A. Avron, On modal systems having arithmetical interpretation, The Journal of Symbolic Logic, vol. 49 (1984), pp. 935--942.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is used for the following well known conservative extensions of K: T = K 2p ) p, K4 = K 2p ) 22p, S4 = K4 2p ) p and Grz = S4 2(2(p ) 2p) p) 2p. Clearly, OE 2 S4 implies OE 2 Grz. We call GT, GS4 and GGrz the cut free versions of the Gentzen style calculi defined in [22, 1] where the sequents are built from multisets of formulas. Moreover, we assume that the contraction and the weakening rules are absorbed in the introduction rules and axioms (see e.g. 26, Section 3.4 and Section 9.1] For instance, the initial sequents of all the Gentzen style calculi used in the ....

....(2 ) where 2 Gamma = f2 : 2 Gammag. The rule (2 ) belongs to the three systems GT, GS4 and GGrz and each rule ( 2) T , 2) S4 and ( 2) Grz belongs respectively to GT, GS4 and GGrz. For each L 2 fT ; S4; Grzg, we know that OE 2 L iff the sequent OE is derivable in GL (see e.g. [22, 1]) Further details can be found in [11] 3 A transformation from Grz into S4 Let f : For Theta f0; 1g For be the following map: ffl for any p 2 For 0 , f(p; 0) f(p; 1) ffl f( OE; i) f(OE; 1 Gamma i) for i 2 f0; 1g ffl f(OE 1 OE 2 ; i) f(OE 1 ; i) f(OE 2 ; i) for i 2 ....

A. Avron. On modal systems having arithmetical interpretations. The Journal of Symbolic Logic, 49(3):935--942, September 1984.

.... Grz (for Grzegorczyk) which admit important arithmetical interpretations as logics of provability [Sol76] see also [Boo93] At first glance, this seems to contradict the fact that DL generalizes Gentzen style calculi since the well known traditional sequent and tableau calculi for these logics [SV80, Lei81, SV82, Val83, Fit83, Avr84, Boo93, Gor99] do enjoy cut elimination. Our contribution By abstracting the aspects common to the characteristic axioms for G and Grz, respectively, we define the notion of a formula generation map F(p) in one propositional variable. Let # be a modal formula and F(#) be a formula built from # using , #, ....

....and no logical constants, such that for # FML, F(#) is obtained from # F by replacing every occurrence of p by # 2. no subformula of the form 2# occurs negatively in # F . F is also written #p.# F . The definition of a formula generation map is actually a restricted form of maps defined in [Avr84]. For instance, no restriction on the polarity of the occurrences of 2 is assumed in [Avr84] For any properly displayable logic L and for any formula generation map F, we write LF to denote the logic obtained from L by addition of the schema 2(F(p) 2p (1) Observe that 2(F(q) 2q is not ....

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A. Avron. On modal systems having arithmetical interpretations. The Journal of Symbolic Logic, 49(3):935--942, 1984. 31

....The other example concerns cut free sequent systems. A cut elimination theorem for K4 can be proved by the standard method due to Gentzen [Gen35] using degree and rank as induction parameters, while the proof for GL first given in Valentini [Val83] uses another parameter width (see also Avron [Avr84]) GL is also obtained by adding Lob s axiom to K4. So, as was asserted in [Smo84] the knowledge of K4 is useful for the discussion of GL. Smorynski treated K4 as a preliminary for the study of GL, where he used the name Basic modal logic instead of K4. Here, in chapter 2 and chapter 6, we ....

A. Avron, On modal systems having arithmetical interpretations, The Journal of Symbolic Logic, 49, 1984, pp. 935--942.

....Definition 9.3. A realization r is normal if all negative occurrences of 2 are realized by proof variables and the corresponding constant specification is injective. Theorem 9.4. If S4 F then LP F r for some normal realization r Proof. Consider a cut free sequent formulation of S4 (cf. [18], 73] with sequents Gamma ) Delta, where Gamma and Delta are finite multisets of modal formulas. Axioms are sequents of the form S ) S, where S is a propositional letter, and the sequent ) Along with the usual structural rules (weakening, contraction, cut) and rules introducing boolean ....

A. Avron, On modal systems having arithmetical interpretation, The Journal of Symbolic Logic, vol. 49 (1984), pp. 935--942.

.... nor occurs in F . However, m( i) and m( i) are defined in view of Lemma 4.4. The calculus ffiLF satisfies conditions (C2) C7) In particular, ffiG satisfies the conditions (C1) C7) The ( 2G ) rule in ffiG is similar to the GLR rule in [SV82] or to the G rule in [Rau83] see also [Avr84]) Analogously, the ( 2 Grz ) rule in ffiGrz is similar to the (GRZc) rule in [BG86] or to the ( 2) rule in [Avr84] An intuitively obvious way to understand the ( 2 LF ) rule is to recall the double nature of the 2 formulae in LF as illustrated by the LF theorem below: 2OE , 2(F(OE) OE) ....

.... conditions (C2) C7) In particular, ffiG satisfies the conditions (C1) C7) The ( 2G ) rule in ffiG is similar to the GLR rule in [SV82] or to the G rule in [Rau83] see also [Avr84] Analogously, the ( 2 Grz ) rule in ffiGrz is similar to the (GRZc) rule in [BG86] or to the ( 2) rule in [Avr84]. An intuitively obvious way to understand the ( 2 LF ) rule is to recall the double nature of the 2 formulae in LF as illustrated by the LF theorem below: 2OE , 2(F(OE) OE) 7 The rule shown below would highlight this double nature even more clearly: X ffl(F(OE) OE) X 2OE ( 2 0 ....

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A. Avron. On modal systems having arithmetical interpretations. The Journal of Symbolic Logic, 49(3):935--942, September 1984.

.... due to Martin Amerbauer [Ame93] who following suggestions of Rautenberg and Gor e also gives systems for KG:2 and KGL (which Amerbauer calls K4:3G) Provability logics have also been studied using Gentzen systems, and ap 56 Rajeev Gor e propriate cut elimination proofs have been given by Avron [Avr84], Bellin [Bel85] Borga [Bor83] Borga and Gentilini [BG86] Sambin and Valentini [SV80, VS83, SV82] and Valentini [Val83, Val86] 4.17 Monomodal Temporal Logics In this section, which is based heavily on [Gor94] we give tableau systems for normal modal logics with natural temporal ....

Arnon Avron. On modal systems having arithmetical interpretations. Journal of Symbolic Logic, 49:935--942, 1984.

....then both 2 Gamma and 2 Delta; 3: if 2 2 Gamma then 2 Gamma, for all ; 2 L2a (L2 ) Trivially, the empty sequent, is S4 saturated. Variants of the notion of saturation for sequents are found throughout the modal and non classical logic literature; see, for example, AS93] [Av84]. This notion is intimately related with the notion of a set of signed formulas as a consistency property in [Fi83] The saturation algorithm below is modelled on that of [AS93] Here, we strengthen the notion of saturation to deal with the [a] operator. Definition 6.1 A sequent Gamma ) Delta ....

Arnon Avron, "On Modal Systems Having Arithmetical Interpretations", Journal of Symbolic Logic 49 (1984) 935-942.

.... by a second order class of modal frames (see e.g. Boo93] the relational translation is not possible unless FOL is augmented with fixedpoint operators (see e.g. NS98] However, dedicated sequent style calculi do exist for provability logics such as G or Grz (for Grzegorczyk) see e.g. [SV80, Lei81, Fit83, Val83, Avr84, Boo93, Gor99]. Display Logic. Display Logic (DL) Bel82] is a proof theoretical framework that generalises the structural language of Gentzen s sequents by using multiple complex structural connectives instead of Gentzen s comma. The term display Visit to ARP supported by an Australian Research Council ....

.... and ffiG are respectively: X ffl(2(OE ) 2OE) ffi OE) X 2OE ( 2Grz ) X ffl(2OE ffi OE) X 2OE ( 2G ) The calculus ffiLF satisfies conditions (C2) C7) In particular, ffiG satisfies the conditions (C1) C7) The ( 2G ) rule in ffiG is similar to the GLR rule in [SV82] see also [Avr84]) Analogously, the ( 2Grz ) rule in ffiGrz is similar to the (GRZc) rule in [BG86] or to the ( 2) rule in [Avr84] An intuitively obvious way to understand the ( 2 LF ) rule is to recall the double nature of the 2 formulae in LF as illustrated by the LF theorem 2OE , 2(F(OE) OE) We use ....

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A. Avron. On modal systems having arithmetical interpretations. Journal of Symbolic Logic, 49(3):935--942, 1984.

....K 2p ) 22p, S4 def = K4 2p ) p and Grz def = S4 2(2(p ) 2p) p) 2p. Numerous variants of the system Grz (having the same set of theorems) can be found in the literature (see for instance [GHH97] We call GT , GS4 and GGrz the cut free versions of the Gentzen style calculi defined in [OM57, Avr84] where the sequents are built from finite sets of formulae. Moreover, the weakening rule is absorbed in the initial sequents. For instance, the initial sequents of all the Gentzen style calculi used in the paper are of the form Gamma; OE Delta; OE where , denotes set union. The common core ....

....of Lemma 7. Each rule ( 2) T , 2)S4 and ( 2)Grz belongs respectively to GT , GS4 and GGrz. For each L 2 fT ; S4; Grzg, we know that for any sequent Gamma Delta, the formula ( V OE2 Gamma OE) W OE2 Delta OE) 2 L iff 5 the sequent Gamma Delta is derivable in GL (see e.g. [OM57, Avr84, Gor99]) Consequently, if Gamma Delta is derivable in GL, then so is Gamma; Gamma 0 Delta; Delta 0 . 3 A transformation from Grz into S4 Let f : For Theta f0; 1g For be the following map: for any p 2 For 0 , f(p; 0) def = f(p; 1) def = p f( OE; i) def = f(OE; 1 Gamma i) ....

A. Avron. On modal systems having arithmetical interpretations. The Journal of Symbolic Logic, 49(3):935--942, 1984.

....for the following well known conservative extensions of K: T def = K 2p ) p, K4 def = K 2p ) 22p, S4 def = K4 2p ) p and Grz def = S4 2(2(p ) 2p) p) 2p. Clearly, OE 2 S4 implies OE 2 Grz. We call GT, GS4 and GGrz the cut free versions of the Gentzen style calculi defined in [22, 1] where the sequents are built from multisets of formulas. Moreover, we assume that the contraction and the weakening rules are absorbed in the introduction rules and axioms (see e.g. 26, Section 3.4 and Section 9.1] For instance, the initial sequents of all the Gentzen style calculi used in the ....

....(2 ) where 2 Gamma def = f2 : 2 Gammag. The rule (2 ) belongs to the three systems GT, GS4 and GGrz and each rule ( 2) T , 2) S4 and ( 2) Grz belongs respectively to GT, GS4 and GGrz. For each L 2 fT ; S4; Grzg, we know that OE 2 L iff the sequent OE is derivable in GL (see e.g. [22, 1]) Further details can be found in [11] 3 A transformation from Grz into S4 Let f : For Theta f0; 1g For be the following map: ffl for any p 2 For 0 , f(p; 0) def = f(p; 1) def = p ffl f( OE; i) def = f(OE; 1 Gamma i) for i 2 f0; 1g ffl f(OE 1 OE 2 ; i) def = f(OE 1 ; i) f(OE 2 ....

A. Avron. On modal systems having arithmetical interpretations. The Journal of Symbolic Logic, 49(3):935--942, September 1984.

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Avron A., On Modal Systems having arithmetical interpretations Journal of Symbolic Logic, vol. 49 (1984), pp. 935-942.

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A. Avron, On modal systems having arithmetical interpretation, The Journal of Symbolic Logic, vol. 49 (1984), pp. 935--942.

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