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I just finished reading Russell's paper "On the Nature of Truth", in which appears a form of the well-known Liar's Paradox. Here is the passage I am interested in:

This argument would be conclusive, I think, if it were certain that a belief can be validly regarded as a single state of mind. There are, however, difficulties in so regarding a belief. The chief of these difficulties is derived from paradoxes analogous to that of the liar, e.g., from the man who believes that all his beliefs are mistaken, and whose other beliefs are certainly all mistaken. If he is mistaken in this belief, then all his beliefs are mistaken, which is what he is believing; therefore he is not mistaken; therefore he is right in believing that all his beliefs are mistaken, and therefore this belief is mistaken. We can escape this paradox if a belief cannot be validly treated as a single thing.

The last sentence of this quotation is what I want to focus on. Sadly, he never goes on to show exactly how belief not being treated as a single thing evaporates the paradox, and try as I may, I cannot see how such an assumption could. Would someone mind helping me fill in the details of what Russell has in mind for escaping this paradox? And if you don't think this assumption would allow one to escape the paradox, I would be interested in hearing your thoughts, too.

  • @GeorgeChen Unless I am not reading the sections you had intended, what you have linked doesn't seem relevant (I'm not really sure where I am supposed to start and stop). In his paper "On the Nature of Knowledge", he seems to be dealing with the ontology or nature of a belief, whereas in his Principia Mathematca, he seems to be approaching the problem from a purely logical standpoint; in fact, I didn't see the word "belief" appear in what I read. So I am not sure how this connects.... – user193319 Oct 13 '17 at 18:49
  • @GeorgeChen ...In his paper, he seems to be saying that if the ontology/nature of beliefs, which is mental, is such that belief is not treated as a single thing (whatever that might mean), then there is no paradox. This doesn't seem to be a logical resolution of the problem but a metaphysical solution. Perhaps you (or anyone, really) could summarize what Russell is saying in his Principia Mathematica that ostensibly sheds light on what he is saying in "On the Nature of Knowledge" – user193319 Oct 13 '17 at 18:49
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    Give me some more time. What I like about this question is that it simultaneously leads to many more questions. – George Chen Oct 13 '17 at 19:43
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    @RamTobolski Enter this link jstor.org/stable/4543744?seq=1#page_scan_tab_contents into this wonderful website sci-hub.io – user193319 Oct 14 '17 at 0:02
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Russell's idea in "on the nature of truth" (1906) seems to be, to escape the Liar paradox through the rather colorful suggestion, that there is no such thing as a false belief. In ordinary language we do recognize, of course, false beliefs. Russell's suggestion is that on reality, "behind" ordinary language, there are no false beliefs. What we ordinarily call false belief is, in reality, a belief in nothing.

In the event of the objects of the ideas standing in the corresponding relation, we shall say that the belief is true, or that it is belief in a fact. In the event of the objects not standing in the corresponding relation, there will be no objective complex corresponding to the belief, and the belief is belief in nothing, though it is not "thinking of nothing", because it is thinking of the objects of the ideas which constitute the belief.

What Russell meant by a "belief in nothing" seem to be this: Consider the belief that the sun is shining (at some time and place). If the belief is true, there is a corresponding something in the world (something that Russell called "a fact") that connects together "the sun" and "shining". It is the fact that the sun is shining. But if the belief that the sun is shining is (what we would ordinarily call) false, there is no corresponding fact in the world. If there is no corresponding entity in the world, a fact, then what do I believe in? How can a false belief have an object, when no corresponding object exist? Russell's suggestion was that, for a false belief has as object the ingredients of the intended fact, but not the fact itself. For example, if I believe that the sun is shining, and it isn't, then my belief has as objects the sun and (the universal property) shininess. The belief will have no fact as object, and to that extent it will be "a belief in nothing". Also, since the belief itself will be different when it is true than when it is false, the belief will not be "a single thing" as Russell puts it.

How does all this help with the Liar paradox? My understanding is that it does that by eliminating the concept of a false (or mistaken) belief. It still doesn't seem to me to work, because it merely replaces the concept of false belief with that of empty belief ("a belief in nothing"). So you can replace, in the formulation of the paradox, each "mistaken belief" with "empty belief" and still you end in a similar paradox. In addition, even if this solution worked, the cost would be high. Because it warps the concept of belief in a strange and unnatural way.

I suppose that Russell realized quite quickly the shortcomings of this attempted solution to the Liar. In Russell's later views, the problem of non-existing objects of beliefs was solved by the theory of descriptions. And the Liar paradox he attempted to solve through the logic of types.

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