Classical Gentzen-type Methods in Propositional Many-Valued Logics

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1 Classical Gentzen-type Methods in Propositional Many-Valued Logics Arnon Avron School of Computer Science Tel-Aviv University Ramat Aviv 69978, Israel Abstract A classical Gentzen-type system is one which employs two-sided sequents, together with structural and logical rules of a certain characteristic form. A decent Gentzentype system should allow for direct proofs, which means that it should admit some useful forms of cut elimination and the subformula property. In this tutorial we explain the main difficulty in developing classical Gentzen-type systems with these properties for many-valued logics. We then illustrate with numerous examples the various possible ways of overcoming this difficulty. Our examples include practically all 3-valued logics, the most important class of 4-valued logics, as well as central infinite-valued logics (like Gödel- Dummett logic, S5 and some substructural logics). 1 Introduction Many-valued logics are, above all, Logics. This means that like any other logic, the main issue with which they deal is: what formulas follow from what sets of premises (under certain assumptions). The many-valued part of their name refer to the type of semantics which determines the answers they give to this basic question. However, any logic needs efficient proof systems in order to actually be applied (as well as for a deeper understanding of its logical properties), and many-valued logics are no exception. The problem of adapting to many-valued logics the usual methods of automated reasoning (like Gentzen-type systems, tableaux, and resolution) has therefore attracted a lot of attention in recent years. The main idea which is used in most of the works on this subject is to use for any particular Ò some Ò-valued counterparts of the structures used in the usual proof systems for classical logic (like sequents with Ò components, tableaux systems with Ò signs, etc.). This fact is well reflected in the two recent survey papers [36] and [16] (see there for extensive list of references). In this paper we concentrate, in contrast, on proofs systems which use the same syntactic structures as the classical ones (with particular emphasis on Gentzen-type systems, on which all other systems are based). This approach has two advantages. First, the implementation of the proofs systems described here can be based on existing systems, because no new data structure is used. Second, two sided sequents (like those used in classical logic) can directly represent the consequence relation of any given logic, and this is, after all, what logics are all about. 2 Gentzen-type Systems A Background In what follows Ä is a propositional language, Ô Õ Ö denote atomic formulas, ³ denote arbitrary formulas (of Ä), and denote finite sets of formulas of Ä. Definition [44, 43] A (Scott) consequence relation (scr for short) for Ä is a binary relation between (finite) sets of formulas of Ä that satisfies the following conditions: R reflexivity: ³ ³ for every ³. M monotonicity: if and ¼, ¼ C cut: then ¼ ¼. if and ¼ ¼ then ¼ ¼. 2. An scr is uniform if, for every uniform substitution and every and, implies µ µ. is consistent (or non-trivial) if there exist non-empty and s.t.. 3. A propositional logic is a pair Ä Ä Ä, where Ä is a propositional language and Ä is a uniform consistent scr for Ä. Some notes concerning this Definition are in order:

2 There are exactly four inconsistent finitary scrs in any given language: the one in which iff and are non-empty; the one in which iff is nonempty; the one in which iff is non-empty; and the one in which for all and. All of them should be considered trivial, and are excluded from our definition of a logic. The notion of an scr is a natural symmetric generalization of the notion of a Tarskian consequence relation (tcr), which is defined similarly, but is a relation between sets of formulas and formulas (the monotonicity condition applies in it therefore only to the l.h.s.). A further useful generalization of the concept of a consequence relation is obtained by giving up the monotonicity condition. There are also generalizations in which the relation is taken to be between multisets (or even sequences) of formulas rather than sets thereof. A Gentzen-type calculus ([30]) over Ä is an axiomatic system which manipulates higher-level constructs called sequents, rather than the formulae themselves. There are several variants of what constitutes a sequent. Here we usually take it to be a construct of the form µ, where are finite sets of formulae of Ä and µ is a new symbol, not occurring in Ä. In other variants may be either multisets or sequences of formulae. There is no real difference, if we assume the following standard structural rules (permutation, contraction, and expansion, respectively): ½ ³ ¾ µ ½ ³ ¾ µ ½ ³ ³ ¾ µ ½ ³ ¾ µ ½ ³ ¾ µ ½ ³ ³ ¾ µ µ ½ ³ ¾ µ ½ ³ ¾ µ ½ ³ ³ ¾ µ ½ ³ ¾ µ ½ ³ ¾ µ ½ ³ ³ ¾ Definition 2.2 A Gentzen-type system is called standard if its set of axioms includes the standard axioms: µ (for all ), and its set of rules includes the standard structural rules, and the following weakening and cut rules: µ ¼ µ ¼ ½ µ ½ ³ ³ ¾ µ ¾ ½ ¾ µ ½ ¾ Definition 2.3 A rule of a Gentzen-type system is called pure ([11]), or multiplicative ([33]), if whenever µ can be inferred by it from µ ( ½ Ò) then, for arbitrary ¼ ¼ ½ Ò ¼ ½ ¼ Ò, the sequent ¼ ½ ¼ Ò µ ¼ ½ ¼ Ò can also be inferred by it from ¼ µ ¼ ( ½ Ò)1. is called pure if all its rules are pure. 1 In other words: a rule is pure if it is context-free. In practice, this means that there are no side conditions on how it can be applied. A Gentzen-type system,, directly defines a Tarskian consequence relation between sequents. Usually, however, it is mainly used as a tool for investigating scrs between the formulas of the underlying language. Any non-trivial standard Gentzen-type system naturally defines in fact an scr : Definition Let be a standard Gentzen-type system. We say that iff µ is a theorem of. is called the standard scr defined by. 2. A Gentzen-type system over a language Ä is sound and complete for a logic Ä Ä Ä if Ä. is weakly sound and complete for Ä if for all, Ä iff µ is provable in. is the most natural consequence relation associated with a given Gentzen-type system. However, it is not the only possible or useful one. Thus according to another important Tarskian consequence relation frequently associated with, a formula follows from the set ³ ½ ³ Ò µ if µ follows in from µ ³ ½ µ µ ³ Ò µ. We next describe a corresponding scr. Proposition 2.5 ([12]) If a Gentzen-type system is standard and pure then ³ ½ ³ Ò µ ½ Ñ iff µ follows in G from µ ³ ½ µ µ ³ Ò µ ½ µµ Ñ µµ This remains true even if is not closed under weakening. An ideal Gentzen-type system (of which the usual systems for classical logic provide the principal examples) is a pure, standard system in which every logical rule is an introduction rule for one connective. Moreover: it should introduce exactly one occurrence of that connective in its conclusion, no other occurrence of that connective or any other connective should be mentioned anywhere else in its formulation, and its side formulas should be immediate subformulas of the principal formula. The next definition formulates this idea in exact terms, and provides a method for describing such rules. Definition 2.6 ([15]) 1. A canonical rule of arity Ò is an expression of the form µ ½Ñ, where Ñ ¼, is either Ô ½ Ô ¾ Ô Ò µ µ or µ Ô ½ Ô ¾ Ô Ò µ for some connective (of arity Ò), and for ½ Ñ, µ is a clause such that Ô ½ Ô ¾ Ô Ò 2 2 By a clause we mean a sequent which consists of atomic formulas only. When propositional clauses are written in this way, resolution and cut amount to the same thing.

3 2. An application of a canonical rule of the form: Notes: µ ½Ñ Ô ½ Ô Ò µ µ is any inference step of the form: µ ½Ñ ½ Ò µ µ where and are obtained from and (respectively) by substituting for Ô (for ½ Ò), are any finite sets of formulas, Ë Ñ ½, and Ë Ñ ½. An application of a canonical rule of the other type is defined similarly. 1. While sequents are written in a metalanguage for Ä (which includes the extra symbol µ), a canonical rule is formulated in a meta-meta language of Ä (which includes one further symbol: ). 2. Definition 2.6 allows the case Ñ ¼, when canonical rules reduce to canonical axioms of the form Ô ½ Ô ¾ Ô Ò µ µ or µ Ô ½ Ô ¾ Ô Ò µ. This amounts to allowing canonical propositional constants. Example 2.7 The two usual introduction rules for classical conjunction can be formulated as the two canonical rules: Ô ½ Ô ¾ µµ Ô ½ Ô ¾ µ Applications of these rules have the form: µ µ µ Ô ½ µ µ Ô ¾ µ µ Ô ½ Ô ¾ µ ¼ µ ¼ ¼ µ ¼ Definition 2.8 A Gentzen-type system is called canonical if its axioms include the standard axioms, its rules include the standard structural rules, cut and weakening, and all its other rules and axioms are canonical. A canonical system is obviously standard and pure. However, in order for to really be an scr, one further condition should be satisfied: Theorem 2.9 ([15]) If is a canonical system then is consistent (and so an scr) iff the following coherence condition is satisfied: whenever Ë ½ Ô ½ Ô ¾ Ô Ò µ µ and Ë ¾ µ Ô ½ Ô ¾ Ô Ò µ are rules of, the set of clauses Ë ½ Ë ¾ is classically inconsistent (and so the empty clause can be derived from it using cuts). Moreover: any canonical system which satisfies this condition admits cut-elimination (i.e.: its cut rule is redundant). Example 2.10 The two classical rules for conjunction described in Example 2.7 form a coherent set of rules, where Ë ½ Ô ½ Ô ¾ µµ, Ë ¾ µ Ô ½ µ µ Ô ¾ µ. Ë ½ Ë ¾ is here the inconsistent set Ô ½ Ô ¾ µµ µ Ô ½ µ µ Ô ¾ µ. 3 A Classical Example The classical examples of canonical Gentzen-type systems are those for Classical Propositional Logic (CPL) and its fragments. We present now one of them, in which the primitive connectives are negation, disjunction, conjunction, and implication, and all the rules are pure. It should be noted that Gentzen s original formulation in [30] uses the additive form of these rules, in which all premises of a rule have the same side formulas. In the presence of the standard structural rules and weakening the two forms are equivalent, but otherwise the additive form is impure. In the present context it has however the advantage of being invertible. This is a very important property, and so in later sections we shall prefer to use the additive form whenever it is both invertible and equivalent to the pure one. THE SYSTEM GCPL Axioms: ³ µ ³ Structural Rules: Cut, Weakening (and the standard rules) Logical Rules: µµ µ µ µµ µµ µµ µ µ µµ µ µ µ ³ ³ µ ³ µ µ ³ ½ µ ½ ³ ¾ µ ¾ ³ ½ ¾ µ ½ ¾ ³ µ µ ³ ³ µ ³ µ ½ µ ½ ³ ¾ µ ¾ ½ ¾ µ ½ ¾ ³ ½ ³ µ ½ ¾ µ ¾ ½ ¾ ³ µ ½ ¾ µ ³ µ ³ The most important theorem concerning this calculus is Gentzen s celebrated cut-elimination theorem ([30, 48]). We present now a strong form of it, together with applications and two methods of proof. Theorem 3.1 Strong Cut-elimination for GCPL ([34, 12]): µ is derivable from ½ µ ½ Ò µ Ò

4 in È Ä iff it has a proof from these assumptions in which all cuts are done on formulas in Ë Ò ½ Ë Ò ½. Corollary 3.2 ÈÄ µ iff it has a cut-free proof. Gentzen s Method of Proof: Show by a double induction on the complexity of the cut-formula ³ and on the sum of the lengths of the proofs of the premises that if ½ µ ½ ³ ³ and ³ ³ ¾ µ ¾ have good proofs, then so does ½ ¾ µ ½ ¾. Semantic Method of Proof (outline): Let Ë be a given set of sequents, and assume Ë ½ µ ½ Ò µ Ò. A sequent µ is called Ë-saturated if: (1) It has no good proof from Ë (2) If ³ ¾ Ò Ë ½ Ò Ë ½ then ³ ¾ (3) If ³ ¾ then ³ ¾ and ¾ If ³ ¾ then either ³ ¾ or ¾ (4) Similar conditions obtain for the other connectives The theorem easily follows from the following two lemmas: Lemma 1: If µ is Ë-saturated, then the following valuation is a model of Ë, but not a model of µ : Ú Ôµ Ø Ô ¾ Ô ¾ In fact, for every ³, ³ ¾ implies Ú ³µ Ø and ³ ¾ implies Ú ³µ. Lemma 2: If µ has no good proof from Ë then it can be extended to an Ë- saturated sequent. The (strong) cut-elimination theorem for È Ä has many applications, like: decidability, the interpolation theorem, and the subformula property. It is also the basis for the two main proof search methods for È Ä: The Tableaux Method: Try to show that ÈÄ by searching for a cut-free proof of µ in È Ä. This is done by applying backwards the invertible versions of the rules. In particular one shows that ³ is valid by searching for a cut-free proof of µ ³. This gives either such a proof, or an equivalent set of clauses (i.e.: sequents consisting of atomic formulas) which can be translated into a CNF for ³. The Resolution Method: Try to show that ³ ½ ³ Ò ÈÄ ½ by showing that the empty sequent µ can be derived in È Ä from the set: µ ³ ½ µ µ ³ Ò µ ½ µµ µµ (In particular, a formula ³ is shown to be valid by proving that ³ µµ ÈÄ µ). For this replace (using tableaux) each µ ³ µ and µµ by an equivalent set of clauses. By the strong cut-elimination theorem, if µ is derivable from the original set of sequents, then it can be derived from the union of the equivalent sets of clauses using only cuts. 4 The Problem with Gentzenizing Many- Valued Logics We start by defining in precise terms what we mean by a many-valued logic : Definition A matrix Å for a language Ä is a triple Å Ç such that: (a) Å is a nonempty set (of truth-values ). (b) is a proper, nonempty subset of Å (the designated values ). 3 (c) Ç is a set of operations on Å, so that for each connective of Ä there is a corresponding operation on Å, and vice versa. 2. Let Å be a matrix for Ä. Å, The consequence relation induced by Å, is defined by: Å iff for every valuation Ú in Å which respects the operations and such that Ú µ ¾ for every ¾, we have that Ú ³µ ¾ for some ³ ¾. 3. Let ½ Ò ¼. A logic Ä Ä Ä is (weakly) Ò-valued if there exists a matrix Å for Ä such that: Notes: (a) Å has exactly Ò elements. (b) Å Ä ( Å ³ iff Ä ³ for every sentence ³). (c) For every natural number there exists a finite submatrix Å of Å such that whenever a sequent µ (a formula ³) contains occurrences of no more than atomic formulas, we have that Ä ( Ä ³) iff Å ( Å ³). 1. The main factor in this definition of a many-valued logic is the existence of a single characteristic matrix. Condition (c), on the other hand, guarantees that every propositional many-valued logic is decidable, and that even ¼ -valued logics may be viewed as semi-finite (in a certain sense). 3 In [49] and elsewhere the only condition concerning is that it should be a subset of Å. We exclude here the two extreme cases ( Å and ) because the corresponding scrs (as defined below) are not consistent.

5 2. A matrix Å is frequently used for defining consequence relations other than Å. In all cases we know the definitions of these alternative consequence relations can be reduced to the validity of certain formulas in Å, and so to Å (for example: the consequence relation usually associated with Łukasiewicz 3-valued logic can be characterized as follows: ³ ½ ³ Ò iff the formula ³ ½ ³ ¾ ³ Ò µ µµ is valid in Łukasiewicz 3-valued matrix). The following theorem is the main obstacle to providing decent Gentzen-type calculi for many-valued logics: Theorem 4.2 ([15]) Let be a consistent canonical calculus. Then either defines a logic which is a fragment of classical logic, or it is not sound and complete for any many-valued logic. It follows that a Gentzen-type calculus for a given manyvalued logic should use at least one of the following: Noncanonical rules Axioms which are neither standard nor canonical Impure rules (i.e.: rules with side conditions on their applications) A nonstandard set of structural rules In the following sections we shall see examples of all these alternatives. 5 The Use of Noncanonical Rules and Axioms The most important feature of canonical rules is that they introduce exactly one new occurrence of a connective at a time. Most of the Gentzen-type systems for many-valued logics give up this feature by allowing rules which introduce two occurrences of connectives at the same time. 5.1 Three-Valued Logics We start with the simplest type of non-classical manyvalued logics: the three-valued ones. We assume (w.l.o.g.) that those three values are Ø and Á, where Ø is designated and is not. The 3-valued logics are accordingly divided into two classes: those in which Á is designated (i.e. Ø Á), and those in which it is not (i.e. Ø). The logics inside each class differ only with respect to the expressive power of their languages. The most famous 3-valued logic with Ø is that of Kleene ([39]). It is based on the classical connectives, and, where the interpretation of is given by: Ø Ø Á Á and, are interpreted as the ÑÜ and ÑÒ operations (respectively) according to the order Á Ø. This logic has no valid sentences and no appropriate implication. One natural way to remedy this (resulting in a language equivalent to that used in the logic LPF of the VDM project ([38]) as well as to the language of Łukasiewicz 3-valued logic Ł ([40])) is to add to the language the following connective from [10], interpreted as: Ø if ¾ if ¾ The standard 3-valued logic with Ø Á is the paraconsistent logic  [19, 8, 41, 24]. Its basic connectives are again, with the interpretations defined identically as above 4. Now according to both interpretations, the classical rules for, and are valid, but one of the rules for is not (they differ w.r.t to which of them is valid and which fails). The way to get useful cut-free Gentzen-type systems for these logics is therefore to replace, first of all, the two classical rules for negation with the following rules (from [10]) for the combination of negation with the connectives of the language: 5 µµ µµ µµ µµ µµ µµ µµ µµ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ Let us call the resulting system Ë. This system is sound for both choices of. In both cases, in order to get corresponding sound and complete systems one needs 4 This means that in the case of the interpretations are not identical, because is different. The two interpretations are however definable in terms of each other using negation and the propositional constant. 5 As we stated above, we use from now on the additive form of a rule (i.e., we assume that the side formulas are the same in all premises of a rule) whenever weakening and the standard structural rules are among the rules of the system under consideration.

6 to add noncanonical axioms as well: ³ ³ µ in case Ø, µ ³ ³ in case Ø Á. The classical proofs of the generalized cut-elimination theorem and the completeness theorem can be now adapted for the two systems with very slight changes. For example, in the case where Ø Á the countermodel of a saturated sequent µ is defined by: Ú Ôµ Ø Ô ¾ Á Ô ¾ Ô ¾ Ô ¾ Unlike the two-valued case, the set of connectives used in Ë is not functionally complete (with either of the interpretaions for used above). In [13] it is proved, however, that by adding the propositional constants and Á (interpreted as the corresponding truth values), one does get a functionally complete set of connectives. To handle these constants, one only needs to add to the systems above some simple axioms. In the case of these are: µ µ What should be taken as the axioms for Á depends, on the other hand, on the choice of. In case Ø these are: Á µ Á µ while in case Ø Á the needed axioms are: µ Á µ Á It is straightforward now to generalize the completeness theorem and the cut-elimination theorem (as well as their proofs) to the extended systems. Since the set of connectives used in them is functionally complete, this allows us to find an adequate set of rules for every connective, and so for every 3-valued logic (provided its language includes ). Thus if is Łukasiewicz implication, then ³ is equivalent to ³ µ ³µ (where is defined like in the case Ø). This leads to the following four rules for : 6 µµ µµ µµ µµ µ ³ µ µ ³ ³ µ µ ³ ³ µ µ ³ ³ µ ³µ µ µ µ ³ µ ³µ 6 Note again that the sequents here do not reflect the consequence relation dictated by using as the official implication. That consequence relation is not even an scr, since contraction fails for it! The (strong) cut-elimination theorem which obtains for the Gentzen-type systems described in this section has the same applications and consequences in the 3-valued case as in the two-valued one. Examples are the interpolation theorem, and an appropriate version of the subformula property (according to which a proof of a sequent contains only subformulas of this sequent, or negations of proper subformulas of it). It also leads to versions of the tableaux and resolution methods which are very similar to those used in the classical case. The differences are as follows: 1. A clause in the present context is a sequent which contains only literals (i.e. atomic formulas or their negations) on both sides. Since all the rules of the systems above are invertible, every sequent can be reduced to an equivalent finite set of clauses (in this sense) by the corresponding tableaux rules. 2. A clause µ is valid not only when, but also when, for some atomic È, È È (in case Ø) or È È (in case Ø Á). 3. For proving ³ ½ ³ Ò ½ using resolution, the ordinary set of clauses obtained from the set µ ³ ½ µ µ ³ Ò µ ½ µµ µµ should be extended with all sequents of the form È È µ (in case Ø) or µ È È (in case Ø Á), where È is any atomic formula occurring in the original sequent. Note: In the tableaux method one should consider 4 types of signed formulas: ̳, ̳, ³, and ³. It might be more convenient to use instead four different signs: Ì, Ì Æ, and Æ. The resulting rules become very similar to the classical ones (but ̳, e.g., is reduced to Æ ³ rather than to ³). A branch is then closed if for some ³, it contains either ̳ and ³, or Ì Æ ³ and Æ ³, or ̳ and Æ ³ (in case Ø), or Ì Æ ³ and ³ (in case Ø Á). 5.2 Four-Valued Logics The methods used above for three-valued logics can be extended with very slight changes to four-valued logics in which there are exactly two designated elements. Let the truth values of such logics be Ø, where Ø and are the classical values, and Ø. is the fourvalued counterpart of the designated Á, while is the counterpart of the non-designated Á. This is the basis of what is known as Belnap four-valued logic ([18, 17]) 7. Intuitively, represents the truth-value of formulas about which there is inconsistent data (such a formula is both true and false ), while is the truth-value of formulas on which no data at 7 Belnap s structure is nowadays known also as the basic bilattice, and its logic as the basic bilattice logic (see [31, 32, 26, 25, 27, 28, 3, 4, 5]).

7 all is available ( neither true nor false ). With these intuitions, the interpretations of the connectives are exactly the same as in the 3-valued case (so,, is defined exactly as above, and, are interpreted as the ÑÜ and ÑÒ operations according to the following partial order: Ø). Now the methods described in the previous subsection can be used (see [4]) to show that the basic system Ë is sound and complete for the resulting logic, and that the (strong) cut-elimination theorem obtains for it. These results can be extended to the case when the language includes the propositional constants and, provided we add to Ë the obvious axioms for these connectives (for these are the axioms used above for Á in case Ø Á, for the axioms for Á in case Ø). Since is a functionally complete set of connectives for 4-valued logics ([13]), this again allows us to find a complete, cut-free Gentzen-type system for any 4-valued logic in which Ø. These results have the same applications here as in the 3-valued case. In fact, both of the resulting tableaux and resolution methods are simpler here than in the 3-valued case, because the set of axioms is simpler in the present system than in its 3-valued counterparts. Except for the class just dealt with, there are two other possible classes of 4-valued logics: those with Ø, and those with Ø. Similar general methods for handling these two classes have recently been found, but at present they are more complicated, and we shall not describe them here Gödel-Dummett Logics Up to now, we have only considered three- or fourvalued logics. In this subsection we show that the same method is also applicable to Ò-valued logics with Ò having bigger values, and even to infinite-valued logics. Our example will be the famous Gödel-Dummett logics. In [35] Gödel introduced a sequence Ò (Ò ¾) of Ò- valued matrices. He used these matrices to show some important properties of intuitionistic logic. An infinite-valued matrix in which all the Ò can be embedded was later introduced by Dummett in [20]. The logic of was axiomatized in the same paper, and has been known since then as Gödel-Dummett s Ä. It is probably the most important intermediate logic, which turns up in several places, such as the provability logic of Heyting s Arithmetics ([50]), and relevance logic ([21]). Recently it has again attracted a lot of attention because of its recognition as one of the three most basic fuzzy logics ([37]). The language of Ä is. The matrix is Æ Ø, where is the usual order on 8 One particular four-valued logic in which Ø belongs to the sequence of logics dealt with in the next subsection. Æ extended by a greatest element Ø, the interpretation of the propositional constant is the number 0, is Ø if and otherwise, is ¼, and and are resp. the ÑÒ and ÑÜ operations on Æ Ø 9. The matrices of Ò are similar, but the set of truth values of Ò is ¼ Ò ¾ Ø. A cut-free Gentzen-type formulation for Ä was first given by Sonobe in [47]. His approach was improved in [6] and [23]. All those systems have, however, the serious drawback of using the following rule, which introduces an arbitrary number of implications, and has an arbitrary number of premises, all of which contain formulas of essential importance for the inference: ³ µ µ Ê Here contains an arbitrary number Ñ ¼ of implicational formulas ³ ½ Ñ may also contain some other kinds of formulas. indicates the multiset consisting of exactly the Ñ implicational formulas of. is just after removal of the th formula ³. An alternative system Ä ÊË for Ä, which does not have a rule with arbitrary number of premises, has been given in [14]. Like the systems for 3-valued and 4-valued logics described above, Ä ÊË uses invertible rules which decompose formulas in a sequent into simpler ones, until a set of clauses equivalent to the original sequent is reached. Here, however, a clause is defined as a sequent consisting only of atomic formulas or implications between atomic formulas. Accordingly, the number of rules needed for each connective (other than ) is six. The rules for, for example, are: µ µ µµ µ µ µµ µ µ µµ µ ³ µ ³ ³ µ µ ³ µ µ ³ ½ µ ³ ¾ µ ³ ½ ³ ¾ µ ³ ½ ³ ¾ µ ³ ½ ³ ¾ µ µ µ ³ ½ ³ ¾ µ ³ ½ ¾ µ ³ ½ µ ³ ¾ µ ³ ½ ¾ µ µ 9 This interpretation is not the one given by Gödel and Dummett, but its dual. We note also that for the application as a fuzzy logic it is more useful ([37]) to use instead of Æ Ø the real interval [0,1], with 1 playing the role of Ø. This makes a difference only when we consider inferences from infinite theories.

8 Like in the 3-valued case, these rules are not sufficient for completeness, and we have to add non-standard axioms as well. Unfortunately, the needed axioms are more complicated than those employed in the 3-valued case. Definition 5.1 Let µ be a clause of Ä ÊË. We say that Ô Õµ ¾ µ µ iff Ô Õµ ¾. We say that Ø Õµ ¾ µ µ iff Õ ¾. We say that Ô Õµ ¾ µ µ iff Õ Ôµ ¾. We say that Õ Øµ ¾ µ µ iff Õ ¾. Let Ô Õ be either atomic formulas or Ø 10. We say that Ô» Õµ ¾ µ µ iff either Ô Õµ ¾ µ µ or Ô Õµ ¾ µ µ. A sequence Õ ½ Õ Ð (where Õ is either atomic or Ø) is called a strictly increasing sequence for µ if Õ» Õ ½ µ ¾ µ µ for ½ Ð ½, and either Õ Õ ½ µ ¾ µ µ for some ½ Ð ½, or Õ ½ Ø Õ Ð. Now the axioms needed in the case of Ä itself are those clauses for which there exists a strictly increasing sequence Õ ½ Õ Ð such that either Õ ½ Õ Ð, r Õ ½ Ø, or Õ Ð. To get a complete, cut-free formulation for a given finite k-valued Gödel-Dummett logic, one needs to have as additional axioms all basic sequents µ having a strictly increasing sequence Õ ½ Õ Ð such that for at least different Õ s, Õ Õ ½ µ ¾ µ µ. Instead of enriching the set of axioms, an alternative approach presented in [14] is to employ special analytic simplification rules. Those needed for the ¼ -valued Ä are: Transitivity: Left maximality: Right maximality: Linearity: Minimality of : Ô Õ Õ Ö Ô Ö µ Ô Õ Õ Ö µ Ô Õ Ô Õ µ Ô Õ Ô µ µ Õ Ô Ô µ Õ Ô Ô Õ µ Õ Ô µ Õ Ô Ô Ô µ Ô µ 10 Note that in this paper Ø is not an official symbol of the language of Ä. Here, however, it is used in the metalanguage. Note: All the logical rules of the systems described in this subsection have what might be called the semi-subformula property: written in Polish notation, every formula in their premises either appears in their conclusion or is obtained from some formula there by deleting some of its symbols. This is not very different from the usual subformula property. The simplification rules, in contrast, do not have this property. Nevertheless, each of them is still analytic in the sense that its possible premises (for any potential conclusion) are determined by its conclusion, they are finite in number, and can be found effectively. These rules are close in nature to what is known as analytic cut (see next section), although they are in fact simpler. 6 Systems For Weak Completeness In this section we present examples of systems which employ impure rules and systems which differ from classical logic in their structural rules. The immediate effect is that usually the consequence relation defined by a Gentzentype system of this kind cannot be identical with a consequence relation induced by a matrix. What can be achieved is weak soundness and completeness (see Definition 2.4). This is usually sufficient, since the consequence relation of the logic is frequently reducible to its set of valid sentences using some deduction theorem. 6.1 Employing Impure Rules: The Modal Ë In this subsection we present one important example of the use of an impure rule: the standard Gentzen-type system for the modal logic Ë. Like most other modal logics, Ë has a well-known (particularly simple) possible-worlds semantics. However, Ë had been shown in [45] to be a many-valued logic (in the strict sense defined above) much before its Kripke-style semantics was discovered. The many-valued semantics of Ë is based on Boolean Algebras. Let ½ ¼ be such an algebra. Define an operation ¾ on as follows: ½ ¾ ¼ ½ ½ Let ½ be the only designated value in the resulting structure. It has been proved in [45] that Ë is sound and complete for the class of these structures. Moreover: if a sentence ³ in the language of Ë is not a theorem of Ë, then it is refutable in some finite structure of this type (with at most ¾ ¾Ò elements, where Ò is the number of atomic formulas occurring in ³) All these finite matrices are embeddable in one denumerable matrix, which is characteristic for Ë.

9 Ë, the standard Gentzen-type system for Ë, is obtained from È Ä by adding to it the following two simple rules: ¾ µ ³ µ ¾³ µ µ ³ µ ¾³ µ ¾ There is, however, a side condition on the application of µ ¾, which makes this rule impure: all the side formulas (i.e., the formulas of and ) should begin with ¾ (an equivalent version demands them only to be essentially modal, in the sense that each occurrence of an atomic formula should be within the scope of a ¾). As a result, the standard consequence relation defined by Ë is different from that induced by the many-valued semantics described above (for example: ³ Ë ¾³ according to that semantics, but the sequent ³ µ ¾³ is not provable in Ë). Nevertheless, we have the following weak completeness theorem: Theorem 6.1 A formula ³ is a theorem of Ë iff Ë µ ³. Another drawback of Ë is the failure of the cut elimination theorem for Ë. Instead we have, however, the possibility to eliminate all non-analytic cuts: Definition 6.2 A cut in a proof of µ is called analytic if the cut formula is a subformula of µ. Theorem 6.3 Ë µ iff there exists a proof of this sequent in which all the cuts are analytic and are performed on formulas of the form ¾. This theorem suffices for the subformula property, and for developing a good tableaux system for Ë. 6.2 Systems without Weakening Substructural logics ([42]) are logics defined by Gentzen-type systems with an irregular set of structural rules (the regular set being the one consisting of the standard structural rules, weakening, and cut). The official consequence relation associated with a logic of this kind is therefore not a Scott consequence relation in the sense of Definition 2.4. Its set of valid formulas may still correspond, however, to some many-valued logic. We present now two substructural logics where this is indeed the case. The system ÊÅÁ Ñ is obtained from È Ä by simply deleting the weakening rule 12. It is easy to prove, using Gentzen s method, that cut-elimination still obtains for this system. Moreover: a strong form of the interpolation theorem holds for it: µ ³ is provable only if there exists 12 In the literature on relevance logic (including [7], from which the results below are taken) the symbols,, Æ, and + are used instead of,,, and (respectively). an interpolant containing only atomic formulas common to ³ and 13. This puts the corresponding logic, ÊÅÁ Ñ, in the family of Relevance logics (in which framework it was originally introduced and investigated). In [7] it was proved that it is also a many-valued logic. The corresponding matrix, called in [7], is the following: Truth Values: Ø Á ½ Á ¾ Á. All elements except are designated. Operations: Ø, Ø, Á Á ½ ½µ. Ø ÓÖ Ø Á Á otherwise. Like, can be viewed as the limit of its finite substructures. Indeed, let Ò be the substructure of consisting of Ø Á ½ Á Ò. Then we have: Theorem 6.4 A formula ³ is valid in (or in all of the finite structures Ò ) iff ÊÅÁÑ µ ³. Moreover, if ³ contains Ò atomic formulas and ÊÅÁÑ µ ³ then there is a valuation which refutes ³ in Ò. Among the Ò s, the three-valued is particularly interesting. It was first introduced and axiomatized by Sobociński in [46]. The corresponding logic is known to be the purely intensional (or multiplicative, in the terminology of [33]) fragment of Dunn-McColl semi-relevant logic ÊÅ ([1, 22]). In [9] it was shown that a corresponding cutfree Gentzen-type system, ÊÅ Ñ, can be obtained from ÊÅÁ Ñ by adding to it the following mingle rule (also called mix in the literature on Linear Logic): ½ µ ½ ¾ µ ¾ ½ ¾ µ ½ ¾ This is of course a new structural rule, which is weaker than weakening. Note: The rules of ÊÅÁ Ñ and ÊÅ Ñ are pure, but they are not invertible. It seems therefore difficult to base a reasonable tableaux system on them (although like in classical logic, there exist semantical arguments for these logics that simultaneously prove completeness and cut elimination). References [1] A. Anderson and N. Belnap. Entailment, volume I. Princeton University Press, Unlike in classical logic, such atomic formulas necessarily exist in case µ ³ is provable in ÊÅ ÁÑ. This is called the variable sharing property in the literature on relevance logics ([1, 2, 22]), and it is the most characteristic feature of these logics.

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