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Harley Eades III, Dominic Orchard
2020 arXiv   pre-print
We give a sequent calculus, natural deduction, and term assignment for Graded Adjoint Logic.  ...  We introduce a new logic that combines Adjoint Logic with Graded Necessity Modalities. This results in a very expressive system capable of controlling when and how structural rules are used.  ...  Graded Adjoint Logic is more general than Graded Modal Logic, because every semiring is a pointed semiring where all operations are defined.  ...

### Multi-adjoint Logic Programming with Continous Semantics [chapter]

Jesús Medina, Manuel Ojeda-Aciego, Peter Vojtaš
2001 Lecture Notes in Computer Science
In this paper we apply this idea to introduce multi-adjoint logic programs as an extension of monotonic logic programs.  ...  Considering different implication operators, such as Lukasiewicz, Gödel or product implication in the same logic program, naturally leads to the allowance of several adjoint pairs in the lattice of truthvalues  ...  Syntax of Multi-Adjoint Logic Programs The definition of multi-adjoint logic program is given, as usual, as a set of rules and facts.  ...

### Page 1417 of Mathematical Reviews Vol. , Issue 96c [page]

1996 Mathematical Reviews
This logic is first-order and is dictated directly by the underlying categorical se- mantics.  ...  It is a graded generalization of a Poisson algebra in which there is a graded commutative associative product and (up to suitable signs) a Lie bracket of degree —1 with a signed Leibniz relation between  ...

### A Multi-adjoint Logic Approach to Abductive Reasoning [chapter]

Jesús Medina, Manuel Ojeda-Aciego, Peter Vojtáš
2001 Lecture Notes in Computer Science
Multi-adjoint logic programs has been recently introduced [9, 10] as a generalization of monotonic logic programs [2, 3] , in that simultaneous use of several implications in the rules and rather general  ...  To model uncertainty in human cognition and real world applications; we use multi-adjoint logic programming to introduce and study a model of abduction problem.  ...  Syntax and Semantics of Multi-Adjoint Logic Programs Multi-adjoint logic programs are constructed from the abstract syntax induced by a multi-adjoint algebra on a set of propositional symbols.  ...

### An algebraic theory of normal forms

Silvio Ghilardi
1995 Annals of Pure and Applied Logic
. * There is a general theorem by Lawvere of existence of left adjoints for algebraic functors which applies to our case.  ...  One may also directly use some adjoint functor theorem (see ). 3 This will be our trend for the rest of the paper: we give full details of the general theory and of its applications to the basic example  ...  for very simple logics, like S4).  ...

### A Multi-Adjoint Approach to Similarity-Based Unification

Jesús Medina, Manuel Ojeda-Aciego, Peter Vojtáš
2002 Electronical Notes in Theoretical Computer Science
A formal model for similarity-based fuzzy unification in multi-adjoint logic programs is presented.  ...  computational model, a similarity-based unification approach is constructed by simply adding axioms of fuzzy similarities and using classical crisp unification which provides a semantic framework for logic  ...  logic programs as follows: Definition 3.5 Let P be a multi-adjoint logic program.  ...

### Page 8070 of Mathematical Reviews Vol. , Issue 2003k [page]

2003 Mathematical Reviews
Propositional calculus under adjointness. (English summary) Possibility theory and fuzzy logic. Fuzzy Sets and Systems 132 (2002), no. 1, 91-106.  ...  We call the resulting logic propositional calculus under adjointness, abbreviated AdjPC. “Most algebraic theorems on such (L,<), A and K are inequal- ities for the posets (L, <) of truth values.  ...

### Logical Foundations of Quantitative Equality (long version) [article]

Francesco Dagnino, Fabio Pasquali
2021 arXiv   pre-print
We introduce a deductive calculus for (Graded) Linear Logic with quantitative equality and the notion of Lipschitz doctrine to give semantics to it.  ...  To overcome this issue, we consider the extension of Linear Logic with graded modalities and use them to write a resource sensitive substitution rule that keeps equality quantitative.  ...  n of Bounded Linear Logic  and, indeed, syntactic doctrines built out from it are N = -graded.  ...

### Flat Coalgebraic Fixed Point Logics [chapter]

Lutz Schröder, Yde Venema
2010 Lecture Notes in Computer Science
Extending this notion to the generic semantic framework of coalgebraic logic enables covering a wide range of logics beyond the standard mu-calculus including, e.g., flat fragments of the graded mu-calculus  ...  The family of such flat fixpoint logics includes, e.g., LTL, CTL, and the logic of common knowledge.  ...  Graded fixed point logics are sublogics of the graded µ-calculus  .  ...

### Towards Biresiduated Multi-adjoint Logic Programming [chapter]

Jesús Medina, Manuel Ojeda-Aciego, Agustín Valverde, Peter Vojtáš
2004 Lecture Notes in Computer Science
Multi-adjoint logic programs were recently proposed as a generalization of monotonic and residuated logic programs, in that simultaneous use of several implications in the rules and rather general connectives  ...  logic programming and sketch some considerations on its fixpoint semantics. 1.  ...  As our approach does not require adjoint conjunctors to be commutative, it would allow the development of a sound and complete graded resolution. 5 (x, y) is not commutative.  ...

### Categorical Accommodation of Graded Fuzzy Topological System, Graded Frame and Fuzzy Topological Space with Graded inclusion [article]

Purbita Jana, Mihir K. Chakraborty
2017 arXiv   pre-print
The notions of graded fuzzy topological system and fuzzy topological space with graded inclusion were obtained via fuzzy geometric logic with graded con- sequence.  ...  A detailed study of graded frame, graded fuzzy topological system and fuzzy topological space with graded inclusion is already done in our earlier paper.  ...  Ext g is the right adjoint to the functor J g . Proof. Follows from the combination of the adjoint situations in Lemmas 2.7, 4.1.  ...

### Similarity-based unification: a multi-adjoint approach

Jesús Medina, Manuel Ojeda-Aciego, Peter Vojtáš
2004 Fuzzy sets and systems (Print)
The aim of this paper is to build a formal model for similarity-based fuzzy uniÿcation in multi-adjoint logic programs.  ...  Speciÿcally, a general framework of logic programming which allows the simultaneous use of di erent implications in the rules and rather general connectives in the bodies is introduced, then a procedural  ...  Let P be a multi-adjoint logic program.  ...

### A Procedural Semantics for Multi-adjoint Logic Programming [chapter]

Jesús Medina, Manuel Ojeda-Aciego, Peter Vojtáš
2001 Lecture Notes in Computer Science
Multi-adjoint logic program generalise monotonic logic programs introduced in  in that simultaneous use of several implications in the rules and rather general connectives in the bodies are allowed.  ...  In this work, a procedural semantics is given for the paradigm of multiadjoint logic programming and completeness theorems are proved.  ...  Definition 2 (Multi-Adjoint Logic Programs).  ...

### An algebraic generalization of Kripke structures

SÉRGIO MARCELINO, PEDRO RESENDE
2008 Mathematical proceedings of the Cambridge Philosophical Society (Print)
As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL, and the ramified temporal logic CTL.  ...  The Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales.  ...  adjoints.  ...

### Towards Simplification Logic for Graded Attribute Implications with General Semantics

Pablo Cordero, Manuel Enciso, Angel Mora, Vilém Vychodil
2018 International Conference on Concept Lattices and their Applications
We present variant of simplification logic for reasoning with if-then dependencies that arise in formal concept analysis of data with graded attributes.  ...  We describe semantics of the rules, axiomatic system of the logic, and prove its soundness and completeness.  ...  The simplification logic was later introduced for graded attribute implications parameterized by hedges in  .  ...
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