Russell held that logic is a science of logical form. But his views on the nature of logical form changed significantly as his “retreat from Pythagoras” widened. The changes are very important to understanding Russell's analysis of logical necessity. They are no less important for understanding Wittgenstein's conception of logical necessity in the Tractatus. Bradley writes that Russell's [logic] is extensionalistic, having no room for modal notions especially the de re (essentialistic) ones, while Wittgenstein's atomism is robustly modal and essentialistic. We shall find that this is far from the case.

Russell's analysis of logical necessity as universal truth

Perhaps one of the most commonly criticized doctrines of Russell's philosophical logic is his analysis of logical necessity in terms of full generality and truth. In Russell's early view, every statement which is fully general and true is logically necessary. Russell's thesis that logical necessity is properly analyzed in terms of full generality and truth must be evaluated with respect to his language for logic. Full generality, after all, is relative to a given formal language for logic. To prepare us to understand how Russell might think the analysis is plausible, consider a naive formal language for a theory of attributes with bindable predicate variables allowed in both subject and predicate positions. (This is called a “second-order logic with nominalized predicates.”) The theory is naive because it assumes a comprehension principle for attributes that yields Russell's paradox. In Russell's view, logical necessity coincides with logical truth.