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Russell’s Principles
Constituting Bertrand Russell’s theory of logical atomism are three interworking parts: the atomic proposition, the atomic fact, and the atomic complex. An atomic proposition, also known as an elemental judgement, is a fundamental statement describing a single entity. Russell refers to this entity as an atomic fact, and recognizes a range of elements within each fact that he refers to as particulars and universals. A particular denotes a signifier such as a name, many of which may apply to a single atomic fact, while a universal groups these particulars according to their relationship. In Russell’s Theory of Acquaintance, awareness of these particulars and universals comes through sense data. Every system consists of many atomic propositions and their corresponding atomic facts, known together as an atomic complex. Rather than decoding the complex in a top-down manner, logical atomism analyzes its propositions individually before considering their collective effect.

Russell’s perspective on belief proved a point of contention between him and Wittgenstein, causing it to shift throughout his career. In logical atomism, belief is a complex that possesses both true and untrue propositions. Initially, Russell plotted belief as the special relationship between a subject and a complex proposition. Later, he amended this to say that belief lacks a proposition, and instead associates with universals and particulars directly. Here, the link between psychological experience - sense data - and components of logical atomism - universals and particulars - causes a breach in the typical logic of the theory; Russell’s logical atomism is in some respects defined by the crossover of metaphysics and analytical philosophy, which characterizes the field of naturalized epistemology. (Kitchner) - {Find Critique against Russell saying that the logical aspect of his theory wavered}

In his theory of Logical Atomism, Russell posited the highly controversial idea that for every positive fact exists a parallel negative fact: a fact that is untrue. The Correspondence Theory maintains that every atomic proposition coordinates with exactly one atomic fact, and that all atomic facts exist. The Theory of Acquaintance says that for any given statement taking the form of an atomic proposition, we must be familiar with the assertion it makes. For example, in the positive statement, “the leaf is green,” we must be acquainted with the atomic fact that the leaf is green, and we know that this statement corresponds to exactly this one fact. Along this same line, the complementary negative statement, “the leaf is not green," is clearly false given what we know about the color of the leaf, but our ability to form a statement of this nature means that a corresponding fact must exist. Regardless of whether the second statement is or isn't true, the connection between its proposition and a fact must itself be true. One central doctrine of Logical Atomism, known as The Logically Perfect Language Principle, enables this conclusion. This principle establishes that everything exists as atomic proposition and fact, and that all language signifies reality. In Russell’s viewpoint, this necessitates the negative fact, whereas Wittgenstein maintained the more conventional Principle of Bivalence, in which the states "P" and "Not (P)" cannot coexist.

Wittgenstein’s Principles
In his Tractatus Logico-Philosophicus, Ludwig Wittgenstein explains his version of logical atomism as the relationship between proposition, state of affairs, object, and complex, often referred to as “Picture theory”. In view of Russell’s version, the propositions are congruent in that they are both unconvoluted statements about an atomic entity. Every atomic proposition is constructed from “names” that correspond to “objects”, and the interaction of these objects generate “states of affairs,” which are analogous to what Russell called atomic facts. Where Russell identifies both particulars and universals, Wittgenstein amalgamates these into objects for the sake of protecting the truth-independence of his propositions; a self-contained state of affairs defines each proposition, and the truth of a proposition cannot be proven by the sharing or exclusion of objects between propositions. In Russell’s work, his concept of universals and particulars denies truth-independence, as each universal accounts for a specific set of particulars, and the exact matching of any two sets implies equality, difference implies inequality, and this act as a qualifier of truth. In Wittgenstein’s theory, an atomic complex is a layered proposition subsuming many atomic propositions, each representing its own state of affairs.

Wittgenstein’s handling of belief was dismissive, and reflects his abstainment from the epistemology that concerned Russell. Because his theory dealt with understanding the nature of reality, and because any item or process of the mind barring positive fact, i.e. something absolute and without interpretation, may become altered and thus divorced from reality, belief exists as a sign of reality but not reality itself. Wittgenstein was decidedly skeptical of epistemology, which tends to value unifying metaphysical ideas while depreciating the casewise and methodological inspection of philosophy that dominates his Tractatus Logico-Philosophicus. Furthermore, Wittgenstein concerned himself with defining the exact correspondence between language and reality wherein any explanation of reality that defies or overburdens these semantic structures, namely metaphysics, becomes unhinged. Wittgenstein’s work bears the exact philosophical determinants that he openly admonished, hence his later abandonment of this theory altogether.