Home Philosophy Neologicist Foundations: Inconsistent Abstraction Principles and Part-Whole
Chapter
Licensed
Unlicensed Requires Authentication

Neologicist Foundations: Inconsistent Abstraction Principles and Part-Whole

  • Paolo Mancosu and Benjamin Siskind
Become an author with De Gruyter Brill
Explore this Subject How to publish with us
Philosophy of Logic and Mathematics
This chapter is in the book Philosophy of Logic and Mathematics

Abstract

Neologicism emerges in the contemporary debate in philosophy of mathematics with Wright’s book Frege’s Conception of Numbers as Objects (1983).Wright’s project was to show the viability of a philosophy of mathematics that could preserve the key tenets of Frege’s approach, namely the idea that arithmetical knowledge is analytic. The key result was the detailed reconstruction of how to derive, within second order logic, the basic axioms of second order arithmetic from Hume’s Principle HP ∀X, Y(#X = #Y ↔ X ≅ Y) (and definitions). This has led to a detailed scrutiny of so-called abstraction principles, of which Basic Law V BLV ∀X, Y(∂X = ∂Y ↔ ∀x (X(x) ↔ Y(x))) and HP are the two most famous instances. As is well known, Russell proved that BLV is inconsistent. BLV has been the only example of an abstraction principle from (monadic) concepts to objects giving rise to inconsistency, thereby making it appear as a sort of monster in an otherwise regular universe of abstraction principles free from this pathology. We show that BLV is part of a family of inconsistent abstractions. The main result is a theorem to the effect that second-order logic formally refutes the existence of any function F that sends concepts into objects and satisfies a ‘part-whole’ relation. In addition, we study other properties of abstraction principles that lead to formal refutability in second-order logic.

Abstract

Neologicism emerges in the contemporary debate in philosophy of mathematics with Wright’s book Frege’s Conception of Numbers as Objects (1983).Wright’s project was to show the viability of a philosophy of mathematics that could preserve the key tenets of Frege’s approach, namely the idea that arithmetical knowledge is analytic. The key result was the detailed reconstruction of how to derive, within second order logic, the basic axioms of second order arithmetic from Hume’s Principle HP ∀X, Y(#X = #Y ↔ X ≅ Y) (and definitions). This has led to a detailed scrutiny of so-called abstraction principles, of which Basic Law V BLV ∀X, Y(∂X = ∂Y ↔ ∀x (X(x) ↔ Y(x))) and HP are the two most famous instances. As is well known, Russell proved that BLV is inconsistent. BLV has been the only example of an abstraction principle from (monadic) concepts to objects giving rise to inconsistency, thereby making it appear as a sort of monster in an otherwise regular universe of abstraction principles free from this pathology. We show that BLV is part of a family of inconsistent abstractions. The main result is a theorem to the effect that second-order logic formally refutes the existence of any function F that sends concepts into objects and satisfies a ‘part-whole’ relation. In addition, we study other properties of abstraction principles that lead to formal refutability in second-order logic.

Chapters in this book

  1. Frontmatter I
  2. Contents V
  3. Preface IX
  4. Part I: Philosophy of Logic
  5. Link’s Revenge: A Case Study in Natural Language Mereology 3
  6. Universal Translatability: An Optimality- Based Justification of (Classical) Logic 37
  7. Invariance and Necessity 55
  8. Translations Between Logics: A Survey 71
  9. On the Relation of Logic to Metalogic 91
  10. Free Logic and the Quantified Argument Calculus 105
  11. Dependencies Between Quantifiers Vs. Dependencies Between Variables 117
  12. Three Types and Traditions of Logic: Syllogistic, Calculus and Predicate Logic 133
  13. Truth, Paradox, and the Procedural Conception of Fregean Sense 153
  14. Wittgenstein and Frege on Assertion 169
  15. Assertions and Their Justification: Demonstration and Self-Evidence 183
  16. Surprises in Logic: When Dynamic Formality Meets Interactive Compositionality 197
  17. Part II: Philosophy of Mathematics
  18. Neologicist Foundations: Inconsistent Abstraction Principles and Part-Whole 215
  19. What Hilbert and Bernays Meant by “Finitism” 249
  20. Wittgenstein and Turing 263
  21. Remarks on Two Papers of Paul Bernays 297
  22. The Significance of the Curry-Howard Isomorphism 313
  23. Reductions of Mathematics: Foundation or Horizon? 327
  24. What Are the Axioms for Numbers and Who Invented Them? 343
  25. Part III: Wittgenstein
  26. Following a Rule: Waismann’s Variation 359
  27. Propositions in Wittgenstein and Ramsey 375
  28. An Unexpected Feature of Classical Propositional Logic in the Tractatus 385
  29. Ontology in Tractatus Logico-Philosophicus: A Topological Approach 397
  30. Adding 4.0241 to TLP 415
  31. Understanding Wittgenstein’s Wood Sellers 429
  32. On the Infinite, In-Potentia: Discovery of the Hidden Revision of Philosophical Investigations and Its Relation to TS 209 Through the Eyes of Wittgensteinian Mathematics 441
  33. Incomplete Pictures and Specific Forms: Wittgenstein Around 1930 457
  34. „Man kann die Menschen nicht zum Guten führen“ – Zur Logik des moralischen Urteils bei Wittgenstein und Hegel 473
  35. Der Status mathematischer und religiöser Sätze bei Wittgenstein 485
  36. Gutes Sehen 499
  37. Wittgenstein’s Conjecture 515
  38. Index of Names 535
  39. Index of Subjects 539
https://doi.org/10.1515/9783110657883-014

Chapters in this book

  1. Frontmatter I
  2. Contents V
  3. Preface IX
  4. Part I: Philosophy of Logic
  5. Link’s Revenge: A Case Study in Natural Language Mereology 3
  6. Universal Translatability: An Optimality- Based Justification of (Classical) Logic 37
  7. Invariance and Necessity 55
  8. Translations Between Logics: A Survey 71
  9. On the Relation of Logic to Metalogic 91
  10. Free Logic and the Quantified Argument Calculus 105
  11. Dependencies Between Quantifiers Vs. Dependencies Between Variables 117
  12. Three Types and Traditions of Logic: Syllogistic, Calculus and Predicate Logic 133
  13. Truth, Paradox, and the Procedural Conception of Fregean Sense 153
  14. Wittgenstein and Frege on Assertion 169
  15. Assertions and Their Justification: Demonstration and Self-Evidence 183
  16. Surprises in Logic: When Dynamic Formality Meets Interactive Compositionality 197
  17. Part II: Philosophy of Mathematics
  18. Neologicist Foundations: Inconsistent Abstraction Principles and Part-Whole 215
  19. What Hilbert and Bernays Meant by “Finitism” 249
  20. Wittgenstein and Turing 263
  21. Remarks on Two Papers of Paul Bernays 297
  22. The Significance of the Curry-Howard Isomorphism 313
  23. Reductions of Mathematics: Foundation or Horizon? 327
  24. What Are the Axioms for Numbers and Who Invented Them? 343
  25. Part III: Wittgenstein
  26. Following a Rule: Waismann’s Variation 359
  27. Propositions in Wittgenstein and Ramsey 375
  28. An Unexpected Feature of Classical Propositional Logic in the Tractatus 385
  29. Ontology in Tractatus Logico-Philosophicus: A Topological Approach 397
  30. Adding 4.0241 to TLP 415
  31. Understanding Wittgenstein’s Wood Sellers 429
  32. On the Infinite, In-Potentia: Discovery of the Hidden Revision of Philosophical Investigations and Its Relation to TS 209 Through the Eyes of Wittgensteinian Mathematics 441
  33. Incomplete Pictures and Specific Forms: Wittgenstein Around 1930 457
  34. „Man kann die Menschen nicht zum Guten führen“ – Zur Logik des moralischen Urteils bei Wittgenstein und Hegel 473
  35. Der Status mathematischer und religiöser Sätze bei Wittgenstein 485
  36. Gutes Sehen 499
  37. Wittgenstein’s Conjecture 515
  38. Index of Names 535
  39. Index of Subjects 539
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 17.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/9783110657883-014/html
Scroll to top button