### 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.... ..."

### Table 1: Basic sequent calculus B Axioms

1997

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### Table 1: Sequent calculus formalisation of the fragment

"... In PAGE 4: ... The other terms are similarly translated to speaker=person(i), car=vehicle(x), and undef=vehicle(x), respec- tively. Table1 shows our fragment of linear logic. We customary let (capital) alphabets range over single formulae and greek letters over the strings of formulae.... ..."

### Table 1: The calculus C6 BWInt

2000

"... In PAGE 4: ...Table 1: The calculus C6 BWInt We will write BCCY Int A to indicate that a sequent BCBC CO A with BCBC AI BC is provable using the rules in Table1 . The natural deduction calculus C6 BWCl for first-order Classical Logic is obtained by replacing the rule BRInt of the calculus C6 BWInt with the rule: BCBN BMA CO BR BC CO A BRCl For the natural deduction calculi, the notions of proof (tree), end-sequent of a proof, subproof of a proof,as well as the notion of depth of a proof AP, denoted by depthBAAPBB, are defined in the usual way (see, e.... In PAGE 4: ...ay (see, e.g., [24, 25]). The variable y in the rules IBK and EBL of Table1 is called proper parameter of the rule. We call free variable of a proof every variable which occurs free in some wff of the proof and does not occur as a proper parameter in such a wff.... ..."

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### Table 2. Program logic part of the ODL sequent calculus.

2006

"... In PAGE 10: ... For rst-order and propositional logic standard rule schemata are listed in Table 1, including an integer induction scheme. Within the rules for the program logic part ( Table2 ), state update rules R29{R30 constitute a peculiarity of ODL and will be discussed after de ning rule applications. Essentially, the ODL inference rules have the e ect of reducing more complex formulas to simpler ones.... ..."

Cited by 7