Related papers: The logic of the reverse mathematics zoo
In this paper, we discuss different models for human logic systems and describe a game with nature. G\"odel`s incompleteness theorem is taken into account to construct a model of logical networks based on axioms obtained by symmetry…
Redox biology underpins signalling, metabolism, immunity, and adaptation, yet lacks a unifying theoretical framework capable of formalising structure, function, and dynamics. Current interpretations rely on descriptive catalogues of…
This paper explores relational syllogistic logics, a family of logical systems related to reasoning about relations in extensions of the classical syllogistic. These are all decidable logical systems. We prove completeness theorems and…
We introduce and investigate a family of consequence relations with the goal of capturing certain important patterns of data-driven inference. The inspiring idea for our framework is the fact that data may reject, possibly to some degree,…
The paper proposes a derivation system for a logic of presuppositions as introduced by P. F. Strawson. It is based on truth-relevant logic described by M. Richard Diaz in 1981. In another paper I outlined a derivation system for t-relevant…
The purpose of this paper is to introduce justification logics based on conditional logics. We introduce a new family of logics, called conditional justification logics, which incorporates a counterfactual conditional in its language. For…
Non-classical negations may fail to be contradictory-forming operators in more than one way, and they often fail also to respect fundamental meta-logical properties such as the replacement property. Such drawbacks are witnessed by intricate…
The outline of a programme for restructuring mathematical logic. We explain what we mean by "restructuring" and carry out exemplary parts of the programme.
One of the major open problems in automata and logic is the following: is there an algorithm which inputs a regular tree language and decides if the language can be defined in first-order logic? The goal of this paper is to present this…
We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present…
We present a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the predicate calculus as well as a formal induction principle. We introduce recursive systems generating…
We examine a number of results of infinite combinatorics using the techniques of reverse mathematics. Our results are inspired by similar results in recursive combinatorics. Theorems included concern colorings of graphs and bounded graphs,…
We present a modification of the superposition calculus that is meant to generate consequences of sets of first-order axioms. This approach is proven to be sound and deductive-complete in the presence of redundancy elimination rules,…
As suggested by the title, the aim of this paper is to uncover the vast computational content of classical Nonstandard Analysis. To this end, we formulate a template $\mathfrak{CI}$ which converts a theorem of 'pure' Nonstandard Analysis,…
There are close similarities between the Weihrauch lattice and the zoo of axiom systems in reverse mathematics. Following these similarities has often allowed researchers to translate results from one setting to the other. However, amongst…
We assume that every element of a matrix has a small, individual error, and model it by an external number, which is the sum of a nonstandard real number and a neutrix, the latter being a convex (external) set having the group property. The…
In the first part of this paper we try to explain to a general mathematical audience some of the remarkable web of conjectures linking representations of Galois groups with algebraic geometry, complex analysis and discrete subgroups of Lie…
Our earlier publications showed semantic tableau admits partial exceptions to the Second Incompleteness Theorem where a formalism recognizes its self consistency and views multiplication as a 3-way relation (rather than as a total…
The program Reverse Mathematics in the foundations of mathematics seeks to identify the minimal axioms required to prove theorems of ordinary mathematics. One always assumes the base theory, a logical system embodying computable…
The logic of abduction involves a collision between deduction and induction, where empirical surprises violate expectations and scientists innovate to resolve them. Here we reformulate abduction as a social process, occurring not only…