Non-normal logics: semantic analysis and proof theory (extended version)
Jinsheng Chen, Giuseppe Greco, Alessandra Palmigiano, Apostolos Tzimoulis
aa r X i v : . [ m a t h . L O ] S e p Non-normal logics: semantic analysis and proof theory
Jinsheng Chen , Giuseppe Greco ⋆ , Alessandra Palmigiano , ⋆⋆ , and ApostolosTzimoulis University of Utrecht, the Netherlands Vrije Universiteit Amsterdam, the Netherlands Department of Mathematics and Applied Mathematics, University of Johannesburg, SouthAfrica
Abstract.
We introduce proper display calculi for basic monotonic modal logic,the conditional logic CK and a number of their axiomatic extensions. Thesecalculi are sound, complete, conservative and enjoy cut elimination and subfor-mula property. Our proposal applies the multi-type methodology in the design ofproper display calculi, starting from a semantic analysis which motivates syntac-tic translations from single-type non-normal modal logics to multi-type normalpoly-modal logics.
Keywords:
Monotonic modal logic · Conditional logic · Proper display calculi. By non-normal logics we understand in this paper those propositional logics alge-braically captured by varieties of Boolean algebra expansions , i.e. algebras A = ( B , F A , G A ) such that B is a Boolean algebra, and F A and G A are finite, possibly empty fami-lies of operations on B in which the requirement is dropped that each operation in F A be finitely join-preserving or meet-reversing in each coordinate and each operation in G A be finitely meet-preserving or join-reversing in each coordinate. Very well-knownexamples of non-normal logics are monotonic modal logic [6] and conditional logic [42,5], which have been intensely investigated, since they capture key aspects of agents’reasoning, such as the epistemic [49], strategic [47,46], and hypothetical [23,39].Non-normal logics have been extensively investigated both with model-theoretictools [34] and with proof-theoretic tools [41,43,26]. Specific to proof theory, the mainchallenge is to endow non-normal logics with analytic calculi which can be modularlyexpanded with additional rules so as to uniformly capture wide classes of axiomaticextensions of the basic frameworks, while preserving key properties such as cut elimi-nation. In this paper, which builds and expands on [8], we propose a method to achieve ⋆ The research of the second author is supported by a NWO grant under the scope of the project“A composition calculus for vector-based semantic modelling with a localization for Dutch”(360-89-070). ⋆⋆ The research of the third and fourth author is supported by the NWO Vidi grant 016.138.314,the NWO Aspasia grant 015.008.054, and a Delft Technology Fellowship awarded to the thirdauthor Chen, Greco, Palmigiano, Tzimoulis this goal. We will illustrate this method for the two specific signatures of monotonicmodal logic and conditional logic.Our starting point is the observation, very well-known e.g. from [34], that, underthe interpretation of the modal connective of monotonic modal logic in neighbourhoodframes F = ( W , ν ), the monotonic ‘box’ operation can be understood as the compo-sition of a normal (i.e. finitely join-preserving) semantic diamond h ν i and a normal (i.e. finitely meet-preserving) semantic box [ ∋ ]. The binary relations R ν and R ∋ cor-responding to these two normal operators are not defined on one and the same do-main, but span over two domains, namely R ν ⊆ W × P ( W ) is s.t. wR ν X i ff X ∈ ν ( w )and R ∋ ⊆ P ( W ) × W is s.t. XR ∋ w i ff w ∈ X (cf. [34, Definition 5.7], see also [36,24]).We refine and expand these observations so as to: (a) introduce a semantic environ-ment of two-sorted Kripke frames (cf. Definition 4) and their heterogeneous algebras(cf. Definition 5); (b) outline a network of discrete dualities and correspondences amongthese semantic structures and the algebras and frames for monotone modal logic andconditional logic (cf. Propositions 2, 10, 14, 17); (c) based on these semantic relation-ships, introduce multi-type normal logics into which the original non-normal logics canbe embedded via suitable translations (cf. Section 4) following a methodology whichwas successful in several other cases [18,19,20,21,29,9,28,30,33,48]; (d) retrieve well-known dual characterization results for axiomatic extensions of monotone modal logicand conditional logics as instances of general algorithmic correspondence theory fornormal (multi-type) LE-logics applied to the translated axioms (cf. Section B); (e) ex-tract analytic structural rules from the computations of the first-order correspondents ofthe translated axioms, so that, again by general results on proper display calculi [31](which, as discussed in [2], can be applied also to multi-type logical frameworks) theresulting calculi are sound, complete, conservative and enjoy cut elimination and sub-formula property. Structure of the paper
In Section 2, we collect well-known definitions and facts aboutmonotone modal logic and conditional logic, their algebraic and state-based seman-tics, and the connection between the two. In Section 3, we introduce the multi-typeenvironment (both in the form of heterogeneous algebras and of multi-type Kripkeframes) which will provide the semantic justification for the two-sorted modal log-ics introduced in Section 4, as well as for the syntactic translation of the original lan-guages of monotone modal logic and conditional logic into suitable (multi-type) normalmodal languages. In Section 5, the theory of unified correspondence is applied to thistwo-sorted environment to establish a Sahlqvist-type correspondence framework formonotone modal logic and conditional logic which encompasses and extends the ex-tant correspondence-theoretic results for these logics. In Section 6, proper (multi-type)display calculi are introduced for the basic two sorted normal modal languages and forsome of their best known extensions. The main properties of these calculi are discussedin Section 7. Conclusions and further directions are discussed in Section 8.
Notation.
Throughout the paper, the superscript ( · ) c denotes the relative complement ofthe subset of a given set. When the given set is a singleton { x } , we will write x c instead on-normal logics: semantic analysis and proof theory 3 of { x } c . For any binary relation R ⊆ S × T , let R − ⊆ T × S be the converse relation of R ,i.e. tR − s i ff sRt . For any S ′ ⊆ S and T ′ ⊆ T , we let R [ S ′ ] : = { t ∈ T | ( s , t ) ∈ R for some s ∈ S ′ } and R − [ T ′ ] : = { s ∈ S | ( s , t ) ∈ R for some t ∈ T ′ } . As usual, we write R [ s ] and R − [ t ]in place of R [ { s } ] and R − [ { t } ], respectively. For any ternary relation R ⊆ S × T × U andsubsets S ′ ⊆ S , T ′ ⊆ T , and U ′ ⊆ U , we also let – R (0) [ T ′ , U ′ ] = { s ∈ S | ∃ t ∃ u ( R ( s , t , u ) & t ∈ T ′ & u ∈ U ′ ) } , – R (1) [ S ′ , U ′ ] = { t ∈ T | ∃ s ∃ u ( R ( s , t , u ) & s ∈ S ′ & u ∈ U ′ ) } , – R (2) [ S ′ , T ′ ] = { u ∈ U | ∃ s ∃ t ( R ( s , t , u ) & s ∈ S ′ & t ∈ T ′ ) } . Any binary relation R ⊆ S × T gives rise to the modal operators h R i , [ R ] , [ R i , h R ] : P ( T ) → P ( S ) s.t. for any T ′ ⊆ T – h R i T ′ : = R − [ T ′ ] = { s ∈ S | ∃ t ( sRt & t ∈ T ′ ) } ; – [ R ] T ′ : = ( R − [ T ′ c ]) c = { s ∈ S | ∀ t ( sRt ⇒ t ∈ T ′ ) } ; – [ R i T ′ : = ( R − [ T ′ ]) c = { s ∈ S | ∀ t ( sRt ⇒ t < T ′ ) } ; – h R ] T ′ : = R − [ T ′ c ] = { s ∈ S | ∃ t ( sRt & t < T ′ ) } . By construction, these modal operators are normal. In particular, h R i is completely join-preserving, [ R ] is completely meet-preserving, [ R i is completely join-reversing and h R ] is completely meet-reversing. Hence, their adjoint maps exist and coincide with[ R − ] h R − i , [ R − i , h R − ] : P ( S ) → P ( T ), respectively. That is, for any T ′ ⊆ T and S ′ ⊆ S , h R i T ′ ⊆ S ′ i ff T ′ ⊆ [ R − ] S ′ , S ′ ⊆ [ R ] T ′ i ff h R − i S ′ ⊆ T ′ , S ′ ⊆ [ R i T ′ i ff T ′ ⊆ [ R − i S ′ h R ] T ′ ⊆ S ′ i ff h R − ] S ′ ⊆ T ′ . Any ternary relation R ⊆ S × T × U gives rise to binary modal operators ⊲ R : P ( T ) × P ( U ) → P ( S ) N R : P ( T ) × P ( S ) → P ( U ) ◮ R : P ( S ) × P ( U ) → P ( T )s.t. for any S ′ ⊆ S , T ′ ⊆ T , and U ′ ⊆ U , – T ′ ⊲ R U ′ : = ( R (0) [ T ′ , U ′ c ]) c = { s ∈ S | ∀ t ∀ u ( R ( s , t , u ) & t ∈ T ′ ⇒ u ∈ U ′ ) } ; – T ′ N R S ′ : = R (2) [ T ′ , S ′ ] = { u ∈ U | ∃ t ∃ s ( R ( s , t , u ) & t ∈ T ′ & s ∈ S ′ ) } ; – S ′ ◮ R U ′ : = ( R (1) [ S ′ , U ′ c ]) c = { t ∈ T | ∀ s ∀ u ( R ( s , t , u ) & s ∈ S ′ ⇒ u ∈ U ′ ) } . The stipulations above guarantee that these modal operators are normal. In partic-ular, ⊲ R and ◮ R are completely join-reversing in their first coordinate and completelymeet-preserving in their second coordinate, and N R is completely join-preserving inboth coordinates. These three maps are residual to each other, i.e. for any S ′ ⊆ S , T ′ ⊆ T ,and U ′ ⊆ U , S ′ ⊆ T ′ ⊲ R U ′ i ff T ′ N R S ′ ⊆ U ′ i ff T ′ ⊆ S ′ ◮ R U ′ . Syntax.
For a countable set of propositional variables
Prop , the languages L ∇ and L > of monotonic modal logic and conditional logic over Prop are defined as follows: L ∇ ∋ ϕ :: = p | ¬ ϕ | ϕ ∧ ϕ | ∇ ϕ L > ∋ ϕ :: = p | ¬ ϕ | ϕ ∧ ϕ | ϕ > ϕ. Chen, Greco, Palmigiano, Tzimoulis
The connectives ⊤ , ∧ , ∨ , → and ↔ are defined as usual. The basic monotone modallogic L ∇ (resp. basic conditional logic L > ) is a set of L ∇ -formulas (resp. L > -formulas)containing the axioms of classical propositional logic and closed under modus ponens,uniform substitution and the following rule(s) M (resp. RCEA and
RCK n for all n ≥ ϕ → ψ M ∇ ϕ → ∇ ψ ϕ ↔ ψ RCEA ( ϕ > χ ) ↔ ( ψ > χ ) ϕ ∧ ... ∧ ϕ n → ψ RCK n ( χ > ϕ ) ∧ ... ∧ ( χ > ϕ n ) → ( χ > ψ ) Algebraic semantics. A monotone Boolean algebra expansion , abbreviated as m-algebra (resp. conditional algebra , abbreviated as c-algebra ) is a pair A = ( B , ∇ A ) (resp. A = ( B , > A )) s.t. B is a Boolean algebra and ∇ A is a unary monotone operation on B (resp. > A is a binary operation on B which is finitely meet-preserving in its second coordinate).Such an m-algebra (resp. c-algebra) is perfect if B is a complete and atomic Booleanalgebra (and, in the c-algebra case, > A is completely meet-preserving in its secondcoordinate). Hence, the underlying Boolean algebra of any perfect m-algebra (resp. c-algebra) can be identified with the powerset algebra P ( W ) for some set W .Interpretation of formulas in algebras under assignments h : L ∇ → A (resp. h : L > → A ) and validity of formulas in algebras (in symbols: A | = ϕ ) are defined as usual. By aroutine Lindenbaum-Tarski construction one can show that L ∇ (resp. L > ) is sound andcomplete w.r.t. the class of m-algebras V m (resp. c-algebras V c ). Canonical extensions.
The canonical extension of an m-algebra (resp. c-algebra) A is A δ : = ( B δ , ∇ σ ) (resp. A δ : = ( B δ , > π )), where B δ (cid:27) P ( Ult ( B )), with Ult ( B ) denoting theset of the ultrafilters of B , is the canonical extension of B [35], and ∇ σ (resp. > π ) is the σ -extension of ∇ A (resp. the π -extension of > A ). Let us recall that for all u , u , u ∈ B δ , ∇ σ u : = _ { ^ {∇ a | a ∈ B and k ≤ a } | k ∈ K ( B δ ) and k ≤ u } , u > π u : = ^ { _ { a > a | a i ∈ B and o i ≤ a i ≤ k i } | k i ∈ K ( B δ ) , o i ∈ O ( B δ ) and k i ≤ u i ≤ o i } , where K ( B δ ) and O ( B δ ) respectively denote the join-closure and the meet-closure of B in B δ under the canonical embedding, mapping each a ∈ B to { U ∈ Ult ( B ) | a ∈ U } .By definition and general results on canonical extensions of maps (cf. [25]), thecanonical extension of an m-algebra (resp. c-algebra) as above is a perfect m-algebra(resp. c-algebra). Frames and models. A neighbourhood frame , abbreviated as n-frame (resp. conditionalframe , abbreviated as c-frame ) is a pair F = ( W , ν ) (resp. F = ( W , f )) s.t. W is a non-empty set and ν : W → P ( P ( W )) is a neighbourhood function ( f : W × P ( W ) → P ( W ) isa selection function ). In the remainder of the paper, even if it is not explicitly indicated,we will assume that n-frames are monotone , i.e. s.t. for every w ∈ W , if X ∈ ν ( w ) and X ⊆ Y , then Y ∈ ν ( w ). For any n-frame (resp. c-frame) F , the complex algebra of F is F ∗ : = ( P ( W ) , ∇ F ∗ ) (resp. F ∗ : = ( P ( W ) , > F ∗ )) s.t. for all X , Y ∈ P ( W ), ∇ F ∗ X : = { w | X ∈ ν ( w ) } X > F ∗ Y : = { w | f ( w , X ) ⊆ Y } . Proposition 1. If F is an n-frame (resp. a c-frame), then F ∗ is a perfect m-algebra(resp. c-algebra). on-normal logics: semantic analysis and proof theory 5 Proof.
Let F = ( W , ν ) be an n-frame. Recall that, by definition, ν ( w ) is an upward-closedcollection of subsets of W . To show that F ∗ is a perfect m-algebra, it is enough to showthat ∇ F ∗ is monotone. Let w ∈ W and X ⊆ Y ⊆ W . Since ν ( w ) is upward-closed, X ∈ ν ( w )implies that Y ∈ ν ( w ). Hence, ∇ F ∗ X = { w | X ∈ ν ( w ) } ⊆ { w | Y ∈ ν ( w ) } = ∇ F ∗ Y .Let F = ( W , f ) be a c-frame. To show that F ∗ is a perfect c-algebra, it is enough toshow that > F ∗ is completely meet-preserving in its second coordinate. For any X ⊆ W , X > F ∗ ⊤ F ∗ = X > F ∗ W = { w | f ( w , X ) ⊆ W } = W = ⊤ F ∗ , and for any X ⊆ P ( W ), X > F ∗ \ X = { w ∈ W | f ( w , X ) ⊆ \ X} = { w | f ( w , X ) ⊆ Y } ∩ { w ∈ W | f ( w , X ) ⊆ Y for any Y ∈ X} = \ { ( X > F ∗ Y ) | Y ∈ X} . (cid:3) Models are pairs M = ( F , V ) such that F is a frame and V : L → F ∗ is a homomor-phism of the appropriate type. Hence, the truth of formulas at states in models is definedas M , w (cid:13) ϕ i ff w ∈ V ( ϕ ), and unravelling this stipulation for ∇ - and > -formulas, we get: M , w (cid:13) ∇ ϕ i ff V ( ϕ ) ∈ ν ( w ) M , w (cid:13) ϕ > ψ i ff f ( w , V ( ϕ )) ⊆ V ( ψ ) . Local validity (notation: F , w (cid:13) ϕ ) is defined as local satisfaction for every valuation V . Global satisfaction (notation: M (cid:13) ϕ ) and frame validity (notation: F (cid:13) ϕ ) are definedin the usual way as local satisfaction / validity at every state. Thus, by definition, F (cid:13) ϕ i ff F ∗ | = ϕ , from which the soundness of L ∇ (resp. L > ) w.r.t. the corresponding class offrames immediately follows from the algebraic soundness. Completeness follows fromalgebraic completeness, by observing that (a) the canonical extension of any algebrarefuting ϕ will also refute ϕ ; (b) canonical extensions are perfect algebras; (c) perfect m-algebras (resp. c-algebras) can be associated with n-frames (resp. c-frames) as follows:for any A = ( P ( W ) , ∇ A ) (resp. A = ( P ( W ) , > A )) let A ∗ : = ( W , ν ∇ A ) (resp. A ∗ : = ( W , f > A ))s.t. for all w ∈ W and X ⊆ W , ν ∇ A ( w ) : = { X ⊆ W | w ∈ ∇ A X } f > A ( w , X ) : = \ { Y ⊆ W | w ∈ X > A Y } . That A ∗ is a monotone n-frame can be proved as follows: if X ∈ ν ∇ ( w ) and X ⊆ Y , thenthe monotonicity of ∇ A implies that ∇ A X ⊆ ∇ A Y and hence Y ∈ ν ∇ A ( w ), as required.Let ϕ ∈ L ∇ (resp. ϕ ∈ L > ). It can be shown by a straightforward induction on ϕ that w ∈ V ( ϕ ) i ff ( A ∗ , V ) , w (cid:13) ϕ for any perfect algebra A and assignment V . Then, A | = ϕ i ff A ∗ (cid:13) ϕ . This completes the argument deriving the frame completeness of L ∇ (resp. L > )from its algebraic completeness. Proposition 2. If A is a perfect m-algebra (resp. c-algebra) and F is an n-frame(resp. c-frame), then ( F ∗ ) ∗ (cid:27) F and ( A ∗ ) ∗ (cid:27) A . Chen, Greco, Palmigiano, Tzimoulis
Proof.
Let F = ( W , ν ) be an n-frame. By definition, ( F ∗ ) ∗ = ( W , ν ∇ F ∗ ), where, for every w ∈ W , ν ∇ F ∗ ( w ) = { X ⊆ W | w ∈ ∇ F ∗ X } = { X ⊆ W | w ∈ { u | X ∈ ν ( u ) }} = { X ⊆ W | X ∈ ν ( w ) } = ν ( w ) , which shows that ( F ∗ ) ∗ = F , as required. Let F = ( W , f ) be a c-frame. By definition,( F ∗ ) ∗ = ( w , f > F ∗ ), where, for every w ∈ W and X ⊆ W , f > F ∗ ( w , X ) = \ { Y ⊆ W | w ∈ X > F ∗ Y } = \ { Y ⊆ W | w ∈ { u ∈ W | f ( u , X ) ⊆ Y }} = \ { Y ⊆ W | f ( w , X ) ⊆ Y } = f ( w , X ) , which shows that ( F ∗ ) ∗ = F , as required. Let A = ( P ( W ) , ∇ A ) be a perfect m-algebra (upto isomorphism). Then ( A ∗ ) ∗ = ( P ( W ) , ∇ ( A ∗ ) ∗ ), where for every X ⊆ W , ∇ ( A ∗ ) ∗ X = { w | X ∈ ν ∇ A ( w ) } = { w | X ∈ { Y ⊆ W | w ∈ ∇ A Y }} = { w | w ∈ ∇ A X } = ∇ A X , which shows that ( A ∗ ) ∗ (cid:27) A , as required. Let A = ( P ( W ) , > A ) be a perfect c-algebra (upto isomorphism). Then ( A ∗ ) ∗ = ( P ( W ) , > ( A ∗ ) ∗ ), where for all X , Y ⊆ W , X > ( A ∗ ) ∗ Y = { w | f > A ( w , X ) ⊆ Y } = { w | \ { Z ⊆ W | w ∈ X > A Z } ⊆ Y } = X > A Y . Let us show the last equality. If w ∈ X > A Y , then Y ∈ { Z ⊆ W | w ∈ X > A Z } , and hence T { Z ⊆ W | w ∈ X > A Z } ⊆ Y . Conversely, let w ∈ W be s.t. T { Z ⊆ W | w ∈ X > A Z } ⊆ Y .Since > A is completely meet-preserving in the second coordinate, this implies that w ∈ \ { X > A Z | Z ⊆ W and w ∈ X > A Z } = X > A \ { Z ⊆ W | w ∈ X > A Z } ⊆ X > A Y , as required. This completes the proof that ( A ∗ ) ∗ (cid:27) A . (cid:3) Axiomatic extensions. A monotone modal logic (resp. a conditional logic ) is any ex-tension of L ∇ (resp. L > ) with L ∇ -axioms (resp. L > -axioms). Below we collect corre-spondence results for axioms that have cropped up in the literature [34, Theorem 5.1][43]. on-normal logics: semantic analysis and proof theory 7 Theorem 3.
For every n-frame (resp. c-frame) F , N F (cid:13) ∇⊤ i ff F | = ∀ w [ W ∈ ν ( w )] P F (cid:13) ¬∇⊥ i ff F | = ∀ w [ ∅ < ν ( w )] C F (cid:13) ∇ p ∧ ∇ q → ∇ ( p ∧ q ) i ff F | = ∀ w ∀ X ∀ Y [( X ∈ ν ( w ) & Y ∈ ν ( w )) ⇒ X ∩ Y ∈ ν ( w )] T F (cid:13) ∇ p → p i ff F | = ∀ w ∀ X [ X ∈ ν ( w ) ⇒ w ∈ X ] F (cid:13) ∇∇ p → ∇ p i ff F | = ∀ w ∀ Y X [( X ∈ ν ( w ) & ∀ x ( x ∈ X ⇒ Y ∈ ν ( x ))) ⇒ Y ∈ ν ( w )] F (cid:13) ∇ p → ∇∇ p i ff F | = ∀ w ∀ X [ X ∈ ν ( w ) ⇒ { y | X ∈ ν ( y ) } ∈ ν ( w )] F (cid:13) ¬∇¬ p → ∇¬∇¬ p i ff F | = ∀ w ∀ X [ X < ν ( w ) ⇒ { y | X ∈ ν ( y ) } c ∈ ν ( w )] B F (cid:13) p → ∇¬∇¬ p i ff F | = ∀ w ∀ X [ w ∈ X ⇒ { y | X c ∈ ν ( y ) } c ∈ ν ( w )] D F (cid:13) ∇ p → ¬∇¬ p i ff F | = ∀ w ∀ X [ X ∈ ν ( w ) ⇒ X c < ν ( w )] CS F (cid:13) ( p ∧ q ) → ( p > q ) i ff F | = ∀ x ∀ Z [ f ( x , Z ) ⊆ { x } ] CEM F (cid:13) ( p > q ) ∨ ( p > ¬ q ) i ff F | = ∀ X ∀ y [ | f ( y , X ) | ≤ ID F (cid:13) p > p i ff F | = ∀ x ∀ Z [ f ( x , Z ) ⊆ Z ] . CN F (cid:13) ( p > q ) ∨ ( q > p ) i ff F | = ∀ z ∀ x ∀ y ∀ X ∀ Y [ x < f ( z , X ) or y < f ( z , Y )] . T F (cid:13) ( ⊥ > ¬ p ) → p i ff F | = ∀ z ∃ x [ x ∈ f ( z , ∅ )] . In the following section we will introduce a semantic environment which will makeit possible to obtain all these correspondence results as instances of a suitable multi-typeversion of unified correspondence theory [10,11], and which will provide the motivationfor the introduction of proper display calculi for the logics axiomatised by some of theseaxioms, namely, those the translation of which is analytic inductive (cf. Section 4).
Structures similar to those below are considered implicitly in [34], and explicitly in[22].
Definition 4. A two-sorted n-frame (resp. c-frame ) is a structure K : = ( X , Y , R ∋ , R = , R ν , R ν c ) (resp. K : = ( X , Y , R ∋ , R = , T f ) ) such that X and Y are nonempty sets, R ∋ , R = ⊆ Y × X and R ν , R ν c ⊆ X × Y and T f ⊆ X × Y × X. Such an n-frame is supported if for everyD ⊆ X, R − ν [( R − ∋ [ D c ]) c ] = ( R − ν c [( R − = [ D ]) c ]) c . (1) For any two-sorted n-frame (resp. c-frame) K , the complex algebra of K is K + : = ( P ( X ) , P ( Y ) , [ ∋ ] K + , h = i K + , h ν i K + , [ ν c ] K + ) (resp. K + : = ( P ( X ) , P ( Y ) , [ ∋ ] K + , [ = i K + , ⊲ K + ) ), s.t. h ν i K + : P ( Y ) → P ( X ) [ ∋ ] K + : P ( X ) → P ( Y ) h = i K + : P ( X ) → P ( Y ) U R − ν [ U ] D ( R − ∋ [ D c ]) c D R − = [ D ][ ν c ] K + : P ( Y ) → P ( X ) [ = i K + : P ( X ) → P ( Y ) ⊲ K + : P ( Y ) × P ( X ) → P ( X ) U ( R − ν c [ U c ]) c D ( R − = [ D ]) c ( U , D ) ( T (0) f [ U , D c ]) c The adjoints and residuals of the maps above (cf. Section 2) are defined as follows:
Chen, Greco, Palmigiano, Tzimoulis[ ν ] K + : P ( X ) → P ( Y ) h∈i K + : P ( Y ) → P ( X ) [ < ] K + : P ( Y ) → P ( X ) D ( R ν [ D c ]) c U R ∋ [ U ] U ( R = [ U c ]) c h ν c i K + : P ( X ) → P ( Y ) [ < i K + : P ( Y ) → P ( X ) ◮ K + : P ( X ) × P ( X ) → P ( Y ) D R ν c [ D ] U ( R = [ U ]) c ( C , D ) ( T (1) f [ C , D c ]) c N K + : P ( Y ) × P ( X ) → P ( X )( U , D ) T (2) f [ U , D ] Complex algebras of two-sorted frames can be recognized as perfect heterogeneousalgebras (cf. [3]) of the following kind:
Definition 5. A heterogeneous m-algebra (resp. c-algebra ) is a structure H : = ( A , B , [ ∋ ] H , h = i H , h ν i H , [ ν c ] H ) (resp. H : = ( A , B , [ ∋ ] H , [ = i H , ⊲ H ) )such that A and B are Boolean algebras, h ν i H , [ ν c ] : B → A are finitely join-preservingand finitely meet-preserving respectively, [ ∋ ] H , [ = i H , h = i H : A → B are finitely meet-preserving, finitely join-reversing, and finitely join-preserving respectively, and ⊲ H : B × A → A is finitely join-reversing in its first coordinate and finitely meet-preservingin its second coordinate. Such an H is complete if A and B are complete Booleanalgebras and the operations above enjoy the complete versions of the finite preser-vation properties indicated above, and is perfect if it is complete and A and B areperfect. The canonical extension of a heterogeneous m-algebra (resp. c-algebra) H is H δ : = ( A δ , B δ , [ ∋ ] H δ , h = i H δ , h ν i H δ , [ ν c ] H δ ) (resp. H δ : = ( A δ , B δ , [ ∋ ] H δ , [ = i H δ , ⊲ H δ ) ), where A δ and B δ are the canonical extensions of A and B respectively [35], moreover [ ∋ ] H δ , [ = i H δ , [ ν c ] H δ , ⊲ H δ are the π -extensions of [ ∋ ] H , [ = i H , [ ν c ] H , ⊲ H respectively, and h ν i H δ , h = i H δ are the σ -extensions of h ν i H , h = i H respectively. Definition 6.
A heterogeneous m-algebra H : = ( A , B , [ ∋ ] H , h = i H , h ν i H , [ ν c ] H ) is sup-ported if h ν i H [ ∋ ] H a = [ ν c ] H h = i H a for every a ∈ A . It immediately follows from the definitions that
Lemma 7.
The complex algebra of a supported two-sorted n-frame is a perfect hetero-geneous supported m-algebra.Proof.
Let K = ( X , Y , R ∋ , R = , R ν , R ν c ) be a supported two-sorted n-frame. Then its com-plex algebra is K + = ( P ( X ) , P ( Y ) , [ ∋ ] K + , h = i K + , h ν i K + , [ ν c ] K + ), which is clearly perfect.Since K is also supported, R − ν [( R − ∋ [ D c ]) c ] = ( R − ν c [( R − = [ D ]) c ]) c for any D ⊆ K . Hence, h ν i K + [ ∋ ] K + D = R − ν [( R − ∋ [ D c ]) c ] = ( R − ν c [( R − = [ D ]) c ]) c = [ ν c ] K + h = i K + D . (cid:3) Definition 8. If H = ( P ( X ) , P ( Y ) , [ ∋ ] H , h = i H , h ν i H , [ ν c ] H ) is a perfect heterogeneous m-algebra (resp. H = ( P ( X ) , P ( Y ) , [ ∋ ] H , [ = i H , ⊲ H ) is a perfect heterogeneous c-algebra),its associated two-sorted n-frame (resp. c-frame) is H + : = ( X , Y , R ∋ , R = , R ν , R ν c ) (resp. H + : = ( X , Y , R ∋ , R = , T f ) ), s.t. on-normal logics: semantic analysis and proof theory 9 – R ∋ ⊆ Y × X is defined by yR ∋ x i ff y < [ ∋ ] H x c , – R = ⊆ Y × X is defined by xR = y i ff y ∈ h = i H { x } (resp. y < [ = i H { x } ), – R ν ⊆ X × Y is defined by xR ν y i ff x ∈ h ν i H { y } , – R ν c ⊆ X × Y is defined by xR ν c y i ff x < [ ν c ] H y c , – T f ⊆ X × Y × X is defined by ( x ′ , y , x ) ∈ T f i ff x ′ < { y } ⊲ H x c . Lemma 9. If H is a perfect supported heterogeneous m-algebra, then H + is a supportedtwo-sorted n-frame.Proof. To show that H + is supported, for every D ⊆ X , R − ν [( R − ∋ [ D c ]) c ] = h ν i H [ ∋ ] H D = [ ν c ] H h = i H D = ( R − ν c [( R − = [ D ]) c ]) c . (cid:3) The duality between perfect BAOs and Kripke frames can be readily extended tothe present two-sorted case. The following proposition collects these well-known facts,the proofs of which are analogous to those of the single-sorted case, hence are omitted.
Proposition 10.
For every heterogeneous m-algebra (resp. c-algebra) H and every two-sorted n-frame (resp. c-frame) K ,1. K + is a perfect heterogeneous m-algebra (resp. c-algebra);2. ( K + ) + (cid:27) K , and if H is perfect, then ( H + ) + (cid:27) H . Every supported heterogeneous m-algebra (resp. c-algebra) can be associated with anm-algebra (resp. a c-algebra) as follows:
Definition 11.
For every supported heterogeneous m-algebra H = ( A , B , [ ∋ ] H , h = i H , h ν i H , [ ν c ] H ) (resp. c-algebra H = ( A , B , [ ∋ ] H , [ = i H , ⊲ H ) ), let H • : = ( A , ∇ H • ) (resp. H • : = ( A ,> H • ) ), where for every a ∈ A (resp. a , b ∈ A ), ∇ H • a = h ν i H [ ∋ ] H a = [ ν c ] H h = i H a (resp. a > H • b : = ([ ∋ ] H a ∧ [ = i H a ) ⊲ H b) . It immediately follows from the stipulations above that ∇ H • is a monotone map(resp. > H • is finitely meet-preserving in its second coordinate), and hence H • is an m-algebra (resp. a c-algebra). Conversely, every complete m-algebra (resp. c-algebra) canbe associated with a complete supported heterogeneous m-algebra (resp. a c-algebra)as follows: Definition 12.
For every complete m-algebra C = ( A , ∇ C ) (resp. complete c-algebra C = ( A , > C ) ), let C • : = ( A , P ( A ) , [ ∋ ] C • , h = i C • , h ν i C • , [ ν c ] C • ) (resp. C • : = ( A , P ( A ) , [ ∋ ] C • , [ = i C • , ⊲ C • ) ), where for every a ∈ A and B ∈ P ( A ) , [ ∋ ] C • a : = { b ∈ A | b ≤ a } h ν i C • B : = _ {∇ C b | b ∈ B } [ = i C • a : = { b ∈ A | a ≤ b } [ ν c ] C • B : = ^ {∇ C b | b < B } B ⊲ C • a : = ^ { b > C a | b ∈ B } h = i C • a : = { b ∈ A | a (cid:2) b } . Lemma 13. If C is a complete m-algebra (resp. complete c-algebra), then C • is acomplete supported heterogeneous m-algebra (resp. c-algebra).Proof. Let C = ( A , ∇ C ) be a complete m-algebra. First we show that C • is a completeheterogeneous m-algebra. For X ⊆ A and Γ ⊆ P ( A ),[ ∋ ] C • ^ X = { b ∈ A | b ≤ ^ X } = \ x ∈ X { b ∈ A | b ≤ x } = \ x ∈ X [ ∋ ] C • x h = i C • _ X = { b ∈ A | _ X ≮ b } = [ x ∈ X { b ∈ A | x ≮ b } = [ x ∈ X h = i C • x h ν i C • [ Γ = _ {∇ C b | b ∈ [ Γ } = _ Y ∈ Γ _ {∇ C b | b ∈ Y } = _ Y ∈ Γ h ν i C • Y [ ν c ] C • \ Γ = ^ {∇ C b | b < \ Γ } = \ Y ∈ Γ ^ {∇ C b | b < Y } = \ Y ∈ Γ [ ν c ] C • Y . Let us show that C • is supported. For every a ∈ A , h ν i C • [ ∋ ] C • a = h ν i C • { b ∈ A | b ≤ a } = _ {∇ C b | b ≤ a } = ∇ C a , [ ν c ] C • h = i C • a = [ ν c ] C • { b ∈ A | a b } = ^ {∇ C b | a ≤ b } = ∇ C a . Hence, h ν i C • [ ∋ ] C • a = [ ν c ] C • h = i C • a .Let C = ( A , > C ) be a complete c-algebra. That [ ∋ ] C • is completely join preservingcan be proved as shown above. As to the remaining connectives, for any X ⊆ A and Γ ⊆ P , [ = i C • _ X = { b ∈ A | _ X ≤ b } = \ x ∈ X { b ∈ A | x ≤ b } = \ x ∈ X [ = i C • x [ Γ ⊲ C • a = ^ { b > C a | b ∈ [ Γ } = ^ Y ∈ Γ ^ { b > C a | b ∈ Y } = ^ Y ∈ Γ ( Y ⊲ C • a ) B ⊲ C • ^ X = ^ { b > C ^ X | b ∈ B } = ^ x ∈ X ^ { b > C x | b ∈ B } = ^ x ∈ X ( B ⊲ C • x ) . (cid:3) Proposition 14. If C is a complete m-algebra (resp. c-algebra), then C (cid:27) ( C • ) • . More-over, if H is a complete supported heterogeneous m-algebra (resp. c-algebra), then H (cid:27) C • for some complete m-algebra (resp. c-algebra) C i ff H (cid:27) ( H • ) • .Proof. For the first part of the statement, by definition, C and ( C • ) • have the sameunderlying Boolean algebra. Moreover, ∇ ( C • ) • a = h ν i C • [ ∋ ] C • a = ∇ C a for every a ∈ C ,the first identity holding by definition, the second one being shown in the proof ofLemma 13.As to the second part, for the left to right direction, assume that H (cid:27) C • for somecomplete m-algebra (resp. c-algebra) C . From the first part of the proposition we knowthat C (cid:27) ( C • ) • . Then H (cid:27) C • (cid:27) (( C • ) • ) • (cid:27) ( H • ) • . For the right to left direction, H • is therequired complete m-algebra (resp. c-algebra). (cid:3) The proposition above characterizes up to isomorphism the supported heteroge-neous m-algebras (resp. c-algebras) which arise from single-type m-algebras (resp. c-algebras). on-normal logics: semantic analysis and proof theory 11
Thanks to the discrete dualities discussed in Sections 2.1 and 3.1, we can transfer thealgebraic characterization of Proposition 14 to the side of frames, as detailed in thissubsection.
Definition 15.
For any n-frame (resp. c-frame) F , we let F ⋆ : = (( F ∗ ) • ) + , and for everysupported two-sorted n-frame (resp. c-frame) K , we let K ⋆ : = (( K + ) • ) ∗ . Spelling out the definition above, if F = ( W , ν ) (resp. F = ( W , f )) then F ⋆ = ( W , P ( W ) , R ∋ , R = , R ν , R ν c ) (resp. F ⋆ = ( W , P ( W ) , R = , R ∋ , T f )) where: – R ν ⊆ W × P ( W ) is defined as xR ν Z i ff Y ∈ ν ( x ); – R ν c ⊆ W × P ( W ) is defined as xR ν c Z i ff Z < ν ( x ); – R ∋ ⊆ P ( W ) × W is defined as ZR ∋ x i ff x ∈ Z ; – R = ⊆ P ( W ) × W is defined as ZR = x i ff x < Z ; – T f ⊆ W × P ( W ) × W is defined as T f ( x , Z , x ′ ) i ff x ′ ∈ f ( x , Z ). Moreover, if K = ( X , Y , R ∋ , R = , R ν , R ν c ) (resp. K = ( X , Y , R ∋ , R = , T f )), then K ⋆ = ( X , ν ⋆ )(resp. K ⋆ = ( X , f ⋆ )) where: – ν ⋆ ( x ) = { D ⊆ X | x ∈ R − ν [( R − ∋ [ D c ]) c ] } = { D ⊆ X | x ∈ ( R − ν c [( R − = [ D ]) c ]) c } ; – f ⋆ ( x , D ) = T { C ⊆ X | x ∈ T (0) f [ { C } , D c ] } . Lemma 16. If F = ( W , ν ) is an n-frame, then F ⋆ is a supported two-sorted n-frame.Proof. By definition, F ⋆ is a two-sorted n-frame. Moreover, for any D ⊆ W , ( R − ν c [( R − = [ D ]) c ]) c = { w | ∀ X ( X < ν ( w ) ⇒ ∃ u ( X = u & u ∈ D )) } = { w | ∀ X ( X < ν ( w ) ⇒ D * X ) } = { w | ∀ X ( D ⊆ X ⇒ X ∈ ν ( w )) } = { w | ∃ X ( X ∈ ν ( w ) & X ⊆ D ) } ( ∗ ) = R − ν [( R − ∋ [ D c ]) c ] . To show the identity marked with ( ∗ ), from top to bottom, take X : = D ; conversely,if D ⊆ Z then X ⊆ Z , and since by assumption X ∈ ν ( w ) and ν ( w ) is upward closed, weconclude that Z ∈ ν ( w ), as required. (cid:3) The next proposition is the frame-theoretic counterpart of Proposition 14.
Proposition 17. If F is an n-frame (resp. c-frame), then F (cid:27) ( F ⋆ ) ⋆ . Moreover, if K is asupported two-sorted n-frame (resp. c-frame), then K (cid:27) F ⋆ for some n-frame (resp. c-frame) F i ff K (cid:27) ( K ⋆ ) ⋆ .Proof. For the first part of the statement,( F ⋆ ) ⋆ = ((((( F ∗ ) • ) + ) + ) • ) ∗ definition of ( − ) ⋆ and ( − ) ⋆ (cid:27) ((( F ∗ ) • ) • ) ∗ Proposition 10.2, ( F ∗ ) • perfect heterogeneous algebra = ( F ∗ ) ∗ Proposition 14, since F ∗ is complete = F . Proposition 2As to the second part, for the left to right direction, assume that K (cid:27) F ⋆ for some m-frame (resp. c-frame) F . From the first part of the statement we know that F (cid:27) ( F ⋆ ) ⋆ .Then K (cid:27) F ⋆ (cid:27) (( F ⋆ ) ⋆ ) ⋆ (cid:27) ( K ⋆ ) ⋆ . For the right to left direction, K ⋆ is the requiredm-frame (resp. c-frame). (cid:3) The two-sorted frames and heterogeneous algebras discussed in the previous sectionserve as semantic environment for the multi-type languages defined below.
Multi-type languages.
For a denumerable set
Prop of atomic propositions, the lan-guages L MT ∇ and L MT > in types S (sets) and N (neighbourhoods) over Prop are de-fined as follows: S ∋ A :: = p | ⊤ | ⊥ | ¬ A | A ∧ A | h ν i α | [ ν c ] α S ∋ A :: = p | ⊤ | ⊥ | ¬ A | A ∧ A | α ⊲ A N ∋ α :: = | | ∼ α | α ∩ α | [ ∋ ] A | h = i α N ∋ α :: = | | ∼ α | α ∩ α | [ ∋ ] A | [ = i A . Algebraic semantics.
Interpretation of L MT ∇ -formulas (resp. L MT > formulas) in hetero-geneous m-algebras (resp. c-algebras) under homomorphic assignments h : L MT ∇ → H (resp. h : L MT > → H ) and validity of formulas in heterogeneous algebras ( H | = Θ ) aredefined as usual. Frames and models. L MT ∇ - models (resp. L MT > - models ) are pairs N = ( K , V ) s.t. K = ( X , Y , R ∋ , R = , R ν , R ν c ) is a supported two-sorted n-frame (resp. K = ( X , Y , R ∋ , R = , T f ) isa two-sorted c-frame) and V : L MT → K + is a heterogeneous algebra homomorphismof the appropriate signature. Hence, truth of formulas at states in models is defined as N , z (cid:13) Θ i ff z ∈ V ( Θ ) for every z ∈ X ∪ Y and Θ ∈ S ∪ N , and unravelling this stipulationfor formulas with a modal operator as main connective, we get: – N , x (cid:13) h ν i α i ff N , y (cid:13) α for some y s.t. xR ν y ; – N , x (cid:13) [ ν c ] α i ff N , y (cid:13) α for all y s.t. xR ν c y ; – N , y (cid:13) [ ∋ ] A i ff N , x (cid:13) A for all x s.t. yR ∋ x ; – N , y (cid:13) h = i A i ff N , x (cid:13) A for some x s.t. yR = x ; – N , y (cid:13) [ = i A i ff N , x A for all x s.t. yR = x ; – N , x (cid:13) α ⊲ A i ff for all y and all x ′ , if T f ( x , y , x ′ ) and N , y (cid:13) α then N , x ′ (cid:13) A . Global satisfaction (notation: N (cid:13) Θ ) is defined relative to the domain of the appro-priate type, and frame validity (notation: K (cid:13) Θ ) is defined as usual. Thus, by definition, K (cid:13) Θ i ff K + | = Θ , and if H is a perfect heterogeneous algebra, then H | = Θ i ff H + (cid:13) Θ . Correspondence theory for multi-type normal logics.
This semantic environment sup-ports a straightforward extension of unified correspondence theory for multi-type nor-mal logics, which includes the definition of inductive and analytic inductive formulasand inequalities in L MT ∇ and L MT > (cf. Section A), and a corresponding version ofthe algorithm ALBA [11] for computing their first-order correspondents and analyticstructural rules. Translation.
Correspondence theory and analytic calculi for the non-normal logics L ∇ and L > and their analytic extensions can be then obtained ‘via translation’, i.e. by re-cursively defining translations τ , τ : L ∇ → L MT ∇ and ( · ) τ : L > → L MT > as follows: τ ( p ) = p τ ( p ) = p p τ = p τ ( ϕ ∧ ψ ) = τ ( ϕ ) ∧ τ ( ψ ) τ ( ϕ ∧ ψ ) = τ ( ϕ ) ∧ τ ( ψ ) ( ϕ ∧ ψ ) τ = ϕ τ ∧ ψ τ τ ( ¬ ϕ ) = ¬ τ ( ϕ ) τ ( ¬ ϕ ) = ¬ τ ( ϕ ) ( ¬ ϕ ) τ = ¬ ϕ τ τ ( ∇ ϕ ) = h ν i [ ∋ ] τ ( ϕ ) τ ( ∇ ϕ ) = [ ν c ] h = i τ ( ϕ ) ( ϕ > ψ ) τ = ([ ∋ ] ϕ τ ∧ [ = i ϕ τ ) ⊲ ψ τ on-normal logics: semantic analysis and proof theory 13 Let τ ( ϕ ⊢ ψ ) : = ϕ τ ⊢ ψ τ if ϕ ⊢ ψ is an L > -sequent, and τ ( ϕ ⊢ ψ ) : = τ ( ϕ ) ⊢ τ ( ψ ) if ϕ ⊢ ψ is an L ∇ -sequent. Proposition 18. If F is an n-frame (resp. c-frame) and ϕ ⊢ ψ is an L ∇ -sequent (resp. an L > -sequent), then F (cid:13) ϕ ⊢ ψ i ff F ⋆ (cid:13) τ ( ϕ ⊢ ψ ) .Proof. When F is an n-frame, the proposition is an immediate consequence of the fol-lowing claim:( F , V ) , w (cid:13) ϕ i ff ( F ⋆ , V ) , w (cid:13) τ ( ϕ ) i ff ( F ⋆ , V ) , w (cid:13) τ ( ϕ ) , which can be proved by induction on ϕ . We only sketch the case in which ϕ : = ∇ ψ . Inthis case, τ ( ∇ ψ ) = h ν i [ ∋ ] τ ( ψ ) and τ ( ∇ ψ ) = [ ν c ] h = i τ ( ψ ). F , V , w (cid:13) ∇ ψ i ff ∃ D ( D ∈ ν ( w ) & D ⊆ V ( ψ ))i ff ∃ D ( wR ν D & ∀ d ( DR ∋ d ⇒ d ∈ V ( ψ )))i ff ∃ D ( wR ν D & ∀ d ( DR ∋ d ⇒ d ∈ V ( τ ( ψ ))) Induction hypothesisi ff F ⋆ , V , w (cid:13) h ν i [ ∋ ] τ ( ψ ) F , V , w (cid:13) ∇ ψ i ff ∃ D ( D ∈ ν ( w ) & D ⊆ V ( ψ ))i ff ∃ D ( wR ν D & ∀ d ( DR ∋ d ⇒ d ∈ V ( ψ )))( ∗ ) i ff ∀ D ( wR ν c D ⇒ ∃ d ( DR = d & d ∈ V ( ψ )))i ff ∀ D ( wR ν c D ⇒ ∃ d ( DR = d & d ∈ V ( τ ( ψ )))) Induction hypothesisi ff F ⋆ , V , w (cid:13) [ ν c ] h = i τ ( ψ ).The equivalence marked by ( ∗ ) follows from Lemma 16.When F is a c -frame, the proposition is an immediate consequence of the followingclaim, which can be shown by induction on ϕ .( F , V ) , w (cid:13) ϕ i ff ( F ⋆ , V ) , w (cid:13) ϕ τ . We only sketch the case in which ϕ : = ϕ > ψ . In this case, ( ϕ > ψ ) τ = ([ ∋ ] ϕ τ ∧ [ = i ϕ τ ) ⊲ ψ τ .( F , V ) , w (cid:13) ϕ > ψ i ff f ( w , V ( ϕ )) ⊆ V ( ψ )i ff ∀ x ( x ∈ f ( w , V ( ϕ )) ⇒ x ∈ V ( ψ ))i ff ∀ x ∀ Y ( x ∈ f ( w , Y ) & Y = V ( ϕ ) ⇒ x ∈ V ( ψ ))i ff ∀ x ∀ Y ( x ∈ f ( w , Y ) & Y = V ( ϕ τ ) ⇒ x ∈ V ( ψ τ )) I.H.i ff ∀ x ∀ Y ( T f ( w , Y , x ) & ( ∀ y ( YR ∋ y ⇒ y ∈ V ( ϕ τ ))) &( ∀ y ( YR = y ⇒ y < V ( ϕ τ ))) ⇒ x ∈ V ( ψ τ ))i ff ( F ⋆ , V ) , w (cid:13) ([ ∋ ] ϕ τ ∧ [ = i ϕ τ ) ⊲ ψ τ . (cid:3) With this framework in place, we are in a position to (a) retrieve correspondenceresults in the setting of non-normal logics, such as those collected in Theorem 3, asinstances of the general Sahlqvist theory for multi-type normal logics, and (b) recognizewhether the translation of a non-normal axiom is analytic inductive, and compute itscorresponding analytic structural rules (cf. Section B).
In this section, we detail how the two-sorted environment introduced and discussed inthe previous sections can be used to establish a Sahlqvist-type correspondence frame-work for classes of non-normal logics (the generality of this approach is further dis-cussed in Section 8) which can be specialized to the signatures of monotone modallogic and conditional logic, encompasses and extends the well-known correspondence-theoretic results for these logics collected in Theorem 3, and brings them into the fold ofunified correspondence theory [10,11]. The unified correspondence approach pivots onthe order theoretic properties of the algebraic interpretation of logical connectives. Aspointed out in [2], when the relevant order theoretic properties hold in a given multi-typesetting such as the one introduced in Section 3, the insights, tools and results of unifiedcorrespondence theory can be straightforwardly transferred to it. Specifically for thepresent cases of monotone modal logic and conditional logic, this means, firstly, that wecan specialize the definition of inductive and analytic inductive inequalities / sequents tothe languages L MT ∇ and L MT > defined in the previous section. This definition is givenin Section A; in the following table, we list the translations of the axioms of Theorem3, and for each, the last column of the table specifies whether its translation is analyticinductive. Axiom Translation Inductive AnalyticN ∇⊤ ⊤ ≤ [ ν c ] h = i⊤ X X P ¬∇⊥ ⊤ ≤ ¬h ν i [ ∋ ] ⊥ X X C ∇ p ∧ ∇ q → ∇ ( p ∧ q ) h ν i [ ∋ ] p ∧ h ν i [ ∋ ] q ≤ [ ν c ] h = i ( p ∧ q ) X X T ∇ p → p h ν i [ ∋ ] p ≤ p X X ∇∇ p → ∇ p h ν i [ ∋ ] h ν i [ ∋ ] p ≤ [ ν c ] h = i p X × ∇ p → ∇∇ p h ν i [ ∋ ] p ≤ [ ν c ] h = i [ ν c ] h = i p X × ¬∇¬ p → ∇¬∇¬ p ¬ [ ν c ] h = i¬ p ≤ [ ν c ] h = i¬h ν i [ ∋ ] ¬ p X × B p → ∇¬∇¬ p p ≤ [ ν c ] h = i¬h ν i [ ∋ ] ¬ p X × D ∇ p → ¬∇¬ p h ν i [ ∋ ] p ≤ ¬h ν i [ ∋ ] ¬ p X X
CS ( p ∧ q ) → ( p > q ) ( p ∧ q ) ≤ (([ ∋ ] p ∧ [ = i p ) ⊲ q ) X X
CEM ( p > q ) ∨ ( p > ¬ q ) ⊤ ≤ (([ ∋ ] p ∧ [ = i p ) ⊲ q ) ∨ (([ ∋ ] p ∧ [ = i p ) ⊲ ¬ q ) X X ID p > p ⊤ ≤ ([ ∋ ] p ∧ [ = i p ) ⊲ p X X
CN ( p > q ) ∨ ( q > p ) ⊤ ≤ ([ ∋ ] p ∧ [ = i p ) ⊲ q ) ∨ (([ ∋ ] q ∧ [ = i q ) ⊲ p X X
T ( ⊥ > ¬ p ) → p (([ ∋ ] ⊥ ∧ [ = i⊥ ) ⊲ ¬ p ) ≤ p X × Remark 1.
The positional translation of L ∇ -axioms / sequents guarantees that a greaternumber of translated axioms are analytic inductive. To illustrate this point, consideraxiom C above; translating it using e.g. only τ yields h ν i [ ∋ ] p ∧ h ν i [ ∋ ] q ≤ h ν i [ ∋ ]( p ∧ q )which is inductive but not analytic, since in −h ν i [ ∋ ]( p ∧ q ) some branches (in fact all)are not good. This trick is not a panacea: occurrences of nested ∇ connectives, as inaxiom 4, 4’, 5 and B, will give rise to McKinsey-type nestings of modal operators alsounder the positional translation, which results in some branches being not good.Secondly, the algorithm ALBA defined in [11] can be straightforwardly adapted to L MT ∇ and L MT > and their algebraic and relational semantics; since the translations ofall the axioms listed above are inductive, by the general theory, ALBA succeeds in elim-inating the propositional variables occurring in them and in equivalently transforming on-normal logics: semantic analysis and proof theory 15 their validity on frames into suitable conditions expressible in the predicate languagescanonically associated with n-frames (resp. c-frames). The ALBA runs on these axiomsare reported in Section B.To further expand on how the correspondence results of Theorem 3 can be obtainedas instances of algorithmic correspondence on two-sorted frames and their complexalgebras, let F be an n-frame (resp. a c-frame) and ϕ ⊢ ψ an L ∇ -sequent (resp. L ∇ -sequent). Let τ ( ϕ ⊢ ψ ) denote τ ( ϕ ) ⊢ τ ( ψ ) or ϕ τ ⊢ ψ τ as appropriate. Let ALBA ( τ ( ϕ ⊢ ψ )) denote the output of ALBA when run on τ ( ϕ ⊢ ψ ), and ST ( ALBA ( τ ( ϕ ⊢ ψ ))) be itsstandard translation in the appropriate predicate language of n-frames (resp. c-frames).Then the following chain of equivalences holds: F (cid:13) ϕ ⊢ ψ i ff F ⋆ (cid:13) τ ( ϕ ⊢ ψ ) Proposition 18i ff ( F ⋆ ) + | = τ ( ϕ ⊢ ψ ) def. of validity on two sorted-framesi ff ( F ⋆ ) + | = ALBA ( τ ( ϕ ⊢ ψ )) two-sorted correspondencei ff F ⋆ | = ST ( ALBA ( τ ( ϕ ⊢ ψ )))i ff F | = ST ( ALBA ( τ ( ϕ ⊢ ψ )))Let us concretely illustrate this proof pattern by applying it to the following axiom: ∇ p ∧ ∇ q ⊢ ∇ ( p ∧ q ) . (2)Let F = ( W , ν ) be a n-frame, and F ⋆ = ( W , P ( W ) , R ∋ , R = , R ν , R ν c ) be its associated two-sorted n-frame, where e.g. wR ν Z i ff Z ∈ ν ( w ) and so on (full details are in Definition 15).By Proposition 18, the validity of axiom (2) on F is equivalent to its translation h ν i [ ∋ ] p ∧ h ν i [ ∋ ] q ⊢ [ ν c ] h = i ( p ∧ q ) (3)being valid on F ⋆ , which, by definition of satisfaction and validity in the two-sortedenvironment, is equivalent to the validity of axiom (3) on the complex algebra ( F ⋆ ) + = ( P ( W ) , PP ( W ) , [ ∋ ] , h = i , h ν i , [ ν c ]).According to Definition 21, axiom (3) is a ( Ω, ǫ )-analytic inductive inequality for p < Ω q and ǫ ( p ) = ǫ ( q ) =
1. Let us now run ALBA on axiom (3). In what follows welet i and i be nominal variables of type N and m be a co-nominal variable of type N .This means that i and i are interpreted as — and hence range in the set of — atoms ofthe second domain PP ( W ) of the perfect heterogeneous c-algebra ( F ⋆ ) + (i.e. singletonsubsets { Z } for Z ⊆ W ), while m ranges over the set of coatoms of PP ( W ), and hence isinterpreted as the collection of subsets { Z } c : = { Y ⊆ W | Y , Z } for an arbitrary Z ⊆ W .As no preprocessing is needed, ALBA performs first approximation, which equiva-lently transforms ∀ p ∀ q [ h ν i [ ∋ ] p ∧ h ν i [ ∋ ] q ≤ [ ν c ] h = i ( p ∧ q )]into the following quasi-inequality: ∀ p ∀ q ∀ i ∀ i ∀ m [( i ≤ [ ∋ ] p & i ≤ [ ∋ ] q & h = i ( p ∧ q ) ≤ m ) ⇒ h ν i i ∧ h ν i i ≤ [ ν c ] m ] . Recall that h∈i and [ ∋ ] form a residuation pair. Hence, i ≤ [ ∋ ] p is equivalent to h∈i i ≤ p and i ≤ [ ∋ ] q is equivalent to h∈i i ≤ q . Then the quasi inequality above is equivalent tothe following quasi-inequality: ∀ p ∀ q ∀ i ∀ i ∀ m [( i ≤ [ ∋ ] p & h∈i i ≤ q & h = i ( p ∧ q ) ≤ m ) ⇒ h ν i i ∧ h ν i i ≤ [ ν c ] m ] . The quasi inequality above is in Ackermann shape, hence the Ackermann rule can beapplied (cf. [11, Lemma 4.2]) to eliminate all occurrences of p and q , yielding thefollowing (pure) quasi inequality in output ∀ i ∀ i ∀ m [ h = i ( h∈i i ∧ h∈i i ) ≤ m ⇒ h ν i i ∧ h ν i i ≤ [ ν c ] m ] , which, for the sake of convenience, applying adjunction, we equivalently rewrite as ∀ i ∀ i ∀ m [ h∈i i ∧ h∈i i ≤ [ < ] m ⇒ h ν i i ∧ h ν i i ≤ [ ν c ] m ] . (4)Let ALBA ( τ ( ∇ p ∧ ∇ q ⊢ ∇ ( p ∧ q ))) denote the quasi inequality above. The soundnessof ALBA on perfect heterogeneous m-algebras and the validity of (3) on ( F ⋆ ) + implythat ALBA ( τ ( ∇ p ∧ ∇ q ⊢ ∇ ( p ∧ q ))) holds in ( F ⋆ ) + . The next step is to translate thisquasi-inequality into a condition on F ⋆ expressible in its appropriate correspondencelanguage.As discussed above, nominal and conominal variables correspond to subsets of W .Moreover, recall that the heterogeneous connectives [ ∋ ] , h = i , h ν i , [ ν c ] are interpretedin ( F ⋆ ) + as heterogeneous operations defined by the following assignments: for any D ∈ P ( W ) and U ∈ PP ( W ) (cf. Definition 4),[ < ] U = ( R = [ U c ]) c h∈i U = R ∋ [ U ] h ν i U = R − ν [ U ] [ ν c ] U = ( R − ν c [ U c ]) c . Let Z , Z , Z ⊆ W and { Z } , { Z } , { Z } c be the interpretations of i , i , m , respectively.Then, writing R ◦ [ Z ] for R ◦ [ { Z } ] for any ◦ ∈ {∋ , = , ν, ν c } , we can translate (4) as follows: ∀ i ∀ i ∀ m [ h∈i i ∧ h∈i i ≤ [ < ] m ⇒ h ν i i ∧ h ν i i ≤ [ ν c ] m ] = ∀ Z ∀ Z ∀ Z [ h∈i{ Z } ∧ h∈i{ Z } ≤ [ < ] { Z } c ⇒ h ν i{ Z } ∧ h ν i{ Z } ≤ [ ν c ] { Z } c ] = ∀ Z ∀ Z ∀ Z [ R ∋ [ Z ] ∩ R ∋ [ Z ] ⊆ ( R = [ { Z } cc ]) c ⇒ R − ν [ Z ] ∩ R − ν [ Z ] ⊆ ( R − ν c [ { Z } cc ]) c ] = ∀ Z ∀ Z ∀ Z [ R ∋ [ Z ] ∩ R ∋ [ Z ] ⊆ ( R = [ Z ]) c ⇒ R − ν [ Z ] ∩ R − ν [ Z ] ⊆ ( R − ν c [ Z ]) c ] . Thus, we have obtained F ⋆ | = ∀ Z ∀ Z ∀ Z [ R ∋ [ Z ] ∩ R ∋ [ Z ] ⊆ ( R = [ Z ]) c ⇒ R − ν [ Z ] ∩ R − ν [ Z ] ⊆ ( R − ν c [ Z ]) c ] . The final step is to translate this condition into a condition on F . Recalling the definitionsof R = , R ∋ , R ν , R ν c in Definition 15, it is easy to see that for any Z ⊆ W , R ∋ [ Z ] = Z = ( R = [ Z ]) c and R − ν [ Z ] = { w ∈ W | Z ∈ ν ( w ) } = ( R − ν c [ Z ]) c . Hence, we get: F | = ∀ Z ∀ Z ∀ Z [ Z ∩ Z ⊆ Z ⇒ ∀ x [( Z ∈ ν ( x ) & Z ∈ ν ( x )) ⇒ Z ∈ ν ( x )]] , which, by uncurrying and then currying again, and suitably distributing quantifiers, isequivalent to F | = ∀ Z ∀ Z ∀ x [( Z ∈ ν ( x ) & Z ∈ ν ( x )) ⇒ ∀ Z [ Z ∩ Z ⊆ Z ⇒ Z ∈ ν ( x )]] , on-normal logics: semantic analysis and proof theory 17 which is equivalent to F | = ∀ Z ∀ Z ∀ x [( Z ∈ ν ( x ) & Z ∈ ν ( x )) ⇒ Z ∩ Z ∈ ν ( x )]] :Indeed, for the top-to-bottom direction, take Z = Z ∩ Z . Conversely, assume that Z ∩ Z ⊆ Z , and that Z ∈ ν ( x ) and Z ∈ ν ( x ). Then, the assumption implies that Z ∩ Z ∈ ν ( x ). Since ν ( x ) is upwards-closed, Z ∩ Z ⊆ Z implies that Z ∈ ν ( x ). This completesthe algorithmic proof of item C of Theorem 3. The remaining items can be obtained bysimilar arguments. In Appendix B we collect the relevant ALBA runs and translationsof their output.Finally, the tools of unified correspondence can be used also for computing analyticrules corresponding to analytic inductive axioms in the given two-sorted languages, soto obtain analytic calculi for some axiomatic extensions of the basic monotone modallogic and basic conditional logic as an application of the theory developed in [31]. Thistreatment yields the analytic calculi defined in the next section. In this section we introduce proper multi-type display calculi for L ∇ and L > and theiraxiomatic extensions generated by the analytic axioms in the table above. Languages.
The language L DMT ∇ of the calculus D.MT ∇ for L ∇ is defined as follows: S ( A :: = p | ⊤ | ⊥ | ¬ A | A ∧ A | h ν i α | [ ν c ] α X :: = A | ˆ ⊤ | ˇ ⊥ | ˜ ¬ X | X ˆ ∧ X | X ˇ ∨ X | h ˆ ν i Γ | [ ˇ ν c ] Γ | h ˆ ∈i Γ | [ˇ < ] Γ N ( α :: = [ ∋ ] A | h = i A Γ :: = α | ˆ1 | ˇ0 | ˜ ∼ Γ | Γ ˆ ∩ Γ | Γ ˇ ∪ Γ | [ˇ ∋ ] X | h ˆ = i X | [ˇ ν ] X | h ˆ ν c i X The language L DMT > of the calculus D.MT > for L > is defined as follows: S ( A :: = p | ⊤ | ⊥ | ¬ A | A ∧ A | α ⊲ AX :: = A | ˆ ⊤ | ˇ ⊥ | ˜ ¬ X | X ˆ ∧ X | X ˇ ∨ X | h ˆ ∈i Γ | Γ ˇ ⊲ X | Γ ˆ N X | [ˇ < i Γ N ( α :: = [ ∋ ] A | [ = i A | α ∩ αΓ :: = α | ˆ1 | ˇ0 | ˜ ∼ Γ | Γ ˆ ∩ Γ | Γ ˇ ∪ Γ | [ˇ ∋ ] X | [ˇ = i X | X ˇ ◮ X Multi-type display calculi.
In what follows, we use X , Y , W , Z as structural S -variables,and Γ, ∆, Σ, Π as structural N -variables. Propositional base.
The calculi D.MT ∇ and D.MT > share the rules listed below. – Identity and Cut: Id S p ⊢ p X ⊢ A A ⊢ Y Cut S X ⊢ Y Γ ⊢ α α ⊢ ∆ Cut N Γ ⊢ ∆ – Pure S -type display rules: ⊥ ⊥ ⊢ ˇ ⊥ ⊤ ˆ ⊤ ⊢ ⊤ ˜ ¬ X ⊢ Y gal S ˜ ¬ Y ⊢ X X ⊢ ˜ ¬ Y gal S Y ⊢ ˜ ¬ X X ˆ ∧ Y ⊢ Z res S Y ⊢ ˜ ¬ X ˇ ∨ Z X ⊢ Y ˇ ∨ Z res S ˜ ¬ Y ˆ ∧ X ⊢ Z – Pure N -type display rules: ˜ ∼ Γ ⊢ ∆ gal N ˜ ∼ ∆ ⊢ Γ Γ ⊢ ˜ ∼ ∆ gal N ∆ ⊢ ˜ ∼ Γ Γ ˆ ∩ ∆ ⊢ Σ res N ∆ ⊢ ˜ ∼ Γ ˇ ∪ Σ Γ ⊢ ∆ ˇ ∪ Σ res N ˜ ∼ ∆ ˆ ∩ Γ ⊢ Σ – Pure S -type structural rules: X ⊢ Y cont S ˜ ¬ Y ⊢ ˜ ¬ X X ⊢ Y ˆ ⊤ X ˆ ∧ ˆ ⊤ ⊢ Y X ⊢ Y ˇ ⊥ X ⊢ Y ˇ ∨ ˇ ⊥ X ⊢ Y W S X ˆ ∧ Z ⊢ Y X ⊢ Y W S X ⊢ Y ˇ ∨ Z X ˆ ∧ X ⊢ Y C S X ⊢ Y X ⊢ Y ˇ ∨ Y C S X ⊢ YX ˆ ∧ Y ⊢ Z E S Y ˆ ∧ X ⊢ Y X ⊢ Y ˇ ∨ Z E S X ⊢ Z ˇ ∨ Y X ˆ ∧ ( Y ˆ ∧ Z ) ⊢ W A S ( X ˆ ∧ Y ) ˆ ∧ Z ⊢ W W ⊢ X ˆ ∧ ( Y ˆ ∧ Z ) A S W ⊢ ( X ˆ ∧ Y ) ˆ ∧ Z – Pure N -type structural rules: Γ ⊢ ∆ cont N ˜ ∼ ∆ ⊢ ˜ ∼ Γ Γ ⊢ ∆ ˆ1 Γ ˆ ∩ ˆ1 ⊢ ∆ Γ ⊢ ∆ ˇ0 Γ ⊢ ∆ ˇ ∪ ˇ0 Γ ⊢ ∆ W N Γ ˆ ∩ Π ⊢ ∆ Γ ⊢ ∆ W N Γ ⊢ ∆ ˇ ∪ Π Γ ˆ ∩ Γ ⊢ ∆ C N Γ ⊢ ∆ Γ ⊢ ∆ ˇ ∪ ∆ C N Γ ⊢ ∆Γ ˆ ∩ ∆ ⊢ Π E N Y ˆ ∩ Γ ⊢ ∆ Γ ⊢ ∆ ˇ ∪ Π E N Γ ⊢ Π ˇ ∪ ∆ Γ ˆ ∩ ( ∆ ˆ ∩ Π ) ⊢ Σ A N ( Γ ˆ ∩ ∆ ) ˆ ∩ Π ⊢ Σ Σ ⊢ Γ ˆ ∩ ( ∆ ˆ ∩ Π ) A N Σ ⊢ ( Γ ˆ ∩ ∆ ) ˆ ∩ Π – Pure S -type logical rules: ˜ ¬ A ⊢ X ¬ ¬ A ⊢ X X ⊢ ˜ ¬ A ¬ X ⊢ ¬ A A ˆ ∧ B ⊢ X ∧ A ∧ B ⊢ X X ⊢ A Y ⊢ B ∧ X ˆ ∧ Y ⊢ A ∧ B Monotonic modal logic.
D.MT ∇ also includes the rules listed below. – Multi-type display rules: h ˆ ν i Γ ⊢ X h ˆ ν i [ˇ ν ] Γ ⊢ [ˇ ν ] X h ˆ ν c i X ⊢ Γ h ˆ ν c i [ ˇ ν c ] X ⊢ [ ˇ ν c ] Γ h ˆ ∈i Γ ⊢ X h ˆ ∈i [ˇ ∋ ] Γ ⊢ [ˇ ∋ ] X h ˆ ∈i Γ ⊢ X h ˆ ∈i [ˇ ∋ ] Γ ⊢ [ˇ ∋ ] X h ˆ = i X ⊢ Γ h ˆ = i [ˇ < ] X ⊢ [ˇ < ] Γ – Logical rules for multi-type connectives: h ˆ ν i α ⊢ X h ν i h ν i α ⊢ X Γ ⊢ α h ν i h ˆ ν i Γ ⊢ h ν i α α ⊢ Γ [ ν c ] [ ν c ] α ⊢ [ ˇ ν c ] Γ X ⊢ [ ˇ ν c ] α [ ν c ] X ⊢ [ ν c ] α h ˆ = i A ⊢ Γ h = i h = i A ⊢ Γ X ⊢ A h = i h ˆ = i X ⊢ h = i A A ⊢ X [ ∋ ] [ ∋ ] A ⊢ [ˇ ∋ ] X Γ ⊢ [ˇ ∋ ] A [ ∋ ] Γ ⊢ [ ∋ ] A Conditional logic.
D.MT > includes left and right logical rules for [ ∋ ], the displaypostulates h ˆ ∈i [ ˇ ∋ ] and the rules listed below. – Multi-type display rules: on-normal logics: semantic analysis and proof theory 19 X ⊢ Γ ˇ ⊲ Y ˆ N ˇ ⊲ Γ ˆ N X ⊢ Y Γ ⊢ X ˇ ◮ Y ˇ ◮ ˇ ⊲ X ⊢ Γ ˇ ⊲ Y X ⊢ [ˇ < i Γ [ˇ < i [ˇ = i Γ ⊢ [ˇ = i X – Logical rules for multi-type connectives and pure G -type logical rules: Γ ⊢ α A ⊢ X ⊲ α ⊲ A ⊢ Γ ˇ ⊲ X X ⊢ α ˇ ⊲ A ⊲ X ⊢ α ⊲ A X ⊢ A [ = i [ = i A ⊢ [ˇ = i X Γ ⊢ [ˇ = i A [ = i Γ ⊢ [ = i A α ˆ ∩ β ⊢ Γ ∩ α ∩ β ⊢ Γ Γ ⊢ α ∆ ⊢ β ∩ Γ ˆ ∩ ∆ ⊢ α ∩ β Axiomatic extensions.
Each rule is labelled with the name of its corresponding axiom. h ˆ = i ˆ ⊤ ⊢ Γ N ˆ ⊤ ⊢ [ ˇ ν c ] Γ ∆ ⊢ [ˇ = ih ˆ ∈i Γ h ˆ ∈i Γ ⊢ X ID ˆ ⊤ ⊢ ( Γ ˆ ∩ ∆ ) ˇ ⊲ X h ˆ = i ( h ˆ ∈i Γ ˆ ∧ h ˆ ∈i ∆ ) ⊢ Θ C h ˆ ν i Γ ˆ ∧ h ˆ ν i ∆ ⊢ [ ˇ ν c ] ΘΓ ⊢ [ˇ ∋ ] ˜ ¬h ˆ ∈i ∆ D h ˆ ν i ∆ ⊢ ˜ ¬h ˆ ν i Γ Γ ⊢ [ˇ ∋ ] ˇ ⊥ P ˆ ⊤ ⊢ ˜ ¬h ˆ ν i Γ Γ ⊢ [ˇ ∋ ][ˇ < i ∆ X ⊢ [ˇ < i ∆ Y ⊢ Z CS X ˆ ∧ Y ⊢ ( Γ ˆ ∩ ∆ ) ˇ ⊲ Z Π ⊢ [ˇ = ih ˆ ∈i Γ Π ⊢ [ˇ = ih ˆ ∈i Θ ∆ ⊢ [ˇ = ih ˆ ∈i Γ ∆ ⊢ [ˇ = ih ˆ ∈i Θ Y ⊢ X CEM ˆ ⊤ ⊢ (( Γ ˆ ∩ ∆ ) ˇ ⊲ X ) ˇ ∨ (( Θ ˆ ∩ Π ) ˇ ⊲ ˜ ¬ Y ) Γ ⊢ [ˇ ∋ ] X T h ˆ ν i Γ ⊢ X Γ ⊢ [ˇ ∋ ][ˇ < i ∆ Γ ⊢ [ˇ ∋ ] Y Θ ⊢ [ˇ ∋ ][ˇ < i Π Θ ⊢ [ˇ ∋ ] X CN ˆ ⊤ ⊢ (( Γ ˆ ∩ ∆ ) ˇ ⊲ X ) ˇ ∨ (( Θ ˆ ∩ Π ) ˇ ⊲ Y ) The calculi introduced above are proper (cf. [50,31]), and hence the general theory ofproper multi-type display calculi guarantees that they enjoy cut elimination and subfor-mula property [17].Let H m (resp. H c ) be the class of all perfect heterogeneous m-algebras (resp. perfectheterogeneous c-algebras). Given a set of analytic sequents R , the extension of D.MT ∇ (resp. D.MT > ) with inference rules obtained by running ALBA on R is denoted byD.MT ∇ R (resp. D.MT > R ). The subclass of H m (resp. H c ) defined by R is denoted by H m ( R ) (resp. H c ( R )). To show the soundness of the rules of D.MT ∇ R (resp. D . MT > R ) w.r.t. H m ( R ) (resp. H c ( R )), it su ffi ces to show that the interpretation of each rule in D.MT ∇ R (resp. D . MT > R ) is valid in H m ( R ) (resp. H c ( R )). The soundness of the rules in D.MT ∇ and D.MT > follows from the definitions of H m and H c , respectively. And the soundness of the rulesfrom R follows from the soundness of ALBA rules on members of H m (resp. H c ), andthe ALBA runs reported in the appendix. Specifically, in what follows, for any perfectm-algebra (resp. c-algebra) H : = ( A , B , ... ), let x range over A and γ , δ , θ range over B .Then the rules on the left-hand side of the squiggly arrows below are interpreted as thequasi-inequalities on the right-hand side: h ˆ ∈i Γ ⊢ X Γ ⊢ [ ˇ ∋ ] X ∀ γ ∀ x [ h∈i γ ≤ x ⇔ γ ≤ [ ∋ ] x ] h ˆ = i ( h ˆ ∈i Γ ˆ ∧ h ˆ ∈i ∆ ) ⊢ Θ h ˆ ν i Γ ˆ ∧ h ˆ ν i ∆ ⊢ [ ˇ ν c ] Θ ∀ γ ∀ δ ∀ θ [ h = i ( h∈i γ ∧ h∈i δ ) ≤ θ ⇒ h ν i γ ∧ h ν i δ ≤ [ ν c ] θ ] Γ ⊢ [ ˇ ∋ ] ˇ ⊥ ˆ ⊤ ⊢ ˜ ¬h ˆ ν i Γ ∀ γ [ γ ≤ [ ∋ ] ⊥ ⇒ ⊤ ≤ ¬h ν i γ ]The validity of ∀ γ ∀ x [ h∈i γ ≤ x ⇔ γ ≤ [ ∋ ] x ] follows from the fact that h∈i and [ ∋ ]form a residuation pair in H . The validity of the quasi-inequalities corresponding to ax-ioms C and P in H m ( { C } ) and H m ( { P } ) respectively follows from the validity-preservingALBA runs reported in the appendix. We report below on the validity-preserving ALBArun for C. C. H | = ∇ p ∧ ∇ q → ∇ ( p ∧ q ) h ν i [ ∋ ] p ∧ h ν i [ ∋ ] q ⊆ [ ν c ] h = i ( p ∧ q ) H | = h ν i [ ∋ ] p ∧ h ν i [ ∋ ] q ⊆ [ ν c ] h = i ( p ∧ q )i ff H | = ∀ γ ∀ δ ∀ θ ∀ pq [ γ ⊆ [ ∋ ] p & δ ⊆ [ ∋ ] q & h = i ( p ∧ q ) ⊆ θ ⇒ h ν i γ ∧ h ν i δ ⊆ [ ν c ] θ ] first approx.i ff H | = ∀ γ ∀ δ ∀ θ ∀ pq [ h∈i γ ⊆ p & h∈i δ ⊆ q & h = i ( p ∧ q ) ⊆ θ ⇒ h ν i γ ∧ h ν i δ ⊆ [ ν c ] θ ] Residuationi ff H | = ∀ γ ∀ δ ∀ θ [ h = i ( h∈i γ ∧ h∈i δ ) ⊆ θ ⇒ h ν i γ ∧ h ν i δ ⊆ [ ν c ] θ ] ( ⋆ ) Ackermann As discussed above, the algorithmic correspondence perspective on the theory of ana-lytic calculi (here in their incarnation as “proper display calculi”) allows for a uniformjustification of the soundness of analytic rules in terms of the soundness of the algorithmALBA used to generate them. These benefits extend also to the uniform justification ofthe completeness of proper display calculi w.r.t. the logics they are intended to capture.Specifically, in [7], an e ff ective procedure is introduced for generating cut free deriva-tions of the translations of each rule and analytic inductive axiom (of any normal latticeexpansion signature) in the corresponding proper display calculus. Below, we illustratethis e ff ective procedure by applying it to the analytic axioms of the present setting.N. ∇⊤ [ ν c ] h = i⊤ P. ¬∇⊥ ¬h ν i [ ∋ ] ⊥ T. ∇ A → A h ν i [ ∋ ] A ⊢ A ˆ ⊤ ⊢ ⊤h ˆ ∋i ˆ ⊤ ⊢ h∋i⊤ N ˆ ⊤ ⊢ [ ˇ ν c ] h∋i⊤ ⊥ ⊢ ˇ ⊥ [ ∋ ] ⊥ ⊢ [ˇ ∋ ] ˇ ⊥ P ˆ ⊤ ⊢ ˜ ¬ [ ∋ ] ⊥ A ⊢ A [ ∋ ] A ⊢ [ˇ ∋ ] A T h ˆ ν i [ ∋ ] A ⊢ A ID. A > A ([ ∋ ] A ∧ [ = i A ) ⊲ A A ⊢ A [ = i A ⊢ [ˇ = i AA ⊢ [ˇ < i [ = i A [ ∋ ] A ⊢ [ˇ ∋ ][ˇ < i [ = i A h ˆ ∈i [ ∋ ] A ⊢ [ˇ < i [ = i A [ = i A ⊢ [ˇ = ih ˆ ∈i [ ∋ ] A A ⊢ A [ ∋ ] A ⊢ [ˇ ∋ ] A h ˆ ∈i [ ∋ ] A ⊢ A ID ˆ ⊤ ⊢ ([ˇ ∋ ] A ˆ ∩ [ˇ = i A ) ˇ ⊲ A on-normal logics: semantic analysis and proof theory 21 CS. ( A ∧ B ) → ( A > B ) ( A ∧ B ) ⊢ ([ ∋ ] A ∩ [ = i A ) ⊲ B A ⊢ A [ = i A ⊢ [ˇ = i AA ⊢ [ˇ < i [ = i A [ ∋ ] A ⊢ [ˇ ∋ ][ˇ < i [ = i A A ⊢ A [ = i A ⊢ [ˇ = i AA ⊢ [ˇ < i [ = i A B ⊢ B CS A ˆ ∧ B ⊢ ([ˇ ∋ ] A ˆ ∩ [ˇ = i A ) ˇ ⊲ B CEM. ( A > B ) ∨ ( A > ¬ B ) ([ ∋ ] A ∩ [ = i A ) ⊲ B ∨ ([ ∋ ] A ∩ [ = i A ) ⊲ ¬ B A ⊢ A [ ∋ ] A ⊢ [ˇ ∋ ] A h ˆ ∈i [ ∋ ] A ⊢ A [ = i A ⊢ [ˇ = ih ˆ ∈i [ ∋ ] A A ⊢ A [ ∋ ] A ⊢ [ˇ ∋ ] A h ˆ ∈i [ ∋ ] A ⊢ A [ = i A ⊢ [ˇ = ih ˆ ∈i [ ∋ ] A A ⊢ A [ ∋ ] A ⊢ [ˇ ∋ ] A h ˆ ∈i [ ∋ ] A ⊢ A [ = i A ⊢ [ˇ = ih ˆ ∈i [ ∋ ] A A ⊢ A [ ∋ ] A ⊢ [ˇ ∋ ] A h ˆ ∈i [ ∋ ] A ⊢ A [ = i A ⊢ [ˇ = ih ˆ ∈i [ ∋ ] A B ⊢ B CEM ˆ ⊤ ⊢ ([ ∋ ] A ˆ ∩ [ = i A ) ˇ ⊲ B ˇ ∨ ([ ∋ ] A ˆ ∩ [ = i A ) ˇ ⊲ ˜ ¬ B C. ∇ A ∧ ∇ B → ∇ ( A ∧ B ) h ν i [ ∋ ] A ∧ h ν i [ ∋ ] B ⊢ [ ν c ] h = i ( A ∧ B )D. ∇ A → ¬∇¬ A h ν i [ ∋ ] A ⊢ ¬h ν i [ ∋ ] ¬ A A ⊢ A [ ∋ ] A ⊢ [ ∋ ] A h ˆ ∈i [ ∋ ] A ⊢ A B ⊢ B [ ∋ ] B ⊢ [ ∋ ] B h ˆ ∈i [ ∋ ] B ⊢ B h ˆ ∈i [ ∋ ] A ˆ ∧ h ˆ ∈i [ ∋ ] B ⊢ A ∧ B h ˆ = i ( h ˆ ∈i [ ∋ ] A ˆ ∧ h ˆ ∈i [ ∋ ] B ) ⊢ h = i ( A ∧ B ) C h ˆ ν i [ ∋ ] A ˆ ∧ h ˆ ν i [ ∋ ] B ⊢ [ ˇ ν c ] h = i ( A ∧ B ) A ⊢ A [ ∋ ] A ⊢ [ˇ ∋ ] A h ˆ ∈i [ ∋ ] A ⊢ A ¬ A ⊢ ˜ ¬h ˆ ∈i [ ∋ ] A [ ∋ ] ¬ A ⊢ [ˇ ∋ ] ˜ ¬h ˆ ∈i [ ∋ ] A D h ˆ ν i [ ∋ ] A ⊢ ˜ ¬h ˆ ν i [ ∋ ] ¬ A CN. ( A > B ) ∨ ( B > A ) ([ ∋ ] A ∩ [ = i A ) ⊲ B ∨ ([ ∋ ] B ∩ [ = i B ) ⊲ A A ⊢ A [ = i A ⊢ [ˇ = i AA ⊢ [ˇ < i [ = i A [ ∋ ] A ⊢ [ˇ ∋ ][ˇ < i [ = i A A ⊢ A [ ∋ ] A ⊢ [ˇ ∋ ] A B ⊢ B [ = i B ⊢ [ˇ = i BB ⊢ [ˇ < i [ = i B [ ∋ ] B ⊢ [ˇ ∋ ][ˇ < i [ = i B B ⊢ B [ ∋ ] B ⊢ [ˇ ∋ ] B CN ˆ ⊤ ⊢ (([ ∋ ] A ˆ ∩ [ = i A ) ˇ ⊲ B ) ˇ ∨ (([ ∋ ] B ˆ ∩ [ = i B ) ˇ ⊲ A ) The (translations of the) rules M, RCEA and RCK n are derivable as follows.M. A ⊢ B ∇ A ⊢ ∇ B A ⊢ B h ν i [ ∋ ] A ⊢ h ν i [ ∋ ] B A ⊢ B [ ∋ ] A ⊢ [ˇ ∋ ] B [ ∋ ] A ⊢ [ ∋ ] B h ˆ ν i [ ∋ ] A ⊢ h ν i [ ∋ ] B h ν i [ ∋ ] A ⊢ h ν i [ ∋ ] B RCEA. A ↔ B ( A > C ) ↔ ( B > C ) A ⊢ B B ⊢ A ([ ∋ ] A ∩ [ = i A ) ⊲ C ⊢ ([ ∋ ] B ∩ [ = i B ) ⊲ C B ⊢ A [ ∋ ] B ⊢ [ˇ ∋ ] A [ ∋ ] B ⊢ [ ∋ ] A A ⊢ B [ = i B ⊢ [ˇ = i A [ = i B ⊢ [ = i A [ ∋ ] B ˆ ∩ [ = i B ⊢ [ ∋ ] ϕ ∩ [ = i A [ ∋ ] B ∩ [ = i B ⊢ [ ∋ ] ϕ ∩ [ = i A C ⊢ C ([ ∋ ] A ∩ [ = i A ) ⊲ C ⊢ ([ ∋ ] B ∩ [ = i B ) ˇ ⊲ C ([ ∋ ] A ∩ [ = i A ) ⊲ C ⊢ ([ ∋ ] B ∩ [ = i B ) ⊲ C RCK n . A ∧ . . . ∧ A n → B ( C > A ) ∧ . . . ∧ ( C > A n ) → ( C > B ) A ∧ . . . ∧ A n ⊢ B ([ ∋ ] C ∩ [ = i C ) ⊲ A ∧ . . . ∧ ([ ∋ ] χ ∩ [ = i C ) ⊲ A n ⊢ ([ ∋ ] C ∩ [ = i C ) ⊲ B To show that the translation of
RCK n is derivable, let us preliminarily show that ([ ∋ ] C ∩ [ = i C ) ˆ N ([ ∋ ] C ∩ [ = i C ) ⊲ A ∧ ([ ∋ ] C ∩ [ = i C ) ⊲ A ⊢ A ∧ A is derivable. C ⊢ C [ ∋ ] C ⊢ [ˇ ∋ ] C [ ∋ ] C ⊢ [ ∋ ] C C ⊢ C [ = i C ⊢ [ˇ = i C [ = i C ⊢ [ = i C [ ∋ ] C ˆ ∩ [ = i C ⊢ [ ∋ ] C ∩ [ = i C [ ∋ ] C ∩ [ = i C ⊢ [ ∋ ] C ∩ [ = i C A ⊢ A ([ ∋ ] C ∩ [ = i C ) ⊲ A ⊢ ([ ∋ ] C ∩ [ = i C ) ˇ ⊲ A W S ([ ∋ ] C ∩ [ = i C ) ⊲ A ˆ ∧ ([ ∋ ] C ∩ [ = i C ) ⊲ A ⊢ ([ ∋ ] C ∩ [ = i C ) ˇ ⊲ A ([ ∋ ] C ∩ [ = i C ) ⊲ A ∧ ([ ∋ ] C ∩ [ = i C ) ⊲ A ⊢ ([ ∋ ] C ∩ [ = i C ) ˇ ⊲ A ([ ∋ ] C ∩ [ = i C ) ˆ N ([ ∋ ] C ∩ [ = i C ) ⊲ A ∧ ([ ∋ ] C ∩ [ = i C ) ⊲ A ⊢ A C ⊢ C [ ∋ ] C ⊢ [ˇ ∋ ] C [ ∋ ] C ⊢ [ ∋ ] C C ⊢ C [ = i C ⊢ [ˇ = i C [ = i C ⊢ [ = i C [ ∋ ] C ˆ ∩ [ = i C ⊢ [ ∋ ] C ∩ [ = i C [ ∋ ] C ∩ [ = i C ⊢ [ ∋ ] C ∩ [ = i C A ⊢ A ([ ∋ ] C ∩ [ = i C ) ⊲ A ⊢ ([ ∋ ] C ∩ [ = i C ) ˇ ⊲ A W S ([ ∋ ] C ∩ [ = i C ) ⊲ A ˆ ∧ ([ ∋ ] C ∩ [ = i C ) ⊲ A ⊢ ([ ∋ ] C ∩ [ = i C ) ˇ ⊲ A ([ ∋ ] C ∩ [ = i C ) ⊲ A ∧ ([ ∋ ] C ∩ [ = i C ) ⊲ A ⊢ ([ ∋ ] C ∩ [ = i C ) ˇ ⊲ A ([ ∋ ] C ∩ [ = i C ) ˆ N ([ ∋ ] C ∩ [ = i C ) ⊲ A ∧ ([ ∋ ] C ∩ [ = i C ) ⊲ A ⊢ A (([ ∋ ] C ∩ [ = i C ) ˆ N ([ ∋ ] C ∩ [ = i C ) ⊲ A ∧ ([ ∋ ] C ∩ [ = i C ) ⊲ A ) ˆ ∧ (([ ∋ ] C ∩ [ = i C ) ˆ N ([ ∋ ] C ∩ [ = i C ) ⊲ A ∧ ([ ∋ ] C ∩ [ = i C ) ⊲ A ) ⊢ A ∧ A C S ([ ∋ ] C ∩ [ = i C ) ˆ N ([ ∋ ] C ∩ [ = i C ) ⊲ A ∧ ([ ∋ ] C ∩ [ = i C ) ⊲ A ⊢ A ∧ A Iterating the previous derivation n − W S issuitably chosen so as to derive the specific instantiation of the end sequent), we obtainthe left premise of the following derivation, which provides the required derivation ofthe conclusion of RCK n from its premise. ... ([ ∋ ] C ∩ [ = i C ) ˆ N (([ ∋ ] C ∩ [ = i C ) ⊲ A ∧ ... ∧ ([ ∋ ] C ∩ [ = i C ) ⊲ A n ) ⊢ A ∧ ... ∧ A n A ∧ ... ∧ A n ⊢ B Cut S ([ ∋ ] C ∩ [ = i C ) ˆ N (([ ∋ ] C ∩ [ = i C ) ⊲ A ∧ ... ∧ ([ ∋ ] C ∩ [ = i C ) ⊲ A n ) ⊢ B ([ ∋ ] C ∩ [ = i C ) ⊲ A ∧ ... ∧ ([ ∋ ] C ∩ [ = i C ) ⊲ A n ⊢ ([ ∋ ] C ∩ [ = i C ) ˇ ⊲ B ([ ∋ ] C ∩ [ = i C ) ⊲ A ∧ ... ∧ ([ ∋ ] C ∩ [ = i C ) ⊲ A n ⊢ ([ ∋ ] C ∩ [ = i C ) ⊲ B To argue that the calculi introduced in Section 6 conservatively extend their correspond-ing Hilbert systems, we follow the standard proof strategy discussed in [31,32]. Let ⊢ L denote the syntactic consequence relation arising from Hilbert systems, and | = H denotethe semantic consequence relation arising from heterogeneous Kripke frames and theircomplex (heterogeneous) algebras. We need to show that, for all formulas A and B ofthe original language of the Hilbert system, if τ ( A ⊢ B ) is derivable in a display calculus, on-normal logics: semantic analysis and proof theory 23 then A ⊢ L B . This claim can be proved using the following facts: (a) the rules of displaycalculi are sound w.r.t. heterogeneous Kripke frames and their complex (heterogeneous)algebras (cf. Section 7.1); (b) Hilbert systems are complete w.r.t. their respective classof algebras; and (c) homogenous algebras are equivalently presented as heterogeneousalgebras (cf. Section 3.2), so that the semantic consequence relations arising from eachtype of structures preserve and reflect the translation (cf. Proposition 18). Then, let A ⊢ B be an entailment between formulas of the language of the original Hilbert systems. If τ ( A ⊢ B ) is derivable in a display calculus, then, by (a), | = H τ ( A ⊢ B ). By (c), this im-plies that A | = V B , where | = V denotes the semantic consequence relation arising fromm-algebras or c-algebras. By (b), this implies that A ⊢ L B , as required. Present contributions.
In the present paper, we have proposed a semantic analysis oftwo well-known non-normal logics (monotone modal logic and conditional logic), andused it to introduce both a uniform correspondence-theoretic framework encompassingand significantly extending various well-known Sahlqvist-type results for these logics,and a proof-theoretic framework modularly capturing not only the basic logics but alsoan infinite class of axiomatic extensions of the basic monotone modal logic and condi-tional logic which includes well-known logics such as coalitional logic [45] and pref-erential logic [51]. The correspondence-theoretic and the proof-theoretic frameworksare closely connected with each other, both because they stem from the same semanticanalysis, and because, more fundamentally, they instantiate results, tools and insightsdeveloped at the interface of correspondence theory and structural proof theory [31].This line of research can be naturally extended in various ways, and in what follows welist some natural further directions.
A modular framework for classical modal logic.
In the present paper, we have consid-ered monotone modal logic and conditional logic because this choice made it possibleto address a significant diversity of order-theoretic behaviour of the non-normal con-nectives with a minimal set of examples: namely a unary monotone operator and abinary operator which is normal (finitely meet-preserving) in its second coordinate andarbitrary in the first coordinate. A natural further direction concerns the systematic ap-plication of these techniques to wider classes of non-normal logics. Even restrictingattention to the signature of L ∇ , a natural direction concerns developing a modular ac-count of classical modal logic [6] and its (monotone, regular) extensions up to normalmodal logic. Of course the translations employed in the present paper for monotonemodal logic do not account for classical modal logic, because monotonicity is in-builtin these translations. The question is then whether one can express monotonicity as an(analytic) inductive condition under a translation similar to the one used in the non-normal coordinate of the conditional logic operator > . From Boolean to distributive lattice-based non-normal logics.
The semantic analysisof the present paper hinges on the embedding of well-known state-based semantics(monotone neighbourhood frames, selection functions) into two-sorted classical Kripke frames and their discrete dualities with perfect (heterogeneous) Boolean algebras. Piv-oting on more general discrete dualities, such as Birkho ff ’s discrete duality betweenperfect distributive lattices and posets, one can develop the systematic theory of e.g. the non-normal counterparts of positive modal logic [14,4] or intuitionistic modal logics[15,16,44]. In particular, it would be interesting to investigate the applicability of thepresent approach for capturing the lattice of non-normal intuitionistic modal logics in-troduced in [12]. Neighbourhood and selection functions as formal tools for context-relativization andcategory-formation.
We plan to investigate alternative (intuitive) interpretations of neigh-bourhood and conditional frames in order to expand the realm of possible applications.A natural option would be to consider a neighbourhood as a context relativising theinterpretation of a term. An obvious application would be in lexical semantics (seee.g. [1]) where the meaning of a word is often context-dependent.A second option would be to consider neighbourhoods as categories. Again, an obviousapplication would be in computational linguistics (see e.g. [38]) where each word isassigned to a syntactical category depending on the role it plays in the formation ofgrammatically correct sentences or phrases.Notice that a word can occur in di ff erent contexts or it can be assigned to di ff erentcategories. Therefore, one may consider generalizations of the framework with multiple(weighed) neighbourhood functions or relations as a way to represent (probabilistic)distributions in a data set.In many machine learning approaches, a system needs both positive and negative evi-dence. For example, a classification system needs examples for each class that it is capa-ble of predicting; if the classification is binary (e.g. the system tries to decide whether anemail is spam or not), it needs to have positive and negative examples. This generalisesto multiple classes (e.g. given a music song, predict the genre of that song). Therefore,one may consider (generalisations of) bi-neighbourhood frames (see e.g. [13]), in whichsets of pairs of neighbourhoods provide independent positive and negative evidence.Finally, each neighbourhood can be endowed with additional structure in order to cap-ture specific behaviour. This refinement would build a bridge between the literaturein non-normal modal logics and the literature on so-called modal logics for structuralcontrol in linguistics and logic (see e.g. [37,40,27,32]). A Analytic inductive inequalities
In the present section, we specialize the definition of analytic inductive inequalities (cf.[31]) to the multi-type languages L MT ∇ and L MT > reported below. S ∋ A :: = p | ⊤ | ⊥ | ¬ A | A ∧ A | h ν i α | [ ν c ] α S ∋ A :: = p | ⊤ | ⊥ | ¬ A | A ∧ A | α ⊲ A N ∋ α :: = | | ∼ α | α ∩ α | [ ∋ ] A | h = i A N ∋ α :: = | | ∼ α | α ∩ α | [ ∋ ] A | [ = i A . An order-type over n ∈ N is an n -tuple ǫ ∈ { , ∂ } n . If ǫ is an order type, ǫ ∂ is its opposite order type; i.e. ǫ ∂ ( i ) = ff ǫ ( i ) = ∂ for every 1 ≤ i ≤ n . The connectives ofthe language above are grouped together into the families F : = F S ∪ F N ∪ F MT and G : = G S ∪ G N ∪ G MT , defined as follows: on-normal logics: semantic analysis and proof theory 25 F S : = {¬} G S = {¬}F N : = {∼} G N : = {∼}F MT : = {h ν i , h = i} G MT : = { [ ∋ ] , [ ν c ] , ⊲ , [ = i} For any f ∈ F (resp. g ∈ G ), we let n f ∈ N (resp. n g ∈ N ) denote the arity of f (resp. g ),and the order-type ǫ f (resp. ǫ g ) on n f (resp. n g ) indicate whether the i th coordinate of f (resp. g ) is positive ( ǫ f ( i ) = ǫ g ( i ) =
1) or negative ( ǫ f ( i ) = ∂ , ǫ g ( i ) = ∂ ). Definition 19 ( Signed Generation Tree ) . The positive (resp. negative ) generation tree of any L MT -term s is defined by labelling the root node of the generation tree of s withthe sign + (resp. − ), and then propagating the labelling on each remaining node asfollows: For any node labelled with ℓ ∈ F ∪ G of arity n ℓ , and for any ≤ i ≤ n ℓ , assignthe same (resp. the opposite) sign to its ith child node if ǫ ℓ ( i ) = (resp. if ǫ ℓ ( i ) = ∂ ).Nodes in signed generation trees are positive (resp. negative ) if are signed + (resp. − ). For any term s ( p , . . . p n ), any order type ǫ over n , and any 1 ≤ i ≤ n , an ǫ -criticalnode in a signed generation tree of s is a leaf node + p i with ǫ ( i ) = − p i with ǫ ( i ) = ∂ .An ǫ - critical branch in the tree is a branch ending in an ǫ -critical node. For any term s ( p , . . . p n ) and any order type ǫ over n , we say that + s (resp. − s ) agrees with ǫ , andwrite ǫ ( + s ) (resp. ǫ ( − s )), if every leaf in the signed generation tree of + s (resp. − s ) is ǫ -critical. We will also write + s ′ ≺ ∗ s (resp. − s ′ ≺ ∗ s ) to indicate that the subterm s ′ inherits the positive (resp. negative) sign from the signed generation tree ∗ s . Finally, wewill write ǫ ( s ′ ) ≺ ∗ s (resp. ǫ ∂ ( s ′ ) ≺ ∗ s ) to indicate that the signed subtree s ′ , with thesign inherited from ∗ s , agrees with ǫ (resp. with ǫ ∂ ). Definition 20 ( Good branch ) . Nodes in signed generation trees will be called ∆ -adjoints , syntactically left residual (SLR) , syntactically right residual (SRR) , and syn-tactically right adjoint (SRA) , according to the specification given in Table 1. A branchin a signed generation tree ∗ s, with ∗ ∈ { + , −} , is called a good branch if it is the con-catenation of two paths P and P , one of which may possibly be of length , such thatP is a path from the leaf consisting (apart from variable nodes) only of PIA-nodes andP consists (apart from variable nodes) only of Skeleton-nodes. Skeleton PIA ∆ -adjoints SRA + ∨ ∪− ∧ ∩ + ∧ ∩ [ ∋ ] [ ν c ] ⊲ [ = i ¬ ∼− ∨ ∪ h ν i h = i ¬ ∼ SLR SRR + ∧ ∩ h ν i h = i ¬ ∼− ∨ ∪ [ ∋ ] [ ν c ] ⊲ [ = i ¬ ∼ + ∨ ∪− ∧ ∩ Table 1: Skeleton and PIA nodes. + Skeleton + p s PIA ≤ − Skeleton + p s PIA
Definition 21 ( Analytic inductive inequalities ) . For any order type ǫ and any irreflex-ive and transitive relation < Ω on p , . . . p n , the signed generation tree ∗ s ( ∗ ∈ {− , + } ) ofan L MT term s ( p , . . . p n ) is analytic ( Ω, ǫ )-inductive if1. every branch of ∗ s is good (cf. Definition 20);2. for all ≤ i ≤ n, every SRR-node occurring in any ǫ -critical branch with leaf p i isof the form ⊛ ( s , β ) or ⊛ ( β, s ) , where the critical branch goes through β and(a) ǫ ∂ ( s ) ≺ ∗ s (cf. discussion before Definition 20), and(b) p k < Ω p i for every p k occurring in s and for every ≤ k ≤ n.An inequality s ≤ t is analytic ( Ω, ǫ )-inductive if the signed generation trees + s and − t are analytic ( Ω, ǫ ) -inductive. An inequality s ≤ t is analytic inductive if is analytic ( Ω, ǫ ) -inductive for some Ω and ǫ . B Algorithmic proof of Theorem 3
In what follows, we show that the correspondence results collected in Theorem 3 canbe retrieved as instances of a suitable multi-type version of algorithmic correspondencefor normal logics (cf. [10,11]), hinging on the usual order-theoretic properties of the al-gebraic interpretations of the logical connectives, while admitting nominal variables oftwo sorts. For the sake of enabling a swift translation into the language of m-frames andc-frames, we write nominals directly as singletons, and, abusing notation, we quantifyover the elements defining these singletons. These computations also serve to prove thateach analytic structural rule is sound on the heterogeneous perfect algebras validatingits correspondent axiom. In the computations relative to each analytic axiom, the linemarked with ( ⋆ ) marks the quasi-inequality that interprets the corresponding analyticrule. This computation does not prove the equivalence between the axiom and the rule,since the variables occurring in each starred quasi-inequality are restricted rather thanarbitrary. However, the proof of soundness is completed by observing that all ALBArules in the steps above the marked inequalities are (inverse) Ackermann and adjunctionrules, and hence are sound also when arbitrary variables replace (co-)nominal variables. N. H | = ∇⊤ ⊤ ⊆ [ ν c ] h = i⊤ P. H | = ¬∇⊥ ⊤ ⊆ ¬h ν i [ ∋ ] ⊥⊤ ⊆ [ ν c ] h = i⊤ ⊤ ⊆ ¬h ν i [ ∋ ] ⊥ i ff ∀ X ∀ w [ h = i⊤ ⊆ { X } c ⇒ { w } ⊆ [ ν c ] { X } c ] i ff ∀ X [ X ⊆ [ ∋ ] ⊥ ⇒ T ⊆ ¬h ν i X ]( ⋆ ) first. app. ( ⋆ ) first. app.i ff ∀ X ∀ w [ X = W ⇒ { w } ⊆ [ ν c ] { X } c ) i ff W ⊆ ¬h ν i [ ∋ ] ∅ ( h∋i⊤ = { W } c )i ff ∀ w [ { w } ⊆ [ ν c ] { W } c ] i ff W ⊆ ¬h ν i{∅} [ ∋ ] ∅ = { Z ⊆ W | Z ⊆ ∅} i ff ∀ w [ { w } ⊆ ( R − ν c [ W ]) c ] i ff W ⊆ { w ∈ W | wR ν ∅} c i ff ∀ w [ { w } ⊆ R − ν [ W ]] i ff ∀ w [ ∅ < ν ( w )].i ff ∀ w [ W ∈ ν ( w )]on-normal logics: semantic analysis and proof theory 27C. H | = ∇ p ∧ ∇ q → ∇ ( p ∧ q ) h ν i [ ∋ ] p ∧ h ν i [ ∋ ] q ⊆ [ ν c ] h = i ( p ∧ q ) h ν i [ ∋ ] p ∧ h ν i [ ∋ ] q ⊆ [ ν c ] h = i ( p ∧ q )i ff ∀ Z Z Z ∀ pq [ { Z } ⊆ [ ∋ ] p & { Z } ⊆ [ ∋ ] q & h = i ( p ∧ q ) ⊆ { Z } c ⇒ h ν i{ Z } ∧ h ν i{ Z } ⊆ [ ν c ] { Z } c ]first approx.i ff ∀ Z Z Z ∀ pq [ h∈i{ Z } ⊆ p & h∈i{ Z } ⊆ q & h = i ( p ∧ q ) ⊆ { Z } c ⇒ h ν i{ Z } ∧ h ν i{ Z } ⊆ [ ν c ] { Z } c ]Residuationi ff ∀ Z ∀ Z ∀ Z [ h = i ( h∈i{ Z } ∧ h∈i{ Z } ) ⊆ { Z } c ⇒ h ν i{ Z } ∧ h ν i{ Z } ⊆ [ ν c ] { Z } c ] ( ⋆ ) Ackermanni ff ∀ Z ∀ Z ∀ Z [( h∈i{ Z } ∧ h∈i{ Z } ) ⊆ [ < ] { Z } c ⇒ h ν i{ Z } ∧ h ν i{ Z } ⊆ [ ν c ] { Z } c ] Residuationi ff ∀ Z ∀ Z ∀ Z [ ∀ x ( xR ∈ Z & xR ∈ Z ⇒ ¬ xR < Z ) ⇒ ∀ x ( xR ν Z & xR ν Z ⇒ ¬ xR ν c Z )]Standard translationi ff ∀ Z ∀ Z ∀ Z [ ∀ x ( x ∈ Z & x ∈ Z ⇒ x ∈ Z ) ⇒ ∀ x ( Z ∈ ν ( x ) & Z ∈ ν ( x ) ⇒ Z ∈ ν ( x ))]Relations interpretationi ff ∀ Z ∀ Z ∀ Z [ Z ∩ Z ⊆ Z ⇒ ∀ x ( Z ∈ ν ( x ) & Z ∈ ν ( x ) ⇒ Z ∈ ν ( x ))]i ff ∀ Z ∀ Z ∀ x [ Z ∈ ν ( x ) & Z ∈ ν ( x ) ⇒ Z ∩ Z ∈ ν ( x ))]. Monotonicity4’. H | = ∇ p → ∇∇ p h ν i [ ∋ ] p ⊆ [ ν c ] h = i [ ν c ] h = i p h ν i [ ∋ ] p ⊆ [ ν c ] h = i [ ν c ] h = i p i ff ∀ Z ∀ x ′ ∀ p [ { Z } ⊆ [ ∋ ] p & [ ν c ] h = i [ ν c ] h = i p ⊆ { x ′ } c ) ⇒ h ν i{ Z } ⊆ { x ′ } c ] first approx.i ff ∀ Z ∀ x ′ ∀ p [ h∈i{ Z } ⊆ p & [ ν c ] h = i [ ν c ] h = i p ⊆ { x ′ } c ) ⇒ h ν i{ Z } ⊆ { x ′ } c ] Residuationi ff ∀ Z ∀ x ′ [[ ν c ] h = i [ ν c ] h = ih∈i{ Z } ⊆ { x ′ } c ⇒ h ν i{ Z } ⊆ { x ′ } c ] Ackermanni ff ∀ Z [ h ν i{ Z } ⊆ [ ν c ] h = i [ ν c ] h = ih∈i{ Z } ]i ff ∀ Z ∀ x [ xR ν Z ⇒ ∀ Z ( xR ν c Z ⇒ ∃ y ( Z R = y & ∀ Z ( yR ν c Z ⇒ ∃ w ( Z R = w & wR ∈ Z ))))]Standard translationi ff ∀ Z ∀ x [ x ∈ ν ( Z ) ⇒ ∀ Z ( Z < ν ( x ) ⇒ ∃ y ( y < Z & ∀ Z ( Z < ν ( y ) ⇒ ∃ w ( w < Z & w ∈ Z ))))]Relations translationi ff ∀ Z ∀ x [ x ∈ ν ( Z ) ⇒ ∀ Z ( Z < ν ( x ) ⇒ ∃ y ( y < Z & ∀ Z ( Z < ν ( y ) ⇒ Z * Z )))]Relations translationi ff ∀ Z ∀ x [ x ∈ ν ( Z ) ⇒ ( ∀ Z ( ∀ y ( ∀ Z ( Z ⊆ Z ⇒ Z ∈ ν ( y )) ⇒ y ∈ Z ) ⇒ Z ∈ ν ( x )))]Contrapositioni ff ∀ Z ∀ x [ x ∈ ν ( Z ) ⇒ ( ∀ Z ( ∀ y ( Z ∈ ν ( y )) ⇒ y ∈ Z ) ⇒ Z ∈ ν ( x )))] Monotonicityi ff ∀ Z ∀ x [ x ∈ ν ( Z ) ⇒ { y | Z ∈ ν ( y ) } ∈ ν ( x )]. Monotonicity4. H | = ∇∇ p → ∇ p h ν i [ ∋ ] h ν i [ ∋ ] p ⊆ [ ν c ] h = i p h ν i [ ∋ ] h ν i [ ∋ ] p ⊆ [ ν c ] h = i p i ff ∀ x ∀ Z ∀ p [ { x } ⊆ h ν i [ ∋ ] h ν i [ ∋ ] p & h = i p ⊆ { Z } c ⇒ { x } ⊆ [ ν c ] { Z } c ] first approx.i ff ∀ x ∀ Z ∀ p [ { x } ⊆ h ν i [ ∋ ] h ν i [ ∋ ] p & p ⊆ [ < ] { Z } c ⇒ { x } ⊆ [ ν c ] { Z } c ] Adjunctioni ff ∀ x ∀ Z [ { x } ⊆ h ν i [ ∋ ] h ν i [ ∋ ][ < ] { Z } c ⇒ { x } ⊆ [ ν c ] { Z } c ] Ackermanni ff ∀ x ∀ Z [( ∃ Z ( xR ν Z & ∀ y ( Z R ∋ y ⇒ ∃ Z ( yR ν Z & ∀ w ( Z R ∋ w ⇒ ¬ wR < Z ))))) ⇒ ¬ xR ν c Z ]Standard translationi ff ∀ x ∀ Z [(( ∃ Z ∈ ν ( x ))( ∀ y ∈ Z )( ∃ Z ∈ ν ( y ))( ∀ w ∈ Z )( w ∈ Z )) ⇒ Z ∈ ν ( x )]Relation translationi ff ∀ x ∀ Z [(( ∃ Z ∈ ν ( x ))( ∀ y ∈ Z )( ∃ Z ∈ ν ( y ))( Z ⊆ Z )) ⇒ Z ∈ ν ( x )]i ff ∀ x ∀ Z ∀ Z [( Z ∈ ν ( x ) & ( ∀ y ∈ Z )( ∃ Z ∈ ν ( y ))( Z ⊆ Z )) ⇒ Z ∈ ν ( x )]i ff ∀ x ∀ Z ∀ Z [( Z ∈ ν ( x ) & ( ∀ y ∈ Z )( Z ∈ ν ( y ))) ⇒ Z ∈ ν ( x )] Monotonicity8 Chen, Greco, Palmigiano, Tzimoulis5. H | = ¬∇¬ p → ∇¬∇¬ p ¬ [ ν c ] h = i¬ p ⊆ [ ν c ] h = i¬h ν i [ ∋ ] ¬ p ¬ [ ν c ] h = i¬ p ⊆ [ ν c ] h = i¬h ν i [ ∋ ] ¬ p i ff ∀ x ∀ Z [[ ν c ] h = i¬h ν i [ ∋ ] ¬ p ⊆ { x } c & h = i¬ p ⊆ { Z } c ⇒ ¬ [ ν c ] { Z } c ⊆ { x } c ] first approx.i ff ∀ x ∀ Z [[ ν c ] h = i¬h ν i [ ∋ ] ¬ p ⊆ { x } c & ¬ [ < ] { Z } c ⊆ p ⇒ ¬ [ ν c ] { Z } c ⊆ { x } c ] Residuationi ff ∀ x ∀ Z [[ ν c ] h = i¬h ν i [ ∋ ] ¬¬ [ < ] { Z } c ⊆ { x } c ⇒ ¬ [ ν c ] { Z } c ⊆ { x } c ] Ackermanni ff ∀ Z [ ¬ [ ν c ] { Z } c ⊆ [ ν c ] h = i¬h ν i [ ∋ ] ¬¬ [ < ] { Z } c ]i ff ∀ Z ∀ x [ xR ν c Z ⇒ ∀ Z ( xR ν c Z ⇒ ∃ y ( Z R = y & ∀ Z ( yR ν Z ⇒ ∃ w ( Z R ∋ w & wR < Z ))))]Standard translationi ff ∀ Z ∀ x [ Z < ν ( x ) ⇒ ( ∀ Z < ν ( x ))( ∃ y < Z )( ∀ Z ∈ ν ( y ))( ∃ w ∈ Z )( w < Z )]Relation translationi ff ∀ Z ∀ x [ Z < ν ( x ) ⇒ ( ∀ Z < ν ( x ))( ∃ y < Z )( ∀ Z ∈ ν ( y ))( Z * Z )]i ff ∀ Z ∀ x [ Z < ν ( x ) ⇒ ∀ Z ((( ∀ y < Z )( ∃ Z ∈ ν ( y ))( Z ⊆ Z )) ⇒ Z ∈ ν ( x ))] Contrapositioni ff ∀ Z ∀ x [ Z < ν ( x ) ⇒ ∀ Z (( ∀ y < Z )( Z ∈ ν ( y )) ⇒ Z ∈ ν ( x ))] Monotonicityi ff ∀ Z ∀ x [ Z < ν ( x ) ⇒ { y | Z ∈ ν ( y ) } c ∈ ν ( x ))] MonotonicityB. H | = p → ∇¬∇¬ p p ⊆ [ ν c ] h = i¬h ν i [ ∋ ] ¬ pp ⊆ [ ν c ] h = i¬h ν i [ ∋ ] ¬ p i ff ∀ x ∀ p [ { x } ⊆ p ⇒ { x } ⊆ [ ν c ] h = i¬h ν i [ ∋ ] ¬ p ] first approx.i ff ∀ x [ { x } ⊆ [ ν c ] h = i¬h ν i [ ∋ ] ¬{ x } ] Ackermanni ff ∀ x [ { x } ⊆ [ ν c ] h = i [ ν ] h∋i{ x } ]i ff ∀ x [ ∀ Z ( xR ν c Y ⇒ ∃ y ( YR = x & ∀ Z ( yR ν Z ⇒ Z R ∋ x )))] Standard translationi ff ∀ x [ ∀ Z ( Z < ν ( x ) ⇒ ∃ y ( x < Z & ∀ Z ( Z ∈ ν ( y ) ⇒ x ∈ Z )))] Relations translationi ff ∀ x [ ∀ Z ( ∀ y ( ∀ Z ( x < Z ⇒ Z < ν ( y )) ⇒ y ∈ Z ) ⇒ Z ∈ ν ( x ))] Contrapositivei ff ∀ x [ ∀ Z ( ∀ y ( { x } c < ν ( y )) ⇒ y ∈ Z ) ⇒ Z ∈ ν ( x ))] Monotonicityi ff ∀ x [ { y | { x } c < ν ( y ) } ∈ ν ( x ))] Monotonicityi ff ∀ x ∀ X [ x ∈ X ⇒ { y | X c < ν ( y ) } ∈ ν ( x )] MonotonicityD. H | = ∇ p → ¬∇¬ p h ν i [ ∋ ] p ⊆ ¬h ν i [ ∋ ] ¬ p h ν i [ ∋ ] p ⊆ ¬h ν i [ ∋ ] ¬ p i ff ∀ Z ∀ Z ′ [ { Z } ⊆ [ ∋ ] p & Z ′ ⊆ [ ∋ ] ¬ p ⇒ h ν i{ Z } ⊆ ¬h ν i Z ′ ] first approx.i ff ∀ Z ∀ Z ′ [ h∈i{ Z } ⊆ p & { Z ′ } ⊆ [ ∋ ] ¬ p ⇒ h ν i{ Z } ⊆ ¬h ν i{ Z ′ } ] Residuationi ff ∀ Z ∀ Z ′ [ { Z ′ } ⊆ [ ∋ ] ¬h∈i{ Z } ⇒ h ν i{ Z } ⊆ ¬h ν i{ Z ′ } ] ( ⋆ ) Ackermanni ff ∀ Z [ h ν i{ Z } ⊆ ¬h ν i [ ∋ ] ¬h∈i{ Z } ]i ff ∀ Z [ h ν i{ Z } ⊆ [ ν ] h∋ih∈i{ Z } ]i ff ∀ Z ∀ x [ xR ν Z ⇒ ∀ Y ( xR ν Y ⇒ ∃ w ( YR ∋ w & wR ∈ Z ))] Standard Translationi ff ∀ Z ∀ x [ Z ∈ ν ( x ) ⇒ ∀ Y ( Y ∈ ν ( x ) ⇒ ∃ w ( w ∈ Y & w ∈ Z ))] Relation translationi ff ∀ Z ∀ x [ Z ∈ ν ( x ) ⇒ ∀ Y ( Y ∈ ν ( x ) ⇒ Y * Z c )]i ff ∀ Z ∀ x [ Z ∈ ν ( x ) ⇒ ∀ Y ( Y ⊆ Z c ⇒ Y < ν ( x ))] Contrapositivei ff ∀ Z ∀ x ∀ Y [ Z ∈ ν ( x ) ⇒ Z c < ν ( x )] Monotonicityon-normal logics: semantic analysis and proof theory 29CS. H | = ( p ∧ q ) → ( p ≻ q ) ( p ∧ q ) ⊆ ([ ∋ ] p ∩ [ = i p ) ⊲ q ( p ∧ q ) ⊆ ([ ∋ ] p ∩ [ = i p ) ⊲ q i ff ∀ x ∀ Z ∀ x ′ ∀ pq [ { x } ⊆ p ∧ q & { Z } ⊆ [ ∋ ] p ∩ [ = i p & q ⊆ { x ′ } c ⇒ { x } ⊆ { Z } ⊲ { x ′ } c ] first. approx.i ff ∀ x ∀ Z ∀ x ∀ p ∀ q [ { x } ⊆ p & { x } ⊆ q & { Z } ⊆ [ ∋ ] p & { Z } ⊆ [ = i p & q ⊆ { x ′ } c ⇒ { x } ⊆ { Z } ⊲ { x ′ } c ]Splitting rulei ff ∀ x ∀ Z ∀ x ′ ∀ p ∀ q [ { x } ⊆ p & { x } ⊆ q & { Z } ⊆ [ ∋ ] p & p ⊆ [ < i{ Z } & q ⊆ { x ′ } c ⇒ { x } ⊆ { Z } ⊲ { x ′ } c ]Residuationi ff ∀ x ∀ Z ∀ x ′ ∀ q [ { x } ⊆ [ < i{ Z } & { x } ⊆ q & { Z } ⊆ [ ∋ ][ < i{ Z } & q ⊆ { x ′ } c ⇒ { x } ⊆ { Z } ⊲ { x ′ } c ]Ackermanni ff ∀ x ∀ Z ∀ x ′ [ { x } ⊆ [ < i{ Z } & { Z } ⊆ [ ∋ ][ < i{ Z } & { x } ⊆ { x ′ } c ⇒ { x } ⊆ { Z } ⊲ { x ′ } c ] ( ⋆ ) Ackermanni ff ∀ x ∀ Z [ { x } ⊆ [ < i{ Z } & { Z } ⊆ [ ∋ ][ < i{ Z } ⇒ { x } ⊆ { Z } ⊲ { x } ]i ff ∀ x ∀ Z [ ¬ xR < Z & ∀ y ( ZR ∋ y ⇒ ¬ yR < Z ) ⇒ ∀ y ( T f ( x , Z , y ) ⇒ y = x )] Standard translationi ff ∀ x ∀ Z [ x ∈ Z & ∀ y ( y ∈ Z ⇒ Z ∈ y ) ⇒ ∀ y ( y ∈ f ( x , Z ) ⇒ y = x )] Relation interpretationi ff ∀ x ∀ Z [ x ∈ Z ⇒ ∀ y ( y ∈ f ( x , Z ) ⇒ y = x )]i ff ∀ x ∀ Z [ x ∈ Z ⇒ f ( x , Z ) ⊆ { x } ]ID. H | = p ≻ p ([ ∋ ] p ∩ [ = i p ) ⊲ p ⊤ ⊆ ([ ∋ ] p ∩ [ = i p ) ⊲ p i ff ∀ ZZ ′ ∀ x ′ p [( { Z } ⊆ [ ∋ ] p & { Z ′ } ⊆ [ = i p & p ⊆ { x ′ } c ) ⇒ ⊤ ⊆ ( { Z } ∩ { Z ′ } ) ⊲ { x ′ } c ] first approx.i ff ∀ ZZ ′ ∀ x ′ p [( h∈i{ Z } ⊆ p & { Z ′ } ⊆ [ = i p & p ⊆ { x ′ } c ) ⇒ ⊤ ⊆ ( { Z } ∩ { Z ′ } ) ⊲ { x ′ } c ] Adjunctioni ff ∀ Z ∀ Z ′ ∀ x ′ [( { Z ′ } ⊆ [ = ih∈i{ Z } & h∈i{ Z } ⊆ { x ′ } c ) ⇒ ⊤ ⊆ ( { Z } ∩ { Z ′ } ) ⊲ { x ′ } c Ackermanni ff ∀ Z ∀ Z ′ [ { Z ′ } ⊆ [ = ih∈i{ Z } ⇒ ∀ x ′ [ h∈i{ Z } ⊆ { x ′ } c ⇒ ⊤ ⊆ ( { Z } ∩ { Z ′ } ) ⊲ { x ′ } c ]] Curryingi ff ∀ Z ∀ Z ′ [ { Z ′ } ⊆ [ = ih∈i{ Z } ⇒ ⊤ ⊆ ( { Z } ∩ { Z ′ } ) ⊲ h∈i{ Z } ] ( ⋆ ) Ackermanni ff ∀ x ∀ Z ∀ Z ′ [ ∀ w ( Z ′ R = w ⇒ ¬ wR ∈ Z ) ⇒ ∀ y ( T f ( x , Z , y ) & Z = Z ′ ⇒ y ∈ Z )] Standard Translationi ff ∀ x ∀ Z ∀ Z ′ ∀ y [ ∀ w ( Z ′ R = w ⇒ ¬ wR ∈ Z ) & ( T f ( x , Z , y ) & Z = Z ′ ⇒ y ∈ Z )]i ff ∀ x ∀ Z ∀ Z ′ ∀ y [ ∀ w ( w < Z ′ ⇒ w < Z ) & ( y ∈ f ( x , Z ) & Z = Z ′ ⇒ y ∈ Z )] Relation interpretationi ff ∀ x ∀ Z ∀ Z ′ ∀ y [ Z ⊆ Z ′ & ( y ∈ f ( x , Z ) & Z = Z ′ ⇒ y ∈ Z )]i ff ∀ x ∀ Z ∀ y [( y ∈ f ( x , Z ) ⇒ y ∈ Z )]i ff ∀ x ∀ Z [ f ( x , Z ) ⊆ Z ]T. H | = ∇ p → p h ν i [ ∋ ] p ⊆ p h ν i [ ∋ ] p ⊆ p i ff ∀ x ∀ Z ∀ p [ p ⊆ { x } c & { Z } ⊆ [ ∋ ] p ⇒ h ν i{ Z } ⊆ { x } c ] first approx.i ff ∀ x ∀ Z ∀ p [ p ⊆ { x } c & h∈i{ Z } ⊆ p ⇒ h ν i{ Z } ⊆ { x } c ] Adjunctioni ff ∀ x ∀ Z [ h∈i{ Z } ⊆ { x } c ⇒ h ν i{ Z } ⊆ { x } c ] ( ⋆ ) Ackermanni ff ∀ Z [ h ν i{ Z } ⊆ h∋i{ Z } ] inverse approx.i ff ∀ x ∀ Z [ xR ν Z ⇒ xR ∋ Z ] Standard translationi ff ∀ x ∀ Z [ Z ∈ ν ( x ) ⇒ x ∈ Z ]. Relation translation0 Chen, Greco, Palmigiano, TzimoulisCEM. H | = ( p ≻ q ) ∨ ( p ≻ ¬ q ) (([ ∋ ] p ∩ [ = i p ) ⊲ q ) ∨ (([ ∋ ] p ∩ [ = i p ) ⊲ ¬ q ) ⊤ ⊆ (([ ∋ ] p ∩ [ = i p ) ⊲ q ) ∨ (([ ∋ ] p ∩ [ = i p ) ⊲ ¬ q )i ff ∀ p ∀ q ∀ X ∀ Y ∀ x ∀ y ( { X } ⊆ [ ∋ ] p ∩ [ = i p & { Y } ⊆ [ ∋ ] p ∩ [ = i p & q ⊆ { x } c & { y } ⊆ q ⇒ ⊤ ⊆ ( { X } ⊲ { x } c ) ∨ ( { Y } ⊲ ¬{ y } ) first approx.i ff ∀ p ∀ q ∀ X ∀ Y ∀ x ∀ y ( { X } ⊆ [ ∋ ] p & { X } ⊆ [ = i p & { Y } ⊆ [ ∋ ] p & { Y } ⊆ [ = i p & q ⊆ { x } c & { y } ⊆ q ⇒ ⊤ ⊆ ( { X } ⊲ { x } c ) ∨ ( { Y } ⊲ ¬{ y } ) ( ⋆ ) Splittingi ff ∀ p ∀ q ∀ X ∀ Y ∀ x ∀ y ( { X } ⊆ [ ∋ ] p & p ⊆ [ < i{ X } & { Y } ⊆ [ ∋ ] p & p ⊆ [ < i{ Y } & q ⊆ { x } c & { y } ⊆ q ⇒ ⊤ ⊆ ( { X } ⊲ { x } c ) ∨ ( { Y } ⊲ ¬{ y } ) Residuationi ff ∀ X ∀ Y ∀ x ∀ y ( { X } ∨ { Y } ⊆ [ ∋ ]([ < i{ X } ∧ [ < i{ Y } ) & { y } ⊆ { x } c ⇒ ⊤ ⊆ ( { X } ⊲ { x } c ) ∨ ( { Y } ⊲ ¬{ y } ) Ackermanni ff ∀ X ∀ Y ∀ x ( { X } ∨ { Y } ⊆ [ ∋ ]([ < i{ X } ∧ [ < i{ Y } ) ⇒ ∀ y ( { y } ⊆ { x } c ⇒ ⊤ ⊆ ( { X } ⊲ { x } c ) ∨ ( { Y } ⊲ ¬{ y } ))Curryingi ff ∀ X ∀ Y ∀ x ( { X } ∨ { Y } ⊆ [ ∋ ]([ < i{ X } ∧ [ < i{ Y } ) ⇒ ⊤ ⊆ ( { X } ⊲ { x } c ) ∨ ( { Y } ⊲ ¬{ x } c ))i ff ∀ X ∀ Y ∀ x [( ∀ y ( XR ∋ y or YR ∋ y ) ⇒ ¬ yR < X & ¬ yR < Y ) ⇒ ∀ y ( ¬ T f ( y , X , x ) or ( ∀ z ( T f ( y , Y , z ) ⇒ z = x )))] Standard translationi ff ∀ X ∀ Y ∀ x [( ∀ y ( y ∈ X or y ∈ Y ) ⇒ y ∈ X & y ∈ Y ) ⇒ ∀ y ( x < f ( y , X ) or ( ∀ z ( z ∈ f ( y , Y ) ⇒ z = x )))] Relation interpretationi ff ∀ X ∀ Y ∀ x [( X ∪ Y ⊆ X ∩ Y ) ⇒ ∀ y ( x < f ( y , X ) or ( ∀ z ( z ∈ f ( y , Y ) ⇒ z = x )))]i ff ∀ X ∀ Y ∀ x [ X = Y ⇒ ∀ y ( x < f ( y , X ) or ( ∀ z ( z ∈ f ( y , Y ) ⇒ z = x )))]i ff ∀ X ∀ x ∀ y [( x < f ( y , X ) or ( ∀ z ( z ∈ f ( y , X ) ⇒ z = x )))]i ff ∀ X ∀ x ∀ y [( x ∈ f ( y , X ) ⇒ f ( y , X ) = { x } )]i ff ∀ X ∀ y [ | f ( y , X ) | ≤ H | = ( p > q ) ∨ ( q > p ) ([ ∋ ] p ∧ [ = i p ) ⊲ q ) ∨ (([ ∋ ] q ∧ [ = i q ) ⊲ p ⊤ ⊆ (([ ∋ ] p ∧ [ = i p ) ⊲ q ) ∨ (([ ∋ ] q ∧ [ = i q ) ⊲ p )i ff ∀ p ∀ q ∀ x ∀ y ∀ X ∀ Y ( { X } ⊆ [ ∋ ] p ∩ [ = i p & { Y } ⊆ [ ∋ ] q ∩ [ = i q & q ⊆ { x } c & p ⊆ { y } c ⇒ ⊤ ⊆ ( { X } ⊲ { x } c ) ∨ ( { Y } ⊲ { y } c ) first approx.i ff ∀ p ∀ q ∀ x ∀ y ∀ X ∀ Y ( { X } ⊆ [ ∋ ] p & { X } ⊆ [ = i p & { Y } ⊆ [ ∋ ] q & { Y } ⊆ [ = i q & q ⊆ { x } c & p ⊆ { y } c ⇒ ⊤ ⊆ ( { X } ⊲ { x } c ) ∨ ( { Y } ⊲ { y } c ) Splittingi ff ∀ p ∀ q ∀ x ∀ y ∀ X ∀ Y ( { X } ⊆ [ ∋ ] p & p ⊆ [ < i{ X } & { Y } ⊆ [ ∋ ] q & q ⊆ [ < i{ Y } & q ⊆ { x } c & p ⊆ { y } c ⇒ ⊤ ⊆ ( { X } ⊲ { x } c ) ∨ ( { Y } ⊲ { y } c ) Residuationi ff ∀ x ∀ y ∀ X ∀ Y ( { X } ⊆ [ ∋ ]([ < i{ X } ∩ { y } c )& { Y } ⊆ [ ∋ ]([ < i{ Y } ∩ { x } c ) & ⇒ ⊤ ⊆ ( { X } ⊲ { x } c ) ∨ ( { Y } ⊲ { y } c ) Ackermanni ff ∀ x ∀ y ∀ X ∀ Y ( { X } ⊆ ( R − ∋ [ R = [ { X } ] ∩ { y } c ]) c & { Y } ⊆ ( R − ∋ [ R = [ { Y } ] ∩ { x } c ]) c & ⇒ ⊤ ⊆ (( T (0) f [ { X } , { x } ]) c ) ∨ (( T (0) f [ { Y } , { y } ]) c ) Standard translationi ff ∀ x ∀ y ∀ X ∀ Y ( { X } ⊆ ( R − ∋ [ X c ∩ { y } c ]) c & { Y } ⊆ ( R − ∋ [ Y c ∩ { x } c ]) c & ⇒ ⊤ ⊆ (( T (0) f [ { X } , { x } ]) c ) ∨ (( T (0) f [ { Y } , { y } ]) c )i ff ∀ x ∀ y ∀ X ∀ Y ( X ⊆ X ∪ { y } & Y ⊆ Y ∪ { x } ⇒ ⊤ ⊆ { z | x ∈ f ( z , X ) } c ∪ { z | y ∈ f ( z , Y ) } c )i ff ∀ z ∀ x ∀ y ∀ X ∀ Y [ x < f ( z , X ) or y < f ( z , Y )]T. H | = ( ⊥ > ¬ p ) → p (([ ∋ ] ⊥ ∧ [ = i⊥ ) ⊲ ¬ p ) ⊆ p (([ ∋ ] ⊥ ∧ [ = i⊥ ) ⊲ ¬ p ) ⊆ p i ff (([ ∋ ] ⊥ ∧ [ = i⊥ ) ⊲ ¬⊥ ) ⊆ ⊥ Variable eliminationi ff ( {∅} ) ⊲ ¬⊥ ) ⊆ ⊥ i ff { z | ∀ X ∀ x ( T f ( z , X , x ) → X ∈ {∅} & x ∈ W ) } ⊆ ⊥ i ff ∀ z ∃ x [ x ∈ f ( z , ∅ )]on-normal logics: semantic analysis and proof theory 31 References
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