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.
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 $\sim a:=1\backslash a$.
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.
$\sim a$ 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 $\sim$.
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 August 26, 2022 at 15:32:22. See the history of this page for a list of all contributions to it.