nLab co-Heyting negation

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

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

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

• $\sim 1=0$ and $\sim 0= 1$.

• $\sim$ can be used to define the co-Heyting boundary operator $\partial :L\to L$ by $\partial a:=a\wedge\sim a$. That $\partial a$ 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 $\sim (a\wedge b)=(\sim a)\vee(\sim b)$, but not $\sim (a\vee b) = (\sim a)\wedge (\sim b)$ in general.

• For $a\in L$ define its core as $\sim\sim a$. Then $a=\partial a\vee(\sim\sim a)$. Call $a$ with $a=\sim\sim a$ 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 $a$ of a co-Heyting algebra has a complement $x$ i.e. $a\vee x = 1$ and $a\wedge x = 0$, then the complement $x$ coincides with $\sim a$. Because from $a\vee x=1$ follows $\sim a\leq x$ since $\sim a$ is the least element with this property; conversely, $\sim a=\sim a\vee (a\wedge x)=(\sim a\vee a)\wedge (\sim a\vee x)=\sim a\vee x$ whence $x\leq \sim a$.

• A complemented element is obviously regular. The converse is not true. Complemented elements and regular elements of a co-Heyting algebra both form Boolean algebras, but in general regular elements do not inherit the meet operation while complemented elements do. See relations between Heyting and Boolean algebras.

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

Related entries

• co-Heyting boundary
• co-Heyting algebra
• negation
• Heyting algebra
• Heyting category
• mereology
• adjoint modality
• paraconsistent logic

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.