Talk:Sequent calculus
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| − | == Quantifiers == |
+ | == Equivalences == |
| − | The presentation does not seem to be completely uniform concerning quantifiers: are first-order quantifiers taken into account? It would be nice. |
+ | Equivalences might deserve a specific page (maybe merged with [[isomorphism]]s and [[equiprovability]]?). |
| − | A few related points: |
+ | We might imagine a page or some pages giving a collection of [[Provable formulas|valid principles]] of linear logic (with appropriate proofs) and specifying which ones correspond to implications, equivalences or isomorphisms. |
| − | * Why a distinction between atomic formulas and propositional variables? |
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| − | * Some mixing between <math>\forall x A</math> and <math>\forall X A</math>. I tried to propose a [[notations#formulas|convention]] on that point, but it does not match here with the use of <math>\alpha</math> for atoms. |
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| − | * Define immediate subformula of <math>\forall X A</math> as <math>A</math>? |
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| − | -- [[User:Olivier Laurent|Olivier Laurent]] 18:37, 14 January 2009 (UTC) |
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| − | :I improved the uniformity for quantifiers: the full system with first and second order quantification is described, only predicate variables with first-order arguments are not described. |
+ | -- [[User:Olivier Laurent|Olivier Laurent]] 10:39, 15 March 2009 (UTC) |
| − | :The distinction between atomic formulas and propositional variables is because there are systems with atomic formulas that are not propositional variables but fixed predicates like equalities. |
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| − | :I found <math>\alpha</math> to be a used notation for atomic formulas in other texts, so I used <math>\xi,\psi,\zeta</math> instead for arbitrary variables. |
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| − | :Using substitution in the definition of subformulas is questionable, but if the only immediate subformula of <math>\forall\xi.A</math> is <math>A</math>, then the ''subformula'' property does not hold. |
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| − | :''Edit:'' Well... my bad. The subformula property does hold if the only immediate subformula of <math>\forall\xi.A</math> is <math>A</math>, substitution is necessary only for <math>\exists\xi.A</math>. I changed it. |
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| − | :-- [[User:Emmanuel Beffara|Emmanuel Beffara]] |
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| − | == Two-sided sequent calculus == |
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| − | I think the terminology "two-sided sequent calculus" should be used for the system where all the connectives are involved and all the rules are duplicated (with respect to the one-sided version) and negation is a connective. |
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| − | In this way, we obtain the one-sided version from the two-sided one by: |
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| − | * quotient the formulas by de Morgan laws and get negation only on atoms, negation is defined for compound formulas (not a connective) |
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| − | * fold all the rules by <math>\Gamma\vdash\Delta \mapsto {}\vdash\Gamma\orth,\Delta</math> |
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| − | * remove useless rules (negation rules become identities, almost all the rules appear twice) |
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| − | A possible name for the two-sided system presented here could be "two-sided positive sequent calculus". |
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| − | -- [[User:Olivier Laurent|Olivier Laurent]] 21:34, 15 January 2009 (UTC) |
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Latest revision as of 21:01, 25 April 2013
[edit] Equivalences
Equivalences might deserve a specific page (maybe merged with isomorphisms and equiprovability?).
We might imagine a page or some pages giving a collection of valid principles of linear logic (with appropriate proofs) and specifying which ones correspond to implications, equivalences or isomorphisms.
-- Olivier Laurent 10:39, 15 March 2009 (UTC)
