Related papers: Implicit Commitment in a General Setting
This introduction begins with a section on fundamental notions of mathematical logic, including propositional logic, predicate or first-order logic, completeness, compactness, the L\"owenheim-Skolem theorem, Craig interpolation, Beth's…
The expression problem describes a fundamental tradeoff between two types of extensibility: extending a type with new operations, such as by pattern matching on an algebraic data type in functional programming, and extending a type with new…
Generic programming (GP) is an increasingly important trend in programming languages. Well-known GP mechanisms, such as type classes and the C++0x concepts proposal, usually combine two features: 1) a special type of interfaces; and 2)…
In much discussed work Artemov has recently shown that, for $\mathrm{PA}$, the consistency schema admits a form of uniform verification via selector proofs, despite the unprovability of the corresponding uniform consistency sentence…
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…
Deep and shallow embeddings of non-classical logics in classical higher-order logic have been explored, implemented, and used in various reasoning tools in recent years. This paper presents a method for the simultaneous deployment of deep…
The famous G\"odel incompleteness theorem says that for every sufficiently rich formal theory (containing formal arithmetic in some natural sense) there exist true unprovable statements. Such statements would be natural candidates for being…
We provide a novel semantics for belief using simplicial complexes. In our framework, belief satisfies the \textsf{KD45} axioms and rules as well as the ``knowledge implies belief'' axiom ($K\phi \lthen B\phi$); in addition, we adopt the…
Godel's First Incompleteness Theorem is generalized to definable theories, which are not necessarily recursively enumerable, by using a couple of syntactic-semantic notions, one is the consistency of a theory with the set of all true…
Is it possible to encompass the full extent of the universe within a theory based on a finite set of first principles and inference rules? The r\^{o}le of observers and observations in physics theories is considered here in the light of…
Large Language Models (LLMs) have demonstrated strong generalization across a wide range of tasks. Reasoning with LLMs is central to solving multi-step problems and complex decision-making. To support efficient reasoning, recent studies…
Belief systems are often treated as globally consistent sets of propositions or as scalar-valued probability distributions. Such representations tend to obscure the internal structure of belief, conflate external credibility with internal…
Across machine learning (ML) sub-disciplines, researchers make explicit mathematical assumptions in order to facilitate proof-writing. We note that, specifically in the area of fairness-accuracy trade-off optimization scholarship, similar…
Abstract algebraic logic is a theory that provides general tools for the algebraic study of arbitrary propositional logics. According to this theory, every logic L is associated with a matrix semantics Mod*(L). This paper is a contribution…
We rely on the strength of linguistic and philosophical perspectives in constructing a framework that offers a unified explanation for presuppositions and existential commitment. We use a rich ontology and a set of methodological principles…
Implicit computational complexity, which aims at characterizing complexity classes by machine-independent means, has traditionally been based, on the one hand, on programs and deductive formalisms for free algebras, and on the other hand on…
Goedel's completeness theorem is concerned with provability, while Girard's theorem in ludics (as well as full completeness theorems in game semantics) are concerned with proofs. Our purpose is to look for a connection between these two…
The need for formal definition of the very basis of mathematics arose in the last century. The scale and complexity of mathematics, along with discovered paradoxes, revealed the danger of accumulating errors across theories. Although,…
The fact that the famous Godel incompleteness theorem and the archetype of all logical paradoxes, that of the Liar, are related closely is, of course, not only well known, but is a part of the common knowledge of logician community.…
This position paper argues that text embedding research should move beyond surface meaning and embrace implicit semantics as a central modeling objective. Text embeddings are a foundational component of modern NLP, underpinning a wide range…