The Paradox

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The Paradox

A Paradox is a reasoning which seems to contradict itself, but really doesn't. During the recorded history a number of people have been active with this phenomenon. One runs across them during life now and again, although they are often not recognised for what they are and thought of as real contradictions, unexplainable phenomena or acts of 'God'. As said paradoxes are not real contradictions, so all predicates mentioned above are really not applicable to the noumenon that was percieved. In that sense paradoxes are untrue thoughts, while the reasoning in it can be correct.

During the ages people have had different standings towards paradoxes. In ancient Greece paradoxes were thought of as inexplicable and a means to ponder the different ways things can take place. In the nineteenth century the paradox was finally defined as the oppositional outcome of a mathematical equasion, while in modernty the paradox is divided into several different categories, implying that the existance of a paradox means having overlooked something.

I will try to give a comprehensible explanation of what paradoxes are by explaining the work of some philosophers in the field in a none historical order. I will note when the specified philosopher lived and died.

Heraclitus
Ca 535-475 BC

One of the first philosophers that is known for his interest in the 'unity of opposites' is the pre-socratic named Heraclitus. In his work numerous contradictions are used to point certain things out.

Sea: the purest and dirtiest water - drinkable and life's necessity for fish, for man undrinkable and lifethreatening.

The contradictions in the above quote are drinkable versus undrinkable and life's necessity versus lifethreatening being predicated of the sea at the same time. It is not a real contradiction because these predications are defined as being valid for respectively fish and man. Both responding differently to the sea's water the water of the sea can be truthfully predicated drinkable and life's necessity for fish and the sea can be truthfully predicated lifethreatening and undrinkable for man. The contradiction therefore exists with a correct reasoning, but the contradiction was predicated for two different situations and therefore does not contradict itself.

Willard van Orman Quine
June 25 1908 - December 25 2000

Quine defined a number of paradoxes. The list makes clear a number of ways someone can accidently create a paradox in their thoughts and how to see through them.

1) A veridical paradox creates a result which seems absurd, but which proves to be true nonetheless. One might say that a certain event has taken place in 2300 (it now being 2008 A.D.), which seems absurd. In Thailand, however, it is now 2551 (after Buddha).
2) A falsidical paradox creates a result which is false, because there is a fallacy in its demonstration. Most of these proofs rely on a devision by 0.
3) An antinomy is a paradox which reaches a contradictory result because of our way of reasoning. Ontological differences can be problematic because several ontological levels may be thought to exist simultaniously, thereby creating several meanings of a given thing.
4) A dialetheia is a paradox in which opposite truths are both true. One might say that a man standing with one foot in a room is inside the room and is outside the room.

Bertrand Russell
May 18 1872 - February 2 1970

Russell came up with a formulation of a paradox in order to study the phenomenon. The Russell Paradox is the question if the set R is a member of itself when the set R consists of A's which are not elements of A. A mathematical formulation might be

The question if R is a member of itself will lead to the following two options:
1) If R is an element of R, then R is not a member of R according to the definition.
2) If R is not a element of R, then according to the definition R is a member of R.

This leads to the conclusion that R is both a member and not a member of R.

This paradox proved the existence of paradoxes. It was often argued to that point that paradoxes were merely present when one had not properly observed a situation, such as in the case of Heraclitus. In this case a closer look had determined that paradoxes indeed did exist.

Ludwig Wittgenstein
April 26 1889 - April 29 1951

Ludwig Wittgenstein was a student of Bertrand Russel. In his work Tractatus Logico-Philosophicus, which he wrote as a soldier and in a prisoner of war camp during the first world war, Ludwig Wittgenstein defines a universal formulation for all paradoxes. The work was published in 1921 by Bertrand Russel, without the knowledge of Wittgenstein. Wittgenstein had moved to Scandinavia saying that he had solved all philosophical problems.

The proof is as follows:

3.333 A function cannot be its own argument, because the functional sign already contains the prototype of its own argument and it cannot contain itself.
If, for example, we suppose that the function F(fx) could be its own argument, then there would be a proposition 'F(F(fx))', and in this the outer function F and the inner function F must have different meanings; for the inner has the form F(fx) and the outer has the form Y(F(fx)). Common to both functions is only the letter 'F', which by itself signifies nothing.
This is at once clear, if instead of 'F(F(u))' we write '($F):F(Fu).Fu=Fu'.
Herewith Russel's Paradox vanishes.

Paradoxes in Easy language.

Paradoxes are caused by the use of similar predicates for different meanings. The easiest example I can think of is the problem 11+1. If one is not fully familiar with what the problem means one might be inclined to say 21 is the answer, not realising that both 1's in the 11 have a different meaning. Had we used different predicates for the decimal units and single units the problem would never have occured. In the roman umerical system X would be used for 10 and I for 1. The problem would then become XI+I=XII; no paradox can be the consequence of it.

Definitions

Noumenon
A thing-in-itself.

Phenomenon
That which is observed of a thing-in-itself.

Thought-object
The mental image containing all properties thought to belong to a certain noumenon.

Perception
The activity of the mind which 'grasps' whats is observed and forms it into thought-objects.

Predicate
A name or judgement given to something percieved; a thought-object.

Paradox

Sources:

Classes by Prof. dr.A. Visser
Wikipedia
Tractatus Logico-Philosophicus

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@Arjen,

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@Grimlock,

Because it is a way of deducing a paradox from a certain mindset, because it shows how the paradox was eventually understood by Wittgenstein and because philosophy is showing the phallacy of the seductions of syntax.

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@Arjen,

What's the difference between a paradox and a conundrum?

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@MITech,

Paradoxes are not designed to reason over. They exist in thoughts because of several causes and some might be overcome by 'lateral thinking' of sorts, although overcoming a paradox is primarily done by the realisation of truth values. Perhaps treating a paradox as a conundrum might benefit the problem solving process, but I do not think so.

In any case the conundra is a purposefully created problem while paradoxes are not purposefully created. The difference between them therefore is one of intent. The paradox has the intent to depict reality, while the conundrum has the intent not to depict reality.

Apart from that I think a conundrum is not necessarily related to namegiving and seperate sets.

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@Arjen,

Hi everyone,

I didn't want my first post to be argumentative but I might have known it would be. Oh well.

I'm not entirely comfortable with Arjen's summary. I don't want to nitpick so will just mention a couple of things. The example of a dialetheia is misleading because it does not describe a true contradiction. A true contradition would be Heraclitus's 'We are and are not.' This seems to me one of the most radical and important statements in philosophy. Perhaps it is a stronger statement of his seemingly paradoxical philosophical position than the one you mention, as it is not clear that the seawater example is a paradox in a philosophical sense.

Sorry to start by not agreeing. What I would agree with is the importance of the topic.

Could we define a true paradox as an intellectual dilemma, or is that a tautology?

Whoever

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@Whoever,

Whoever wrote:

Sorry to start by not agreeing.

Don't be. Disagreement is the force behind intellectual advancement.

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