Bibliographic record
Abstract
introduction A surprising feature of Russell's work in logic is that he began and ended with a theory of types. This chapter begins with a summary of the 1903 theory of types and then proceeds to the much more complex ramified theory of types that emerged from Russell's intense work on the foundations of logic from 1903 to 1907. After discussing the problems connected with the Axiom of Reducibility, the chapter concludes with the simple theory of types, and the later history of type theory, after the demise of the logicist programme. the 1903 theory of types Russell’s early theory of types, presented in Appendix B to the Principles of Mathematics , already contains many of the basic features of the mature system given in his fundamental paper of 1908 and in Principia Mathematica . In 1901, Russell had begun writing out the derivation of mathematics from logic, employing the methods of Peano and his school. This led him to examine Cantor’s proof that there is no greatest cardinal number. This result conflicted with his assumption that there is a universal class, having all objects as members, which ought to have the greatest cardinal number. Close analysis of the diagonal argument used in Cantor’s proof led to the discovery of the paradox of the class of all classes that are not members of themselves, now called “Russell’s paradox,” but which Russell called “the Contradiction.” The logical paradoxes emerged at an awkward moment, when Russell had already written most of the penultimate draft of the Principles . Rather than hold up its publication indefinitely, he took the manuscript of his book to the printer in May 1902 before finding a solution. His initial reaction was that the Contradiction was of a somewhat trivial character, and that it could be avoided by a simple modification of the primitive propositions of logic.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.004 |
| Science and technology studies | 0.003 | 0.016 |
| Scholarly communication | 0.007 | 0.015 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.013 | 0.005 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".