This means that there must be a flaw in the argument that there is no good reason to switch — in the "no switching argument". But what, precisely, is wrong with it? Suppose you pick door A, and then Monty shows you that door C has a goat behind it. A key premise in the no switching argument is that after door C is revealed as a goat door, it is as likely that the car is behind door A as it is that it is behind door B; the two possibilities are equally likely.
To find a fallacy in the no switching argument and thereby resolve the paradox, it is this premise, I believe, that must be successfully rebutted. If Monty's intent had been simply to open one of the three doors at random, and this intent had resulted in his opening door C, and revealing a goat behind it, then it would be equally likely that the car was behind door A and that it was behind door B. But Monty's choice was in fact restricted to door B or door C, and his intent was to choose a goat door.
So you know something about door B that you do not know about door A; there is an asymmetry in your knowledge of the two doors. In a choice between door B and another door, where the intent was to choose a goat door, door B was not chosen. How is this knowledge relevant to the assignment of probabilities? Since you knew that Monty's intent was to open a goat door, and that he could do this, his opening door C and revealing it to be a goat door does not change this probability.
But if Monty's choice was forced, then the car is behind door B. If this analysis is correct, then we have dissolved the paradox, dispelled the illusion, by showing that one of the premises in one of the paradox-generating arguments is false. Of course, the reasoning is subtle; the flaw in the no switching argument is not easy to discern.
Indeed, some mathematicians and probability theorists have been initially taken in by the no switching argument, although none has persisted in defending it. Further illustrations of the resolution of a paradox must await the more detailed treatment of individual paradoxes. But, at this point, some words of caution are in order. First, to be satisfactory, the resolution of a paradox should be robust: it should stand up to strengthened versions of the paradox. For instance, the paradox-generating argument may initially be presented with an extremely strong premise, a premise that makes a very broad, sweeping claim.
If so, it may take no great acumen to point out counter-examples to the premise. However, before declaring the paradox vanquished, we need to be sure that it cannot simply be reinstated when a suit-ably weakened version of the critical assumption is provided. To be robust, the attack on a paradoxical argument should be focused on the strongest, most impregnable version of the argument available. A related point concerns different versions of the same paradox. The Achilles and the tortoise paradox, for example, seems to be essentially the same as the racetrack paradox see the Appendix.
If so, then any solution to the one should also be applicable, with the appropriate changes, to the other. An attempted resolution of a paradox that cannot be applied successfully to every version of the paradox must be off the mark in that it focuses on some inessential feature of the paradox. This is not to say that it is always evident whether one paradox is a variant of another. In fact, the criteria for two arguments being versions of the same paradox are far from obvious. One can even imagine cases in which the fact that a solution applies to one argument, but not to the other, would be cited as reason for denying that these are just two versions of the same paradox.
Nonetheless, as we shall also see, there are many cases in which it is entirely clear that different scenarios are all versions of the same paradox, and thus require a unified solution. What does not count as a resolution of a paradox? The negative may be almost as significant as the positive here. One very common and natural response to a stated paradox is to present another argument. More specifically, the response is an attempt to present an even more compelling or persuasive argument for or against the conclusion one of the conclusions involved in the original paradox.
Consider the Monty Hall paradox, for example. Suppose a mathematician friend, having announced that she has a solution to the paradox, proceeds to give a very clear, very explicit, and very powerful argument for the conclusion that one ought to switch doors. Whatever the merits of her argument and the worth of her contribution, they do not constitute a solution to the paradox, for the argument in favour of not switching is left untouched. So there is still an apparent conflict of reasons: two ostensibly strong arguments for inconsistent conclusions.
Consequently, there is still a sense of confusion, of being befuddled, which can be cleared up only by an analysis of an error or flaw in one of the arguments. A paradox is not unravelled by attempting, however successfully, to prove that one "side" in the conflict is correct. Later chapters will provide examples of philosophers responding to the challenge of a paradox in this way. The ideal, in treating a paradox, is to puncture the illusion of letter-perfect reasoning leading to clear absurdity. But short of achieving this ideal, there are still worthwhile contributions one can make.
The mathematician's argument alluded to in the previous paragraph may convince us that the rational response in the Monty Hall scenario is indeed to switch, when previously we had been uncertain. While such an argument does not suffice to dissolve the paradox, it may convert a previously controversial paradox to the status of uncontroversial.
Assuming the argument to be correct, this constitutes an advance in the understanding of the problem, and progress in the search for a solution. If it is known that the correct strategy is to switch doors in the Monty Hall game, then the focus must be squarely on the no switching argument, and the attempt to locate a flaw in it.
The range of possible solutions has been narrowed. One other way to move a controversial type I paradox into the uncontroversial category is worth mentioning here and will be illustrated in Chapter 4. Suppose we construct an argument that is strongly analogous to the original paradoxical argument, but that leads to a conclusion even more preposterous or bizarre than that of the paradoxical argument; so bizarre, in fact, that it is completely clear that the conclusion, and therefore the argument, have to be rejected.
Since the new argument is strongly analogous to the original paradoxical argument, it is now apparent that the original argument must also be rejected. Again, the set of possible options for a solution has contracted.
Please, please revisit the major's focus trees before adding any more new ones
There is, finally, one other way to make progress on a paradox, short of resolving it or narrowing the range of possible solutions: progress can be made by clarifying one of the central arguments. This may be achieved in a variety of ways.
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Among the more significant are: analysing one of the key concepts; setting out, fully, rigorously and explicitly, the premises necessary for the argument; ensuring that the premises are just as strong as needed, but no stronger, so that the argument is as immune to criticism as possible; and making clear exactly what the inferential steps are that take us from the premises to the conclusion, so that any logical gaps in reasoning will be more apparent.
This section has considered how to resolve a paradox, how not to resolve a paradox and how to make progress on a paradox short of resolution.
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To conclude, let us consider the question of why we feel a pressing need to untangle a paradox, why we care about paradoxes. One answer has already been suggested. An unresolved paradox is a threat to the trustworthiness of reason. How can reason command our respect if it leads to absurdities?
But another motivation stems from the fact that the proper resolution of a paradox may give us greater philosophical knowledge. Faulty assumptions concerning, for instance, justified belief, or rational action, may be uncovered in the unravelling of the paradox. Of course, some paradoxes, of which the barber is one, have little, or no, philosophical punch.
The ship of Theseus, on the other hand, may reveal a good deal about the principles governing our concept of the identity of a physical object. The depth of a paradox is generally considered to be a function of the sort of philosophical impact its resolution will have. At one extreme of the spectrum, a paradox may reveal an incoherence that necessitates a fundamental revision of our conceptual scheme; at the other end of the spectrum there is the barber.
Unfortunately, a proper appreciation of the depth of a paradox frequently must await its resolution. Can time have a beginning? Which came first, the chicken or the egg? Riddles, paradoxes, conundrums--for millennia the human mind has found such knotty logical problems both perplexing and irresistible. Now Roy Sorensen offers the first narrative history of paradoxes, a fascinating and eye-opening account that extends from the ancient Greeks, through the Middle Ages, the Enlightenment, and into the twentieth century.
When Augustine asked what God was doing before He made the world, he was told: "Preparing hell for people who ask questions like that. Organized chronologically, the book is divided into twenty-four chapters, each of which pairs a philosopher with a major paradox, allowing for extended consideration and putting a human face on the strategies that have been taken toward these puzzles.
Readers get to follow the minds of Zeno, Socrates, Aquinas, Ockham, Pascal, Kant, Hegel, and many other major philosophers deep inside the tangles of paradox, looking for, and sometimes finding, a way out.
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Filled with illuminating anecdotes and vividly written, A Brief History of the Paradox will appeal to anyone who finds trying to answer unanswerable questions a paradoxically pleasant endeavor. Philosophers map our conceptual scheme through a logical study of semantics. Quine agreed that philosophers are more apt to use the strategy of semantic ascent: they love to switch the topic from the things that puzzle us to the words we use in describing those puzzling things.
But Quine thinks that the use of semantic ascent is only a rough mark of philosophy. The physicist Albert Einstein engaged in semantic ascent when trying to resolve anomalies about the nature of simultaneity. And metaphysicians sometimes appeal to empirical results to solve philosophical problems. Quine has fostered this naturalistic turn in philosophy. He maintains that philosophy differs from science in degree rather than kind. Philosophy should heed biology just as biology heeds physics and vice versa.
Philosophy takes the further step of trying to organize the results of science into an overall view of the universe. But as we have seen with Brandon Carter's case for human extinction, cosmologists also take a wide perspective. Lord Kelvin contrasted the unclarity of metaphysics with the rigor of physics by claiming that "In science there are no paradoxes.
Many of the scientific paradoxes have been solved. But the same can be said about philosophical paradoxes such as those made famous by Zeno. Philosophy is like an expedition to the horizon. Under one interpretation, the venture is hopeless. We cannot reach the destination because what counts as the horizon constantly shifts.
But becoming a pessimist on the basis of this tautology is like adopting a here-and-now philosophy on the strength of "Tomorrow never comes. Understandably, we look at the history of philosophy from the vantage point of the present. We are impressed by the resiliency of its issues and the broken ambitions of past thinkers. But an accurate measure of progress requires the adoption of an historical perspective.
By this, I do not mean simply looking at the past. I mean looking from the past. The twenty-first-century conception of philosophy will itself become a tonic to the vacuous pessimism of future generations. Given that I have correctly gauged the merits of Carter's doomsday argument, some philosopher in the distant future will find this book aging away in the remote corner of a library. He will know that many of the "paradoxes" discussed in this book are nowdefinitively answered by physics or mathematics or by some hitherto unconceived field.
This future reader will wonder why philosophers tried to answer those questions. As he reads this final sentence, I remind him that he stands at a new horizon, inaccessible to the author of this book. Truth Without Paradox by David A. In the first chapter, Johnson argues against the general acceptability of the schema "if p, then it is true that p," claiming thereby to resolve the paradoxes of the liar and of the sorites.
In the second chapter, he clarifies what was settled by Quine about "truth by convention. Excerpt: Metaphysics is the study of fundamental things, of what are sometimes called "first things. Almost certainly not to be included here are such things as aardvarks, toothpaste, prostitution, and laughter. And then there are borderline cases, such as holes. In practical terms, one might say that the "fundamental things" are the things concerning which there would be rather something wrong with someone who had no interest in them. If you have no interest in aardvarks, then, fine, you have no interest in aardvarks; but if you have no interest in logic, or in morality, then there is something wrong with you.
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Some of the fundamental things have been fruitfully studied, so much so that sciences thereof have come into being, such as logic, mathematics, physics, and biology. But deep mysteries remain even about these well-studied things; I shall call them the Remaining Mysteries. If you wish to know something important about numbers, or about space and time, you may learn from the mathematician and the physicist.
But if you wish to know something important about truth, or similitude, or morality, or the other. Remaining Things, I am afraid that humanly speaking you have no re-course but to consult a philosopher, since there is no other kind of human being who can help you. I shall call it academic metaphysics.
The Remaining Mysteries and the Remaining Things are too many, and too complex, and too much the subject of recalcitrant myth, to be profitably surveyed in a single book. The reader who is looking merely for a quick and perforce hazy overview of the field, and who wishes at all costs to avoid what I suppose the dust jackets would call "burdensome de-tails" so often consigned to graduate school, or to a future life , must look elsewhere. Academic metaphysics is a complex, subtle, and difficult subject so is Greek, or Topology, or Quantum Physics , and an introduction to academic metaphysics which goes easy on the details is about as valuable as an introduction to Latin which goes easy on the participles.
In writing this book, we have tried to avoid two extremes. We have tried to avoid giving an overly tedious treatise of renormalization theory and the obscure intricacies of Feynman graphs. The other extreme is a shallow approach to theoretical physics, where many vital concepts are deleted because they are considered too difficult for the student.
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