Notations

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* Web of the space <math>X</math>: <math>\web X</math>
  
* Web of the space <math>X</math>: <math>\web X</math>
 
* Coherence relation of the space <math>X</math>: large <math>\coh_X</math> and strict <math>\scoh_X</math>
  
* Coherence relation of the space <math>X</math>: large <math>\coh_X</math> and strict <math>\scoh_X</math>
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=== [[Finiteness spaces]] ===
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* Web of the space <math>\mathcal A</math>: <math>\web {\mathcal A}</math>
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* Finiteness structure of the space <math>\mathcal A</math>: <math>\mathfrak F(\mathcal A)</math>

Revision as of 16:41, 6 July 2009

Contents

Logical systems

For a given logical system such as MLL (for multiplicative linear logic), we consider the following variations:

Notation Meaning Connectives
MLL propositional without units X,{\tens},{\parr}
MLLu propositional with units only \one,\bot,{\tens},{\parr}
MLL0 propositional with units and variables X,\one,\bot,{\tens},{\parr}
MLL1 first-order without units X\vec{t},{\tens},{\parr},\forall x A,\exists x A
MLL01 first-order with units X\vec{t},\one,\bot,{\tens},{\parr},\forall x A,\exists x A
MLL2 second-order propositional without units X,{\tens},{\parr},\forall X A,\exists X A
MLL02 second-order propositional with units X,\one,\bot,{\tens},{\parr},\forall X A,\exists X A
MLL12 first-order and second-order without units X\vec{t},{\tens},{\parr},\forall x A,\exists x A,\forall X A,\exists X A
MLL012 first-order and second-order with units X\vec{t},\one,\bot,{\tens},{\parr},\forall x A,\exists x A,\forall X A,\exists X A

Formulas and proof trees

Formulas

  • First order quantification: \forall x A with substitution A[t / x]
  • Second order quantification: \forall X A with substitution A[B / X]
  • Quantification of arbitrary order (mainly first or second): \forall\xi A with substitution A[τ / ξ]

Rule names

Name of the connective, followed by some additional information if required, followed by "L" for a left rule or "R" for a right rule. This is for a two-sided system, "R" is implicit for one-sided systems. For example: \wedge_1 \text{add} L.

Semantics

Coherent spaces

  • Web of the space X: \web X
  • Coherence relation of the space X: large \coh_X and strict \scoh_X

Finiteness spaces

  • Web of the space \mathcal A: \web {\mathcal A}
  • Finiteness structure of the space \mathcal A: \mathfrak F(\mathcal A)
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