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On the real numbers, the notions of a semi-decidable relation and that of an effectively enumerable relation differ. The second only seems to be adequate to express, in an algorithmic way, non deterministic physical theories, where…

We employ the theory of canonical extensions to study residuation algebras whose associated relational structures are functional, i.e., for which the ternary relations associated to the expanded operations admit an interpretation as…

We propose a new definition of actual causes, using structural equations to model counterfactuals.We show that the definitions yield a plausible and elegant account ofcausation that handles well examples which have caused problems forother…

The notion of a non-deterministic logical matrix (where connectives are interpreted as multi-functions) extends the traditional semantics for propositional logics based on logical matrices (where connectives are interpreted as functions).…

Abstract axiomatic formulation of mathematical structures are extensively used to describe our physical world. We take here the reverse way. By making basic assumptions as starting point, we reconstruct some features of both geometry and…

We argue that Godel's completeness theorem is equivalent to completability of consistent theories, and Godel's incompleteness theorem is equivalent to the fact that this completion is not constructive, in the sense that there are some…

Constructor theory seeks to express all fundamental scientific theories in terms of a dichotomy between possible and impossible physical transformations - those that can be caused to happen and those that cannot. This is a departure from…

Recursive relational specifications are commonly used to describe the computational structure of formal systems. Recent research in proof theory has identified two features that facilitate direct, logic-based reasoning about such…

We define constructive truth for arithmetic and for intuitionistic analysis, and investigate its properties. We also prove that the set of constructively true (first order) arithmetical statements is Pi-1-2 and Sigma-1-2 hard, and we…

Classical set theory constructs the continuum via the power set P(N), thereby postulating an uncountable totality. However, constructive and computability-based approaches reveal that no formal system with countable syntax can generate all…

In this paper, we argue that formal systems of first order Arithmetic that admit Goedelian undecidable propositions validly are abnormally non-constructive. We argue that, in such systems, the strong representation of primitive recursive…

All constructive methods employed in modern mathematics produce only countable sets, even when designed to transcend countability. We show that any constructive argument for uncountability -- excluding diagonalization techniques --…

Cantor's ordinal numbers, a powerful extension of the natural numbers, are a cornerstone of set theory. They can be used to reason about the termination of processes, prove the consistency of logical systems, and justify some of the core…

We survey the logical structure of constructive set theories and point towards directions for future research. Moreover, we analyse the consequences of being extensible for the logical structure of a given constructive set theory. We…

Understanding the predictions made by deep learning models remains a central challenge, especially in high-stakes applications. A promising approach is to equip models with the ability to answer counterfactual questions -- hypothetical…

The recent trend in mathematics is towards a framework of abstract mathematical objects, rather than the more concrete approach of explicitly defining elements which objects were thought to consist of. A natural question to raise is whether…

Galles and Pearl claimed that "for recursive models, the causal model framework does not add any restrictions to counterfactuals, beyond those imposed by Lewis's [possible-worlds] framework." This claim is examined carefully, with the goal…

An age-old controversy in mathematics concerns the necessity and the possibility of constructive proofs. The controversy has been rekindled by recent advances which demonstrate the feasibility of a fully constructive mathematics. This…

This article was motivated by the discovery of a potential new foundation for mainstream mathematics. The goals are to clarify the relationships between primitives, foundations, and deductive practice; to understand how to determine what…

There is knowledge. There is belief. And there is tacit agreement.' 'We may talk about objects. We may talk about attributes of the objects. Or we may talk both about objects and their attributes.' This work inspects tacit agreements on…