Polarized Intuitionistic Logic

Transcription

1 Polarized Intuitionistic Logic Chuck Liang 1 and Dale Miller 2 1 Department of Computer Science, Hofstra University, Hempstead, NY chuck.c.liang at hofstra.edu 2 INRIA & LIX/Ecole Polytechnique, Palaiseau, France dale.miller at inria.fr Abstract. We introduce Polarized Intuitionistic Logic, which allows intuitionistic and classical logic to mix. The logic is based on a new analysis of the intuitionistic distinction between left and right as a form of polarity information. In contrast to double-negation translations, classical logic is transparently captured. The logic is given a Kripke-style semantics and is presented as a sequent calculus that admits cut elimination. We discuss the impact of this logic on traditional intuitionistic concepts such as Glivenko s theorem and Markov s principle. 1 Introduction It is well established that much constructive content can be found in certain types of classical proofs which cannot be captured directly in traditional intuitionistic logic. We seek a new logic that preserves the essential characteristics of intuitionistic logic, including its possible-world semantics, yet contains classical logic as a fragment. An important property of intuitionistic logic is its ability to embed classical logic using the double-negation translations of Kolmogorov, Gödel, Gentzen, and others. This ability suggests that intuitionistic logic already contains the potential to serve as a platform for combining intuitionistic and classical reasoning. However, the traditional double-negation style translations leave much to be desired in this regard. Typically, such a translation encodes classical formulas such as A A as (A A). In the context of Gentzen s sequent calculus LJ, such a translation leads to a sequent of the form A A false. Leaving false as the conclusion restricts the possible inclusion of such sequents in larger intuitionistic proofs in a meaningful way. One might observe that using A A to hide the law of excluded middle on the left side of a sequent in fact changes little. The L introduction rule looks much like the R rule. Of course what really has changed is the ability to use contraction on the formula. One is tempted to say that what has changed, therefore, is not from a disjunction to a conjunction but rather to a different version of disjunction, one that is subject to contraction. We introduce a polarized intuitionistic logic (PIL) that allows intuitionistic and classical logic to mix. Our approach to integrating classical logic does not rely on a double-negation translation but rather on a larger set of connectives. The connectives are distinguished by their polarity, which dictates the applicability of structural rules. The connectives are allowed to mix freely with few restrictions. Classical logic results from choosing a particular combination of connectives and the form of end-sequent.

2 In other words, there s no need to use a double-negation to change the polarity of a formula: one can choose the desired polarity to begin with. The content of this paper is organized as follows. Section 2 defines the syntax and polarity scheme of PIL. Section 3 introduces the sequent calculus LP. The core of this paper is Section 4, which defines a Kripke-style semantics for PIL. From the soundness and completeness of the semantics it is also shown that the propositional fragment of this logic is decidable despite borrowing important elements from linear logic. The admissibility of cut also follows from semantic completeness. Section 5 illustrates how classical and intuitionistic logics can be seen as fragments of LP. In section 6 we discuss the impact of the new logic on classical and intuitionistic principles including the disjunction and existence properties, the law of excluded middle, Glivenko s theorem, and Markov s principle. In Section 7, we briefly discuss an important related work: PIL is in fact a fragment of a larger, unified logic that also contains a linear logic component. Although PIL is presented independently because of its unique semantics and sequent calculus, the larger system requires a separate paper. This section also discusses other related efforts including Girard s LU [6] and our own LKU [10] systems. 2 Polarities and Connectives The left-right distinction is crucial for intuitionistic sequent calculus (similar to the input and output distinction for the λ-calculus) and our left and right polarities are defined to be consistent with the traditional concept. However, instead of distinguishing left from right-hand sides of sequents, left and right are now attributes of formulas. The logical connectives of PIL, along with their polarity classifications, are as follows. Positive Right: +, +,, 0, 1, positive-right literals. +R polarity. Negative Left:,,,,, negative-left literals. L polarity. Positive Left:, l, positive-left literals. +L polarity. Negative Right:, r, negative-right literals. R polarity. We assume there are countably many atomic formulas of each polarity. The polarization scheme is also illustrated by the diagram here. The polarity of a formula is determined entirely by its top-level connective. Intuitionistic implication is decomposed into the dual connectives (for right-occurrences) and (for left-occurrences). The two versions of disjunction, and +,, l +, +,, 1, 0 +L +R L R,,,,, r give rise to two versions of false: and 0. Consequently, PIL semantics must account for two notions of consistency. The term literal above refers to an atomic formula a or its De Morgan negation a. The De Morgan (classical) negation of formulas is defined by the following dualities: 1/, 0/, /, l / r, + /, + /, /, a/a for all literals a. All formulas are written in negation normal form, with pushed to literals. There is another form of negation, namely, intuitionistic negation, A 0 (for A 0), which we write as A. 2

3 De Morgan negation is incompatible with the left-versus-right distinction in traditional intuitionistic logic. In PIL, however, connectives are encoded with their relationship to structural rules (i.e., their polarity), and thus De Morgan duality becomes a viable alternative to the intuitionistic negation. In this paper, the positive/negative distinction between formulas is little used because the sequent calculus we present is not focused. We retain this distinction in part to be consistent with related papers. However, the richer distinction in polarities also allows for a clearer analysis of the classical and intuitionistic components. The +R/ L axis can be called the classical axis while the +L/ R axis contains the most essential intuitionistic connectives for implication and universal quantification (i.e., the pure fragment of intuitionistic logic that directly corresponds to lambda calculus). Full intuitionistic logic, with disjunction, existential quantification and false, already incorporates elements from both axes. PIL is an extension of intuitionistic logic that completes the integration of these polarities into a single logic. A Restriction on Formulas. The connectives of PIL are allowed to freely mix except for the following restriction. In a formula of the form A B, at least one of A or B must be a left-polarity formula. Dually, in a formula of the form A B, at least one immediate subformula must be of right-polarity. We often write A B for A B with the understanding that A is a right-polarity formula (and thus A is of left-polarity). In case B is also left-polarized, A B will be equivalent to B A (i.e., the contrapositive holds in this case). The dual of A B is A B. The principal reason for this restriction is that it allows for the interpretation of PIL in linear logic (presented in a larger context elsewhere). The connectives for intuitionistic implication, and its dual, are based on Girard s decomposition of intuitionistic implication in terms of and. It is possible to define PIL without any restriction on the composition of formulas by using as a primitive and making its dual likewise non-commutative. The Kripke-style semantics is invariant under this alteration. However, this minor restriction allows us to consider PIL as a fragment of an even larger, unified logic (see Section 7). It is always possible to write equivalent formulas that satisfy the restriction by enclosing left-polarity formulas as A + 1 or A + 0. In particular, under the restriction the intuitionistic negation is technically only applicable to rightpolarity formulas. However, one can use the equivalent (A + 0). To avoid notational clutter, we assume that such a device has been applied, if needed, when writing A. 3 Sequent Calculus We present the sequent calculus LP for PIL in Figure 1. In all rules, Γ and Θ are multisets of formulas, L is any left-polarity formula, R and S are right-polarity formulas, and a is any literal. Although our use of polarization eliminates the need for two-sided sequents, we nevertheless choose this style in presenting LP. A one-sided calculus is possible but is less readable. The disadvantage of a two-sided calculus is that we must be careful to distinguish left or right polarity from left- or right-hand side. We use the symbols and to represent two modes of proof. This essential distinction of LP separates it from some related efforts such as bi-intuitionistic logic [12] which also 3

4 Γ L Γ L Lift Structural Rules A, Γ Θ Γ A, Θ Store A, Γ A A, Γ Decide Right-Right Introduction Rules a, Γ a I Γ A Γ B Γ A + B + R Γ A i Γ A 1 + A 2 + R R, Γ B Γ R B R Right-Left Introduction Rules A, B, Γ R A + B, Γ R + L A, Γ R B, Γ R A + B, Γ R + L S B, Γ S B, Γ R S B, Γ R L Left-Right Introduction Rules Γ A Γ B Γ A B R Γ A, B Γ A B R Γ R Γ B Γ R B R Rules for Quantifiers Γ A[t/x] Γ x.a R Γ A Γ r y.a r R A, Γ R y.a, Γ R L A[t/x], r x.a, Γ R r x.a, Γ R r L Γ A[t/x] Γ l x.a l R Γ 1 1R Γ A Γ y.a R Here, y is not free in Γ and R. Rules for Constants Γ R Γ 1, Γ R 1L 0, Γ R 0L Γ R Γ R Fig. 1. The LP proof system contains a dual to intuitionistic implication. We write when it can be either or. End-sequents of LP have the form Γ A, where A can be of any polarity. PIL can be defined as consisting of all formulas A such that A is provable. Below, we provide some further explanation and comments on these inference rules. Structural Rules. An essential aspect of LP is that its structural rules are sensitive to the difference between and. The Decide and Lift rules allow a formula from the left-side to become the conclusion in the mode. A combination of Decide and Store also gives us an inverse form to Lift for right-polarity formulas R: R, Γ R R Decide, Γ Γ R Store If the Θ is removed from Store, then this rule is none other than reductio ad absurdum. This sequent calculus has the characteristic that all introduction rules take place in the 4

5 presence of a single conclusion. The mode also requires exactly one formula on the right side of the sequent. These conventions allow us to write fewer rules. Right-right and Right-left Rules. A designation such as right-left refers to rightpolarity formulas on the left-hand side. These rules correspond to regular introduction rules in intuitionistic sequent calculus, and use the mode. It would be wrong, however to simply associate with classical deduction and with intuitionistic deduction. Classical deduction also requires the mode, which is enabled after the embedded contraction in Decide. Classical and intuitionistic logics share the connectives + and +. The connectives and also exist in intuitionistic logic, although they usually do not appear as such in traditional syntax. In PIL, however, (a b) c is technically (a b ) c. The aim of PIL is precisely to no longer hide the occurrences of such connectives: they can be used to compose formulas with virtually no restriction. There is no equivalent to the formula a + (b c) in traditional intuitionistic logic. In the right-left rules, the invertible (asynchronous) nature of left-side occurrences of + and + means that no contraction is required for these rules. We have chosen to use multisets in this representation, although a representation using sets is also possible. Left-Polarity Introduction Rules. The left-right rule R rule must be followed from above by a Store rule before another introduction rule can take place. The left-polarity introduction rules, combined with the structural rules, expand PIL beyond traditional intuitionistic logic. In particular, the law of excluded middle becomes provable in the form A A. However, the disjunction property is also provable in the form A + B. This distinction is possible because end-sequents use. In the context of, right-polarity formulas such as A + B are subject to contraction (via Store) and thus becomes equivalent to the multiplicative A B. See Section 6 for further discussion on the impact of PIL. The missing category of left-left rules is not needed because the Decide rule, which embodies a contraction, can be used to move a left-polarity formula to the right side of the sequent. The introduction rules for quantifiers and constants follow the same schema as the propositional connectives. Admissible Rules. Contraction and weakening do not appear as explicit rules in LP except for the embedded contraction in Decide. Instead, contraction and weakening are admissible by the following lemma, which is proved by induction on the structure of proofs. Lemma 1. If A, A, Γ Θ is provable, then A, Γ Θ is provable. If Γ Θ is provable, then A, Γ Θ is provable A lemma that follows from weakening relates the and modes of proof: Lemma 2. If Γ A is provable then Γ A is also provable. Furthermore, the sequent A is provable if and only if A A is provable. The admissible cut rules of LP can be unified into a single form: Γ A A, Γ B Γ Γ Cut B 5

6 Using structural rules such as Lift, several other forms of cut can be derived. Although cut-elimination can be proved in the usual way, in this paper we derive the result semantically in Section 4. We consider it to be more meaningful to give a procedural proof of cut-elimination in the context of a focused sequent calculus, and we do so in a larger context elsewhere. However, it is worthwhile to comment here that cut-elimination in LP is critically dependent on the sensitivity of the two sequent forms and to polarity information. The inductive cut-elimination proof is tedious because it involves many cases, but one that requires special attention occurs when the cut formula is subject to contraction. A critical invariant of LP is that no left-polarity introduction rule is possible in the mode. This invariant is similar to the promotion rule in linear logic, which also admits cuts when some but not all formulas are subject to contraction. Two Levels of Consistency. The two forms of negation and false naturally imply two notions of consistency in PIL. We say that a set of formulas Γ is 0-consistent if Γ 0 is not provable for any finite subset Γ of Γ, and that the set is -consistent if Γ is not provable for any finite subset Γ of Γ. If a set is -consistent, then it is also 0-consistent, but the reverse does not hold. The same phenomenon is found in linear logic: {a, a } does not prove all formulas, only all?-formulas. The set {a, a } is not -consistent, but it is 0-consistent. As will be seen in Section 5, in traditional intuitionistic logic such a set is not a valid left-side context in a sequent, so one cannot even speak of -inconsistency. In classical logic, 0 is equivalent to, so again the problem is nullified. The discrepancy appears because we allow classical and intuitionistic polarities to mix freely in PIL. With PIL, it becomes possible to pose the question are there other ways to bridge the gap between the two notions of consistency. We demonstrate the impact of this question in Section 6. 4 Model Theoretic Semantics Two factors complicate the formulation of a Kripke-style semantics for PIL. The first is that we must define satisfiability (or forcing ) for left-polarity formulas. The second factor is the discrepancy between -consistency and 0-consistency. In a standard Kripke model, the terminal nodes of the ordering relation represent classical worlds. Because of the two levels of consistency, there will need to be worlds above the classical ones. To account for the richer notion of provability in PIL, we allow possible worlds that can be inconsistent in terms of. The use of such worlds is not without precedent [13, 8] (though not in the context of two levels of consistency). They have been referred to as exploded worlds. We prefer the term imaginary world in analogy to 1 being an imaginary number. These worlds also distinguish PIL from some other efforts to semantically combine classical and intuitionistic logic [2]. We define a Kripke hybrid model as a structure W,, C, = where (W, ) forms a non-empty Kripke frame of possible worlds W with a transitive and reflexive relation. C is a subset of W. Elements of C are used to interpret classical connectives. The relation = maps each element of W to a set of literals. We shall extend the meaning of this symbol to the general satisfiability or forcing relation between possible worlds and formulas. The key idea here is that a left-polarity formula is satisfied at a world u if it is satisfied in all classical worlds above u. 6

7 We use u to represent the set of all k C such that u k. The terminology of - and 0-consistency is extended to describe possible worlds. If u is empty then u is -inconsistent (imaginary), but all worlds are 0-consistent. The following conditions for the satisfiability relation are required for all u W and c C. 1. u = 1, c =, u = 0 (0-consistent), and c = ( -consistent) 2. u = a and u v implies v = a for all literals a 3. c = a if and only if c = a for all literals a 4. u = E iff for all c u, c = E: for every left-polarity formula E 5. u = A + B iff u = A or u = B 6. u = A + B iff u = A and u = B 7. u = A B (A B) for left-polarity formula A iff for all v u, v = A implies v = B 8. c = A B for right-polarity formula A iff for some v c, v = A and v = B 9. c = A B iff c = A or c = B 10. c = A B iff c = A and c = B The quantifiers can be added in the usual way, with r treated in similar manner to. The propositional case already meaningfully extends traditional Kripke semantics. Conditions 1 and 3 state the classical consistency and completeness of elements in C. Condition 7 is equivalent to the usual rule for implication in Kripke models. The critical condition 4 represents a shift to the classical worlds: this rule is also notable in that if u is empty, then all left-polarity formulas are satisfied vacuously in u. The left-polarity connectives are interpreted at the classical worlds in rules 8 through 10. These rules are duals of the right-polarity cases (5 through 7). The dualities exclude the middle from the classical worlds. The following important properties are provable: Lemma 3. In a Kripke hybrid model, for every u, v W and every c C: a. if u v, then u = A implies v = A (monotonicity) b. c = A iff c = A (excluded middle) c. u = A and u = A for some formula A iff u is empty (u is imaginary). The first two properties are proved by induction on formulas. The third property follows from condition 4 and the first two properties. In particular, by monotonicity there can be no classical world above an imaginary world, and therefore imaginary worlds satisfy all left-polarity formulas. Another notable property of hybrid models is that for any world u that s properly below the classical worlds, it holds that u = A implies u = A but not vice versa. For any imaginary world i properly above the classical worlds, it s the reverse that holds: i = A implies i = A but not vice versa. For satisfiability in a model M, we say M = A if u = A for every u W. The notation M = (Γ A) is equivalent to M = Γ A, where Γ is the + - conjunction of formulas in Γ. Sequents Γ A 1,..., A n are included in this definition as Γ A 1... A n (so Γ is treated as Γ ). The soundness and completeness results of this semantics are given with respect to cut-free LP proofs. 7

8 Theorem 4. (Soundness) If Γ A is provable, then M = (Γ A) for every Kripke hybrid model M. The soundness direction is proved by induction on the structure of proofs. We present the highlights of our completeness proof. The organization of this proof follows that of Fitting [4]. However, our proof is given directly for a primarily singleconclusion system as opposed to a multiple-conclusion one (i.e., the Beth-Fitting tableau system). First, we modify LP as given by using sets (on the left-hand side) as opposed to multisets in the representation of sequents, which means that some redundant contractions are not avoided: in particular, the principle formula in L is contracted. The modified version is equivalent to the original by the admissibility of weakening and contraction. Using sets will also more directly yield a decidability result for the propositional fragment of PIL. We concentrate on the propositional fragment below, but leave open the possibility of infinite models. We define an antisequent to be an expression of the form Γ A or Γ. An antisequent is closed if it is of the form (a, Γ a), (a, a, Γ ), (Γ 1), (, Γ ), or (0, Γ Θ). Otherwise, it is open. An antisequent is defined to be consistent if the sequent in question is not provable. All closed antisequents are clearly inconsistent. Given a finite set of formulas Γ, let S Γ label a non-empty set of consistent anti-sequents of the forms Γ A or Γ. All antisequents in S Γ share the same context Γ. We use a set of antisequents in place of a multiple-conclusion sequent (or multiple F -signed formluas in a tableau proof). We define a Hintikka pair as a pair of sets (H, K), where H is a non-empty set of the form {S Γ1, S Γ2,..., S Γi,...} and K is a designated (and possibly empty) subset of H of the form {K Γ 1, K Γ 2,...}. Each set of consistent antisequents in H and K must satisfy the following properties. Here, E is a left-polarity formula: 1. if A + B Γ, then A Γ and B Γ for each S Γ H. 2. if A + B Γ, then A Γ or B Γ for each S Γ H. 3. if A B Γ, for any S Γ H, then either (Γ A) S Γ or B Γ. 4. if (Γ A + B) S Γ then either (Γ A) S Γ or (Γ B) S Γ. 5. if (Γ A + B) S Γ then both (Γ A) S Γ and (Γ B) S Γ. 6. if (Γ A B) S Γ then there exists a S Γj H such that A, Γ Γ j and (Γ j B) S Γj. 7. if (Γ E) S Γ then there exists a K Γ i K such that E, Γ Γ i and (Γ i ) K Γ i. 8. if E Γ i and (Γ i ) K Γ i, then (Γ i E ) K Γ i. 9. if (Γ i E) K Γ i then E Γ i and (Γ i ) K Γ i. Properties 7 and 9 are consistent: property 9 declares that for each K Γ, the K Γ i specified by property 7 is itself. Note that the rules for the right-polarity connectives +, + and ( ) are the same for the K sets. These rules cover derivations such as A, B, A + B, Γ A + B, Γ A, B Store 2 A + B, Γ A B A + B, Γ A B Lift A + B, Γ Decide and A B, B, Γ A B, Γ A A B, Γ B Store R A B, Γ A B A B, Γ A B Lift Decide A B, Γ 8

9 Introduction rules for the left-polarity connectives are thus also accounted for by these properties. Completeness of provability is preserved by aggressively applying Store - at worst we use a few more contractions than necessary. The following lemma shows that a Hintikka pair forms a model that is consistent with its antisequents. Lemma 5. Given a Hintikka pair (H, K), define S Γa S Γb if Γ a Γ b. Define S Γ = d if d Γ for all right-polarity literals d, and define K Γ = e if e Γ for left-polarity literals e. Extend = according to the rules of forcing in hybrid models, using K for the classical worlds. Then H,, K, = is a hybrid model, and 1. if (Γ A) S Γ then S Γ = A for all formulas A. 2. if A Γ then S Γ = A for all formulas A. This important lemma is proved by simultaneous induction on the structure of formulas. What remains of the completeness proof is to show that a Hintikka pair exists for every unprovable sequent. We define an associate set of S Γ as {A, Γ B} if (Γ A B) S Γ. We also define a k-associate set of S Γ as {E, Γ } if (Γ E) S Γ for some left-polarity formula E. Let SA(S Γ ) be the set of all associate sets of S Γ and KA(S Γ ) be the set of all k-associate sets. Given S Γ, we show that there exists a S Γ, the reduced set of S Γ, as follows. Suppose (A + B, Γ D) S Γ, then since the antisequent is assumed consistent, at least one of A, Γ D and B, Γ D is also consistent, so one of the formulas that preserves consistency can be added to Γ. Suppose (Γ A + B) S Γ, then add both Γ A and Γ B to S Γ. The other purely intuitionistic cases are similarly handled. If (E, Γ ) S E,Γ (K i Γ ) for some left-polarity formula E, then add E, Γ E to S E,Γ. If (Γ E) S Γ and (Γ ) S Γ (S Γ = K Γ ) then add E to Γ. SΓ is the closure of S Γ under these rules. Each SΓ = S Γ for some Γ Γ. Every such closure remains consistent and represents a possible world in the model. Let Γ A be an unprovable sequent and let S Γ = {Γ A}. Let S 0 = SΓ. Let SA(S 0 ) = A 1,... A i and let KA(S 0 ) = K i+1,... K m. Let S 1 = A 1,..., S i = A i, S i+1 = Ki+1,..., S m = Km. Now let SA(S 1 ) = A m+1,... A j and KA(S 1 ) = K j+1,..., K n. Let S m+1 = A m+1,..., S n = Kn. Repeat the process exhaustively. Let H be the set of all S i, and let K consist of all Kj of the collection. (i.e., all sets generated from k-associate sets). Since every set of the collection is generated from a single antisequent, the K sets are uniquely identified by the presence of Γ. It is easily verified that (H, K) forms a Hintikka pair. Now if A is not provable, form the model from { A}. By Lemma 5, S 0 = A. Theorem 6. (Completeness) if a formula is satisfied in all Kripke hybrid models, then it is provable. The procedure suggested by the proof of the existence of the closures Sk and the sets S 0, S 1,... given above operates sequentially. In the propositional case the sequence is finite by the subformula property. This procedure can be transformed into a decision 9

10 procedure for the propositional fragment of PIL by generating a finite tree instead of a sequence. Without assuming consistency in this variation, we begin with { A}. At each step in the generation of SΓ where alternatives are encountered, such as when A + B Γ, we branch to include both alternatives. When a closed antisequent is encountered, that branch of the tree is terminated. Since we are using sets instead of multisets, it is immediate that the number of possible sets of antisequents consisting of subformulas of A (and their duals) is bounded. By König s Lemma, the tree is finite because every branch is finite. In the resulting tableau, each leaf of the tree either contains a closed (inconsistent) antisequent, or remains open. By the completeness proof, if all leaves are closed, then A has no consistent model and thus A is provable. On the other hand, if A is provable ( A is inconsistent) then we can show, by an induction on the height of proofs, that for every branch of the tree there is a set on the branch that contains a closed antisequent. If there is an open leaf then A is not provable. Any path from an open leaf to the root represents a countermodel. Thus we have: Corollary 7. The propositional fragment of PIL is decidable. The soundness and completeness of the semantics also yield a trivial proof of the admissibility of cut: if M = (Γ A) and M = (A, Γ B), then it follows directly that M = (Γ Γ B). Corollary 8. The Cut rule is admissible in LP. Examples of Countermodels We give two simple examples of unprovable sequents and their countermodels. Our models are consistent with standard Kripke models so one can use the same countermodels for intuitionistically unprovable formulas. Both of the following examples involve sequents that cannot be expressed in traditional intuitionistic logic. The sequent a a is not provable. Despite using the classical, this version of the excluded middle is not provable in PIL precisely because there may be imaginary worlds above the classical worlds. A countermodel is s 1 : {a, a } s 2 : {a } k : {a } The notation is intended to indicate that the classical world k does not force a (so it must force a ), but k also does not force a 0 since there is a world s 1 above k that forces a. k does not force a because no world above s 2 forces a. The sequent p q p is not provable. A countermodel is: k : {p, q} s : {} Although every classical world above s satisfies p and q, s does not satisfy either. Such models also provide a semantic explanation of the LP invariant that no left-polarity introduction rule is allowed in the mode, an invariant critical to cut-elimination. 10

11 5 Intuitionistic and Classical Logic as Fragments of PIL Intuitionistic and classical logics are found as transparent fragments of LP. They result from restrictions on the composition of formulas and the form of end-sequent, and not on how proofs are to be constructed. The completeness of LP with respect to traditional intuitionistic logic is relatively straightforward. The right-right and right-left introduction rules correspond precisely to those of a typical intuitionistic sequent calculus. What needs to be carefully distinguished is the composition of well-formed formulas. Traditional intuitionistic logic inside PIL is restricted to the following classes of formulas. R := R 1 + R 2 R 1 + R 2 L R x.r r y.r 1 0 r L := L 1 L 2 L 1 L 2 R L x.l l y.l r Here, R defines essentially right formulas and L defines essentially left formulas. r is any right-polarity atomic formula. Traditional intuitionistic implication thus takes the form of R 1 R 2. Sequents are restricted to the form Γ R where Γ consists of only essentially right formulas. We emphasize that the embedding of traditional intuitionistic logic inside LP is adequate not only at the level of provability but at the level of derivation steps: it is easy to show that nothing outside of the right-right and right-left introduction rules will ever be applicable to well-formed intuitionistic sequents. Classical logic is also embedded in LP. The classical fragment consists of all connectives and literals of polarities +R and L. Classical end-sequents can be of the form A 1... A n or A 1,..., A n. In this fragment, + and, as well as + and, are provably equivalent. One can show, for example, that A B A + B is provable for each possible polarity combination of A and B. Importantly, 0 and are also equivalent classically. The classical completeness of this fragment can be proved semantically; it can also be proved by showing, with cut-elimination, the admissibility of the rules of LK. What makes the argument slightly non-trivial is that our version of classical provability also uses the mode. The classical fragment of LP bears closer resemblance to Girard s LC [5] than to LK. LC introduced the polarities that we call +R/ L. However, the polarity of a LC formula is dependent on the polarity of its subformulas. When A and B are both positive (meaning +R), A B is interpreted as A + B in PIL; otherwise, it is A B (and dually for ). LC sequents with a non-empty stoup correspond to the mode while those without a stoup correspond to. LC introduction rules on the stoup formula correspond to right-right introduction rules in LP; the introduction rules for negative ( L) connectives in the presence of a stoup correspond to rightleft rules while those without a stoup correspond to left-right rules. The following are representative examples of LC rules and their equivalents in LP. Here, P is positive (+R) and N is negative ( L). The formula to right of ; is the stoup. Γ, N, P ; S Γ, N P ; S Γ, N, P ; Γ, N P ; Γ ; P, N; Γ ; P N Γ, P, N S Γ, P + N S + L Γ N, P Γ N P R Γ N Γ P Γ N Lift Γ P + N + R 11

12 The structural rules of LC, including weakening and contraction outside of the stoup, are admissible in LP by Lemmas 1 and 2. In particular, the splitting of the context in the LC -rule is only of stylistic significance. Except for different styles in the management of sequents, LC is a fragment of LP. Because the classical fragment also uses the mode, the Kripke-style semantics of Section 4 also applies to classical logic in a meaningful way. In particular, the LC invariant that no positive introduction rule can be applied outside of the stoup is in fact a restricted form of the LP invariant that no left-polarity introduction rule is possible in the mode 1. This invariant is explained by the distinction between classical worlds and other possible worlds in our model theory. 6 The Impact of PIL on Classical and Intuitionistic Principles The ability to mix connectives, the choice of the form of negation (A or A), and the two levels of consistency, can all have a significant impact on various traditional principles related to intuitionistic logic. 6.1 The Existence and Disjunction Properties and the Law of Excluded Middle The end-sequent of LP, Γ A with no polarity restriction on A, is an important designation. By examining the structure of cut-free proofs, it is easily verified that PIL admits the disjunction property in terms of A + B and the existence property in terms of x.a. In particular, the law of excluded middle (LEM) is not provable in PIL as A + A nor as A + A. The LEM is provable in the forms A A and A + A. In the classical mode, the two versions of disjunction become equivalent. Even with the classical disjunction, however, the LEM is not provable when expressed with intuitionistic negation: A A. Unlike +, intuitionistic implication does not lose its force even in the context of : this is explained by the imaginary worlds in our semantics. 6.2 Peirce s Formula. The formula ((p q) p) p is not intuitionistically provable but has been shown to be constructively meaningful. If we wrote this formula as a traditional intuitionistic formula inside PIL, ((p q) p) p with p and q both right-polarity atoms, then it will not be provable. For any such LP proof would immediately also be an LJ proof. What we can do in PIL is to choose different versions of logical connectives to rewrite the formula. The purely classical formula ((p q) + p) p is certainly provable. But we can also find other mixtures of classical and intuitionistic polarities: this is the flexibility that PIL offers. If we assumed that p and q are both left-polarity literals, then we can also proof the formula in PIL. It is also possible to change to but keep. The choice of combination affects the structure of proofs. 1 It so happens that all right-polarized classical connectives are positive, so the LC version of the invariant accidentally resembles focusing. Completely focusing LP is much more difficult. 12

13 6.3 Markov s Principle This principle is stated in various forms, including that S S for all S that does not contain or. It is admissible but not a provable formula in intuitionistic logic. The polarity analysis at the core of PIL helps to explain the admissibility of Markov s principle. If S does not contain the intuitionistic and r then it is in fact a purely classical formula, for which the principle obviously holds. However, using the De Morgan form of negation would trivialize the principle. To tap the potential constructive content embedded in this principle, we can consider the following rules, which are named after similar rules of Herbelin [7]. S, Γ S Γ S catch S, Γ S Γ C throw Here, S is a purely +R formula (all subformulas of S are +R, which precludes r and ). It can also be assumed that Γ S is a pure LJ sequent in the sense of Section 5. Herbelin showed how admitting such rules are equivalent to accepting Markov s principle, and how they can model variations of the exception-handling mechanism found in programming languages. The catch rule, which perhaps could also be called a constructive contraction, is admissible in PIL: since S is purely +R, S is (purely) L. A left-polarity introduction rule on S cannot be applied until a polarity switch occurs between and. This is not possible if S is hereditarily +R and Γ is a traditional intuitionistic (essentially right) context. However, the presence of S in the left-side context can represent an uncaught exception. Since S, Γ S is not a well-formed LJ sequent, the admissibility of this rule in PIL does not contradict any property of traditional intuitionistic logic. To admit the throw rule (as it appears above) would require a partial resolution of the gap between the two notions of consistency. This can be done by accepting the axiom schema (S S) for all purely +R formulas S. The expanded language of PIL allows us to consider such an axiom. The axiom allows throw and catch to both be admissible. Significantly, the addition of this axiom preserves the disjunction and existence properties in the following sense: if a (cut-free) proof of A + B is a conclusion of an application of the axiom, we then have: S S S S A + B S S A + B L It can be shown that the right premise must be the conclusion of a + -introduction rule. This argument can be strengthened and simplified by using a focused sequent calculus. In a focused proof, if the axiom is applied beneath the introduction of the disjunction, then since S is purely positive and is synchronous on the left-hand side, a proof of S must be completed in one uninterrupted phase, which leads to contradiction. 6.4 Glivenko s Theorem Using the De Morgan form of negation does not allow us to duplicate Glivenko s Theorem: if a propositional formula A is provable in classical logic, then A is provable 13

14 in intuitionistic logic. Clearly, if A is equated with A, a classical proof of A does not imply a proof of A. But we can prove a version of Glivenko s theorem for PIL by using the intuitionistic negation A = A 0. In fact, we can expand the range of the result by including formulas that contain as well ( is the only classical connective that does not have a right-polarity equivalent). Purely classical formulas in PIL are composed of +R/ L connectives and literals. However, each propositional connective has a +R equivalent in the classical fragment. Assume that a classical formula A is provable in LP. Since A cannot be a lone literal, we can also assume that A is of +R polarity, which means that the proof must conclude with A A A Decide A Store Because A must be of left-polarity, it is also easily verified that A 0 A is provable. Now by the admissibility of Cut (Section 4), the provability of A follows: A 0 A A A Cut A 0 A 0 0 A 0 0 (A 0) 0 R L 6.5 Double-Negation Translation of Classical Logic The simplicity of the above argument stems from the fact that no elaborate doublenegation embedding of classical logic is required in PIL. To show that A is provable in the traditional intuitionistic fragment of LP, (i.e., Glivenko s Theorem proper), we would still need to apply a double-negation style of transformation at least to atoms, and then argue inductively on the structure of the classical proof. In contrast, the embedding of classical logic in PIL is transparent. To lift a +R formula A to the classical level, one only needs to enclose it as A. 7 Related Work As already alluded to, PIL is a fragment of a larger unified logic currently being developed, one that contains a third axes of polarization: that of multiplicative-additive linear logic. Polarity information eliminates the need for the exponential operators. Polarization along this axis is not defined in terms of the traditional structural rules, but rather of the distinction between synchronous and asynchronous connectives as defined by Andreoli [1]. This notion of focusing can be extended to the classical and intuitionistic cases as well. A focused sequent calculus is defined for this logic, one that is fully adequate for each of the fragments classical, intuitionistic, linear, and PIL. That is, proofs stay naturally within each fragment. One consequence of this unity is that a single cut-elimination proof applies to all of its fragments. This unified system has many of the same motivations as Girard s LU. An important difference between our approach and that of LU (and subsequent works on polarization 14

15 including those of [3], [9] and our own [10]) is the distinction between left-right polarization and positive-negative polarization (especially for the polarities +L/ R). LU contains classical, intuitionistic and linear logics as fragments, but its ability to mix connectives from different logics in the same formula is limited due to insufficient polarity distinctions. The possibility of identifying new logics within LU was never seriously considered (in fact the De Morgan dualities would not always hold for arbitrary mixtures). Our own system LKU was also limited by a lack of enough polarities. In addition to LP, we have developed a focused sequent calculus, a multiple conclusion, tableau-style system that is a closer match to the semantics than LP, and a translation of PIL (with some further restrictions) into linear logic, which serves as an alternate semantics. For example, the sequent p q p, which was invalidated with a countermodel in Section 4, is emulated by the linear sequent?(!p!q ),!p, which is also not provable. We point the reader to a draft of a longer paper [11] for these alternative presentations of PIL. Acknowledgment This work has been supported by INRIA through the Equipes Associées Slimmer. References 1. Jean-Marc Andreoli. Logic programming with focusing proofs in linear logic. J. of Logic and Computation, 2(3): , Luis F. Del Cerro and Andreas Herzig. Combining classical and intuitionistic logic, or: intuitionistic implication as a conditional. In Frontiers of Combining Systems: 1st International Workshop, pages Kluwer Academic Publishers, Vincent Danos, Jean-Baptiste Joinet, and Harold Schellinx. A new deconstructive logic: Linear logic. Journal of Symbolic Logic, 62(3): , Melvin C. Fitting. Intuitionistic Logic Model Theory and Forcing. North-Holland, Jean-Yves Girard. A new constructive logic: classical logic. Math. Structures in Comp. Science, 1: , Jean-Yves Girard. On the unity of logic. Annals of Pure and Applied Logic, 59: , Hugo Herbelin. An intuitionistic logic that proves markov s principle. In Symposium on Logic in Computer Science, pages IEEE, Danko Ilik, Gyesik Lee, and Hugo Herbelin. Kripke models for classical logic. Annals of Pure and Applied Logic, 161(11): , Olivier Laurent. Etude de la polarisation en logique. PhD thesis, Université Aix-Marseille II, March Chuck Liang and Dale Miller. A unified sequent calculus for focused proofs. In LICS: 24th Symp. on Logic in Computer Science, pages , Chuck Liang and Dale Miller. Generalizing intuitionistic logic. Unpublished Report (contact authors for draft), C. Rauszer. A formalization of the propositional calculus of H-B logic. Studia Logica, 33:23 34, Wim Veldman. An intuitionistic completeness theorem for intuitionistic predicate logic. Journal of Symbolic Logic, 41(1): ,