F H Bradley's book The Principles of Logic is unconventional in that it is devoid of theorems, axioms, and rules, which are typically central to formal logic. Like Mill, Bradley was opposed to the formalization of reasoning because, as he expresses in The Principles of Logic, it detaches inquiry from the practical activity of obtaining scientific knowledge. He also has a tendency to blur boundaries between the fields of logic, metaphysics, epistemology, and phenomenology. Because he rejected formalized logic in a treatise on logic, it is often only used in contemporary philosophy for historic purposes, but much of his thought had an unnoticed impact on logic through its acceptance by Bertrand Russell. For one, it brought forth themes such as meaning and reference as concepts that can also be subjected to logical scrutiny.
One important argument he makes is that judgement as it applies to logic, rather than being a conjunction of two separate ideas as philosophers traditionally asserted, is actually an abstract and universal event. Rather than judgments being a combining of separate ideas, ideas are actually an abstraction of already complete judgments. Essentially, objects cannot be divided from their concepts. He uses examples such as proper names, which he claims are just disguised general terms, and specific places in space or moments in time, which only individuate objects and events in a given spatiotemporal series but do not differentiate across different spatiotemporal series. Reference is not fixed in the abstract but is inseparable from the reality of experience.
This is where his lasting influence on Russell lies: the argument that the logical form of universal sentences is hypothetical. For example, the sentence 'All tigers have stripes' can be understood as 'If anything is a tiger then it has stripes.'
When he moves from judgment to inference, he continues to reject traditional Aristotelian logic by claiming that reasoning only occurs through generalities, making universals essential to inference.