Related papers: G\"odel Mirror: A Formal System For Contradiction-…
Large language models are often described as capable of reflective reasoning, yet recursive self-evaluation without external feedback frequently yields reformulation rather than progress. We test this prediction in a cross-provider study of…
Logical paradoxes and inconsistent information pose deep challenges in epistemology and the philosophy of logic. Classical systems typically handle contradictions only through external checks or by altering the logical framework, as in…
This paper discusses limitations of reflexive and diagonal arguments as methods of proof of limitative theorems (e.g. G\"odel's theorem on Entscheidungsproblem, Turing's halting problem or Chaitin-G\"odel's theorem). The fact, that a formal…
This paper proposes a modal typing system that enables us to handle self-referential formulae, including ones with negative self-references, which on one hand, would introduce a logical contradiction, namely Russell's paradox, in the…
While Large language models (LLMs) have the capability to iteratively reflect on their own outputs, recent studies have observed their struggles with knowledge-rich problems without access to external resources. In addition to the…
We first partly develop a mathematical notion of stable consistency intended to reflect the actual consistency property of human beings. Then we give a generalization of the first and second G\"odel incompleteness theorem to stably…
In this paper we explore several fundamental relations between formal systems, algorithms, and dynamical systems, focussing on the roles of undecidability, universality, diagonalization, and self-reference in each of these computational…
A new computational method that uses polynomial equations and dynamical systems to evaluate logical propositions is introduced and applied to Goedel's incompleteness theorems. The truth value of a logical formula subject to a set of axioms…
Propositional G\"odel logic extends intuitionistic logic with the non-constructive principle of linearity $A\rightarrow B\ \lor\ B\rightarrow A$. We introduce a Curry-Howard correspondence for this logic and show that a particularly simple…
As automated reasoning systems advance rapidly, there is a growing need for research-level formal mathematical problems to accurately evaluate their capabilities. To address this, we present Formal Conjectures, an evolving benchmark of…
Formal mathematics and computer science proofs are formalized using Hilbert-Russell-style logical systems which are designed to not admit paradoxes and self-refencing reasoning. These logical systems are natural way to describe and reason…
The human-like automatic deductive reasoning has always been one of the most challenging open problems in the interdiscipline of mathematics and artificial intelligence. This paper is the third in a series of our works. We built a…
We identify a structural property of term-rewriting proof systems called operational inexpressibility: no derivation depends on a specified input dimension and also constrains the target question. The canonical instance is direct…
Formal, automated theorem proving has long been viewed as a challenge to artificial intelligence. We introduce here a new approach to computer theorem proving, one that employs specialized language models for Lean4 proof generation combined…
There is an increasing interest in applying recent advances in AI to automated reasoning, as it may provide useful heuristics in reasoning over formalisms in first-order, second-order, or even meta-logics. To facilitate this research, we…
This paper develops an algorithmic-based approach for proving inductive properties of propositional sequent systems such as admissibility, invertibility, cut-elimination, and identity expansion. Although undecidable in general, these…
Inconsistency Robustness is performance of information systems with pervasively inconsistent information. Inconsistency Robustness of the community of professional mathematicians is their performance repeatedly repairing contradictions over…
G\"odel's Dialectica has been introduced and developed in the tradition of the so-called functional interpretations. Only recently has it been related with the a priori unrelated notion of differentiation, by taking a program-theoretic…
G\"odel's second incompleteness theorem is standardly understood as showing that no sufficiently strong, consistent theory of arithmetic can prove its own consistency, a result typically interpreted against a model-theoretic background in…
We present the design and implementation of the Small Scale Reflection proof methodology and tactic language (a.k.a. SSR) for the Lean 4 proof assistant. Like its Coq predecessor SSReflect, our Lean 4 implementation, dubbed LeanSSR,…