Wednesday, October 17, 2012

26. The Vicious-Circle Principle & The theory of logical types




Harold R. (Hal) Foster’s Prince Valiant
© Respective copyright/trademark holders.

CHAPTER II.

THE THEORY OF LOGICAL TYPES

The theory of logical types, to be explained in the present Chapter, recommended itself to us in the first instance by its ability to solve certain contradictions, of which the one best known to mathematicians is Burali-Forti's concerning the greatest ordinal. But the theory in question is not wholly dependent upon this indirect recommendation: it has also a certain consonance with common sense which makes it inherently credible. In what follows, we shall therefore first set forth the theory on its own account, and then apply it to the solution of the contradictions.

I.  The Vicious-Circle Principle.

An analysis of the paradoxes to be avoided shows that they all result from a certain kind of vicious circle. The vicious circles in question arise from supposing that a collection of objects may contain members which can only be defined by means of the collection as a whole. Thus, for example, the collection of propositions will be supposed to contain a proposition stating that "all propositions are either true or false." It would seem, however, that such a statement could not be legitimate unless "all propositions" referred to some already definite collection, which it cannot do if new propositions are created by statements about "all propositions." We shall, therefore, have to say that statements about "all propositions' are meaningless. More generally, given any set of objects such that, if we suppose the set to have a total, it will contain members which presuppose this total, then such a set cannot have a total. By saying that a set has "no total," we mean, primarily, that no significant statement can be made about "all its members." Propositions, as the above illustration shows, must be a set having no total. The same is true, as we shall shortly see, of propositional functions, even when these are restricted to such as can significantly have as argument a given object a. In such cases, it is necessary to break up our set into smaller sets, each of which is capable of a total. This is what the theory of types aims at effecting.

The principle which enables us to avoid illegitimate totalities may be stated as follows: "Whatever involves all of a collection must not be one of the collection"; or, conversely: "If, provided a certain collection had a total, it would have members only definable in terms of that total, then the said collection has no total." We shall call this the "vicious-circle principle," because it enables us to avoid the vicious circles involved in the assumption of illegitimate totalities. Arguments which are condemned by the vicious-circle principle will be called "vicious-circle fallacies." Such arguments, in certain circumstances, may lead to contradictions, but it often happens that the conclusions to which they lead are in fact true, though the arguments are fallacious. Take, for example, the law of excluded middle, in the form "all propositions are true or false." If from this law we argue that, because the law of excluded middle is a proposition, therefore the law of excluded middle is true or false, we incur a vicious-circle fallacy. "All propositions" must be in some way limited before it becomes a legitimate totality, and any limitation which makes it legitimate must make any statement about the totality fall outside the totality.

Similarly, the imaginary skeptic, who asserts that he knows nothing, and is refuted by being asked if he knows that he knows nothing, has asserted nonsense, and has been fallaciously refuted by an argument which involves a vicious-circle fallacy. In order that the skeptic's assertion may become significant, it is necessary to place some limitation upon the things of which he is asserting his ignorance, because the things of which it is possible to be ignorant form an illegitimate totality. But as soon as a suitable limitation has been placed by him upon the collection of propositions of which he is asserting his ignorance, the proposition that he is ignorant of every member of this collection must not itself be one of the collection. Hence any significant skepticism is not open to the above form of refutation.

The paradoxes of symbolic logic concern various sorts of objects: propositions, classes, cardinal and ordinal numbers, etc. All these sorts of objects, as we shall show, represent illegitimate totalities, and are therefore capable of giving rise to vicious-circle fallacies. But by means of the theory (to be explained in Chapter III) which reduces statements that are verbally concerned with classes and relations to statements that are concerned with propositional functions, the paradoxes are reduced to such as are concerned with propositions and propositional functions. The paradoxes that concern propositions are only indirectly relevant to mathematics, while those that more nearly concern the mathematician are all concerned with propositional functions. We shall therefore proceed in the next post to the consideration of propositional functions.

Script: Stan Lee  Pencils: Al Williamson Inks: Roy Krenkle  © Respective copyright holder.


The vicious-circle principle goes very far to explain so many of society’s greatest problems; and especially those that seek to place people into categories convenient to political, economic, religious or criminal purposes. Reading this post through a few times is useful to begin to see how the principle applies in the world you live in. It helps to see ahead, to contradictions and to error in lines of thought; to help decide which are worth pursuing, and which should be abandoned.
Right and wrong, true and false are subjective parameters in framing any complex proposition. It can easily be seen that with these forming the basis of any proposition, the measure of such will alter in small or great degree in each individual’s estimation. Therefore right and wrong, true and false or as are normally held to be ‘moral’ considerations are dangerous to take into account in propositional equations, because they will inevitably lead to error or ‘nonsense’ that is, propositions that cannot be proven true or false, but which inevitably lead to error.
I would not care to end this post with an impression that faith, religion, belief in God or any devotion is wrong and a certain source of error. This is not the case, and indeed without faith no one would endeavor to learn and practice a system such as Whitehead and Russell set forth in Principia Mathematica. Only by clarity do we see that thoughts contain errors that we cannot trace, and most often they arise from vicious-circle fallacies.  





Script, Pencils and Inks: Stan Sakai   Usagi Yojimbo  © Respective copyright/trademark holders




Harold R. (Hal) Foster’s Prince Valiant
© Respective copyright/trademark holders.
 

No comments:

Post a Comment