Buridan's ass#Application to digital logic: metastability

The paradox predates Buridan; it dates to antiquity, being found in Aristotle's On the Heavens.{{cite book

| last = Rescher

| first = Nicholas

| title = Cosmos and Logos: Studies in Greek Philosophy

| publisher = Walter de Gruyter

| date = 2013

| pages = 93–99

| url = https://books.google.com/books?id=d-erjyNt2iAC

| isbn = 978-3937202655

}} Aristotle, in ridiculing the Sophist Anaximander's idea that the Earth is stationary simply because it is spherical and any forces on it must be equal in all directions,{{rp|p.67}} says that is as ridiculous as saying that

{{Quotation|...a man, being just as hungry as thirsty, and placed in between food and drink, must necessarily remain where he is and starve to death.|Aristotle, On the Heavens 295b, c. 350 BC }}

However, the Greeks only used this paradox as an analogy in the context of the equilibrium of physical forces. The 12th-century Persian scholar and philosopher Al-Ghazali discusses the application of this paradox to human decision making, asking whether it is possible to make a choice between equally good courses without grounds for preference. He takes the attitude that free will can break the stalemate.

{{Quotation|Suppose two similar dates in front of a man, who has a strong desire for them but who is unable to take them both. Surely he will take one of them, through a quality in him, the nature of which is to differentiate between two similar things.|Abu Hamid al-Ghazali, The Incoherence of the Philosophers, c. 1100{{cite book |last=Kane |first=Robert |title=A Contemporary Introduction to Free Will |year=2005 |publisher=Oxford |location=New York|pages=37}}}}

Andalusian philosopher Averroes (1126–1198), in commentary on Ghazali, takes the opposite view.

Although Buridan nowhere discusses this specific problem,{{rp|p.67}} its relevance is that he did advocate a moral determinism whereby, save for ignorance or impediment, a human faced by alternative courses of action must always choose the greater good. In the face of equally good alternatives Buridan believed a rational choice could not be made.{{cite book | last = Kinniment | first = David J. | title = Synchronization and Arbitration in Digital Systems | publisher = John Wiley & Sons | date = 2008 | pages = 3 | url = https://books.google.com/books?id=_1p-BA4yriIC&pg=PA3

| isbn = 978-0470517130}}{{cite web

| last = Zupko

| first = Frank

| title = John Buridan

| work = Stanford Encyclopedia of Philosophy

| website =

| publisher = Dept. of Philosophy, Stanford University

| date = 2024

| url = https://plato.stanford.edu/entries/buridan/

| format =

| doi =

| issn = 1095-5054

| accessdate = 21 May 2025}}

{{Quotation|Should two courses be judged equal, then the will cannot break the deadlock, all it can do is to suspend judgement until the circumstances change, and the right course of action is clear.|Jean Buridan, c. 1340}}

Later writers satirised this view in terms of an ass which, confronted by both food and water, must necessarily die of both hunger and thirst while pondering a decision.

Many later philosophers have addressed this problem of "choice without preference". In his Ethics ({{circa|1661}}), Baruch de Spinoza accepts that his determinist philosophy implies that such a state of indecision could happen, but that this should be classed with other irrational behavior:Clemens, Justin, "Spinoza's Ass", in {{cite book

| last = Vardoulakis

| first = Demitris

| title = Spinoza Now

| publisher = University of Minnesota Press

| date = 2011

| location =

| pages =

| language =

| url = https://www.google.com/books/edition/Spinoza_Now/naBlCKjmKfsC?hl=en&gbpv=1&bsq=%22buridan%27s%20ass%22%20paradox

| archive-url=

| archive-date=

| doi =

| id =

| isbn = 9780816672806

| mr =

| zbl =

| jfm =}}{{rp|p.71}} {{Quotation|[I]t may be objected, if man does not act from free will, what will happen if the incentives to action are equally balanced, as in the case of Buridan's ass? [In reply,] I am quite ready to admit, that a man placed in the equilibrium described (namely, as perceiving nothing but hunger and thirst, a certain food and a certain drink, each equally distant from him) would die of hunger and thirst. If I am asked, whether such a one should not rather be considered an ass than a man; I answer, that I do not know, neither do I know how a man should be considered, who hangs himself, or how we should consider children, fools, madmen, &c.|Baruch Spinoza|Ethics, book 2, proposition 49, scholium}}

Pierre Bayle (1647–1706) in his Dictionnaire authored some of the most comprehensive discussions of the problem. He recognized explicitly that if humans have the ability to make a decision between choices with no reason for preference, this means that humans have free will, and bears on man's rationalization of God's choices. He regarded the previous examples of the ass or dates as artificial, and pointed out there are many real instances in everyday life in which a person must make a choice in which the choice doesn't matter to him, and that this presents no problem.

Gottfried Wilhelm Leibniz (1646–1716) believed, from his principle of sufficient reason, that if a person's preferences were really balanced no decision could be made, but that petites perceptions, small perceived preferences below the threshold of consciousness that are always present, explains why people are able to make such a choice.

Some proponents of hard determinism such as Spinoza have granted the unpleasantness of the scenario, but have denied that it illustrates a true paradox, since one does not contradict oneself in suggesting that a man might die between two equally plausible routes of action.

Other writers{{who|date=September 2014}} have opted to deny the validity of the illustration. A typical{{citation needed|date=September 2014}} counter-argument is that rationality as described in the paradox is so limited as to be a straw man version of the real thing, which does allow the consideration of meta-arguments. In other words, it is entirely rational to recognize that both choices are equally good and arbitrarily (randomly) pick one instead of starving; although the decision that they are sufficiently the same is also subject to Buridan's ass. The idea that a random decision could be made is sometimes used as an attempted justification for faith or intuitivity (called by Aristotle noetic or noesis). The argument is that, like the starving ass, we must make a choice to avoid being frozen in endless doubt. Other counter-arguments exist.{{specify|date=August 2014}}

According to Edward Lauzinger, Buridan's ass fails to incorporate the latent biases that humans always bring with them when making decisions."Thought and Process", Lauzinger, Edward, 1994{{full|date=July 2013}}

Social psychologist Kurt Lewin's field theory treated this paradox experimentally.{{Cite journal|url=https://doi.org/10.1007/BF03393765|title=On the Origin of Conflict Types|first=Noel W.|last=Smith|date=Apr 1, 1968|journal=The Psychological Record|volume=18|issue=2|pages=229–232|accessdate=May 15, 2023|via=Springer Link|doi=10.1007/BF03393765|url-access=subscription}} He demonstrated that lab rats experience difficulty when choosing between two equally attractive (approach–approach) goals. The typical response to approach–approach decisions is initial ambivalence, though the decision becomes more decisive as the organism moves towards one choice and away from another.

The situation of Buridan's ass was given a mathematical basis in a 1984 paper by American computer scientist Leslie Lamport, in which Lamport presents an argument that, given certain assumptions about continuity in a simple mathematical model of the Buridan's ass problem, there is always some starting condition under which the ass starves to death, no matter what strategy it takes.{{cite journal

| last = Lamport

| first = Leslie

| author-link =

| title = Buridan's principle

| journal = Foundations of Physics

| volume = 42

| issue = 8

| pages = 1056-1066

| publisher = Springer

| location =

| date = 2012

| language =

| url = https://link.springer.com/article/10.1007/s10701-012-9647-7

| jstor =

| issn =

| doi = 10.1007/s10701-012-9647-7

| id =

| mr =

| zbl =

| jfm =

| access-date = 21 May 2025}} Original December 1984 version published by Digital Equipment Corp [http://research.microsoft.com/users/lamport/pubs/buridan.pdf archived here] He further illustrates the paradox with the example of a driver stopped at a railroad crossing trying to decide whether he has time to cross before a train arrives. He proves that regardless of how "safe" the policy the driver adopts, because indecision can cause an indefinite delay in action a small percentage of drivers will be hit by the train.

Lamport calls this result "Buridan’s principle":

:A discrete decision based upon an input having a continuous range of values cannot be made within a bounded length of time.

He points out that just because we do not see asses or people starving to death through indecision, or other examples of Buridan's undecided states in real life, does not disprove the principle. The persistence of a Buridan's undecided state for a perceptible length of time may just be sufficiently improbable that it has not been observed.

{{main|Metastability in electronics}}

A version of Buridan's principle occurs in electrical engineering.{{cite book

| last1 = Xanthopoulos

| first1 = Thucydides

| title = Clocking in Modern VLSI Systems

| publisher = Springer Science & Business Media

| date = 2009

| pages = 191

| url = https://books.google.com/books?id=UZZGAAAAQBAJ&q=buridan&pg=PA191

| isbn = 978-1441902610

}}{{cite book

| last1 = Niederman

| first1 = Derrick

| title = The Puzzler's Dilemma

| publisher = Penguin

| date = 2012

| pages = 130

| url = https://books.google.com/books?id=RvhGaqZ_oNUC&q=%22buridan%27s+ass%22&pg=PT130

| isbn = 978-1101560877

}}{{cite book

| last1 = Zbilut

| first1 = Joseph P.

| title = Unstable Singularities and Randomness

| publisher = Elsevier

| date = 2004

| pages = 7

| url = https://books.google.com/books?id=YVid9QybwbUC&q=%22buridan%27s+ass%22+computer&pg=PA7

| isbn = 0080474691

}}{{cite book

| last1 = Kinniment

| first1 = David J.

| title = Synchronization and Arbitration in Digital Systems

| publisher = John Wiley & Sons

| date = 2008

| pages = 2–6

| url = https://books.google.com/books?id=_1p-BA4yriIC&q=%22buridan%27s+ass%22&pg=PA2

| isbn = 978-0470517130

}} Specifically, the input to a digital logic gate must convert a continuous voltage value into either a 0 or a 1, which is typically sampled and then processed. If the input is changing and at an intermediate value when sampled, the input stage acts like a comparator. The voltage value can then be likened to the position of the ass, and the values 0 and 1 represent the bales of hay. As in the situation of the starving ass, there exists an input on which the converter cannot make a proper decision, and the output remains balanced in a metastable state between the two stable states for an undetermined length of time, until random noise in the circuit makes it converge to one of the stable states.

The metastability problem is a significant issue in digital circuit design, and metastable states are a possibility wherever asynchronous inputs (digital signals not synchronized to a clock signal) occur. The ultimate reason the problem is manageable is that the probability of a metastable state persisting longer than a given time interval t is an exponentially declining function of t. In electronic devices, the probability of such an "undecided" state lasting longer than a matter of nanoseconds, while always possible, can be made negligibly low.

In asynchronous system design, arbiter circuits have been developed to make such decisions. They can reduce to insignificance the probability that a metastable state occurs, at the cost of increased decision time.{{cite journal

|last1 =Sutherland

|first1 =Ivan E.

|author-link1 =Ivan Sutherland

|last2 =Ebergen

|first2 =Jo

|title =Computers Without Clocks

|journal =Scientific American

|volume =287

|issue =2

|date =August 2002

|pages =62–69

|doi =10.1038/scientificamerican0802-62

|pmid =12140955

|bibcode =2002SciAm.287b..62S

}}

{{inpopularculture|date=May 2023}}

• Buridan's Donkey, a 1932 French comedy film, is named after the paradox.
• Lewis Cass, the Democratic candidate for the U.S. presidency in 1848, was contrasted with Buridan's ass by Abraham Lincoln: "Mr. Speaker, we have all heard of the animal standing in doubt between two stacks of hay, and starving to death. The like would never happen to General Cass; place the stacks a thousand miles apart, he would stand stock still midway between them, and eat them both at once, and the green grass along the line would be apt to suffer some too at the same time."Carl Sandburg (1954), Abraham Lincoln: The Prairie Years, 1960 reprint, New York: Dell, Vol. 1, Ch. 7, "Congressman Lincoln", p. 173. (This being a reference to Cass's support for "popular sovereignty" in the run-up to the Civil War.)
• In the Doctor Who novel The Eight Doctors, the Fifth and Eighth Doctors are confronted by a Raston Warrior Robot. The Doctors stand exactly the same distance away from the Robot as it approaches them; unable to decide which to attack first (since the Robot attacks by sensing brain patterns, which are identical in the two Doctors), the Robot shuts down.
• In episode 7 of the 10th season of The Big Bang Theory, Sheldon and Amy discuss the history of Buridan's ass (renamed donkey), and its application to their lives. Amy resolves the paradox (of Sheldon desiring to live in different apartments) by creating a more desirable option by engaging Sheldon in a discussion of the theory and its history.

{{Div col|colwidth=25em}}

• Analysis paralysis
• Catch-22
• Dining philosophers problem
• Fredkin's paradox
• Hobson's choice
• Lagrangian point
• Morton's fork
• {{slink|Principle of sufficient reason|Leibniz's view}}
• Search cost
• Spontaneous symmetry breaking

{{div col end}}

{{reflist}}

• {{cite book |title=The Columbia Encyclopedia |year=2006 |edition=6th }}
• {{cite book |author=Knowles, Elizabeth |title=The Oxford Dictionary of Phrase and Fable |year=2006 }}
• {{cite book |author=Mawson, T.J. |title=Belief in God |publisher=Oxford University (Clarendon) Press |location=New York, NY |year=2005 |page=201 }}
• {{cite journal |author=Rescher, Nicholas |title=Choice Without Preference: A Study of the History and of the Logic of the Problem of 'Buridan's Ass' |journal=Kant-Studien |volume=51 |issue=1–4 |pages=142–75 |year=1959 |doi=10.1515/kant.1960.51.1-4.142 |s2cid=171037127 }}
• {{cite book |author=Zupko, Jack |title=John Buridan: Portrait of a Fourteenth-Century Arts Master |publisher=University of Notre Dame Press |location=Notre Dame, Indiana |year=2003 |pages=258, 400n71 }}
• {{cite journal |first1=E. |last1=Ullmann-Margalit |first2=S. |last2=Morgenbesser |title=Picking and Choos-ing |journal=Social Research |volume=44 |year=1977 |pages=757–785 }}

{{Commonscatinline}}

• {{cite journal |author=Vassiliy Lubchenko |title=Competing interactions create functionality through frustration |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=105 |issue=31 |pages=10635–6 |date=August 2008 |doi=10.1073/pnas.0805716105 |pmid=18669666 |pmc=2504771|bibcode=2008PNAS..10510635L |doi-access=free }}

{{Decision theory paradoxes}}

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