# On the axiomatic behavior of the principle of sufficient reason

Have a look at the most controversial principle popularized as the principle of sufficient reason (PSR):

I mean that the concept of PSR, which has been introduced by Gottfried Wilhelm Leibniz, remains as an axiom. Otherwise, including the question I am asking, how does the view of reason-seeking become a necessary attempt ?

In this way, Can I make a conclusion that the 'everything must have a cause' is not a rational necessity? Or is it an appeal to intution? Any philosophical or logical clarification would be greatly appreciated. Thanks in advance.

• 'behavior' = 'status' ? Nov 1, 2022 at 11:49
• Leibniz may have written the most famous formalization of the principle, but very similar principles were used for centuries before his time. I think the consensus today is that it is literally false because there are quantum events that have no sufficient reason (that is, there is no physical cause for certain events to come out one way instead of another), but that it is a useful working rule at the macroscopic scale. Nov 1, 2022 at 14:44
• @DavidGudeman I think the quantum nature of such events mainly due to probabilistic position of electrons. Can it be ever probabilistic? Or there is a 'chance' to make it predictable like the position of a stone thrown up using the Newton's laws of motion. Nov 2, 2022 at 1:07
• Consider a pile of radioactive material at a certain time. The theory is that although you can give a probability function for how long it will be until the next nucleus fissions, you can't predict which nucleus it will be. The theory is that there is no physical variable that determines in advance which will be the next nucleus to fission. There is no reason why it was that nucleus rather than another one. Nov 2, 2022 at 3:06
• @DavidGudeman There are deterministic versions of quantum mechanics. Dec 24, 2022 at 3:34

## 2 Answers

On the axiomatic behavior of the principle of sufficient reason

It looks like your question asks for some proof that PSR is not simply an axiom, but is provably true. If my assumption is incorrect, let me know in the comments.

Here is part of my attempt to prove the validity of inductive reasoning by showing the truth of the uniformity principle. This is the section on continuity, which is the PSR. The full paper is “Induction and the Uniformity Principle; derivation of the principle from fundamental axioms,” available at academia.edu.

Assume there has been an observation of “Event P and Event Q”; here, Event P, Event Q, and the relation between them each will not change without a sufficient reason. The relation between Event P and Event Q might be that of cause and effect, or there might be no relation at all beyond randomness. But whatever that relation might be, it will remain constant until acted upon by a third event. If there is nothing to alter the relationship between two events, that relationship will continue uninterrupted; the relationship will be uniform over time. In this sense the Principle of Sufficient Reason is similar to Isaac Newton’s First Law of Motion.

The Principle of Sufficient Reason is simply stated: “For every fact F, there must be an explanation why F is the case” (Melamed and Lin 2016, §1). The principle is most closely associated with Gottfried Wilhelm Leibniz (Melamed and Lin 2016, §3), although forms of the principle first appeared in antiquity (Melamed and Lin 2016, §4).

Leibniz joined the principles of noncontradiction and sufficient reason:

1. Our reasonings are grounded upon two great principles, that of contradiction, in virtue of which we judge false that which involves a contradiction, and true that which is opposed or contradictory to the false;
1. And that of sufficient reason, in virtue of which we hold that there can be no fact real or existing, no statement true, unless there be a sufficient reason, why it should be so and not otherwise, although these reasons usually cannot be known by us. (Leibniz, cites omitted; Melamed and Lin 2016, §3)

The relation of “P thus Q” is so, and not otherwise, because there is no intervening event making it otherwise.

David Hume denies the validity of the Principle of Sufficient Reason. He considers several arguments in its support, including those from Thomas Hobbes and John Locke (Hume, THN, I,3, 3). Although this principle is said to be “impossible for men in their hearts really to doubt”, Hume finds “no mark of any such intuitive certainty” (Hume, THN, I, 3, 3)

[A]s all distinct ideas are separable from each other, and as the ideas of causeand effect are evidently distinct, it will be easy for us to conceive any event to be non-existent this moment, and existent the next, without conjoining to it the distinct idea of acause or productive principle. (Hume, THN, I, 3, 3)

The assumption that an event can exist one moment and be non-existent the next, without any reason for such a change, destroys any possibility of continuity; a discontinuous result may always be conceived. “[A]ll distinct ideas are separable from each other”, says Hume. Thought would become impossible if this statement were true. The Principle of Sufficient Reason enables a connection between events over time. Without this axiom, each event stands isolated at one instant. For example, the equation “7 + 5 = 12” ceases to have a meaning. The number 12 bears no relation to “7 + 5”, because 7 and 5 are not sufficient reasons for 12 to exist.

If the Principle of Sufficient Reason is false, then there is no content even to the “distinct ideas” themselves. The number 5 is not the result of 1+1+1+1+1, because the series of 1’s has no further meaning beyond a series of unrelated 1’s. When the Principle of Sufficient Reason is denied, such conclusions become rational.

Suppose research shows that three A's, when combined, produce B. But later research cannot reproduce this result. Instead, new research shows that no combination of A's can ever produce B. Thus the first conclusion appears wrong. But assume that events and relations can change for no reason. If relationships are nonuniform, or if the presence of uniformity is unknown, then both conclusions are meaningless. In the absence of some assumption that present and future events will be similar in similar circumstances, the two results cannot be compared. Rather, the conclusions, nominally contradictory, become two unrelated statements.

Thus the Principle of Sufficient Reason is valid because its denial produces absurd results.

Sources: Hume, David. A treatise of human nature (“Hume THN”). http://www.gutenberg.org/files/4705/4705-h/4705-h.htm#link2H_4_0023

Melamed, Yitzhak and Lin, Martin, "Principle of Sufficient Reason", The Stanford Encyclopedia of Philosophy (Spring 2016 Edition), ed. Edward N. Zalta. URL accessed 15 June 2016, 5 December 2019. http://plato.stanford.edu/entries/sufficient-reason/.

• Your argument seems to assume that if PSR is not true, then no change has a cause, but that is going too far. It is possible that PSR is true for some classes of events but not for others, Nov 2, 2022 at 0:15
• @DavidGudeman. That’s an interesting thought. But for what classes of events would PSR not be true? Nov 2, 2022 at 0:19
• I think most physicists and philosophers today would say it is not true for quantum events. There are quantum events which only have a probability of occurring--there is no deterministic prior state which leads to the following state. For PSR to be true, there would have to be hidden variables which make the events deterministic, and it is the general consensus that such variables do not exist. Nov 2, 2022 at 0:24
• @DavidGudeman It seems 'reasonable' to me that quantum events could have no specific prior cause from our point of view. It is like searching for the shoreline, and if you zoom in closer, when you see grains of sand sticking out of the water, it is no longer a continuous edge. Reality has to come into existence somehow (moment to moment). Anything else would be question-begging. Things can only be explained in terms of what they are not already. Causal streams have to be explained acausally. Yes? Maybe logic can only be explained alogically. Nov 2, 2022 at 10:34

The principle of sufficient reason is really just the principle that humans believe something if they have sufficient reason to believe it.

We will only stop believing it if we find a sufficient reason to instead disbelieve it.

This is what it means to be reasonable, and this is therefore what reasonable people do, in particular reasonable scientists.

This is reasonable because this is how logic works.

So for example, A and A → B together are sufficient reason to believe B:

A ∧ (A → B) ⊢ B

But if you now drop A → B, say you just realised that A → B is in fact false, i.e., ¬(A → B) is true, then A and not A → B together are not enough and you no longer have a sufficient reason to believe B:

A ∧ ¬(A → B) ⊬ B

This does not mean that if you have sufficient reason to believe what you believe, then what you believe is true. It just means that it is reasonable to believe it.

We may choose to be unreasonable, but this is what it means to be reasonable.

To be unreasonable is literally to believe something without a sufficient reason. Sounds indeed unreasonable to me.

Is that even a little bit controversial? No.

EDIT

I have to complement my answer with a bit of trivia about the principle of sufficient reason. So here is Leibniz's own understanding of it, in his own words:

Our reasonings are grounded upon two great principles, that of contradiction, in virtue of which we judge false that which involves a contradiction, and true that which is opposed or contradictory to the false; And that of sufficient reason, in virtue of which we hold that there can be no fact real or existing, no statement true, unless there be a sufficient reason, why it should be so and not otherwise, although these reasons usually cannot be known by us. — Leibniz, Monadology

This at least is Leibniz's understanding of the principle. The question presents the principle as having been introduced by Gottfried Wilhelm Leibniz. The principle in fact predates Leibniz but it is usually understood as the idea that there must be a sufficient explanation for things being what and as they are.

Consequently, we can say that the notion of reason in "sufficient reason" is the ordinary notion of reason as defined in standard dictionaries.

Given this, the principle of sufficient reason cannot possibly be understood as a physical law.

All this to come to one comment to this answer:

This is not what the principle of sufficient reason is. The principle of reason is not a law of logic or human reasoning; it is intended to be a physical law from which one can derive other physical laws such as the law of conservation of energy or Newton's law that an object will continue at rest or in motion in a straight line unless acted on by an outside force. – David Gudeman

The principle of reason (...) is intended to be a physical law.

I will let the reader appreciate which is which.

• Someone could also be 'unreasonable' by not accepting something when they do have sufficient reason. I think we see that fairly often. Nov 1, 2022 at 14:31
• This is not what the principle of sufficient reason is. The principle of reason is not a law of logic or human reasoning; it is intended to be a physical law from which one can derive other physical laws such as the law of conservation of energy or Newton's law that an object will continue at rest or in motion in a straight line unless acted on by an outside force. Nov 1, 2022 at 14:48
• @MessiLio "A bit circular?" You assume that people reason using logic alone. False assumption. Reasoning also requires premises and premises are not logic. 2. "what is the sufficient reason" My own reason is what I understand of how humans reason. Nov 1, 2022 at 17:33
• @ScottRowe "by not accepting something when they do have sufficient reason" How would you know? More likely, they would have unbeknownst to you another reason motivating their rejection of whatever seems obvious to you. Nov 1, 2022 at 17:38
• Perhaps the OP has some reason unbeknownst to you for rejecting your answer? How would we ever know? How would we ever agree? Thus the history of 3 millennia of philosophy. And politics. And science. And... This is why I declined to take more philosophy courses in college: it doesn't converge because people are not reasonable :-) Nov 1, 2022 at 23:51