Tag Archives: Paradox

The Stoic theory of universals, as compared to Platonic and Aristotelian theories

By Tim Harding

The philosophical problem of universals has endured since ancient times, and can have metaphysical or epistemic connotations, depending upon the philosopher in question.  I intend to show in this essay that both Plato’s and the Stoics’ theories of universals were not only derived from, but were ‘in the grip’ of their epistemological and metaphysical philosophies respectively; and were thus vulnerable to methodological criticism.  I propose to first outline the three alternative theories of Plato, Aristotle and the Stoics; and then to suggest that Aristotle’s theory, whilst developed as a criticism of Plato’s theory, stands more robustly on its own merits.

According to the Oxford Companion to Philosophy, particulars are instances of universals, as a particular apple is an instance of the universal known as ‘apple’.  (An implication of a particular is that it can only be in one place at any one time, which presents a kind of paradox that will be discussed later in this essay).   Even the definition of the ‘problem of universals’ is somewhat disputed by philosophers, but the problem generally is about whether universals exist, and if so what is their nature and relationship to particulars (Honderich 1995: 646, 887).

Philosophers such as Plato and Aristotle who hold that universals exist are known as ‘realists’, although they have differences about the ontological relationships between universals and particulars, as discussed in this essay.  Those who deny the existence of universals are known as ‘nominalists’.  According to Long and Sedley (1987:181), the Stoics were a type of nominalist known as ‘conceptualists’, as I shall discuss later.

Plato’s theory of universals (although he does not actually use this term) stems from his theory of knowledge.  Indeed, it is difficult to separate Plato’s ontology from his epistemology (Copleston 1962: 142).  In his Socratic dialogue Timaeus, Plato draws a distinction between permanent knowledge gained by reason and temporary opinion gained from the senses.

That which is apprehended by intelligence and reason is always in the same state; but that which is conceived by opinion with the help of sensation and without reason, is always in a process of becoming and perishing and never really is (Plato Timaeus 28a).

According to Copleston (1962: 143-146), this argument is part of Plato’s challenge to Protagoras’ theory that knowledge is sense-perception.  Plato argues that sense-perception on its own is not knowledge.  Truth is derived from the mind’s reflection and judgement, rather than from bare sensations.  To give an example of what Plato means, we may have a bare sensation of two white surfaces, but in order to judge the similarity of the two sensations, the mind’s activity is required.

Plato argues that true knowledge must be infallible, unchanging and of what is real, rather than merely of what is perceived.  He thinks that the individual objects of sense-perception, or particulars, cannot meet the criteria for knowledge because they are always in a state of flux and indefinite in number (Copleston 1962: 149).  So what knowledge does meet Plato’s criteria?  The answer to this question leads us to the category of universals.  Copleston gives the example of the judgement ‘The Athenian Constitution is good’.  The Constitution itself is open to change, for better or worse, but what is stable in this judgement is the universal quality of goodness.  Hence, within Plato’s epistemological framework, true knowledge is knowledge of the universal rather than the particular (Copleston 1962: 150).

We now proceed from Plato’s epistemology to his ontology of universals and particulars.  In terms of his third criterion of true knowledge being what is real rather than perceived, the essence of Plato’s Forms is that each true universal concept corresponds to an objective reality (Copleston 1962: 151).  The universal is what is real, and particulars are copies or instances of the Form.  For example, particulars such as beautiful things are instances of the universal or Form of Beauty.

…nothing makes a thing beautiful but the presence and participation of beauty in whatever way or manner obtained; for as to the manner I am uncertain, but I stoutly contend that by beauty all beautiful things become beautiful (Plato Phaedo, 653).

Baltzly (2106: F5.2-6) puts the general structure of Plato’s argument this way:

What we understand when we understand what justice, beauty, or generally F-ness are, doesn’t ever change.

But the sensible F particulars that exhibit these features are always changing.

So there must be a non-sensible universal – the Form of F-ness – that we understand when we achieve episteme (true knowledge).

Plato’s explanation for where this knowledge of Forms comes from, if not from sense-perceptions, is our existence as unembodied souls prior to this life (Baltzly 2106: F5.2-6).  To me, this explanation sounds like a ‘retrofit’ to solve a consequential problem with Plato’s theory and is a methodological weakness of his account.

Turning now to Aristotle’s theory, whilst he shared Plato’s realism about the existence of universals, he had some fundamental differences about their ontological relationship to particulars.  In terms of Baltzly’s abovementioned description of Plato’s general argument, Plato thought that the universal, F-ness, could exist even if there were no F particulars.  In direct contrast, Aristotle held that there cannot be a universal, F-ness, unless there are some particulars that are F.  For example, Aristotle thought that the existence of the universal ‘humanity’ depends on there being actual instances of particular human beings (Baltzly 2106: F5.2-8).

As for the reality of universals, Aristotle agreed with Plato that the universal is the object of science.  For instance, the scientist is not concerned with discovering knowledge about particular pieces of gold, but with the essence or properties of gold as a universal.  It follows that if the universal is not real, if it has no objective reality, there is no scientific knowledge.  By Modus Tollens, there is scientific knowledge, and if scientific knowledge is knowledge of reality; then to be consistent, the universal must also be real (Copleston 1962: 301-302).  (Whilst it is outside the scope of this essay to discuss whether scientific knowledge describes reality, to deny that there is any scientific knowledge would have major implications for epistemic coherence).

This is not to say that universals have ‘substance’, meaning that they consist of matter and form.  Aristotle maintains that only particulars have substance, and that universals exist as properties of particulars (Russell 1961: 176).  Russell quotes Aristotle as saying:

It seems impossible that any universal term should be the name of a substance. For…the substance of each thing is that which is peculiar to it, which does not belong to anything else; but the universal is common, since that is called universal which is such as to belong to more than one thing.

In other words, Aristotle thinks that a universal cannot exist by itself, but only in particular things.  Russell attempts to illustrate Aristotle’s position using a football analogy.  The game of football (a universal) cannot exist without football players (particulars); but the football players would still exist even if they never actually played football (Russell 1961: 176).

In almost complete contrast to both Plato and Aristotle, the Stoics denied the existence of universals, regarding them as concepts or mere figments of the rational mind.  In this way, the Stoics anticipated the conceptualism of the British empirical philosophers, such as Locke (Long and Sedley 1987:181).

The Stoic position is complicated by their being on the one hand materialists, and on the other holding a belief that there are non-existent things which ‘subsist’, such as incorporeal things like time and fictional entities such as a Centaur.  Their ontological hierarchy starts with the notion of a ‘something’, which they thought of as a proper subject of thought and discourse, whether or not it exists.  ‘Somethings’ can be subdivided into material bodies or corporeals, which exist; and incorporeals and things that are neither corporeal or incorporeal such as fictional entities, which subsist (Long and Sedley 1987:163-164).  Long and Sedley (1987:164) provide colourful examples of the distinction between existing and subsisting by saying:

There’s such a thing as a rainbow, and such a character as Mickey Mouse, but they don’t actually exist.

A significant exclusion from the Stoic ontological hierarchy is universals.  Despite the subsistence of a fictional character like Mickey Mouse, the universal man neither exists nor subsists, which is a curious inconsistency.  Stoic universals are dubbed by the neo-Platonist philosopher Simplicius (Long and Sedley 1987:180) as ‘not somethings’:

(2) One must also take into account the usage of the Stoics about generically qualified things—how according to them cases are expressed, how in their school universals are called ‘not-somethings’ and how their ignorance of the fact that not every substance signifies a ‘this Something’ gives rise to the Not-someone sophism, which relies on the form of expression.

Long and Sedley (1987:164) surmise from this analysis that for the Stoics, to be a ‘something’ is to be a particular, whether existent or subsistent.  Stoic ontology is occupied exclusively by particulars without universals.  In this way, universals are relegated to a metaphysical limbo, as far as the Stoics are concerned.  Nevertheless, they recognise the concept of universals as being not just a linguistic convenience but as useful conceptions or ways of thinking.  For this reason, Long and Sedley (1987:181-182) classify the Stoic position on universals as ‘conceptualist’, rather than simply nominalist.  (Nominalists think of universals simply as names for things that particulars have in common).  In a separate paper, Sedley (1985: 89) makes the distinction between nominalism and conceptualism using the following example:

After all the universal man is not identical with my generic thought of man; he is what I am thinking about when I have that thought.

One of the implications of a particular is that it can only be in one place at any one time, which gives rise to what was referred to above by Simplicius as the ‘Not-someone sophism’.  Sedley (1985: 87-88) paraphrases this sophism in the following terms:

If you make the mistake of hypostatizing the universal man into a Platonic abstract individual-if, in other words you regard him as ‘someone’-you will be unable to resist the following evidently  fallacious syllogism.  ‘If someone  is in Athens, he is not in Megara.  But man is in Athens. Therefore man is not in Megara.’ The improper step  here is clearly  the substitution of ‘man’ in the minor premiss for ‘someone’ in the major premiss. But it can be remedied only by the denial that the  universal man  is ‘someone’.  Therefore the universal man is not-someone.

Baltlzly (2016: F5.2-15) makes that point that the same argument would serve to show that time is a not-something, yet the Stoics inconsistently accept that time subsists as an incorporeal something.

I have attempted to show above that Plato and the Stoics are locked into their theories about universals as a result of their prior philosophical positions.  Although to argue otherwise could make them vulnerable to criticisms of inconsistency, they at the same time have methodological weaknesses that place them on shakier ground than Aristotelian realism.  However, I am also of the view that apart from these methodological issues, Aristotelian Realism is substantively a better theory than Platonic Realism or Stoic Conceptualism or Nominalism.  In coming to this view, I have relied mainly on the work of the late Australian Philosophy Professor David Armstrong.

Armstrong argues that there are universals which exist independently of the classifying mind.  No universal is found except as either a property of a particular or as a relation between particulars.  He thus rejects both Platonic Realism and all varieties of Nominalism (Armstrong 1978: xiii).

Armstrong describes Aristotelian Realism as allowing that particulars have properties and that two different particulars may have the very same property.  However, Aristotelian Realism rejects any transcendent account of properties, that is, an account claiming that universals exist separated from particulars (Armstrong 1975: 146).  Armstrong argues that we cannot give an account of universality in terms of particularity, as the various types of Nominalism attempt to do.  Nor can we give an account of particulars in terms universals, as the Platonic Realists do.  He believes that ‘while universality and particularity cannot be reduced to each other, they are interdependent, so that properties are always properties of a particular, and whatever is a particular is a particular having certain properties’ (Armstrong 1975: 146).

According to Armstrong, what is a genuine property of particulars is to be decided by scientific investigation, rather than simply a linguistic or conceptual classification (Armstrong 1975: 149).  Baltzly (2016: F5.2-18) paraphrases Armstrong’s argument this way:

  1. There are causes and effects in nature.

  2. Whether one event c causes another event e is independent of the classifications we make.

  3. Whether c causes e or not depends on the properties had by the things that figure in the events.

  4. So properties are independent of the classifications that we make and if this is so, then predicate nominalism and conceptualism are false.

Baltzly (2016: F5.2-18, 19) provides an illustration of this argument based on one given by Armstrong (1978: 42-43).  The effect of throwing brick against a window will result from the physical properties of the brick and window, in terms of their relative weight and strength, independently of how we name or classify those properties.  So in this way, I would argue that the properties of particulars, that is universals, are ‘real’ rather than merely ‘figments of the mind’ as the Stoics would say.

As for Platonic Realism, Armstrong argues that if we reject it then we must reject the view that there are any uninstantiated properties (Armstrong 1975: 149); that is, the view that properties are transcendent beings that exist apart from their instances, such as in universals rather than particulars.  He provides an illustration of a hypothetical property of travelling faster than the speed of light.  It is a scientific fact that no such property exists, regardless of our concepts about it (Armstrong 1975: 149).  For this reason, Armstrong upholds ‘scientific realism’ over Platonic Realism, which he thinks is consistent with Aristotelian Realism – a position that I support.

In conclusion, I have attempted to show in this essay that the Aristotelian theory of universals is superior to the equivalent theories of both Plato and the Stoics.  I have argued this in terms of the relative methodologies as well as the substantive arguments.  I would choose the most compelling argument to be that of epistemic coherence regarding scientific knowledge, that is, that the universal is the object of science.  It follows that if the universal is not real, if it has no objective reality, then there is no scientific knowledge.  There is scientific knowledge, and if scientific knowledge is knowledge of reality; then to be consistent, the universal must also be real.

Bibliography

Armstrong, D.M. ‘Towards a Theory of Properties: Work in Progress on the Problem of Universals’ Philosophy, (1975), Vol.50 (192), pp.145-155.

Armstrong, D.M. ‘Nominalism and Realism’ Universals and Scientific Realism Volume 1, (1978) Cambridge: Cambridge University Press.

Baltzly, D. ATS3885: Stoic and Epicurean Philosophy Unit Reader (2016). Clayton: Faculty of Arts, Monash University.

Copleston, F. A History of Philosophy Volume 1: Greece and Rome (1962) New York: Doubleday.

Honderich, T. Oxford Companion to Philosophy (1995) Oxford: Oxford University Press.

Long A. A. and Sedley, D. N. The Hellenistic Philosophers, Volume 1 (1987). Cambridge: Cambridge University Press.

Plato, Phaedo in The Essential Plato trans. Benjamin Jowett, Book-of-the-Month Club (1999).

Plato, Timaeus in The Internet Classics Archive. http://classics.mit.edu//Plato/timaeus.html
Viewed 2 October 2016.

Russell, B. History of Western Philosophy. 2nd edition (1961) London: George Allen & Unwin.

Sedley, D. ‘The Stoic Theory of Universals’ The Southern Journal of Philosophy (1985) Vol. XXIII. Supplement.

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Fitch’s paradox of knowability

Fitch’s paradox of knowability is one of the fundamental puzzles of epistemic logic. It provides a challenge to the knowability thesis, which states that every truth is, in principle, knowable. The paradox is that this assumption implies the omniscience principle, which asserts that every truth is known. Essentially, Fitch’s paradox asserts that the existence of an unknown truth is unknowable. So if all truths were knowable, it would follow that all truths are in fact known. The paradox is of concern for verificationist or anti-realist accounts of truth, for which the knowability thesis is very plausible, but the omniscience principle is very implausible.

A formal proof of the paradox is as follows.

Suppose p is a sentence which is an unknown truth; that is, the sentence p is true, but it is not known that p is true. In such a case, the sentence “the sentence p is an unknown truth” is true; and, if all truths are knowable, it should be possible to know that “p is an unknown truth”. But this isn’t possible, because as soon as we know “p is an unknown truth”, we know that p is true, rendering p no longer an unknown truth, so the statement “p is an unknown truth” becomes a falsity. Hence, the statement “p is an unknown truth” cannot be both known and true at the same time. Therefore, if all truths are knowable, the set of “all truths” must not include any of the form “something is an unknown truth”; thus there must be no unknown truths, and thus all truths must be known.

The proof has been used to argue against versions of anti-realism committed to the thesis that all truths are knowable. For clearly there are unknown truths; individually and collectively we are non-omniscient. So, by the main result, it is false that all truths are knowable. The result has also been used to draw more general lessons about the limits of human knowledge. Still others have taken the proof to be fallacious, since it collapses an apparently moderate brand of anti-realism into an obviously implausible and naive idealism.

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Poe’s Law

Poe’s law is a kind of paradox which states that, without a clear indicator of the author’s intent (even just a smiley), it is impossible to tell the difference between a sincere expression of extreme views and a parody of those views.  This is because the actual views and their parody both seem equally irrational or absurd.

Poe’s Law was formulated by the writer Nathan Poe in August 2005. The law emerged at the online Creation & Evolution forum.  Like most such places, it had seen a large number of creationist parody postings. These were usually followed by at least one user starting a flame war (a series of angry and offensive personal attacks) thinking it was a serious post and taking it at face value.

The law caught on and has since slowly become an Internet meme. Over time it has been extended to include not just creationist parody but any parody of extreme ideology, whether it be religious, secular, anti-science, conspiracy theorist or just totally bonkers.  We even find it a problem in the Skeptics in Australia Facebook group, where it is sometimes difficult to tell whether the poster is joking or not.  That’s why we ask people to include their own comment instead of just providing a link.

12079469_1072701489408101_1921845550481321952_n cropped

A ‘Poe’ as a noun has become almost as ubiquitous as Poe’s Law itself. In this context, a Poe refers to either a person, post or news story that could cause Poe’s Law to be invoked. In most cases, this is specifically in the sense of posts and people who are taken as legitimate, but are probably a parody.  The use of the term is most common in online skeptical and science-based communities.  Many blogs, forums and wikis will often refer to the law when dealing with cranks of any stripe.

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The Barbershop Paradox

The Barbershop paradox was proposed by Lewis Carroll in a three-page essay titled “A Logical Paradox,” which appeared in the July 1894 issue of Mind. The name comes from the ‘ornamental’ short story that Carroll uses to illustrate the paradox (although it had appeared several times in more abstract terms in his writing and correspondence before the story was published). Carroll claimed that it illustrated “a very real difficulty in the Theory of Hypotheticals” in use at the time. Modern logicians would not regard it as a paradox but simply as a logical error on the part of Carroll.

Briefly, the story runs as follows: Uncle Joe and Uncle Jim are walking to the barber shop. There are three barbers who live and work in the shop—Allen, Brown, and Carr—but not all of them are always in the shop. Carr is a good barber, and Uncle Jim is keen to be shaved by him. He knows that the shop is open, so at least one of them must be in. He also knows that Allen is a very nervous man, so that he never leaves the shop without Brown going with him. Uncle Joe insists that Carr is certain to be in, and then claims that he can prove it logically. Uncle Jim demands the proof. Uncle Joe reasons as follows.

Suppose that Carr is out. If Carr is out, then if Allen is also out Brown would have to be in, since someone must be in the shop for it to be open. However, we know that whenever Allen goes out he takes Brown with him, and thus we know as a general rule that if Allen is out, Brown is out. So if Carr is out then the statements “if Allen is out then Brown is in” and “if Allen is out then Brown is out” would both be true at the same time.

Uncle Joe notes that this seems paradoxical; the two “hypotheticals” seem “incompatible” with each other. So, by contradiction, Carr must logically be in.

However, the correct conclusion to draw from the incompatibility of the two “hypotheticals” is that what is hypothesised in them (– that Allen is out) must be false under our assumption that Carr is out. Then our logic simply allows us to arrive at the conclusion “If Carr is out, then Allen must necessarily be in”.

In modern logic theory this scenario is not a paradox. The law of implication reconciles what Uncle Joe claims are incompatible hypotheticals. This law states that “if X then Y” is logically identical to “X is false or Y is true” (¬X ∨ Y). For example, given the statement “if you press the button then the light comes on”, it must be true at any given moment that either you have not pressed the button, or the light is on.

In short, what obtains is not that ¬C yields a contradiction, only that it necessitates A, because ¬A is what actually yields the contradiction.

In this scenario, that means Carr doesn’t have to be in, but that if he isn’t in, Allen has to be in.

A more detailed discussion of this apparent paradox may be found on Wikipedia.


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Catch-22

A Catch-22 is a paradoxical situation from which an individual cannot escape because of contradictory rules.

The term originates from satirical novel by the American author Joseph Heller.  In this novel, a World War II US Air Force bombardier called Yossarian was caught in the original Catch-22. Stationed in Italy and afraid of being shot down, he wanted to be declared insane and sent home.

But military rules said that fear of death was a rational response, so anyone who asked to be grounded could not possibly be truly crazy. And those who were insane would not be aware of the fact, and therefore would be unable to ask to be grounded. Therefore, Catch-22 ensures that no airman can ever be grounded for being insane even if he is.

catch22-finaledit-1

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The Paradox of Thrift

The paradox of thrift (or paradox of saving) is a paradox of economics generally attributed to John Maynard Keynes, although it had been stated as early as 1714 in The Fable of the Bees and similar sentiments dating to antiquity.

Keynes argued that consumer spending contributes to the collective good, because one person’s spending is another person’s income.  Thus, when individuals save too much instead of spending, they can cause collective harm because businesses do not earn as much and have to lay off employees who are then unable to save.  The paradox is that total savings may fall even when individual savings attempt to rise.  In this way, individual savings rather than spending can worsen a recession, and therefore be collectively harmful to the economy.

Consider the following example:

thrift boxes

In the above example, one consumer increased his savings by $100, but this cause no net increase in total savings.  Increased savings reduced income for other economic participants, forcing them to cut their savings. In the end, no new savings was generated while $200 income was lost.

This paradox is related to the fallacy of composition, which falsely concludes what is true of the parts must be true of the whole.  It also represents a prisoner’s dilemma, because saving is beneficial to each individual but deleterious to the general population.

The paradox of thrift is a central component of Keynesian economics, and has formed part of mainstream economics since the late 1940s, though it is disputed on a number of grounds by non-Keynesian economists such as Friedrich Hayek.  One of the main arguments against the paradox of thrift is that when people increase savings in a bank, the bank has more money to lend, which will generally decrease interest rates and thus spur lending and spending.

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Determinism vs ethics

This is a paradox about the relationship between determinism and ethics. Simply stated, the paradox is as follows:

Determinism affords no reason for doing anything, and is therefore ethically irrelevant. Therefore it does not render ethics irrelevant, since if it did, it would be ethically relevant.

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Buridan’s Ass

Buridan’s Ass is the name give to an apparent paradox related to the free will paradox; although there is some debate amongst philosophers as to whether it actually is a paradox (see below).

The paradox is named after the French priest and philosopher Jean Buridan (c.1300-1358CE), who studied under William of Ockham.  It refers to a hypothetical situation where a donkey finds itself exactly halfway between two equally big and delicious bales of hay.  There is no way of distinguishing between these two bales – they appear to be identical.  Because the donkey lacks a reason or cause to choose one over the other, it cannot decide which one to eat, and so starves to death. This tale is usually taken as demonstrating that there is no free will.

The corollary to this argument is that if the donkey eats one of the bales of hay, then the donkey is making a choice.  If the donkey is making a choice, then it must have free will, because there is no causal mechanism to make it choose one bale over another. And if donkeys have free will, then so must humans.

Deliberations_of_Congress

Deliberations of Congress (Source: Wikimedia Commons)

The paradox actually predates Buridan – it dates to antiquity, being found in Aristotle’s On the Heavens.[2] Aristotle, in ridiculing the Sophist idea that the Earth is stationary simply because it is circular and any forces on it must be equal in all directions, says that is as ridiculous as saying that:

…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, (c.350 BCE)

The 12th century Persian Islamic 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.

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.1100CE)

Professor Hauskeller of Exeter University takes a scientifically sceptical view of this paradox, using the donkey scenario:

If we could find a donkey which was dumb enough to starve between two piles of hay, we would have evidence against free will, at least as far as donkeys are concerned (or at least that particular donkey). But that’s not very likely. No matter how artfully we arrange the situation, a donkey will not hesitate very long, if at all, and will soon choose one of the piles of hay. He doesn’t care which, and he certainly won’t starve. However, even if we conducted thousands of experiments like this, and no donkey ever starved, we would still not have proved the existence of free will, because the reason no donkey ever starves in front of two equally attractive piles of hay may simply be that those piles aren’t really equally attractive. Perhaps in real life there aren’t any situations where the weighted reasons for a choice are equal.[1]

So Hauskeller’s suggested solution to the paradox is that the piles of hay are not equal in practice – the donkey detects a slight difference which causes it to choose one pile over the other. This solution is not very convincing when one considers the hypothetical possibility of the two piles of hay being exactly equal in appearance. So it seems that we still have a problem here.

Some proponents of hard determinism have acknowledged the difficulty the scenario creates for determinism, but have denied that it illustrates a true paradox, since a deterministic donkey could recognize that both choices are equally good and arbitrarily (randomly) pick one instead of starving. For example, there are deterministic machines that can generate random numbers, although there is some dispute as to whether such numbers are truly random.

References:

[1] Hauskeller, M. (2010) Why Buridan’s Ass Doesn’t Starve Philosophy Now, London. http://philosophynow.org/issues/81/Why_Buridans_Ass_Doesnt_Starve

[2] Rescher, N. (2005). Cosmos and Logos: Studies in Greek Philosophy . Ontos Verlag. pp. 93–99.

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The Liar’s Paradox

In philosophy and logic, the liar’s paradox  is the statement “this sentence is false.” Trying to assign to this statement a classical binary truth value leads to a contradiction (see paradoxes).

If “this sentence is false” is true, then the sentence is false, which is a contradiction. Conversely, if “this sentence is false” is false, then the sentence is true, which is also a contradiction.

This paradox is obliquely related to the City of Lies or Truth puzzle.

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Russell’s paradox

In the foundations of mathematics, Russell’s paradox (also known as Russell’s antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naive set theory created by Georg Cantor led to a contradiction. The same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl, and other members of the University of Göttingen.

According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell’s paradox.

 

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