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Axiomatizing Category Theory in Free Logic
[article]

2018
*
arXiv
*
pre-print

Our axiom sets have been formalized

arXiv:1609.01493v5
fatcat:omccnmwqgfhshlzhffvwp5zot4
*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 ...##
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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. ...

##
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On Constructive Axiomatic Method
[article]

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. ...

##
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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 ...

##
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The IFF Foundation for Ontological Knowledge Organization

2003
*
Cataloging & Classification Quarterly
*

This paper discusses an

doi:10.1300/j104v37n01_13
fatcat:tp556yxpd5e2vm6kxagdwrxz3u
*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*...##
###
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

doi:10.1093/logcom/12.1.211
fatcat:yowegpu725gxpbpi7eabhc6caa
*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. ...##
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The IFF Foundation for Ontological Knowledge Organization
[article]

2018
*
arXiv
*
pre-print

This paper discusses an

arXiv:1810.04773v1
fatcat:tqrkz7qsuva6rck547nkpbdd4a
*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. ...##
###
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 ...

##
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Semantic Integration in the Information Flow Framework
[article]

2018
*
arXiv
*
pre-print

various institutions

arXiv:1810.08236v1
fatcat:vyr6drjcazg73k2ebgsmj2hnyu
*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. ...##
###
The Information Flow Framework: A Descriptive Category Metatheory
[article]

2011
*
arXiv
*
pre-print

The Information Flow Framework (IFF) is a descriptive

arXiv:1108.4133v1
fatcat:fokkbpvyyrhbhhrf6ixzsac7lm
*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. ...##
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Coherence and Confluence
[article]

2005
*
arXiv
*
pre-print

Proofs of coherence

arXiv:math/0506310v3
fatcat:wbi4fb24end4lieh6bur72hsza
*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. ...##
###
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. ...##
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How to Make a New Logic
[article]

2021
*
arXiv
*
pre-print

We consider certain

arXiv:2108.05934v1
fatcat:r2eb4ltmvrcqpgqb7kqix6s3oe
*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 . ...
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