On Urquhart s C Logic

On Urquhart s C Logic Agata Ciabattoni Dipartimento di Informatica Via Comelico, 39 20135 Milano, Italy ciabatto@dsiunimiit Abstract In this paper we investigate the basic many-valued logics introduced by Urquhart in [15] and [16], here referred to as and Ò Û, respectively We define a cut-free hypersequent calculus for Ò Û and show the following results: (1) and Ò Û are distinct versions of Gödel logic without contraction (2) Ò Û is decidable (3) In Ò Û the family of axioms µ µµ µ µ, with ¾, is in fact redundant 1 Introduction The logic was introduced by Urquhart in the chapter devoted to many-valued logic of the Handbook of Philosophical Logic [15] turns out to be a basic many-valued logic being contained in the most important formalizations of fuzzy logic [7], namely infinite-valued Gödel, Łukasiewicz and product logic (see [3]) In [9, 10] was shown to be a particular Gödel logic without contraction A cut-free calculus for was defined in [3] This calculus uses hypersequents that are a natural generalization of Gentzen sequents Due to semantical motivations, in the second edition of his handbook paper [16], Urquhart presented a new version of We shall refer to this logic as Ò Û Semantically, Ò Û is characterized by model structures on ordered abelian monoids and is therefore contained in Łukasiewicz logic which is characterized by ordered Abelian groups [4, 16] In this paper we introduce a cut-free hypersequent calculus for Ò Û As a corollary, we prove that Ò Û is decidable Moreover, for each ¾, we show the redundancy in Ò Û of the axiom µ µµ µ µµ, where stands for µ µµ ßÞ Ð Ø Ñ In order to define a calculus for Ò Û we shall consider three different contraction-free versions of Avron s hypersequent calculus for Gödel logic [1] and we shall relate them with their corresponding logics, here after named and and, already investigated in [3], are respectively Urquhart s logic and its residuated version As we shall show, turns out to coincide with Ò Û 2 Urquhart s logics and Ò Û Urquhart s basic many-valued logic consists of the following axioms (see [15]): Ax1 µ Ax2 µ µ µ Ax3 µ µ Ax4 µ Ax5 µ Ax6 µ Ax7 µ Ax8 µ Ax9 µ µ µ Lin µ µ together with the rule of modus ponens In [16] Urquhart introduced a new version of We shall refer to this logic as Ò Û Semantically, Ò Û is characterized by model structures on ordered commutative monoids More precisely, let ¼µ be a (totally) ordered commutative monoid and a total ordering of satisfying the condition Ü Ý implies Ü Þ Ý Þ for all Ü Ý Þ ¾ Let ¼ be the set Ü ¾ Ü ¼ Definition 21 ([16]) A model structure over consists of a subset È ¼ for each atomic formula È in the language The sets È are required to be increasing (a subset is increasing if Ü ¾ and Ü Ý imply that Ý ¾ ) Given a model structure over, we define truth at a point Ü in ¼ by the following inductive stipulation:

1 Ü È iff Ü ¾ È ; Product logic Lukasiewicz logic Goedel Logic 2 Ü µ iff Ü and Ü ; 3 Ü µ iff either Ü or Ü ; BL 4 Ü µ iff, for all Ý ¾ ¼, Ý implies Ü Ý A formula is valid iff ¼ in every model structure over an ordered commutative monoid In [16] a Hilbert-style axiomatization of Ò Û is defined by adding to the above axioms ܾ Ü Ü Lin both: U1 µ µµ µµ U2 µ µµ µ µ for every ¾, where stands for µ µµ ßÞ Ð Ø Ñ Urquhart proved that a formula is valid in the model theory of Definition 21 if and only if Ò Û Remark 22 If in Definition 21 one makes the assumption that the underlying algebra is an ordered abelian group, one obtains model structures characterizing infinite-valued Łukasiewicz logic [4, 16] To put the logics and Ò Û into a broader context, let us introduce the following axioms: Res1 µ µ Res2 µ µ Abs Taking Modus Ponens as the only rule of derivation, we consider the logics: Á ܽ Ü Abs Á Lin Á Á Í ½ Á Lin Á Á Res1 Res2 Á Lin Á Á and have been investigated in [3, 5] Á, Á and Á, are different versions of intuitionistic logic ÁÄ without contraction Notice that Res1 is already derivable in Á In fact, this axiom turns out to be redundant in the presence of ܽ ܾ and Ü Á and Á are incomparable and they strictly extend Á (Corollary 313) Á leads to ÁÄ by adding axiom Res2 to the former coincides with and are incomparable versions of Gödel logic without contraction (Corollary 317) and they strictly extend In Section 4 we show that Ò Û Figure 1 shows proper inclusions among all the previously mentioned logics as well as of other well known logics (Corollary 46) C* a-mll IL C # C + I -* I-# I - Figure 1 Inclusions among various logics Indeed, by extending Á with axiom Ò (involutivity of negation), ie, µ µ one gets the affine multiplicative fragment of linear logic a-mll 1 [6] As shown in [13], Łukasiewicz logic [8] is obtainable by adding to a-mll the axiom µ µ µ µ Hájek s Basic Logic Ä was introduced in [7] as the logical counterpart of continuous Ø-norms As proved in [3], by adding to the axiom µ µ one gets Ä If we add to Ä axiom Ò we get Łukasiewicz logic, while by extending Ä with both axioms µ µµ and µµ we obtain product logic (see [7]) Each of, and Ä yields Gödel logic by adding the axiom µ µ (contraction) Remark 23 Henceforth we shall abuse the names Ò Û and to refer to their proper extensions Ò Û and (ie, µ, respectively 3 Sequent and Hypersequent Calculi In this section we define sequent (and hypersequent) calculi for Á and Hypersequent calculi are a simple and natural generalization of Gentzen sequent calculi See [2] for an overview Definition 31 A hypersequent is an expression of the form ½ ½ Ò Ò, where for all ½ Ò, is an ordinary sequent The intended meaning of the symbol is disjunctive Hypersequent calculi are particularly useful to formalize logics containing axiom Ä Ò (see [3, 5]) 1 In fact, we get a-mll extended with the additive disjunction

In [3] cut-free calculi for the logics Á Á and have been introduced They are called Ä Ä À and À, respectively Ä and Ä turn out to be different contraction-free fragments of the Ä sequent calculus for intuitionistic logic, while À and À are their hypersequent versions with Avron s communication rule [1] More precisely, let us consider the Ä sequent calculus for intuitionistic logic, ie, Ü ÓÑ Ùص ËØÖÙØÙÖ Ð ÊÙÐ Ûµ µ ÄÓ Ð ÊÙÐ Öµ Öµ µ ½ µ ¾ µ ½ ¾ µ µ Öµ µ µ ½ ¾ ½ µ ¾ µ ½ ¾ µ ½ µ ¾ µ ½ ¾ When investigating contraction-free fragments of ÄÂ, we can consider the following alternative formulations for the Öµ and rules: ¼ Öµ µ µ ¼ е ½ ¾ µ ÓÖ ½ ¾ The above rules are called additive, while the corresponding ones in Ä are called multiplicative (see [14]) As is well known, using µ and Ûµ, the additive rules are interderivable with the multiplicative ones, while in absence of µ, they are distinct Remark 32 See [12] for a general investigation on intuitionistic logic without contraction Let us consider the calculi obtained substituting the rules for conjunction in the contraction-free Ä calculus by one of the four possible combinations of the above additive and multiplicative rules The calculus defined by choosing and ¼ Öµ as rules for conjunction shows undesirable properties: it allows to derive contraction, and cuts are not eliminable (see [14]) The calculi obtained using the pairs of rules ¼ е Öµ and Öµ are respectively Ä and Ä of [3] Henceforth we shall investigate the fourth possible system, which we shall call Ä In Ä the rules for conjunction are ¼ е and ¼ Öµ and the remaining ones are those of LJ except contraction µ Let us introduce the hypersequent versions of the above calculi In À and À axioms, internal structural rules and logical rules are directly obtained from those of Ä and ÄÂ, respectively: the only difference is the presence of dummy contexts and ¼ representing (possibly empty) hypersequents For instance, the rules for implication are Öµ ½ µ ¼ ¾ µ ¼ ½ ¾ Additional structural rules are external weakening (EW) and contraction (EC) Ï µ µ µ ¼ µ ¼ together with Avron s communication rule [1] ½ ½ µ µ µ ¼ ¾ ¾ µ ¼ ½ ¾ µ ½ ¾ µ µ See [3] for a detailed description on these calculi The hypersequent version of Ä shall be called À In À axioms, internal structural rules and logical rules are obtained from those of ÄÂ, as seen for À and À Additional structural rules are (EW), (EC) and (com) Remark 33 By adding the µ rule to À À or À one gets Avron s Ä calculus [1] for Gödel logic 31 Cut-elimination In [3] it was proved that the cut-elimination theorem holds for Ä Ä À and À This is true also for À Indeed Theorem 34 Ä admits cut-elimination Proof: See, eg, [12] Definition 35 A sequent calculus is single-conclusion iff in all the involved sequents µ, the succedent contains at most one formula Theorem 36 If a single-conclusion sequent calculus admits cut-elimination then its hypersequent version with as additional rule admits cut-elimination Proof: See [3, 5]

Corollary 37 À admits cut-elimination Definition 38 Let ½ Ò µ be a sequent Then the generic interpretation Á of ½ Ò µ is defined as follows: Á µ µ Á ½ Ò µ µ ½ Ò µ µ Á ½ Ò µµ ½ Ò µ µ Let Ë ½ Ë Ò be a hypersequent Its generic interpretation is defined by Á Ë ½ Ë Ò µ Á Ë ½ µ Á Ë Ò µ Definition 39 Sequent (or hypersequent) rules Ë Ë and are called sound for a (Hilbert-style) calculus À iff À Á ˵ Á Ë µ and À Á Ì µ Á Ì ¼ µ Á Ì µµ, respectively If all the rules of a sequent (or hypersequent) calculus Ä Ä are sound for À and À Á ˵ for all the axioms Ë of Ä Ä, then Ä Ä is said to be sound for À A sequent (or hypersequent) calculus Ä Ä is complete for À iff Ä Ä µ µ whenever À Corollary 310 Ì Ì 1 Ä and Ä are proper extensions of ÄÂ Ì ¼ 2 Ä and Ä are incomparable 3 À is a proper extension of À 4 À is a proper extension of À 5 À and À are incomparable which evaluates to Ú È µ ½, for instance, in Gödel logic if Ú µ ¼ Since for each hypersequent Ë which is derivable in À, Á ˵ must be valid in Gödel logic, this concludes the proof 4 Follows by the fact that axiom Res2 is not derivable in À (see [3]) 5 It is immediate to see that À µ Ò À µ Res2 However, neither À derives, nor À derives Res2 The proofs proceed as in cases 3 and 4 above, respectively Corollary 311 Derivability in Ä and À is decidable Proof: (Sketch) Proof search based decision algorithms for sequent calculi without contraction are well known (see, eg, [11]) In the case of hypersequents first note that by the occurrence of the external structural rules, any cut-free proof can be transformed in an equivalent one in which every hypersequent has at most two equal components Thus a bound for the number of hypersequents occurring in the cut-free proofs, as well as for the number of the components in the involved hypersequents, can be obtained in terms of the size of the end hypersequent 32 Correspondences In [3] it was proved that the calculi Ä Ä À À are sound and complete for Á Á and, respectively In this section we relate the logics Á and, as given by their Hilbert-style axiomatizations, to the calculi Ä and À Proof: 1 It is easy to see that Theorem 312 Ä is sound and complete for Á Ä µ Ò Ä µ Res2 By the cut-free completeness of Ä it follows immediately that the above two axioms are not derivable in Ä 2 This trivially follows from the non-derivability of Res2 in Ä and of in Ä 3 It suffices to prove that the formula cannot be derived in À Indeed, À µ if and only if the hypersequent µ µ is derivable in À By Definition 39, this hypersequent translates to the formula È given by µ µ Proof: (Soundness) The axioms of Ä translate into and, respectively The corresponding derivation in Á is straightforward In order to show the soundness of the rules of Ä it suffices to prove that their generic interpretations are derivable in Á As an example we consider the ¼ Öµ rule Its generic interpretation µ is: ½ Ò µ µµ ½ Ò µ µµ ½ Ò µ µµµ For Ò ½ this formula coincides with axiom Ü Thus let Ò ¾ In axiom Í ½ let us replace with ½, with ¾ Ò µ µµ and with ¾ Ò µ µµ Let us denote with Í ½ the resulting formula By induction on Ò one can derive the formula µ Í ½ ½ ¾ Ò

µ µ ½ ¾ Ò µ µ ½ Ò µ µµ For the basic step Ò ½, the derivation is trivial For the inductive step, assuming the claim is true for Ò, applying axioms ܽ ܾ Ü together with modus ponens we can derive µ for Ò ½ Then µ follows by applying to µ both axioms Í ½ and Ê ½ together with modus ponens For the remaining rules see either [3] or Corollary 281 in [12] (Completeness) Observe that Modus Ponens the only rule of Á corresponds to the derivability of and of the cut rule It thus suffices to show that all the axioms of Á are derivable in Ä This is straightforward Corollary 313 The logics Á and Á are incomparable extensions of Á Proof: Immediate from soundness and completeness of the Ä calculus with respect to Á (see [3]), together with Corollary 310 and Theorem 312 Lemma 314 À µ Ä Ò Proof: The derivation proceeds as follows (see [1, 3, 5]): µ µ µ µ µ µ µ µ µ µ µ µ µ µ Theorem 315 À is sound and complete for Proof: (Soundness) In the light of Theorem 312 we must only take care of the external structural rules and of the rule µ corresponds to (EC) µ corresponds to (EW) These formulas are already derivable in Á, and hence also in For the rule, see [3] (Completeness) From Theorem 312 and Lemma 314 Corollary 316 is decidable Corollary 317 The logics and are incomparable versions of Gödel logic without contraction Proof: The proof follows by soundness and completeness of the À calculus with respect to (see [3]), together with Theorem 315 and Corollary 310 4 Ò Û coincides with In this section we prove that Urquhart s Ò Û logic coincides with the contraction-free fragment of Gödel logic Proposition 41 Ò Û Í ¾ Proof: It is immediate to see that Ò Û Í ¾ The converse directly follows from the derivability of axioms ܽ in Ò Û Hence, Ò Û Í ¾ Proposition 42 For ¾, the family of axioms Í ¾ can be derived in À Proof: For all Ò ¼, let us denote with ÒÈ the (possibly empty) set of formulas È È (Ò times) Then, for each ¾, we have the following derivation of axiom Í ¾ in À : µ µ µ µ µ µ µ µ µµ µ µ where, all the (¾ ) upper sequents in µ obtained applying the rule as many times as possible, have the form µ µ Ò Ñ Û Ø Ò Ñ All these sequents are provable in À To see this we shall argue by cases as follows: If Ò we proceed as follows: ½ ½µ µ µ The case Ñ is similar ¼ е If ¼ Ñ Ò, let us denote with È Ò Ñ the sequent È Ò Ñ We can write

µ Ñ ¼ Ñ Ò Ò ¼ Ò Ò ¼ Ñ ¼ Ñ Ò Ò ¼ ½ Ò ½ Ñ ½ Ò Ñ ½ Ò Ñ Ò Ñ µ µ Ò Ñ where µ and are respectively given by: and µ ¼ е Ò ½ Ò ½ ¼ Ò Ò ¼ Ò Ò ¼ Ò Ò ¼ Ñ ¾ ¼ Ñ ¾ Ò Ò ¼ Ñ ½ ¼ Ñ ½ Ò Ò ¼ Ò Ò ¼ Ñ ½ ¼ Ñ ½ Ò Ò ¼ The case ¼ Ò Ñ is similar This concludes the proof Theorem 43 Ò Û coincides with Proof: By the previous propositions Corollary 44 Ò Û and are distinct versions of Gödel logic without contraction Proof: By the previous theorem and Corollary 317 [2] A Avron The method of hypersequents in the proof theory of propositional nonclassical logics In Logic: from Foundations to Applications, European Logic Colloquium, pages 1 32 Oxford Science Publications Clarendon Press Oxford, 1996 [3] M Baaz, A Ciabattoni, C Fermüller, and H Veith Proof theory of fuzzy logics: Urquart s C and related logics In Mathematical Foundations of Computer Science (MFCS98), Lectures Notes in Computer Science, vol 1450, pages 203 212, 1998 [4] C C Chang Algebraic analysis of many-valued logics Trans of the Amer Math Soc, 88:467 490, 1958 [5] A Ciabattoni Proof-theoretic techniques in many-valued logics PhD thesis, 1999 University of Milan, Italy [6] J Y Girard Linear logic Theoretical Computer Science, 50:1 101, 1987 [7] P Hájek Metamathematics of Fuzzy Logic Kluwer, 1998 [8] J Lukasiewicz Philosophische bemerkungen zu mehrwertigen systemen der aussagenlogik Comptes Rendus de la Societé des Science et de Lettres de Varsovie, pages 51 77, 1930 [9] J Mendéz and F Salto Urquhart s C with intuitionistic negation: Dummett s LC without the contraction axiom Notre Dame J of Formal Logic, 36(3):407 413, 1995 [10] J Mendéz and F Salto A natural negation complementation of urquhart s many-valued logic C J of Philosophical Logic, 27:75 84, 1998 [11] H Ono Proof-theoretic methods in nonclassical logic an introduction In M Takahashi, M Okada, and M Dezani- Ciancaglini, editors, Theories of Types and Proofs, pages 207 254 Mathematical Society of Japan, 1998 [12] H Ono and Y Komori Logics without the contraction rule J of Symbolic Logic, 50:169 201, 1985 [13] A Prijatelj Bounded contraction and gentzen style formulation of Lukasiewicz logics Studia Logica, 57:437 456, 1996 [14] A Troelstra and H Schwichtenberg Basic Proof Theory Cambridge University Press, 1996 [15] A Urquhart Many-valued logic In D Gabbay and F Guenthner, editors, Handbook of Philosophical Logic, Vol III: Alternatives in Classical Logic, pages 71 116 Reidel, Dordrecht, 1986 [16] A Urquhart Basic many-valued logic In D Gabbay and F Guenthner, editors, Handbook of Philosophical Logic: Alternatives in Classical Logic Reidel, Dordrecht, To appear Corollary 45 Ò Û is decidable Proof: Follows by Corollary 316 and Theorem 43 Corollary 46 All inclusions of logics in Figure 1 are proper References [1] A Avron Hypersequents, logical consequence and intermediate logics for concurrency Annals of Mathematics and Artificial Intelligence, 4:225 248, 1991