Related papers: The Potential in Frege's Theorem

Bayesian reasoning plays a significant role both in human rationality and in machine learning. In this paper, we introduce transfinite modal logic, which combines modal logic with ordinal arithmetic, in order to formalize Bayesian reasoning…

The general completeness problem of Hoare logic relative to the standard model $N$ of Peano arithmetic has been studied by Cook, and it allows for the use of arbitrary arithmetical formulas as assertions. In practice, the assertions would…

The investigations on higher-order type theories and on the related notion of parametric polymorphism constitute the technical counterpart of the old foundational problem of the circularity (or impredicativity) of second and higher order…

In this paper we introduce the concept of an infinite loop mod $n$ and discuss the properties that these objects have. In particular, we show that a real number $\alpha$ is a counterexample to the $p$-adic Littlewood Conjecture if and only…

A probabilistic propositional logic, endowed with an epistemic component for asserting (non-)compatibility of diagonizable and bounded observables, is presented and illustrated for reasoning about the random results of projective…

I argue that John Norton's notions of empirical, hypothetical, and counterfactual possibility can be successfully used to analyze counterintuitive examples of physical possibility and align better with modal intuitions of practicing…

We consider the question of extending propositional logic to a logic of plausible reasoning, and posit four requirements that any such extension should satisfy. Each is a requirement that some property of classical propositional logic be…

Elimination of quantifiers is shown to fail dramatically for a group of well-known mathematical theories (classically enjoying the property) against a wide range of relevant logical backgrounds. Furthermore, it is suggested that only by…

Many of the theorems of real analysis, against the background of the ordered field axioms, are equivalent to Dedekind completeness, and hence can serve as completeness axioms for the reals. In the course of demonstrating this, the article…

Being mathematics a natural language to Mankind and to physics, it must be constantly adapted to our necessities and our natural perception. Then, mathematical concepts are not absolute to reality. Although mathematical theories are…

Tim Maudlin has claimed that EPR's Reality Criterion is analytically true. We argue that it is not. Moreover, one may be a subjectivist about quantum probabilities without giving up on objective physical reality. Thus, would-be detractors…

In this paper we prove Chaitin's ``heuristic principle'', {\it the theorems of a finitely-specified theory cannot be significantly more complex than the theory itself}, for an appropriate measure of complexity. We show that the measure is…

In this paper I aim to defend one version at least of Hume's dictum: roughly, the idea that possibility is determined by ontology through something like independent variation. My defence is broadly pragmatic, in the sense that adherence to…

In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…

No quantitative theory describing all physical phenomena can be made if any arbitrary standard spacetime structure is assumed. This statement is a consequence of transforming the Peano arithmetic axioms into sentences with a physical…

We formally define a "mathematical object" and "set". We then argue that expressions such as "(Ax)F(x)", and "(Ex)F(x)", in an interpretation M of a formal theory P, may be taken to mean "F(x) is true for all x in M", and "F(x) is true for…

Propositional term modal logic is interpreted over Kripke structures with unboundedly many accessibility relations and hence the syntax admits variables indexing modalities and quantification over them. This logic is undecidable, and we…

We study elementary modal logics, i.e. modal logic considered over first-order definable classes of frames. The classical semantics of modal logic allows infinite structures, but often practical applications require to restrict our…

Crispin Wright in his 1982 paper argues for strict finitism, a constructive standpoint that is more restrictive than intuitionism. In its appendix, he proposes models of strict finitistic arithmetic. They are tree-like structures, formed in…

In this short note we give an alternative proof of Glivenko's Theorem, stating that a formula $\phi$ is provable in classical propositional logic if and only if $\neg\neg\phi$ is provable in intuitionistic propositional logic. We work in…