Related papers: On sequents of $\Sigma$ formulas
One of the main claims of the paper is that Dirac's calculus and broader theories of physics can be treated as theories written in the language of Continuous Logic. Establishing its true interpretation (model) is a model theory problem. The…
Foundation models are subject to an ongoing heated debate, leaving open the question of progress towards AGI and dividing the community into two camps: the ones who see the arguably impressive results as evidence to the scaling hypothesis,…
We explore the implications of restricting the framework of quantum theory and quantum computation to finite fields. The simplest proposed theory is defined over arbitrary finite fields and loses the notion of unitaries. This makes such…
It is useful to have a criterion for when the predictions of an operational theory should be considered classically explainable. Here we take the criterion to be that the theory admits of a generalized-noncontextual ontological model.…
This paper proposes an alternative to standard first-order logic that seeks greater naturalness, generality, and semantic self-containment. The system removes the first-order restriction, avoids type hierarchies, and dispenses with external…
We study the implications of model completeness of a theory for the effectiveness of presentations of models of that theory. It is immediate that for a computable model $\mathcal A$ of a computably enumerable, model complete theory, the…
We demonstrate that many Naturalness tests of particle theories discussed in the literature can be reformulated as straightforward algorithmic finetuning assessments in the matching of Wilsonian effective theories above and below particle…
Expectation is a central notion in probability theory. The notion of expectation also makes sense for other notions of uncertainty. We introduce a propositional logic for reasoning about expectation, where the semantics depends on the…
This paper presents a plausible reasoning system to illustrate some broad issues in knowledge representation: dualities between different reasoning forms, the difficulty of unifying complementary reasoning styles, and the approximate nature…
In this paper we give a brief review of semiparametric theory, using as a running example the common problem of estimating an average causal effect. Semiparametric models allow at least part of the data-generating process to be unspecified…
In this paper we develop cyclic proof systems for the problem of inclusion between the least sets of models of mutually recursive predicates, when the ground constraints in the inductive definitions belong to the quantifier-free fragments…
More often than not, recently popular structuralist interpretations of physical theories leave the central concept of a structure insufficiently precisified. The incipient causal sets approach to quantum gravity offers a paradigmatic case…
There are many ways we can not know. Even in systems that we created ourselves, as, for example, systems in mathematical logic, Go\"edel and Tarski's theorems impose limits on what we can know. As we try to speak of the real world, things…
We consider natural $\Sigma^1_2$ definable analogues of many of the classical statements that have been shown to be equivalent to CH. It is shown that these $\Sigma^1_2$ analogues are equivalent to that all reals are constructible. We also…
We deal with the fragment of modal logic consisting of implications of formulas built up from the variables and the constant `true' by conjunction and diamonds only. The weaker language allows one to interpret the diamonds as the uniform…
The purpose of this work is to demonstrate how an arbitrarily chosen background of the Universe can be made a solution of a simple geometric sigma model. Geometric sigma models are purely geometric theories in which spacetime coordinates…
Natural philosophy integrates scientific observation with abstract frameworks, often using a mathematical Ansatz to hypothesise about physical phenomena. Exploring the possibility of other universes, however, challenges assumptions that…
Axiomatizing mathematical structures and theories is an objective of Mathematical Logic. Some axiomatic systems are nowadays mere definitions, such as the axioms of Group Theory; but some systems are much deeper, such as the axioms of…
Self-similar models are important in general relativity and other fundamental theories. In this paper we shall discuss the ``similarity hypothesis'', which asserts that under a variety of physical circumstances solutions of these theories…
The aim of this paper is to give natural examples of $\mathbf{\Sigma}_1^1$-complete and $\mathbf{\Pi}_1^1$-complete sets. In the first part, we consider ideals on $\omega$. In particular, we show that the Hindman ideal $\mathcal{H}$ is…