(a) Explain Russell's notion of incomplete symbols. Also explain how this notion leads to the doctrine of logical atomism. 20…

(a) Russell's incomplete symbols and logical atomism

In Russell’s philosophy, an incomplete symbol is an expression that has no meaning in isolation but contributes to the meaning of the proposition in which it occurs. In Introduction to Mathematical Philosophy, Russell illustrates this through definite descriptions, such as "the present King of France," as well as class symbols. When analyzing "The present King of France is bald," Russell shows that ordinary grammar misleads us into treating the descriptive phrase as a genuine proper name pointing to an entity. Logically translated, the phrase unpacks into three distinct assertions: an existential claim, a uniqueness claim, and a property attribution. This is the theory of descriptions of 'On Denoting' (1905): the symbol is defined in use (contextual definition), not given a meaning of its own. By demonstrating that the description disappears upon formal analysis, Russell avoids Alexius Meinong’s bloated ontology of non-existent objects.

This method directly generates Russell’s doctrine of Logical Atomism. Guided by his supreme maxim—"wherever possible, substitute constructions out of known entities for inferences to unknown entities"—philosophical analysis systematically eliminates grammatical illusions until it reaches ultimate, unanalyzable constituents. These basic elements are logical simples: particulars (apprehended by immediate acquaintance and denoted by logically proper names) and universals (predicates and relations). Simples combine to form atomic facts, which are mirrored by logically independent atomic propositions. Molecular propositions are truth-functions of atomic ones, and in 'The Philosophy of Logical Atomism' (1918) Russell also admits general and negative facts. Thus, the theory of incomplete symbols serves as the functional mechanism that strips away linguistic confusions to reveal the isomorphic correspondence between logical structure and metaphysical reality.

(b) Meaningfulness for the logical positivists

According to logical positivists like A.J. Ayer (Language, Truth and Logic), the sentences "All objects are either red or not red" and "This page is white" are both cognitively meaningful, but they are not meaningful in the same way. The verification principle divides all significant assertions into two distinct classes: analytic (tautological) and synthetic (empirical).

The sentence "All objects are either red or not red" is an analytic proposition grounded in the logical law of excluded middle (P ∨ ¬ P). It is logically true (L-true) solely in virtue of the definitions of its constituent terms and logical form. Its validity is independent of empirical experience; it is known a priori and cannot be refuted by observation, though it yields zero factual information about the empirical world.

Conversely, "This page is white" is a synthetic proposition. Its truth or falsity cannot be deduced from the meanings of the terms alone; rather, it depends on empirical facts. It is meaningful because sense-experience can verify or falsify it a posteriori. Thus, while both statements avoid the positivist charge of metaphysical meaninglessness, the former derives its meaning from formal-linguistic conventions, whereas the latter derives its meaning from experiential verification. In the Tractatus idiom the tautology is 'senseless' (sinnlos), saying nothing about the world, but not 'nonsense' (unsinnig) like metaphysics. Quine's later attack on the analytic–synthetic distinction challenges this neat two-fold division.

(c) Rationalist accounts of mind and body, and free will

Among the rationalists, Gottfried Wilhelm Leibniz provides the account of the mind-body relation most compatible with human freedom and free will.

René Descartes gives the strongest statement of libertarian freedom: the will is infinite and experienced as free, and interactionism gives the mind real causal power over the body. His dualism locates mind-body contact in the pineal gland, but fails to reconcile how an immaterial soul (res cogitans) can freely exert causal force upon a mechanically determined physical body (res extensa) without violating physical conservation laws. Baruch Spinoza resolves dualism through substance monism and psycho-physical parallelism, where mind and body are attributes of one infinite Substance (God or Nature). However, Spinoza completely rejects free will as an illusion born of ignorance of causes; human freedom is reduced strictly to the intellectual recognition and acceptance of necessity (amor Dei intellectualis).

Leibniz’s doctrine of pre-established harmony best preserves human freedom through rational self-determination. For Leibniz, reality consists of non-physical, "windowless monads" that do not causally interact with one another. Because a human mind-monad is unaffected by external mechanical pushes, all its actions arise from its own internal principle of development—its perceptions and appetitions. Free actions are those where the agent acts spontaneously with reason and deliberation, guided by clear and distinct perceptions rather than confused passions.

Critically, Leibniz faces the objection that his doctrine of the "complete individual concept"—which holds that every predicate true of an individual is contained in its concept from eternity—leads to fatalism. Leibniz resolves this by distinguishing between absolute (geometrical) necessity, where the contrary involves a logical contradiction, and hypothetical or moral necessity, where an individual acts according to what seems best. Because human choices do not involve logical contradiction in their opposites, they incline the will without necessitating it, thereby safeguarding genuine moral responsibility and compatibilist free will.