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The cut rule for linear logic used to be stated as
If $\Gamma \vdash A$ and $A \vdash \Delta$, then $\Gamma \vdash \Delta$.
I don’t think this is general enough, so I corrected it to
If $\Gamma \vdash A, \Phi$ and $\Psi,A \vdash \Delta$, then $\Psi,\Gamma \vdash \Delta,\Phi$.
No objection, but I guess it would depend on the precise rules. In MLL which has duals, if we have rules which allow us to derive
$\frac{\Gamma \vdash \Sigma,\Phi}{\Gamma, \Phi^\ast \vdash \Sigma}, \qquad \frac{\Psi, \Sigma \vdash \Delta}{\Psi \vdash \Delta, \Sigma^\ast}$where $\Phi^\ast$ is the list of duals of formulas of $\Phi$, and $A^{\ast\ast} = A$, I thought we would be able to derive your more general cut rule from the more specific cut rule:
$\array{ \arrayopts{\rowlines{solid}} \Gamma \vdash A, \Phi\;\;\;\;\;\;\;\;\; \Psi, A \vdash \Delta \\ \Gamma, \Phi^\ast \vdash A \;\;\;\;\;\;\;\;\; A \vdash \Psi^\ast, \Delta \\ \Gamma, \Phi^\ast \vdash \Psi^\ast, \Delta \\ \Psi, \Gamma \vdash \Delta, \Phi }$Yes, you’re right. But I think it’s better to formulate the rule in a way that remains correct in fragments without involutive negation.
Understood (and agreed); I was just explaining the likely reason the original form was the way it was (I’m guessing I wrote that).
I think the use of ’categorial semantics’, when the section talks about a ’categorical semantics’ was a typo. If it isn’t, it might be good to explain that there are schools of thought that prefer the adjective ’categorial’ to ’categorical’, and strongly so. Personally, I prefer ’categorical’, but some of my best friends insist on ’categorial’.
Valeria de Paiva
I do expect it was a typo. But it might still be good to explain issues with the terminology, either on this page or on a dedicated page!
I feel that most usage of “categorical” in mathematics is, or originates in, a careless search for terminology that is really looking for “category theoretic”. Compare the analogous clear difference between “numerical” and “number theoretic”.
It almost surely is not a typo – it’s Toby’s preferred word.
Searching MathSciNet, “categorical” is more popular than “categorial” by an order of 20 (11822 vs 591 matches).
Furthermore, “categorial” matches mostly philosophy, logic, and computer science, with almost no actual category theory papers.
Somebody please just add a discussion of terminology, conventions and issues to this entry – or better to categorical logic and then link to it from here.
It sure must look like typos to any outside reader: There was 16 “categorical” vs 4 “categorial” before Valeria’s latest edit, some next to each other, notably there was
- Categorial semantics. We discuss the categorical semantics…
I see from the edit history that the “categorial”s are Toby’s intentional ideosyncrasy. As per #15 I probably agree with the logic behind this, but if mathematical terminology were a matter of logic, most of it would be much different. The goal of an entry must be to communicate with the reader, not to irritate them.
But if we had a decent section “On terminology” at categorical logic we could link to this from the very top of all relevant pages – say: “On the terminology of ’categori(c)al’ in the following see at categorical logic. ” – and then the issue would be dealt with.
So I went ahead and wrote a comment on terminology myself: here (in a new section of the entry categorical semantics).
Maybe best to have any further discussion of this point there, in the thread for the entry on categorical semantics.
Re #15: dictionaries list two different meanings for “categorical”, for example, Wiktionary says
Absolute; having no exception.
Of, pertaining to, or using a category or categories.
Clearly, the second meaning makes “categorical” perfectly appropriate here. I do not understand how or why the interpretation that “categorical” necessarily refers to 1 above, whereas “categorial” means 2, came to be.
Re #20: I don’t think it’s that. I think Toby wants “categorial” to avoid a conflict with the model theorists’ usage of the word “categorical”. My own opinion is that we don’t need such deferment.
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