# Principle of sufficient reason

The principle of sufficient reason states that everything must have a reason or a cause. The modern[1] formulation of the principle is usually attributed to Gottfried Leibniz,[2] although the idea was conceived of and utilized by various philosophers who preceded him, including Anaximander,[3] Parmenides, Archimedes,[4] Plato and Aristotle,[5] Cicero,[5] Avicenna,[6] Thomas Aquinas, and Spinoza.[7] Some philosophers have associated the principle of sufficient reason with "ex nihilo nihil fit".[8][9] Hamilton identified the laws of inference modus ponens with the "law of Sufficient Reason, or of Reason and Consequent" and modus tollens with its contrapositive expression.[10]

## Formulation

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.

It is an open question whether the principle of sufficient reason can be applied to axioms within a logic construction like a mathematical or a physical theory, because 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

Leibniz identified two kinds of truth, necessary and contingent truths. He believed necessary mathematical truths to be derived from the law of identity (and the principle of non-contradiction): "Necessary truths are those that can be demonstrated through an analysis of terms, so that in the end they become identities, just as in Algebra an equation expressing an identity ultimately results from the substitution of values [for variables]. That is, necessary truths depend upon the principle of contradiction."[11] Leibniz states that the sufficient reason for necessary truths is that their negation is a contradiction.[4]

Leibniz admitted contingent truths on the basis of infinitary reasons, to which God had access but humans did not:

In contingent truths, even though the predicate is in the subject, this can never be demonstrated, nor can a proposition ever be reduced to an equality or to an identity, but the resolution proceeds to infinity, God alone seeing, not the end of the resolution, of course, which does not exist, but the connection of the terms or the containment of the predicate in the subject, since he sees whatever is in the series.[12]

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)

Leibniz also used the principle of sufficient reason to refute the idea of absolute space:

I say then, that if space is an absolute being, there would be something for which it would be impossible there should be a sufficient reason. Which is against my axiom. And I prove it thus. Space is something absolutely uniform; and without the things placed in it, one point in space does not absolutely differ in any respect whatsoever from another point of space. Now from hence it follows, (supposing space to be something in itself, beside the order of bodies among themselves,) that 'tis impossible that there should be a reason why God, preserving the same situation of bodies among themselves, should have placed them in space after one particular manner, and not otherwise; why everything was not placed the quite contrary way, for instance, by changing East into West.[13]

## 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 19th centuries. It was influential in the thinking of Leo Tolstoy, amongst others, in the elevated form that history could not be accepted as random.

A sufficient reason is sometimes described as the coincidence of every single thing that is needed for the occurrence of an effect (i.e. of the so-called necessary conditions).[14] Such view could perhaps be also applied to indeterministic systems, as long as randomness is in a way incorporated in the preconditions.

### Hamilton's fourth law: "Infer nothing without ground or reason"

Here is how Hamilton, circa 1837–1838,[15] expressed his "fourth law" in his LECT. V. LOGIC. 60–61:

"I now go on to the fourth law.
"Par. XVII. Law of Sufficient Reason, or of Reason and Consequent:
"XVII. The thinking of an object, as actually characterized by positive or by negative attributes, is not left to the caprice of Understanding – the faculty of thought; but that faculty must be necessitated to this or that determinate act of thinking by a knowledge of something different from, and independent of; the process of thinking itself. This condition of our understanding is expressed by the law, as it is called, of Sufficient Reason (principium Rationis Sufficientis); but it is more properly denominated the law of Reason and Consequent (principium Rationis et Consecutionis). That knowledge by which the mind is necessitated to affirm or posit something else, is called the logical reason ground, or antecedent; that something else which the mind is necessitated to affirm or posit, is called the logical consequent; and the relation between the reason and consequent, is called the logical connection or consequence. This law is expressed in the formula – Infer nothing without a ground or reason.1
"Relations between Reason and Consequent: The relations between Reason and Consequent, when comprehended in a pure thought, are the following:
1. When a reason is explicitly or implicitly given, then there must ¶ exist a consequent; and, vice versa, when a consequent is given, there must also exist a reason.
1 See Schulze, Logik, §19, and Krug, Logik, §20, – ED.[16]
"2. Where there is no reason there can be no consequent; and, vice versa, where there is no consequent (either implicitly or explicitly) there can be no reason. That is, the concepts of reason and of consequent, as reciprocally relative, involve and suppose each other.
"The logical significance of this law: The logical significance of the law of Reason and Consequent lies in this, – That in virtue of it, thought is constituted into a series of acts all indissolubly connected; each necessarily inferring the other. Thus it is that the distinction and opposition of possible, actual and necessary matter, which has been introduced into Logic, is a doctrine wholly extraneous to this science."

## 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.[17]

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.[18]

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.[19] 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."[20]

"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."[21]

Fourth Form: The Principle of Sufficient Reason of Acting (principium rationis sufficientis agendi); briefly known as the law of motivation.[22] "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."[23]

## Proposed proofs of universal validity

Several proofs have been prepared in order to demonstrate that the universe is at bottom causal, i.e. works in accord with the principle in question; perhaps not in every single case (randomness might still play a part here and there), but that causality must be the way it works at least in general, in most of what we see; and that our minds are aware of the principle even before any experience. The two famous arguments or proofs were proposed by Immanuel Kant (from the form of Time, temporal ordering of events and "directionality" of time) and by Arthur Schopenhauer (by demonstrating how all perception depends on causality and the intellect).

Once it is agreed (e.g. from a kind of an "arrow of time") that causal interconnections, as a form of principle of sufficient reason, indeed must in general exist everywhere in the universe (at least in the large scale), backwards causality in general might then be precluded using a form of the paradox of free will (i.e. an event that has a future source might cause us to remove that source quick enough and thus causality would not work[24]).

## References

1. From Hamilton 1860:67" In modern times, the attention of philosophers was called to this law of Leibnitz, who, on the two principles of Reason and of Contradiction, founded the whole edifice of his philosophy.3 3 See Théodicée, § 44. Monadologie, §§ 81, 82. —ED."
2. 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.
3. Freeman, Charles (1999). The Greek Achievement: The Foundation of the Western World. Allen Lane. p. 152. ISBN 0-7139-9224-7.
4. "Principle of Sufficient Reason".
5. Hamilton 1860:66
6. Richardson, Kara (June 2014). "Avicenna and the Principle of Sufficient Reason". The Review of Metaphysics. 67 (4): 743–768.
7. Della Rocca, Michael (2008). Spinoza. New York: Routledge. pp. 8–9. ISBN 0415283302..
8. Alexander R. Pruss (2007) "Ex Nihilo Nihil Fit: Augments new and old for the Principle of Sufficient Reason" in Explication Topic in Contemporary Philosophy Ch. 14
9. Hamilton attributes this expression to Cicero; Hamilton 1860:66
10. From Hamilton 1860:241–242: “2°, "If the essential nature of an Hypothetical Syllogism consist in this, – that the subsumption affirms or denies one or other of the two parts of a thought, standing to each other in the relation of the thing conditioning and the thing conditioned, it will be the law of an hypothetical syllogism, that, – If the condition or antecedent be affirmed, so also must be the conditioned or consequent, and that if the conditioned or consequent be denied, so likewise must be the condition or antecedent. But this is manifestly nothing else than the law of Sufficient Reason, or of Reason and Consequent." 1 The principle of this syllogism is thus variously enounced: Posita conditione, ponitur conditionatum, sublato conditionato, tlitur conditio. Or, otherwise, a ratione ad rationatum, a negatione rationati ad negationem rationis, valet consequentia. The one alternative of either rule being regulative of modus ponens, the other of the modus tollens. 2 1 Esser, Logik, I 91, p. 174. —ED. 2 See Kant, Logik §§ 75–76 . Krug, Logik, § 82. —ED." See in particular Hamilton's discussion that leads to this quote starting at page 239ff.
11. Muhit, Abdul. "Leibniz on Necessary and Contingent Truths". Retrieved 22 April 2014.
12. Ariew, Roger and Daniel Garber, eds and trans, eds. (1989). G. W. Leibniz: Philosophical Essays. Indianapolis: Hackett.CS1 maint: uses editors parameter (link), p. 94, On Freedom (1689?).
13. Alexander, H.G. (1956). The Leibniz-Clarke Correspondence. New York, N.Y.: Barnes and Noble.
14. See e.g. T. Hobbes, Quaestiones de libertate et necessitate, contra Doctorem Bramhallum, 7. Quoted in: A. Schopenhauer, On the Freedom of the Will, c. 4. See also: John Bramhall
15. From the Preface: "The Lectures on Logic, like those on Metaphysics, were chiefly composed during, the session in which they were first delivered (1837–8)." The lectures were assembled, with added footnotes marked by "—ED." by Mansel and Veitch and published in 1860.
16. From the index: "SCHULZE, G. E., KRUG, W. T." These are philosophers Gottlob Ernst Schulze (23 August 1761 – 14 January 1833) and Wilhelm Traugott Krug (22 June 1770 – 12 January 1842).
17. Arthur Schopenhauer, On The Fourfold Root of the Principle of Sufficient Reason, S 20, trans. E. Payne, (Open Court Publishing Company, 1997), 4.
18. Arthur Schopenhauer, On The Fourfold Root of the Principle of Sufficient Reason, S 29, trans. E. Payne, (Open Court Publishing Company, 1997), 5.
19. Arthur Schopenhauer, On The Fourfold Root of the Principle of Sufficient Reason, S 36, trans. E. Payne, (Open Court Publishing Company, 1997), 6.
20. Arthur Schopenhauer, On The Fourfold Root of the Principle of Sufficient Reason, S 38, trans. E. Payne, (Open Court Publishing Company, 1997), 7.
21. 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.
22. Arthur Schopenhauer, On The Fourfold Root of the Principle of Sufficient Reason, S 43, trans. E. Payne, (Open Court Publishing Company, 1997), 9.
23. Arthur Schopenhauer, On The Fourfold Root of the Principle of Sufficient Reason, S 43, trans. E. Payne, (Open Court Publishing Company, 1997), 10.
24. Likewise, announcing prophecies so that they will still be correct requires, in general, a lot of high-level research of human psychics, because sometimes they will be in accord with human determination and will be welcome, but sometimes announcing them without interference with the prophesied outcome is just impossible. The requirement of such high-level research, in every single case, seems in general to rule out the possibility of backwards causality in physics.