Hi,


You can talk about the arity of a function or an operation - something like addition could have an arity of 2, and negation usually has an arity of 1.


A paper I am reading is talking about positive arities and negative arities, and I don't understand what this means. The author gives an example in classical logic: he says disjunction and conjuction have a positive arity of 2, while negation has a negative arity of 1. I suppose I am reading a bit above my level here, but I don't get why there needs to be a distinction between these.


The paper in question:


<http://drops.dagstuhl.de/opus/volltexte/2010/2649/>


Definition 2.1 and Example 2.3 are the relevant bits.


Can anyone lend any insight here?



Positive & Negative Arity
While I have never seen this notion before (it may be common or this may be the first paper that uses those terms), Comment 1 basically explains the idea. A relation $r$ in $\mathcal{R}\_{n,m}$ is an $n+m$-arity relation (on propositional formulas I believe), where the first $n$ inputs are considered "positive", and the last $m$ are "negative".


To see what these mean, consider the following case. Say $r$ is a $\mathcal{R}\_{1,0}$ relation, i.e. the only input is positive. Further, say $B$ is deducible from $A$. (You will have to look in the paper to see what deducible exactly means here.) Then since the arity is positive, $r(B)$ is deducible from $r(A)$. An example would be the trivial relation $r(A) := A$. Another example with a positive arity of $2$ is conjunction in classical logic. If $B\_0$ and $B\_1$ are deductible from $A\_0$ and $A\_1$ respectively, then $B\_0 \wedge B\_1$ is deducible from $A\_0 \wedge A\_1$. 


In the negative case, it is the opposite. Say $r$ is a $\mathcal{R}\_{0,1}$ relation, i.e. the only input is negative. Again, say $B$ is deducible from $A$. Then since the arity is negative, $r(A)$ is deducible from $r(B)$---the opposite direction as before. An example would be the negation relation in classical logic; if $B$ is deducible from $A$, then $\neg A$ is deducible from $\neg B$. (The contrapositive.)


These can be combined so that a relation has both positive and negative arities. An example is implication in classical logic, which is a $\mathcal{R}\_{1,1}$ relation, i.e. it has one positive arity and one negative arity. If $B\_0$ and $B\_1$ are deducible from $A\_0$ and $A\_1$ respectively, then $A\_0 \rightarrow B\_1$ is deductible from $B\_0 \rightarrow A\_1$. (This is basically $A\_0$ implies $B\_0$ implies $A\_1$ implies $B\_1$.) The antecedent is the negative input, while the consequent is the positive one.


I hope this helps.


