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Axiomatizing Category Theory in Free Logic [article]

Christoph Benzmüller, Dana S. Scott
2018 arXiv   pre-print
Our axiom sets have been formalized in the Isabelle/HOL interactive proof assistant, and this formalization utilizes a semantically correct embedding of free logic in classical higher-order logic.  ...  Starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory.  ...  Our purpose is not to make or claim any contribution to category theory but rather to show how formalizations involving the kind of logic required (free logic) can be validated within modern proof assistants  ... 
arXiv:1609.01493v5 fatcat:omccnmwqgfhshlzhffvwp5zot4

Page 1664 of Mathematical Reviews Vol. , Issue 91C [page]

1991 Mathematical Reviews  
Concepts familiar in universal algebra, such as free algebras and polynomials over a structure, are explained in the language of category theory.  ...  Chapter 0 includes basic concepts and techniques in mathemati- cal logic (natural deduction systems for first-order logic), the theory of integers (axiomatization and semantics of Peano arithmetic ex-  ... 

Page 2815 of Mathematical Reviews Vol. , Issue 93e [page]

1993 Mathematical Reviews  
Category theory offers a unifying framework for a wide variety of models, re- vealing the authors’ simpler axiomatization in terms of dualizing objects, and its relationship to previous axiomatizations  ...  A Petri category is defined as a strictly symmetric monoidal category in which the monoid of objects is free. The axioms describing a Petri net as a Petri category are discussed.  ... 

On Constructive Axiomatic Method [article]

Andrei Rodin
2015 arXiv   pre-print
In this last version of the paper one may find a critical overview of some recent philosophical literature on Axiomatic Method and Genetic Method.  ...  types for free.  ...  Landry argues back that these notions are used only for a genetic underpinning of Category theory but play no role in the logical axiomatic foundation of this theory stricto sensu.  ... 
arXiv:1408.3591v3 fatcat:donspj4q7jhwtptukfs4u6mg4q

Page 9254 of Mathematical Reviews Vol. , Issue 2001M [page]

2001 Mathematical Reviews  
In the current paper, he gives a finer analysis of the group axioms by giving a description of the free theories in varieties axiomatized For the web version of Mathematical Reviews, see http: //www.ams.org  ...  In the 1990s, some interest arose among computer scientists in the formulation of relations in a universal algebra style, that is, in the definition of an algebra (or equivalently, of a free category which  ... 

The IFF Foundation for Ontological Knowledge Organization

Robert E. Kent
2003 Cataloging & Classification Quarterly  
This paper discusses an axiomatic approach for the semantic integration of ontologies, an approach that extends to first order logic a previous approach (Kent 2000) based on information flow.  ...  This axiomatic approach is represented in the Information Flow Framework (IFF), a metalevel framework for organizing the information that appears in digital libraries, distributed databases and ontologies  ...  logic log(T) over a theory T has the following meaning: any theory morphism g from a theory T to the component theory th(L) is equivalent to a logic infomorphism ĝ from the free logic log(T) to the logic  ... 
doi:10.1300/j104v37n01_13 fatcat:tp556yxpd5e2vm6kxagdwrxz3u

Special Issue: Fixpoints in Computer Science 2000

2002 Journal of Logic and Computation  
Along the way of proving these results, we give a concrete description of the free algebras in the corresponding variety of ordered iteration algebras.  ...  In more generality, for -continuous endofunctors of locally finitely presentable categories the analogous result holds: here the ordering is considered on the hom-sets ÓÑ´ ̵ for all finitely presentable  ...  Future issues A Foundation for Metareasoning Part I: The Proof Theory G. Criscuolo, F. Giunchiglia and L. Serafini Two Logical Theories of Plan Recognition W.  ... 
doi:10.1093/logcom/12.1.211 fatcat:yowegpu725gxpbpi7eabhc6caa

The IFF Foundation for Ontological Knowledge Organization [article]

Robert E. Kent
2018 arXiv   pre-print
This paper discusses an axiomatic approach for the integration of ontologies, an approach that extends to first order logic a previous approach (Kent 2000) based on information flow.  ...  This axiomatic approach is represented in the Information Flow Framework (IFF), a metalevel framework for organizing the information that appears in digital libraries, distributed databases and ontologies  ...  Apart from the use of set theory or category theory, model theory is agnostic about the kinds of things that exist. The latter are described and axiomatized in object-level ontologies.  ... 
arXiv:1810.04773v1 fatcat:tqrkz7qsuva6rck547nkpbdd4a

Page 7202 of Mathematical Reviews Vol. , Issue 96m [page]

1996 Mathematical Reviews  
Logic 76 (1995), no..2, 169-200. The author axiomatizes, in P.  ...  The other half is con- cerned with the solution of basic geometric constructions (called “problems” in the text), a re-axiomatization of the four geome- tries in higher-level type theory, with computer  ... 

Semantic Integration in the Information Flow Framework [article]

Robert E. Kent
2018 arXiv   pre-print
various institutions in which semantic integration has a natural expression as the colimit of theories.  ...  The main application of the IFF is institutional: the notion of institutions and their morphisms are being axiomatized in the upper metalevels of the IFF, and the lower metalevel of the IFF has axiomatized  ...  Free models and logics exist.  ... 
arXiv:1810.08236v1 fatcat:vyr6drjcazg73k2ebgsmj2hnyu

The Information Flow Framework: A Descriptive Category Metatheory [article]

Robert E. Kent
2011 arXiv   pre-print
The Information Flow Framework (IFF) is a descriptive category metatheory. It is an experiment in foundations, which follows a bottom-up approach to logical description.  ...  The IFF represents meta-logic, and as such operates at the structural level of ontologies. In the IFF, there is a precise boundary between the metalevel and the object level.  ...  The theory shall consist of axioms and definitions stated in a logic-based language, such as those in the Common Logic (CL) framework.  ... 
arXiv:1108.4133v1 fatcat:fokkbpvyyrhbhhrf6ixzsac7lm

Coherence and Confluence [article]

K. Dosen, Z. Petric
2005 arXiv   pre-print
Proofs of coherence in category theory, starting from Mac Lane's original proof of coherence for monoidal categories, are sometimes based on confluence techniques analogous to what one finds in the lambda  ...  calculus, or in term-rewriting systems in general.  ...  This free category will always exist if our axiomatization is purely equational.  ... 
arXiv:math/0506310v3 fatcat:wbi4fb24end4lieh6bur72hsza

Page 4859 of Mathematical Reviews Vol. , Issue 84m [page]

1984 Mathematical Reviews  
, free categorial grammars, etc.  ...  From the introduction: “We have shown [same journal 28 (1982), 229-238; MR 83m:03045] that the categorial grammars based on Lambek’s calculus (of product-free types) or on its finite axiomatic extensions—here  ... 

Page 1200 of Mathematical Reviews Vol. , Issue 97B [page]

1997 Mathematical Reviews  
It has brought together ideas from logic, topology, computability, and category theory in a most fruitful way.  ...  Chapter 6 gives a general treatment of powerdomains in terms of free dcpo-algebras. Chap- ter 7, one of the most celebrated parts of the handbook, presents domain theory in logical form.  ... 

How to Make a New Logic [article]

Takao Inoué
2021 arXiv   pre-print
We consider certain logics in this respect.  ...  The operation to make a new logic from S is the following: for all the axioms and rules of S, change the direction of all the arrows occurred in the sequents of the axiom or the rule to the opposite side  ...  The real novelty of Stahl's formulation from the standpoint of axiomatic rejection consists in the axiomatization of all the contradictions free from the notion of provability 9 .  ... 
arXiv:2108.05934v1 fatcat:r2eb4ltmvrcqpgqb7kqix6s3oe
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