Man as ParadoxRandley Kearl L. CuraBSED 3F1

What’s a Paradox?

A statement or proportion that seems self contradictory It makes the situation impossible A self-contradictory and false proportion

Is generally a puzzling conclusion we seem to be driven towards by our reasoning.

One usually means a statement claiming something which goes beyond ( or even against) “common opinion” (what is usually believed or held)

The Liar Paradox is among the simplest of paradoxes. It an be traced back at least as far as Eubulides of Miletus, a fourth-century B.c greek philosopher.

Eubulide’s version of the paradox is this: A man says that he is lying: is what he says true or false.

However we answer this question. Difficulties arise.

If we suggest that what the man says is true, then we end in contradiction: if the man’s claim that he is lying is true, then he is lying, in which case what he says is false.

If we suggest that what the man says is false, then we are no better off: if the man’s claim that he is lying is false, then he is not lying, in which case what he says is true.

Both answers are logical contradictions: it cannot be the case either that what the man says is true or that what the man says is false.

Sentences can be constructed that cannot consistently be assigned a truth value even though they are completely in accord with grammar and semantic rules.

Three Laws of Thought (Aristotle)

An object is what is ( Law of identity) Everything is the same as itself: of a statement cannot remain

the same and change its truth value.

A cannot be both A and B ( Law of contradiction or no Proposition can be both true and not true)

Nothing can both exist and not exist at the same time.

Every statement is either true or false: there is nothing in between( Law of the excluded middle)

Kinds of paradox

DOUBLE LIAR PARADOX (JOURDAIN'S PARADOX) This version of a famous paradox was presented by

English mathematician P. E. B. Jourdain in 1913.The following is written on opposite sides of a card:

Back side:THE SENTENCE ON THE OTHER SIDE OF THIS CARD IS TRUE.

Face side:THE SENTENCE ON THE OTHER SIDE OF THIS CARD IS FALSE.

BARBER PARADOX (RUSSELL'S PARADOX)

Analogue paradox to the 'liar paradox' formulated by English logician, philosopher and mathematician Bertrand Russell.

In a village, the barber shaves everyone who does not shave himself/herself, but no one else.Who shaves the barber?

LAZY-BONES PARADOX

If destiny designed a master plan which defines everything that is to happen, isn't it useless, for example, to go to a doctor? If I am ill and it is my destiny to regain health, then I will regain health whether I visit a doctor or not. If it is my destiny to not regain my health, then seeing a doctor can't help me.How could you question the presented opinion?

The four main paradoxes attributed to Eubelides who lived in the fourth century Bc, were

The LiarThe Hooded ManThe Heap The Horned Man

The Horned Man is a version of the "When did you stop beating your wife?" puzzle. This is not a simple question, and needs a carefully phrased reply, to avoid the inevitable come-back to "I have not." How is one to understand this denial, as saying you continue to beat your wife, or that you once did but do so no longer, or that you never have, and never will? It is a question of what the "not," or negation means, in this case. If "stopped beating" means "beat before, but no longer," then "not stopped beating" covers both "did not beat before" and "continues to beat." And in that case "I haven't" is an entirely correct answer to the question, if you in fact did not beat your wife. However, your audience might still need to be taken slowly through the alternatives before they clearly see this. Likewise with the Horned Man, which arises if someone wants to say, for instance, "what you have not lost you still have." In that case they will maybe have to accept the unwelcome conclusion "I still have horns," if they admit "I have not lost any horns." Here, if "lost" means "had, but do not still have" then "not lost" would cover the alternative "did not have in the first place" as well as "do still have" -- in which case what you have not lost you do not necessarily still have.

The Heap is nowadays commonly referred to as the Sorites Paradox, and concerns the possibility that the borderline between a predicate and its negation need not be finely drawn. We would all say that a man with no hairs on his head was bald, and that a man with, say, 10,000 hairs on his head was hirsute, that is not bald, but what about a man with only 1,000 hairs on his head, which are, say, evenly spread? It is not too clear what we should say, although maybe some would still want to say positively "bald," while others would want to say "not bald." The learned treatment of this issue, in recent years, has been very extensive, with "the lazy solution" not being the only one favoured, by any means. The lazy solution says that any lack of certainty about what to say is merely a matter of us not having yet decided upon, or even having the need to make up our mind about, a "precisification" of the concept of baldness. There are objectors to this "epistemic" way of seeing the matter, some of whom would prefer to think, for instance (see, e.g. Sainsbury 1995), that there was something essentially "fuzzy" about baldness, so it is a "vague predicate" by the nature of things, instead of just through lack of effort, or need. (For recent work in this area, see, for instance, Williamson 1994, and Keefe 2001).

The Hooded Man is about the concept of knowledge, and in other versions has again been much studied in recent years, as we shall see. In its original version the problem is this: maybe you would be prepared to say that you know your brother, yet surely someone might come in, who was in fact your brother, but with his head covered, so you did not know who it was. One aspect of this paradox is that the verb "know" is ambiguous, and in fact is translated by two separate terms in several other languages than English -- French, for instance, has "connâitre" and "savoir." There is the sense of "being acquainted with," in other words, and the sense of "knowing a fact about something." Perhaps these two senses are inter-related, but distinguishing them provides one way out of the Hooded Man. For we can distinguish being acquainted with your brother from knowing that someone is your brother. Although you do not know it, you are certainly acquainted with the hooded man, since he is your brother, and you are acquainted with your brother. But that does not entail that you know that the hooded man is your brother, indeed, evidently you do not. We could also say, in that case, that you did not recognize your brother, for the notion of recognition is close to that of knowledge. And that points to another aspect of the problem, and another way of resolving the paradox -- showing, in addition, that there needn't just be one solution, or way out. Thus you might well be able to recognise your brother, but that does not require you can always do so, it merely means you can do better at this than those people who cannot do so. If we re-phrase the case: "you can recognise your brother, but you did not recognise him when he had his head covered," then there is not really a paradox.

The last of Eubulides' paradoxes mentioned above was The Liar, which is perhaps the most famous paradox in the "self-reference" family. The basic idea had several variations, even in antiquity. There was, for instance, The Cretan, where Epimenides, a Cretan, says that all Cretans are liars, and The Crocodile, where a crocodile has stolen someone's child, and says to him "I will return her to you if you guess correctly whether I will do so or not" -- to which the father says "You will not return my child"! Indeed a whole host of complications of The Liar have been constructed, especially in the last century, as we shall see. Now in The Cretan there is no real antinomy -- it may simply be false that all Cretans are liars; but if someone says just "I am lying," the situation is different. For if it is true that he is lying then seemingly what he says is false; but if it is false that he is lying then what he is saying may seem to be true. A pedant might say that "lying" was strictly not telling an untruth, but telling merely what one believes to be an untruth. In that case there is not the same difficulty with the person's remark being true: maybe he is indeed lying, although he does not believe it. The pedant, however, misses the point that his verbal nicety can be circumvented, and the paradox re-constructed in another

Zeno’s paradox Goals

In the early fifth century B.C.E., Parmenides emphasized the distinction between appearance and reality. Reality, he said, is a seamless unity that is unchanging and can not be destroyed, so appearances of reality are deceptive. Our ordinary observation reports are false; they do not report what is real.

The Paradoxes of PluralityThe Argument from DensenessIf there are many, they must be as many as they are and

neither more nor less than that. But if they are as many as they are, they would be limited. If there are many, things that are are unlimited. For there are always others between the things that are, and again others between those, and so the things that are are unlimited. (Simplicius(a) On Aristotle's Physics, 140.29)

The Argument from Finite Size  if it should be added to something else that exists, it would not

make it any bigger. For if it were of no size and was added, it cannot increase in size. And so it follows immediately that what is added is nothing. But if when it is subtracted, the other thing is no smaller, nor is it increased when it is added, clearly the thing being added or subtracted is nothing. (Simplicius(a) On Aristotle's Physics,139.9)But if it exists, each thing must have some size and thickness, and part of it must be apart from the rest. And the same reasoning holds concerning the part that is in front. For that too will have size and part of it will be in front. Now it is the same thing to say this once and to keep saying it forever. For no such part of it will be last, nor will there be one part not related to another. Therefore, if there are many things, they must be both small and large; so small as not to have size, but so large as to be unlimited. (Simplicius(a) On Aristotle's Physics, 141.2)

The Argument from Complete Divisibility

 whenever a body is by nature divisible through and through, whether by bisection, or generally by any method whatever, nothing impossible will have resulted if it has actually been divided …  though perhaps nobody in fact could so divide it.What then will remain? A magnitude? No: that is impossible, since then there will be something not divided, whereas ex hypothesi the body was divisible through and through. But if it be admitted that neither a body nor a magnitude will remain … the body will either consist of points (and its constituents will be without magnitude) or it will be absolutely nothing. If the latter, then it might both come-to-be out of nothing and exist as a composite of nothing; and thus presumably the whole body will be nothing but an appearance. But if it consists of points, it will not possess any magnitude. (Aristotle On Generation and Corruption, 316a19)

The Paradoxes of MotionThe Dichotomy

The first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal. (Aristotle Physics, 239b11)

This paradox is known as the ‘dichotomy’ because it involves repeated division into two (like the second paradox of plurality). Like the other paradoxes of motion we have it from Aristotle, who sought to refute it

Achilles and the Tortoise The [second] argument was called “Achilles,” accordingly, from

the fact that Achilles was taken [as a character] in it, and the argument says that it is impossible for him to overtake the tortoise when pursuing it. For in fact it is necessary that what is to overtake [something], before overtaking [it], first reach the limit from which what is fleeing set forth. In [the time in] which what is pursuing arrives at this, what is fleeing will advance a certain interval, even if it is less than that which what is pursuing advanced … . And in the time again in which what is pursuing will traverse this [interval] which what is fleeing advanced, in this time again what is fleeing will traverse some amount … . And thus in every time in which what is pursuing will traverse the [interval] which what is fleeing, being slower, has already advanced, what is fleeing will also advance some amount. (Simplicius(b) On Aristotle's Physics, 1014.10)

The Arrow

The third is … that the flying arrow is at rest, which result follows from the assumption that time is composed of moments … . he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always in a now, the flying arrow is therefore motionless. (Aristotle Physics, 239b.30)Zeno abolishes motion, saying “What is in motion moves neither in the place it is nor in one in which it is not”. (Diogenes Laertius Lives of Famous Philosophers, ix.72)

The Stadium

The fourth argument is that concerning equal bodies which move alongside equal bodies in the stadium from opposite directions—the ones from the end of the stadium, the others from the middle—at equal speeds, in which he thinks it follows that half the time is equal to its double…. (Aristotle Physics, 239b33)

Two More ParadoxesThe Paradox of Place

Zeno's difficulty demands an explanation; for if everything that exists has a place, place too will have a place, and so on ad infinitum. (Aristotle Physics, 209a23)

The Grain of Millet

 Zeno's reasoning is false when he argues that there is no part of the millet that does not make a sound; for there is no reason why any part should not in any length of time fail to move the air that the whole bushel moves in falling. (Aristotle Physics, 250a19)

Thank you!