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Frege's paper "On Sense and Reference" (1892) is seminal, containing Frege's puzzles about identity and advancing a mediated reference theory. Frege points out the reference of "the Morning Star" and "the Evening Star" is the same: both refer to the planet Venus. Therefore, substituting one term for the other doesn't change the truth value (salva veritate). However, they differ in what Frege calls cognitive value or the mode of presentation. One has to distinguish between two notions of meaning: the reference of a term and the sense of a term. As Frege points out, "the Morning Star is the Morning Star" is uninformative, but "the Morning Star is the Evening Star" is informative; thus, the two expressions must differ in a way other than reference. A related puzzle is also known as Frege's puzzle, concerning intensional contexts and propositional attitude reports. Consider the statement "The ancients believed the morning star is the evening star." This statement might be false. However, the statement "The ancients believed the morning star is the morning star" is obviously true. Here again, the morning star and the evening star have different meanings, despite having the same reference. In Frege's paper "On Concept and Object" (1892) he distinguishes between a concept which is the reference of a predicate, and an object which is the reference of a proper name.

==== Thought ====

The paper "The Thought: A Logical Inquiry" (1918) reflects Frege's anti-idealism. He argues for a Platonist account of propositions or thoughts. Frege claims propositions are intangible, like ideas; yet publicly available, like an object. In addition to the physical, public "first realm" of objects and the private, mental "second realm" of ideas, Frege posits a "third realm" of Platonic propositions, such as the Pythagorean theorem.

== Revolt against idealism == British philosophy in the nineteenth century saw a revival of logic started by Richard Whately, in reaction to the anti-logical tradition of British empiricism. The major figure of this period is mathematician George Boole. Other figures include Scottish metaphysician William Hamilton, mathematician Augustus De Morgan, economist William Stanley Jevons, diagram namesake John Venn, Alice's Adventures in Wonderland author Lewis Carroll, Scottish mathematician Hugh MacColl, and American pragmatist Charles Sanders Peirce. However, British philosophy in the late nineteenth century was dominated by British idealism, a neo-Hegelian movement, as taught by philosophers such as F. H. Bradley and T. H. Green. Bradley's work Appearance and Reality (1893) exemplified the school.

Analytic philosophy in the narrower sense of twentieth-century anglophone philosophy is usually thought to begin with Cambridge philosophers Bertrand Russell and G. E. Moore's rejection of Hegelianism for being obscure; or the "revolt against idealism." Russell summed up Moore's common sense influence:

"G. E. Moore...took the lead in rebellion, and I followed, with a sense of emancipation. Bradley had argued that everything common sense believes in is mere appearance; we reverted to the opposite extreme, and that everything is real that common sense, uninfluenced by philosophy or theology, supposes real. With a sense of escaping from prison, we allowed ourselves to think that grass is green, that the sun and stars would exist if no one was aware of them, and also that there is a pluralistic timeless world of Platonic ideas." Russell and Moore contributed to the philosophy of perception with a naïve realism and sense-data theory. In America, the New Realists opposed idealism.

=== Logical atomism === An important aspect of Hegelianism and British idealism was logical holism—the belief that aspects of the world can be known only by knowing the whole world. This is closely related to the doctrine of internal relations, the belief that relations between items are internal relations, or essential properties the items have by nature. Russell and Moore in response promulgated logical atomism and the doctrine of external relations—the belief that the world consists of independent facts.

== Russell ==

In 1901, Russell famously discovered the paradox in Basic Law V (also known as unrestricted comprehension), which undermined Frege's set theory. However, Russell was still a logicist, and in The Principles of Mathematics (1903), he also argued for Meinongianism.

=== Theory of descriptions === During his early career, Russell adopted Frege's predicate logic as his primary philosophical method, thinking it could expose the underlying structure of philosophical problems. This was done most famously in his theory of definite descriptions in "On Denoting", published in Mind in 1905. The essay has been called a "paradigm of philosophy." In this essay, Russell responds to both Meinong and Frege. Russell uses his analysis of descriptions to solve ascriptions of nonexistence, such as with "the present King of France". He argues all proper names (aside from demonstratives like this or that) are disguised definite descriptions; for example, "Walter Scott" can be replaced with "the author of Waverley". This position came to be called descriptivism.

Russell presents his own version of Frege's second puzzle. "If a is identical with b, whatever is true of the one is true of the other, and either may be substituted for the other without altering the truth or falsehood of that proposition. Now George IV wished to know whether Scott was the author of Waverley; and in fact Scott was the author of Waverley. Hence we may substitute “Scott” for “the author of Waverley” and thereby prove that George IV wished to know whether Scott was Scott. Yet an interest in the law of identity can hardly be attributed to the first gentleman of Europe.” The essay also illustrates the concept of scope ambiguity by showing how denying "The present King of France is bald" can mean either "There is no King of France" or "The present King of France is not bald". Russell quips "Hegelians, who love a synthesis, will probably conclude that he wears a wig." For Russell, there was knowledge by description and, from sense-data theory, knowledge by acquaintance.