PARADOXES

• Zeno's Paradoxes

One can never reach the end of a racecourse, for in order to do so one would first have to reach the halfway mark, then the halfway mark of the remaining half, then the halfway mark of the final fourth, then of the final eighth, and so on ad infinitum. Since this series of fractions is infinite, one can never hope to get through the entire length of the track (at least not in a finite time).

• An arrow in flight is really at rest. For at every point in its flight, the arrow must occupy a length of space exactly equal to its own length. After all, it cannot occupy a greater length, nor a lesser one.

• But the arrow cannot move within this length it occupies. It would need extra space in which to move, and it of course has none. So at every point in its flight, the arrow is at rest. And if it is at rest at every moment in its flight, then it follows that it is at rest during the entire flight.

Sorites, or Paradox of the Heap

• (1) If someone has only $1, they are not rich.

• (2) If someone is not rich, then one more dollar won't make them rich.

• The sorites paradox is the name given to a class of paradoxical arguments, also known as little-by-little arguments, which arise as a result of the indeterminacy surrounding limits of application of the predicates involved.

• For example the concept of a heap appears to lack sharp boundaries and, as a consequence of the subsequent indeterminacy surrounding the extension of the predicate ‘is a heap’, no one grain of wheat can be identified as making the difference between being a heap and not being a heap.

• Given then that one grain of wheat does not make a heap, it would seem to follow that two do not, thus three do not, and so on. In the end it would appear that no amount of wheat can make a heap. We are faced with paradox since from apparently true premises by seemingly uncontroversial reasoning we arrive at an apparently false conclusion.

Skepticism

• Defending the rational world from a rising tide of nonsense;

• Applying common sense and the scientific method to the paranormal

• Suspending judgment until the evidence is in

• Skeptics are always willing to give others the benefit of the doubt.

• A skeptic will unpack the studies and data and look for the errors. This is known as doing good science. When a result contradicts accepted theory, or a fundamental assumption of natural science a good researcher, a good skeptic, will give it more than glancing attention

• The barber paradox

• In a certain village there is a man, so the paradox runs, who is a barber; this barber shaves all and only those men in the village who do not shave themselves. Query: Does the barber shave himself?

• Any man in this village is shaved by the barber if and only if he is not shaved by himself. Therefore in particular the barber shaves himself if and only if he does not. We are in trouble if we say the barber shaves himself and we are in trouble if we say he does not.

• What are we to say to the argument that goes to prove this unacceptable conclusion? Happily it rests on assumptions. We are asked to swallow a story about a village and a man in it who shaves all and only those men in the village who do not shave themselves.

• This is the source of our trouble; grant this and we end up saying, absurdly, that the barber shaves himself only if he does not. The proper conclusion to draw is just that there is no such barber. We are confronted with nothing more than what logicians have been referring to for a couple of thousand years as a reductio ad absurdum.

• We disprove the barber by assuming him and deducing the absurdity that he shaves himself if and only if he does not. The paradox is simply a proof that no village can contain a man who shaves all and only those men in it who do not shave themselves