Stumbling across a hole in accepted positivist logic is a somewhat disconcerting experience; perhaps one to remain censorious towards. Such holes prove there are phenomena in this world that human intellect cannot comprehend. But paradoxes are equally as perplexing as they are disturbing, for the simple reason that they cannot be explained in both material and theoretical terms; and thus fear of the unknown has a tendency to trump most other tangible horrors. It is disputed as to exactly what a paradox is, but a generally accepted definition might be a situation that can be explained perfectly rationally in mathematical or positivist theory, but cannot be translated in material reality whilst keeping within the physical rules of our supposedly finite universe. This article, on the admittedly rocky basis of understudied (on the part of the editor) logic theory, discusses some popular paradoxes by nature of time, rationality and reference, before looking at the facets of their self-contradiction, and then potential critiques of the systems in which they are formulated.

**Time – ‘The Mad Scientist’ and ‘Achilles and the Tortoise’**

Professor Stephen Hawking employs a very famous paradox in his popular discussions on the nature of cosmology, time and travel across various dimensions of physical and chronological existence. Hawking describes how a ‘mad scientist’ creates a time portal, which visibly stretches a minute into the past. Through the portal, the scientist can see himself a minute previously, and, given his ‘mad’ psyche, the scientist shoots himself fatally in order to create a paradox. Of course, if his person a minute previously is dead before he could move to shoot, then who fired the shot? This, by nature, should not be possible and the situation is therefore a paradox. The event reveals, in Hawking’s theory, that the world is based on a physical rule: that causes happen before effects and never the other way around; i.e. there is indeed a linear progression to time, and that attempting to mess that progression up is what creates a paradox. Perhaps it would be physically impossible for the scientist to kill his minute-younger self; or perhaps it is simply impossible for humans to comprehend such an event happening.

This paradox is interesting because, according to Hawking, a portal of this kind is not in fact as far fetched a concept as it sounds. The physical world is full of divots and holes in time and space, too miniscule to be of any immediate consequence, but existing nonetheless, potentially devastating to the accepted and orthodox dynamic of existence should such portals be harnessed in a dangerous manner. Hawking, however, comes to the conclusion that some form of natural regulation will step in should the scientist ever get into the position by which he could potentially shoot himself and cause a paradox. There are other concepts by which a natural regulator might intervene too, such as near-light-speed travel or the destruction of dark matter, to prevent paradoxes tearing the world apart.

Another famous paradox, Zeno’s paradox, or Achilles and the Tortoise, adds further to the discussion on time. In this situation, Achilles (A) and the Tortoise (T) are in a running race. A begins 100 metres behind T, and runs considerably faster, so that by the time A has run 100m, T has only run 10. But then again, once A has run another 10, T has run 1, and so on and so forth. By this mathematical logic, whenever A reaches a point at which T once stood, T has moved on, and thus A can never reach T because there are, theoretically, infinite points at which T has once stood.

The problem with this paradox is that is attempts to break down the concept of time into infinity, which is impossible in a finite universe. This tells us that time is not only linear and cannot be reversed, but that its speed is constant too, or at least, constant in acceleration. These are both examples of paradoxes in the conception of time, and its translation from theory to practice. They are the subjects of many popular works, and the uncertainty of their outcome makes fiction an avid fan.

**Rationality – ‘The Hungry Man’**

Hawking’s paradoxes probe the human understanding of a natural phenomenon. Other paradoxes, however, are concerned, not with the human desire to tear apart the world in which we exist, but with the way in which we make ‘rational’ decisions within it. This is effectively the basis of much positivist philosophy and rational choice theory, which informs the majority of economic, political and other social science in modern academia. An example runs as follows. A man, starving hungry and in need of sustenance, is positioned exactly midway between two plates of identical food, with identical conditions filling the space between himself and both plates. The paradox says that, by rational choice theory, because the man has no reason whatsoever to choose one plate over the other, he will not be able to choose either, and will starve right there on the stop despite the provision of both plates.

Of course, this paradox highlights the faults in concepts such as rational choice theory, and thus some of the holes in contemporary microeconomic and political rationality. While the man may not, in rational terms, be able to select one plate over another, his need for food will overcome the logic of rational choice in favour of one particular option. Rational choice theorists would argue that he has to abstract in order to pick a plate. While this may simply seem like common sense, it is slightly disconcerting that much material policy and economic modelling is built on this ‘rational choice’ model, and yet there are holes such as this one in its logic.

**Reference – ‘The Unexpected Hanging’**

Finally, paradoxes occur in Set Theory, which refers to how logic collects information together and references it, and how by extension, we cancel out options and ideas on the basis of such reference. In the ‘Unexpected Hanging’ paradox, a judge convicts a criminal to death by the noose at some point in the following week. The judge sets the following conditions: firstly, that the criminal will not know on which day he is to be hanged until the prison guard knocks on his door at noon on the fateful day; and secondly, that the hanging will be a surprise. The prisoner goes to his cell to ponder his fate, and concludes that he will survive the week on the following principle.

The prisoner argues that he cannot be hanged on Friday, because if he hasn’t been hanged by noon on Thursday, he will know that he is to die the following day and it will not longer be a surprise. Having eliminated Friday as an option, he then applies the same logic to Thursday, stipulating that if he hasn’t been hanged by noon on Wednesday then to die on Thursday would no longer be a surprise either. The prisoner continues this logic back to Monday, before concluding that there is no day on which he can be hanged within the parameters of the Judge’s rules. Of course, thinking that he is safe, the prison guard comes to hang the prisoner on Wednesday (or any other day in fact) and it is, of course, a complete surprise.

The paradox here is the connection between the days of the week, and also that by attempting to work out a way in which he cannot be hanged, the prisoner brings about his own fate. Firstly, just because in one situation Friday can be eliminated does mean that such conditions can be maintained and carried over to the next situation. This is the case with all of the days that the prisoner eliminates. Secondly, it could be argued that the only way in which the prisoner could have avoided being hanged was if he expected to be hanged every day of the week, instead of attempting to find a logical way of escaping death. This is the paradox that haunts his rationale.

**Conclusion**

Paradoxes prove the occasional faults in mathematical logic and its relationship with supposed human rationality. Some have been used to support huge scientific theories on relativity, such as Hawking’s cosmological time theses; others make superb subjects of books and films. Examples include Joseph Heller’s *Catch-22*; the books title has become synonymous with a situation in which a desired outcome can only be achieved by an undesired cause, touching again on the cause/effect relationship probed by Hawking’s theories on time travel. The Ancient Greeks were obsessed with paradoxes, making enough advances in logic and mathematical theory to find holes in their own understanding. It is now up to scientists and philosophers to attempt to manifest such paradoxes in the physical world and see what would actually happen in a mad scientist attempted to shoot himself through a time portal. I’m sure we’d all love to see it.