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I read a bit of logical atomism by Russell, but would appreciate if someone explains with examples of what is meant by it. For example, it says that "[a]ccording to logical atomism, all truths are ultimately dependent upon a layer of atomic facts, which consist either of a simple particular exhibiting a quality or multiple simple particulars standing in a relation".

What is meant by the simple particular exhibiting a quality? Does it refer to things like water which has the quality of being clear, or a sphere which is round, or perhaps the sun which is yellow? If yes, this makes sense and doesn't seem very novel. So can someone shed more light on what is meant by logical atomism (with examples) and explain it to a beginner in philosophy?

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    Logical atomism was a theory which held that the 'root' in any given word contained a segment of 'meaning' which could be measured and accumulated with other 'atoms' to build epistemological certainties. When Russel and Wittgenstein recognized that there were no such things in language, they abandoned their joint project, the "Tractatus". Russell went back to Math, claiming that philosophy was impossible and Wittgenstein went and taught primary school. Wittgenstein eventually returned to philosophy and focused his work on 'natural' language'. (See his 'Blue' and 'Brown' books.) CMS – Charles M Saunders Jan 8 at 17:53
  • "simple particular exhibiting a quality" : the rose is red; "multiple simple particulars standing in a relation" : the pencil is on the table". – Mauro ALLEGRANZA Jan 8 at 19:32
  • Section 3 of the SEP article on Russell's logical atomism suggests he thought that our ordinary names for objects (like balls, the sun, roses etc.) were "logical fictions" and wouldn't be the subjects of the basic atomic sentences, and that he instead advocated beginning with "the uncontroversial doctrine of a certain science, such as mathematics or physics, largely because he held that these theories are the most likely to be true, or at least nearly true, and hence make the most appropriate place to begin the process of analysis." – Hypnosifl Jan 8 at 22:37
  • Also, section 4.2 of that article talks about how he assumed the world was composed of basic units or "simples" of some kind, so presumably logical atoms would concern only those simples--note the quote "I confess it seems obvious to me (as it did Leibniz) that what is complex must be composed of simples". – Hypnosifl Jan 8 at 22:44
  • What "simple particulars" are is quite murky in both Russell and Wittgenstein. They were supposed to be the primitives into which everything decomposes, but Russell later admitted that we can never know if something is simple, and stated that “the whole question whether there are simples to be reached by analysis is unnecessary” because one can relativize "simples" to a given level of discourse (say, chemical atoms or elementary particles). The idea then is that one can analyze all the complexity of language into very primitive atomic sentences: names of "simples" in a relation. – Conifold Jan 8 at 23:04
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Russell's preferred example of what he means by a "simple particular" is a sense-datum (an object of sensory experience), but it could apply to any object of awareness that is the sort of thing having properties. Importantly, what we often intend to call "particular" are not really particulars: they are in fact more complex. This includes any object that is not in our field of awareness, like objects in the past beyond our memory and objects that we are not (or never are) aware of, including ordinary objects of common sense. As Russell puts it:

The simplest imaginable facts are those which consist in the possession of a quality by some particular thing. Such facts, say, as "This is white."...What pass for [proper] names [i.e., words for particulars] in language, like "Socrates," "Plato," and so forth...are really abbreviations for descriptions; not only that, but what they describe are not particulars but complicated systems of classes or series. A name, in the narrow logical sense of a word whose meaning is particular, can only be applied to a particular with which the speaker is acquainted, because you cannot name anything you are not acquainted with...It is only when you use "this" quite strictly, to stand for an actual object of sense, that it is really a proper name. (The Philosophy of Logical Atomism, Lecture II, pages 522-525)

So ordinary objects like desks are not particulars, that is, they are not logical atoms. They are logical fictions, that is, series of particulars having common properties:

Now the essential point is this: What is the empirical reason that makes you call a number of appearances, appearances of the same desk? What makes you say on successive occasions, I am seeing the same desk?...There is something given in experience which makes you call it the same desk, and having once grasped that fact, you can go on and say, it is that something (whatever it is) that makes you call it the same desk which shall be defined as constituting the same desk...In that way the desk is reduced to being a logical fiction... (The Philosophy of Logical Atomism, Lecture VIII, pages 369-370)

So Russell thinks that many ordinary objects are series of particulars having some common experienced property (like the same color, shape, odor, and so on). The real particulars are the relatively short-lived objects of our immediate experiences.

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