### Table 1 The sequent calculus for L/MILL with commutativity implicit

"... In PAGE 3: ...ig. 1. Example sequent derivation 2.1 Sequent Calculus Table1 shows the sequent calculus for the Lambek calculus L, first proposed by Lambek (1958). The commutative version, the Lambek-van Benthem calculus LP, is also known as the multiplicative fragment of intuitionistic linear logic MILL.... ..."

### 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 #.... ..."