### Table 3: the sequent calculus BCT

1995

"... In PAGE 10: ... This should motivate the rules of the sequent calculus BCT, the \By Cases Theory quot; given in the next de nition. In Table3 we use the vector-notation described earlier: in rule ( !), the terms ~ A are of types such that (f ~ A) has product type; and in rule ByCases, the terms ~ P are of types such that (h ~ P ) has sum type. De nition 5.... ..."

Cited by 10

### Table 1: Sequent calculus I.

2000

Cited by 23

### Table 2. The focussed sequent calculus of permutative logic

"... In PAGE 14: ...Table2 . Its negative logical inferences are identical to those of the standard sequent calculus.... ..."

### Table 1 Focalized sequent calculus for MNL

2000

"... In PAGE 19: ... Note that we omit the sym- bol turnstileleft at the beginning of sequents, since it is useless in one-sided sequents. The rules of the sequent calculus are given in Table1 As we are interested in proof search, we only deal with cut-free sequent calculus. Observe that a crucial rule of NL, entropy, does not appear explicitely in Table 1 As we have already said in the introduction, entropy is a source of non-determinism in proof search.... In PAGE 19: ... The rules of the sequent calculus are given in Table 1 As we are interested in proof search, we only deal with cut-free sequent calculus. Observe that a crucial rule of NL, entropy, does not appear explicitely in Table1 As we have already said in the introduction, entropy is a source of non-determinism in proof search. In Table 1, it is included in the rule for circledot, the only place where it is actually necessary: this is not trivial, but a consequence of the results in the previous section, and the rest of the present section is devoted to proving that this optimized sequent calculus is actually equivalent to the original one in [1,14] or in Appendix 9.... In PAGE 19: ... Observe that a crucial rule of NL, entropy, does not appear explicitely in Table 1 As we have already said in the introduction, entropy is a source of non-determinism in proof search. In Table1 , it is included in the rule for circledot, the only place where it is actually necessary: this is not trivial, but a consequence of the results in the previous section, and the rest of the present section is devoted to proving that this optimized sequent calculus is actually equivalent to the original one in [1,14] or in Appendix 9. We do this by proving adequacy and sequentialization w.... ..."

### Table 1: Focussing sequent calculus for MANL with constants

"... In PAGE 6: ...2 on, we shall need to explicitly manipulate the elements of the support set, which we shall call places. The inferences of the focussing sequent calculus are presented in Table1 . The present calculus differs slightly from that given in [14] where the entropy rule is implicitly combined with the rules for the tensor connectives, and optimised so as to introduce only the minimal entropy needed by the tensor.... ..."

### Table 1. The sequent calculus of permutative logic.

"... In PAGE 4: ... If = f(a; b); (c); (d; e)g; 2, then its genus is given by the couple (2; 3) and rk( ) = 6. The multiplicative permutative calculus is recalled in Table1 ; moreover, the involutive duality is given by De Morgan rules: (A O B)? = B? A? ([A)? = #A? }? = h ?? = 1 (A B)? = B? O A? (#A)? = [A? h? = } 1? = ?: By the fact that basic commutations are not provable keeping the lowest topo- logical complexity, PL turns out to be an inference system able to deal with logical noncommutativity. As suggested by some of the next propositions, basic commutations can be recovered throughout the two permutative modalities [ and #.... ..."

### Table 1 Sequent Calculus Rules for S4

"... In PAGE 5: ... More abstractly formu- lated, we will have what can be described category-theoretically as a monad, or proof-theoretically as an S4 modal operator. We should recall that S4 modalities are given by the rules in Table1 ; classically or intuitionistically, these rules give the usual modal logic [34, Section 9.1], but they can equally well be added to linear logic and they satisfy the usual proof-theoretic properties (cut elimination and so on) [18].... In PAGE 8: ...Table 2 The System LL ?; A ` @L ?; @A ` ?; @A ` B; @R1 ?; @A ` @B; ? ` A; B; @R2 ? ` A; @B; ?; A ` B; L1 ?; A ` B; ?; A; @B ` L2 ?; A; @B ` ? ` A; R ? ` A; So we are led to consider modalities given by the rules in Table 2; we will call these strong modalities (the category-theoretic counterpart of (4) is called a strength). Conversely, the usual S4 rules ( Table1 ) will be called monoidal modalities (since (2) makes a monoid in an appropriate category of endofunc- tors).This system, based on classical linear logic together with a strong modality, will be called LL ; it will be our point of departure.... In PAGE 37: ... Proof We make an induction on the length of the proof of ` ; we go by cases according to the last rule of the proof. We use a presentation of our basic system, LL 00, given in Table1 0; we can easily prove a dual form of Proposition 9 to show that they are equivalent to the rules of Table varModalRules. Axiom The sequent is of the form A ` A, so the result is clear.... In PAGE 48: ... The top diagram, for example, expresses the equality A A = A A; now the rst is given by the proof Ax A A ` A R A ` A L A ` A L A ` A A A ` A cut A ` A whereas the second is given by the proof AxA A ` A R A ` A L A ` A L A ` A R A ` A L A ` A A A ` A cut A ` A We want these two proofs to be equal. We need similar equalities for the other diagrams in Table1 1, and for the other diagrams de ning a ?-autonomous category. With this de nition of equality between proofs, then, we have de ned a category, F, in which the objects are linear logic formulae and in which the morphisms are proofs of entailments.... ..."