Related papers: Sort logic and foundations of mathematics
We lift the SCL calculus for first-order logic without equality to the SCL(T) calculus for first-order logic without equality modulo a background theory. In a nutshell, the SCL(T) calculus describes a new way to guide hierarchic resolution…
Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of…
Succinctness is a natural measure for comparing the strength of different logics. Intuitively, a logic L_1 is more succinct than another logic L_2 if all properties that can be expressed in L_2 can be expressed in L_1 by formulas of…
We study the expressive power of successor-invariant first-order logic, which is an extension of first-order logic where the usage of an additional successor relation on the structure is allowed, as long as the validity of formulas is…
Many-valued logics in general, and fuzzy logics in particular, usually focus on a notion of consequence based on preservation of full truth, typical represented by the value 1 in the semantics given the real unit interval [0,1]. In a recent…
Matching logic is a general formal framework for reasoning about a wide range of theories, with particular emphasis on programming language semantics. Notably, the intermediate language of the K semantics framework is an extension of…
Category theory can be used to state formulas in First-Order Logic without using set membership. Several notable results in logic such as proof of the continuum hypothesis can be elegantly rewritten in category theory. We propose in this…
First-order logic is the basis for many knowledge representation formalisms and methods. Providing technological support for learning to write first-order formulas for natural language specifications requires methods to test formulas for…
We study the model-checking problem for first- and monadic second-order logic on finite relational structures. The problem of verifying whether a formula of these logics is true on a given structure is considered intractable in general, but…
We present a simpler way than usual to deduce the completeness theorem for the second-oder classical logic from the first-order one. We also extend our method to the case of second-order intuitionistic logic.
Probability theory as extended logic is completed such that essentially any probability may be determined. This is done by considering propositional logic (as opposed to predicate logic) as syntactically suffcient and imposing a symmetry…
In deduction modulo, a theory is not represented by a set of axioms but by a congruence on propositions modulo which the inference rules of standard deductive systems---such as for instance natural deduction---are applied. Therefore, the…
In this paper we consider first-order logic theorem proving and model building via approximation and instantiation. Given a clause set we propose its approximation into a simplified clause set where satisfiability is decidable. The…
This is a survey of the model theory of second order logic.
Classification theory of elementary classes deals with first order (elementary) classes of structures (i.e. fixing a set T of first order sentences, we investigate the class of models of T with the elementary submodel notion). It tries to…
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…
There is a growing need for abstractions in logic specification languages such as FO(.) and ASP. One technique to achieve these abstractions are templates (sometimes called macros). While the semantics of templates are virtually always…
Sorting is one of the most used and well investigated algorithmic problem [1]. Traditional postulation supposes the sorting data archived, and the elementary operation as comparisons of two numbers. In a view of appearance of new processors…
Many theorems of mathematics have the form that for a certain problem, e.g. a differential equation or polynomial (in)equality, there exists a solution. The sequential version then states that for a sequence of problems, there is a sequence…
We consider a language together with the subword relation, the cover relation, and regular predicates. For such structures, we consider the extension of first-order logic by threshold- and modulo-counting quantifiers. Depending on the…