Frege, Boolos, and Logical Objects

  title={Frege, Boolos, and Logical Objects},
  author={David J. Anderson and Edward N. Zalta},
  journal={Journal of Philosophical Logic},
In this paper, the authors discuss Frege's theory of “logical objects” (extensions, numbers, truth-values) and the recent attempts to rehabilitate it. We show that the ‘eta’ relation George Boolos deployed on Frege's behalf is similar, if not identical, to the encoding mode of predication that underlies the theory of abstract objects. Whereas Boolos accepted unrestricted Comprehension for Properties and used the ‘eta’ relation to assert the existence of logical objects under certain highly… 

Relations Versus Functions at the Foundations of Logic: Type-Theoretic Considerations

There is an interesting system having a logic that can be properly characterized in relational type theory but not in functional type theory (FTT), which shows that RTT is more general than FTT.

A Defense of Logicism∗

We argue that logicism, the thesis that mathematics is reducible to logic and analytic truths, is true. We do so by (a) developing a formal framework with comprehension and abstraction principles,

What is the Purpose of Neo-Logicism ?

OT was originally developed as a formal theory of fictional, abstract, and intensional objects inspired by the work of Meinong’s student Ernst Mally and the current bible of second-order logic is (Shapiro, 1991).

Avoiding Russell-Kaplan Paradoxes : Worlds and Propositions Set Free ∗ ( DRAFT )

The authors first address two paradoxes in the theory of possible worlds and propositions stemming from Russell and Kaplan and show that these paradoxes don’t affect the object-theoretic analysis of

What is Neologicism?

According to one well-received view, logicism was replaced by a very different account of the foundations of mathematics, in which mathematics was seen as the study of axioms and their consequences in models consisting of the sets described by Zermelo-Fraenkel set theory (ZF).

Typed Object Theory

It is shown how typed object theory compares with other intensional type theories, and a number of natural language contexts without requiring the technique of ‘type-raising’ are analyzed.

Reflections on Mathematics ∗

Though the philosophy of mathematics encompasses many kinds of questions, my response to the five questions primarily focuses on the prospects of developing a unified approach to the metaphysical and

What did Frege take Russell to have proved?

In 1902 there arrived in Jena a letter from Russell laying out a proof that shattered Frege’s confidence in logicism, which is widely taken to be the doctrine according to which every truth of

Essence and Modality

  • E. Zalta
  • Philosophy, Computer Science
  • 2006
The principal axiom for abstract objects, described in more detail below, is a comprehension principle that asserts the conditions under which abstract objects exist and encode properties: for any expressible condition 4 that is satisfiable by properties F, there exists an abstract object that encodes exactly the properties F satisfying.

Foundations for Mathematical Structuralism

We investigate the form of mathematical structuralism that acknowledges the existence of structures and their distinctive structural elements. This form of structuralism has been subject to



Intensional Logic and the Metaphysics of Intentionality

In this book, Edward N. Zalta tackles the issues that arise in connection with intensional logic - a formal system for representing and explaining the apparent failures of certain important

Consistent Fragments of Grundgesetze and the Existence of Non-Logical Objects

It is shown that both Heck and TΔ prove the existence of infinitely many non-logical objects (TΔ deriving, moreover, the nonexistence of the value-range concept), and some implications concerning the interpretation of Frege's proof of referentiality and the possibility of classifying any of these subsystems as logicist are discussed.

Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege"s Grundgesetze in Object Theory

The author derives the Dedekind–Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects, with the exception of the claim that every number has a successor, which is derived from a modal axiom that (philosophical) logicians implicitly accept.

The Consistency of predicative fragments of frege’s grundgesetze der arithmetik

As is well-known, the formal system in which Frege works in his Grundgesetze der Arithmetik is formally inconsistent, Russell’s Paradox being derivable in it.This system is, except for minor


Mathematics1 begins with the question: how did the serpent of inconsistency enter Frege's paradise? In the section of that chapter called 'How the serpent entered Eden' Dummett says, 'The secondorder

On the Consistency of the Δ11-CA Fragment of Frege's Grundgesetze

This paper shows that the more encompassing Δ11-comprehension schema would already be inconsistent, and proves the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicative formulae.

Twenty-five basic theorems in situation and world theory

All twenty-five theorems seem to be basic, reasonable principles that structure the domains of properties, relations, states of affairs, situations, and worlds in true and philosophically interesting ways.

The development of arithmetic in Frege's Grundgesetze der arithmetik

Frege's use of the inconsistent Axiom V may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establishment of Logicism), but it should not be allowed to obscure his mathematical accomplishments and his contribution to the authors' understanding of arithmetic.

Frege's philosophy of mathematics

Volume I: Frege's Philosophy in Context Part 1. Frege's Life and Work Part 2. Frege and Other Philosophers Part 3. Frege's Epistemology and Metaphysics Volume II: Frege's Philosophy of Logic Part 4.

Fregean Extensions of First-Order Theories

  • J. Bell
  • Philosophy, Mathematics
    Math. Log. Q.
  • 1994
A stronger version of Frege's logical system in the Grundgesetze der Arithmetic is proved for arbitrary first-order theories and a natural attempt to further strengthen this result runs afoul of Tarski's theorem on the undefinability of truth.