficient Reason of Being (principium rationis sufficientis essendi); the law whereby the parts of space and time determine one another as regards those relations. Example in arithmetic: Each number presupposes the preceding numbers as grounds or reasons of its being; "I can reach ten only by going through all the preceding numbers; and only by virtue of this insight into the ground of being, do I know that where there are ten, so are there eight, six, four."
“Now just as the subjective correlative to the first class of representations is the understanding, that to the second the faculty of reason, and that to the third pure sensibility, so is the subjective correlative to this fourth class found to be the inner sense, or generally self-consciousness.”
Fourth Form: The Principle of Sufficient Reason of Acting (principium rationis sufficientis agendi); briefly known as the law of motivation. “Any judgment that does not follow its previously existing ground or reason” or any state that cannot be explained away as falling under the three previous headings “must be produced by an act of will which has a motive.” As his proposition in 43 states, “Motivation is causality seen from within.”
See also 
- Deterministic system (philosophy)
- Law of thought
- Gödel's incompleteness theorems
- Nothing comes from nothing
- Principle of Insufficient Reason
- There are numerous anticipations. One often pointed to is in Anselm of Canterbury: his phrase quia Deus nihil sine ratione facit and the formulation of the ontological argument for the existence of God. A clearer connection is with the cosmological argument for the existence of God. The principle can be seen in both Thomas Aquinas and William of Ockham. Leibniz formulated it, but was not an originator. See chapter on Leibniz and Spinoza in A. O. Lovejoy, The Great Chain of Being.
- "Principle of Sufficient Reason".
- Freeman, Charles (1999). The Greek Achievement: The Foundation of the Western World. Allen Lane. p. 152. ISBN 0-7139-9224-7.
- Della Rocca, Michael (2008). Spinoza. New York: Routledge. pp. 8–9. ISBN 0415283302..
- Alexander R. Pruss (2007) "Ex Nihilo Nihil Fit: Augments new and old for the Principle of Sufficant Reason" in Explication Topic in Contemporary Philosophy Ch. 14
- Arthur Schopenhauer, On The Fourfold Root of the Principle of Sufficient Reason, S 20, trans. E. Payne, (Open Court Publishing Company, 1997), 4.
- Arthur Schopenhauer, On The Fourfold Root of the Principle of Sufficient Reason, S 29, trans. E. Payne, (Open Court Publishing Company, 1997), 5.
- Arthur Schopenhauer, On The Fourfold Root of the Principle of Sufficient Reason, S 36, trans. E. Payne, (Open Court Publishing Company, 1997), 6.
- Arthur Schopenhauer, On The Fourfold Root of the Principle of Sufficient Reason, S 38, trans. E. Payne, (Open Court Publishing Company, 1997), 7.
- Arthur Schopenhauer, On The Fourfold Root of the Principle of Sufficient Reason, page 212, S 42, trans. E. Payne, (Open Court Publishing Company, 1997), 8.
- Arthur Schopenhauer, On The Fourfold Root of the Principle of Sufficient Reason, S 43, trans. E. Payne, (Open Court Publishing Company, 1997), 9.
- Arthur Schopenhauer, On The Fourfold Root of the Principle of Sufficient Reason, S 43, trans. E. Payne, (Open Court Publishing Company, 1997), 10.
- Principle of Sufficient Reason entry by Yitzhak Melamed, Martin Lin in the Stanford Encyclopedia of Philosophy
- The Principle of Sufficient Reason: A Reassessment by Alexander R. Pruss