nLab
co-Heyting negation

Contents

Context

Category theory

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

In a co-Heyting algebra it is possible to define a negation operator \sim which validates the law of excluded middle but invalidates the law of non-contradiction dual to the negation in a Heyting algebra.

Definition

Let aa be an element of a co-Heyting algebra LL with subtraction \\backslash. The co-Heyting negation of aa , called non a (Lawvere 1991) or the supplement of aa, is defined as a:=1\a\sim a:=1\backslash a.

Remark

A bi-Heyting algebra is naturally equipped with two negation operators: the Heyting negation ¬\neg and the co-Heyting supplement \sim. Both coincide in a Boolean algebra considered as a bi-Heyting algebra.

In applications, such co-occuring pairs of negation operators ¬\neg and \sim are the most interesting cases as their combination give rise to adjoint modalities e.g. in mereology they yield thickened boundaries (Stell&Worboys 1997). Beside mereology they have found applications in linguistics, intuitionistic logic and physics.

Properties

  • a\sim a minimally supplements aa to truth in the sense that a\sim a is the least xx with ax=1a\vee x=1. This follows from the adjointness of \\backslash which unwinds as ax\sim a\leq x iff 1=ax1=a\vee x.

  • 1=0\sim 1=0 and 0=1\sim 0= 1.

  • \sim can be used to define the co-Heyting boundary operator :LL\partial :L\to L by a:=aa\partial a:=a\wedge\sim a. That a\partial a is not necessary trivial is dual to the non-validity of the tertium non datur for general Heyting algebras and already points to the utility of the co-Heyting negation for paraconsistent logic.

  • The co-Heyting supplement satisfies the dual de Morgan rule (ab)=(a)(b)\sim (a\wedge b)=(\sim a)\vee(\sim b), but not (ab)=(a)(b)\sim (a\vee b) = (\sim a)\wedge (\sim b) in general.

  • For aLa\in L define its core as a\sim\sim a. Then a=a(a)a=\partial a\vee(\sim\sim a). Call aa with a=aa=\sim\sim a regular. Lawvere (1986) proposes in the vein of classical mereology e.g. Tarski 1927 on regions as regular open sets, to consider only regular subbodies as bodies in the full sense.

  • Suppose the element aa of a co-Heyting algebra has a complement xx i.e. ax=1a\vee x = 1 and ax=0a\wedge x = 0, then the complement xx coincides with a\sim a. Because from ax=1a\vee x=1 follows ax\sim a\leq x since a\sim a is the least element with this property; conversely, a=a(ax)=(aa)(ax)=ax\sim a=\sim a\vee (a\wedge x)=(\sim a\vee a)\wedge (\sim a\vee x)=\sim a\vee x whence xax\leq \sim a.

  • A complemented element is obviously regular. Conversely, a regular element aa is complemented since 0=1=(aa)=(a)(a)=aa0=\sim 1=\sim (a\vee\sim a)=(\sim a )\wedge (\sim\sim a)=\sim a\wedge a.

  • In a bi-Heyting algebra: ¬a=¬a1=¬a(aa)=(¬aa)(¬aa)=0(¬aa)=¬aa\neg a=\neg a\wedge 1=\neg a\wedge (a\vee\sim a)=(\neg a\wedge a)\vee (\neg a\wedge \sim a)= 0\vee (\neg a\wedge\sim a)=\neg a\wedge\sim a hence ¬aa\neg a\leq\sim a and we see that ¬\neg is more strongly negative than \sim.

References

  • A. Döring, Topos-based Logic for Quantum Systems and Bi-Heyting Algebras , arXiv:1202.2750 (2013). (pdf)

  • Y. Gauthier, A Theory of Local Negation: The Model and some Applications , Arch. Math. Logik 25 (1985) pp.127-143. (gdz)

  • F. W. Lawvere, Introduction , pp.1-16 in Lawvere, Schanuel (eds.), Categories in Continuum Physics , LNM 1174 Springer Heidelberg 1986.

  • F. W. Lawvere, Intrinsic Co-Heyting Boundaries and the Leibniz Rule in Certain Toposes , pp.279-281 in Springer LNM 1488 (1991).

  • M. La Palme Reyes, J. Macnamara, G. E. Reyes, H. Zolfaghari, The non-Boolean logic of natural language negation , Phil. Math. 2 no.1 (1994) pp.45-68.

  • M. La Palme Reyes, J. Macnamara, G. E. Reyes, H. Zolfaghari, Models for non-Boolean negation in natural languages based on aspect analysis , pp.241-260 in Gabbay, Wansing (eds.), What is Negation?, Kluwer Dordrecht 1999.

  • M. La Palme Reyes, G. E. Reyes, H. Zolfaghari, Generic Figures and their Glueings , Polimetrica Milano 2004.

  • M. Menni, C. Smith, Modes of Adjointness , J. Philos. Logic 43 no.3-4 (2014) pp.365-391.

  • T. Mormann, Heyting Mereology as a Framework for Spatial Reasoning , Axiomathes 23 no.1 (2013) pp.237-264. (draft)

  • C. Rauszer, Semi-Boolean algebras and their applications to intuitionistic logic with dual operations, Fund. Math. 83 no.3 (1974) pp.219-249. (pdf)

  • G. E. Reyes, H. Zolfaghari, Bi-Heyting Algebras, Toposes and Modalities , J. Phi. Logic 25 (1996) pp.25-43.

  • J.G. Stell, M.F. Worboys, The algebraic structure of sets of regions , pp.163-174 in Hirtle, Frank (eds.), Spatial Information Theory, Springer LNCS 1329 (1997).

  • H. Wansing, Negation , pp.415-436 in Goble (ed.), The Blackwell Guide to Philosophical Logic , Blackwell Oxford 2001.

Last revised on July 29, 2016 at 17:33:11. See the history of this page for a list of all contributions to it.