Related papers: Algebraic Semantics for the Logic of Proofs
Artificial intelligence assisted mathematical proof has become a highly focused area nowadays. One key problem in this field is to generate formal mathematical proofs from natural language proofs. Due to historical reasons, the formal proof…
A grammar logic refers to an extension to the multi-modal logic K in which the modal axioms are generated from a formal grammar. We consider a proof theory, in nested sequent calculus, of grammar logics with converse, i.e., every modal…
Computability logic is a formal theory of computational tasks and resources. Its formulas represent interactive computational problems, logical operators stand for operations on computational problems, and validity of a formula is…
We introduce hybrid algebras as algebraic semantics for hybrid languages with nominals and, possibly, the satisfaction operator. We establish a duality between hybrid algebras and the descriptive two-sorted general frames of Ten Cate. We…
A new computational method that uses polynomial equations and dynamical systems to evaluate logical propositions is introduced and applied to Goedel's incompleteness theorems. The truth value of a logical formula subject to a set of axioms…
We provide a mathematical theory and methodology for synthesising equational logics from algebraic metatheories. We illustrate our methodology by means of two applications: a rational reconstruction of Birkhoff's Equational Logic and a new…
In a recent work we have shown how to construct an information algebra of coherent sets of gambles defined on general possibility spaces. Here we analyze the connection of such an algebra with the set algebra of subsets of the possibility…
It is proved that every prevariety of algebras is categorically equivalent to a "prevariety of logic", i.e., to the equivalent algebraic semantics of some sentential deductive system. This allows us to show that no nontrivial equation in…
The logic of definitions is a family of logics for encoding and reasoning about judgments, which are atomic predicates specified by inference rules. A definition associates an atomic predicate with a logical formula, which may itself depend…
In this paper we provide two new semantics for proofs in the constructive modal logics CK and CD. The first semantics is given by extending the syntax of combinatorial proofs for propositional intuitionistic logic, in which proofs are…
The author has recently introduced an abstract algebraic framework of analogical proportions within the general setting of universal algebra. The purpose of this paper is to lift that framework from universal algebra to the strictly more…
By limiting the range of the predicate variables in a second-order language one may obtain restricted versions of second-order logic such as weak second-order logic or definable subset logic. In this note we provide an infinitary strongly…
The Boolean logic of subsets, usually presented as `propositional logic,' is considered as being "classical" while intuitionistic logic and the many sublogics and off-shoots are "non-classical." But there is another mathematical logic, the…
There are many different semantics for general logic programs (i.e. programs that use negation in the bodies of clauses). Most of these semantics are Turing complete (in a sense that can be made precise), implying that they are undecidable.…
The fundamental duality theories relating algebra and geometry that were discovered in the mid-20th century can also be applied to logic via its algebraization under categorical logic. They thereby result in known and new completeness…
Mathematical proofs are often said to justify their conclusions by indicating the existence of a corresponding formal derivation. We argue that this widespread view relies on an under-examined notion of correspondence, or what it means for…
The paper introduces the notion of off-line justification for Answer Set Programming (ASP). Justifications provide a graph-based explanation of the truth value of an atom w.r.t. a given answer set. The paper extends also this notion to…
We try to bring to light some combinatorial structure underlying formal proofs in logic. We do this through the study of the Craig Interpolation Theorem which is properly a statement about the structure of formal derivations. We show that…
The field of proof-theoretic semantics (P-tS) offers an alternative approach to meaning in logic that is based on inference and argument (rather than truth in a model). It has been successfully developed for various logics; in particular,…
Dynamic evidence logics are logics for reasoning about the evidence and evidence-based beliefs of agents in a dynamic environment. In this paper, we introduce a family of logics for reasoning about relational evidence: evidence that…