Related papers: Logic and Paradox

An inductive logic can be formulated in which the elements are not propositions or probability distributions, but information systems. The logic is complete for information systems with binary hypotheses, i.e., it applies to all such…

Defining various dishonest notions in a formal way is a key step to enable intelligent agents to act in untrustworthy environments. This review evaluates the literature for this topic by looking at formal definitions based on modal logic as…

We show that self-reference can be formalized in Basic logic by means of the new connective @, called "entanglement". In fact, the property of non-idempotence of the connective @ is a metatheorem, which states that a self-entangled sentence…

It is well known that the resolution method (for propositional logic) is complete. However, completeness proofs found in the literature use an argument by contradiction showing that if a set of clauses is unsatisfiable, then it must have a…

Reasoning is central to human intelligence. However, fallacious arguments are common, and some exacerbate problems such as spreading misinformation about climate change. In this paper, we propose the task of logical fallacy detection, and…

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…

We describe a graph-theoretic syntax for self-referential formulas as well as a four-valued logic to include contradictory and independent formulas. We then explore the degree to which generalized truth tables can be realized in our theory,…

After examining the {\bf P} versus {\bf NP} problem against the Kleene-Rosser paradox of the $\lambda$-calculus [94], it was found that it represents a counter-example to NP-completeness. We prove that it contradicts the proof of Cook's…

Since the diagonal lemma plays a key role in the proof of the main limitative theorems of logic, its proof could shed light on the very essence of these fundamental theorems. Yet the lemma is often characterized as one of those important…

A logic is defined that allows to express information about statistical probabilities and about degrees of belief in specific propositions. By interpreting the two types of probabilities in one common probability space, the semantics given…

Different notions of the consistency of obligations collapse in standard deontic logic. In justification logics, which feature explicit reasons for obligations, the situation is different. Their strength depends on a constant specification…

We discuss the problems of incompleteness and inexpressibility. We introduce almost self-referential formulas, use them to extend set theory, and relate their expressive power to that of infinitary logic. We discuss the nature of proper…

Reiter's original definition of default logic allows for the application of a default that contradicts a previously applied one. We call failure this condition. The possibility of generating failures has been in the past considered as a…

Formal logic enables computers to reason in natural language by representing sentences in symbolic forms and applying rules to derive conclusions. However, in what our study characterizes as "rulebreaker" scenarios, this method can lead to…

A logic family is a bunch of logics that belong together in some way. First-order logic is one of the examples. Logics organized into a structure occurs in abstract model theory, institution theory and in algebraic logic. Logic families…

A semantics is given to possibilistic logic, a logic that handles weighted classical logic formulae, and where weights are interpreted as lower bounds on degrees of certainty or possibility, in the sense of Zadeh's possibility theory. The…

The use of meta-rules in logic, i.e., rules whose content includes other rules, has recently gained attention in the setting of non-monotonic reasoning: a first logical formalisation and efficient algorithms to compute the (meta)-extensions…

As the etymology of the word shows, logic is intimately related to language, as exemplified by the work of philosophers from Antiquity and from the Middle-Age. At the beginning of the XX century, the crisis of the foundations of mathematics…

As Large Language Models become more ubiquitous across domains, it becomes important to examine their inherent limitations critically. This work argues that hallucinations in language models are not just occasional errors but an inevitable…

The article proposes a new technique for proving the undefinability of logical connectives through each other and illustrates the technique with several examples. Some of the obtained results are new proofs of the existing theorems, others…