Combining Classical Logic and Intuitionistic Logic Double Negation, Polarization, Focusing, and Semantics Chuck Liang and Dale Miller Challenges of Combining Intuitionistic Logic with Classical Logic Using a Double Negation Translation I Intuitionistic implication should not collapse into classical implication. Consider A _e (B ⊃ C) −→∼(∼A^ ∼(B ⊃ C)) I How to distinguish the introduction of a translated classical ”connective” from intuitionistic introductions. Introduction of classical disjunction A _e B: ∼A; ∼B; Γ ` ∼A^ ∼B; Γ ` Γ `∼(∼A^ ∼B) No guarantee that sequence won’t be interrupted by other rules. I How to recognize classical ”dualities”. How is ∼(∼A^ ∼B) the ”dual” of (∼A^ ∼B) in an intuitionistic sense, given that ∼∼P 6≡ P. Challenges Continued ... I How to distinguish classical from intuitionistic equivalence. I Classical versus Intuitionistic Cut Elimination. A; Γ ` B ∼A; Γ ` B Γ ` B Admissible in classical logic but not in intuitionistic logic. How do we simulate this cut in an intuitionistic proof system? I What is the meaning of mixed formulas such as A _e (B ⊃ (C _ D))?. Outline and Overview: I Goal: Combine LJ and LC into Polarized Intuitionistic Logic I LC does not include intuitionistic implication I Start with Intuitionistic Logic with a designated atom ?. I ? is just minimal ”false” - this logic predates ICL. I ICL is a stand-alone logic I PIL combine logics I Assign labels, i.e., ”polarities” to formulas. I Define Double-Negation translation. I Use focusing (focalization) to isolate ”classical connectives” I Derive Unified Sequent Calculus I Define Kripke/Algebraic Semantics Syntax and Colors I Formulas freely generated from atoms, ^, _, ⊃, 0 and designated atom ?. I Define :A = A ⊃ ?;(A ⊃ 0 = ∼A) I Formulas are Red or Green as follows: I A ^ B, A _ B, 0, and A ⊃ B where B 6= ? are red. I All atoms are red, except ?, which is green. I :A (A ⊃ ?) is green. 2n I : (R), n > 0, are reddish green (also includes ?) 2n+1 I : (R) are solidly green I Red and Green formulas can be logically equivalent: (A ^ B) ⊃ ? ≡ A ⊃ (B ⊃ ?) I This polarization is not same as duality in linear logic: ?X−◦!Y Recovering Classical ”Dualities” ? I M = :M for red or reddish-green M ? I (:M) = M Syntactic Identity: A?? = A ? I A is convenient way to refer to doubly-negated formulas ? I A is not a connective. ? ? I if A ≡ B, then A is only classically ≡ to B ((A ^ B) ⊃ ?)? = A ^ B, (A ⊃ (B ⊃ ?))? = :(A ⊃ :B) Double Negation as Macro Expansion R red and E green e ? ? ? ? ? I A _ B = (A ^ B ) = :(A ^ B ) e ? ? ? ? ? I A ^ B = (A _ B ) = :(A _ B ) ? ? I 1 = ? = ? ⊃ ?; > = 0 = 0 ⊃ ? To complete the definition of A?, we need missing link: ? I A / B = :(A ⊃ B ) includes special case: (R / 1) = ::R These are not yet new connectives, just labels The following holds: ? ? I 1 = ?; > = 0 e ? ? ? I (A _ B) = A ^ B e ? ? ? I (A ^ B) = A _ B ? ? I (A / B) ≡ A ⊃ B (mod :::P ≡ :P) Caution: do not equate “green” with “classical.” Classical fragment will use _ and _e, ^ and ^e, 0 and ?, 1 and >. The classical fragment will be more LC than LK. Intuitionistic Sequent Calculus LJ A; B; Γ ` D A; Γ ` DB; Γ ` D A ⊃ B; Γ ` AB; Γ ` D ^L _L ⊃L A ^ B; Γ ` D A _ B; Γ ` D A ⊃ B; Γ ` D Γ ` A Γ ` B Γ ` Ai A; Γ ` B ^R _R ⊃R Γ ` A ^ B Γ ` A1 _ A2 Γ ` A ⊃ B Id 0L 1R a; Γ ` a 0; Γ ` D Γ ` 1 ? is considered a special atom _e, ^e and / as Synthetic Connectives in LJ? Distinguish between sequents Γ ` A and Γ ` ?: Correspond to sequents with and without a stoup Introduction of a green formula :A = A ⊃ ?: Id A; Γ ` ? A ⊃ ?; Γ ` A ?; Γ ` ? ⊃R ⊃L Γ ` A ⊃ ? A ⊃ ?; Γ ` ? But LJ is not good enough We don’t want the following: B ⊃ C; Γ ` BC; Γ ` A ⊃ ? A ⊃ ?; Γ ` A ?; Γ ` B B ⊃ C; Γ ` A ⊃ ? A ⊃ ?; Γ ` B Looking for one-to-one mapping between proofs Derive new introduction rules for A _e B = :(A? ^ B?): Γ ` A; B A _e B; Γ ` A? A _e B; Γ ` B? • e ◦ ◦ e e _ R e _ L Γ ` A _ B A _ B; Γ `• Want this to correspond one-to-one with the following fragments: A?; B?; Γ ` ? :(A? ^ B?); Γ ` A? :(A? ^ B?); Γ ` B? A? ^ B?; Γ ` ? :(A? ^ B?); Γ ` A? ^ B? ?; Γ ` ? Γ ` :(A? ^ B?) :(A? ^ B?); Γ ` ? Need focused intuitionistic sequent calculus (LJF) even for unfocused synthetic introduction rules A new dimension of polarization I Atoms are “positive,” except ?, which is “negative” + I _, ^ , 1 and 0 are positive − I ^ , ⊃, are negative I Positives are “synchronous” on the right; Negatives are synchronous on the left I Asynchronous rules are always invertible I Synchronous (and asynchronous) rules can be stringed together into a single phase. e ? + ? ? I A _ B = (A ^ B ) I Caution: Do not confuse positive with red polarities: A ⊃ B is red but negative (red=positive only in LC) Use Delays to Fine-Tune Focusing @+(A) = A ^+ 1; @−(A) = 1 ⊃ A F F ` (left) F r (right) atomic a @−(a) a 0 @−(0) 0 1 @−(1) 1 A ^ B @+(A`) ^− @+(B`) @+(Ar ^− Br ) A _ B @−(A` _ B`) @−(Ar ) _ @−(Br ) A ⊃ B @−(Ar ) ⊃ @+(B`) @+(A` ⊃ Br ) F F ` (left) F r (right) ? ? ? a?, atomic a :(@−(a)) :@−(a) A ^e B :(@−(A?r ) _ @−(B?r )) :@−(A?` _ B?`) A _e B :(@−(A?r ) ^+ @−(B?r )) :@−(A?` ^+ B?`) A / B :(A` ⊃ B?r ) :(@−(Ar ) ⊃ @+(B?`)) Deriving the Sequent Calculus LP Different modes of sequents: ∼ ? ? ∼ I Γ `• A1;:::; An = A1 ;:::; An ; Γ ` ? (Γ `• = Γ ` ?) ∼ I Γ `◦ A = Γ ` A Structural Rules (R red, E green) ? Γ ` E A ; Γ `◦ A • Load=Go Signal=Stop ? Γ `◦ E A ; Γ `• =∼ [Γ;@−(A)] −! [?] [A ⊃ ?; Γ] −! A ? [Γ;@−(A)] −! ? [A ⊃ ?; Γ] −A! [A ⊃ ?; Γ] −! [?] [Γ];@−(A) −! ? [A ⊃ ?; Γ] A−!⊃⊥ [?] [Γ] −! @−(A) ⊃ ? [A ⊃ ?; Γ] −! [?] Γ `◦ A Γ `• B /R Γ `• A / B # [:::; Γ];@+(B?`) −! [?] [:::; Γ] −! @−(Ar ) + ?` R` Rr @ (B ) [:::; Γ] −@−(Ar )! [:::; Γ] −! [?] ⊃L @−(Ar )⊃@+(B?`) [:::; Γ] −! [?] Lf [@−(Ar ) ⊃ @+(B?`); Γ] −! [?] Correspondence between focusing phases and synthetic introduction rules must be relaxed: A / B ≡ (A ⊃ B?) ⊃ ?, which is − followed by + −+, +− are OK, but not + − +. Sequent Calculus LP Structural Rules and Identity ? Γ `• E A; Γ `• Θ A ; Γ `◦ A Signal ? Store ? Load I Γ `◦ E Γ `• A ; Θ A ; Γ `• a; Γ `◦ a Right-Red Introduction Rules Γ `◦ A Γ `◦ B Γ `◦ Ai A; Γ `◦ B ^R _R ⊃R Γ `◦ A ^ B Γ `◦ A1 _ A2 Γ `◦ A ⊃ B Left-Red Introduction Rules A; B; Γ `◦ R A; Γ `◦ RB; Γ `◦ R A ⊃ B; Γ `◦ AB; Γ `◦ R ^L _L ⊃L A ^ B; Γ `◦ R A _ B; Γ `◦ R A ⊃ B; Γ `◦ R Right-Green Introduction Rules Γ ` Γ ` Γ ` A; B • A • B e • e Γ `◦ A Γ `• B e ^ R e _ R /R Γ `• A ^ B Γ `• A _ B Γ `• A / B Rules for Constants Γ `◦ R Γ `• 1R 1L 0L ?R >R Γ `◦ 1 1; Γ `◦ R 0; Γ `◦ R Γ `• ? Γ `• > Extends to First Order Rules for Quantifiers Γ `◦ A[t=x] Γ `◦ A A; Γ `◦ R A[t=x]; Πx:A; Γ `◦ R 9R ΠR 9L ΠL Γ `◦ 9x:A Γ `◦ Πy:A 9y:A; Γ `◦ R Πx:A; Γ `◦ R Γ `• A[t=x] Γ `• A ΣR 8R Here, y is not free in Γ and R. Γ `• Σx:A Γ `• 8y:A Why not remove delays and get focused LPF? Possible, but first ... LC Inside LP ` Γ; N; P; S Γ; P; N `◦ S ^L ` Γ; N _ P; S 7−! Γ; P ^ N `◦ S ` Γ; N; P; Γ ` N; P • e e _ R ` Γ; N _ P; 7−! Γ `• N _ P Γ ` N • Signal ` Γ; P ` ∆; N; Γ `◦ P Γ `◦ N ^R ` Γ∆; P ^ N 7−! Γ `◦ P ^ N LC invariant: no “positive” introductions outside of the stoup ...subsumed by LP invariant: no green introduction in `◦ mode LC is accidentally almost focused, but not LP Independence from Double Negation Translation e e I _ , ^ , /, ? and > are now first-class green connectives and constants. ? I A is now De Morgan negation, defined by “dualities:” _e/^, ^e/_, //⊃, ?/1, >/0, ? I “dual atoms” a/a ; Formulas are in negation normal form. ? ? If A `◦ B is provable, then B `• A is provable. I Reclassification of some formulas: 1 and R ⊃ ? are now red. (R ⊃ ?)? = R / 1 ? I Every green formula is of the form R for some red R. Given A and A?, one is red, the other is green. Kripke Semantics Hybrid Model (Propositional Case): hW; ; C; j=i Requirements and definitions: I is a transitive, reflexive ordering on non-empty set W of “possible worlds.” I j= is a monotonic relation between elements of W and sets of atomic formulas. I C ⊆ W (“classical worlds”) I 4u = fk j k 2 C and u kg (”classical cover” of u) I required: 4k = fkg for all k 2 C. (for propositional models) I if 4u = ; then u is imaginary. Every Kripke Model for IL is immediately a Hybrid Model, with a more structured interpretation of possible worlds. Rules of j= for u; v 2 W; c; k 2 C, green E: I u j= 1 and u 6j= 0 I u j= A _ B iff u j= A or u j= B I u j= A ^ B iff u j= A and u j= B I u j= A ⊃ B iff for all v u, v j= A implies v j= B I u j= E iff for all k 2 4u, k j= E ? I c j= E iff c 6j= E E.g., c j= A / B iff c 6j= A ⊃ B? iff for some v c, v j= A and v 6j= B?.