Relational semantics for flat Heyting-Lewis Logic

Relational semantics for flat He yting-Le wis Logic Jim de Groot † & T adeusz Litak ‡ † Univ ersity of Bern Bern, Switzerland jim.degroot@unibe.ch ‡ Univ ersity of Naples Federico II, Naples, Italy tadeusz.litak@fau.de tadeusz.litak@unina.it Abstract W e introduce relational semantics for “flat Heyting-Lewis logic” HLC ♭ . This logic arises as the extension of intuitionistic logic with a Lewis-style strict implication modality that, contrary to its “sharp” counterpart HLC ♯ , does not turn meets into joins in its first argument. W e pro ve completeness and the finite model property for HLC ♭ and for sev eral extensions with additional axioms. 1 Introduction Recent years hav e seen a revi v al of the interest in intuitionistic modal logics [ 2 , 9 , 25 , 29 , 31 , 36 ], including extensions with a Lewisian strict implication J [ 26 , 27 ] and various types of conditional implications [ 8 , 42 ]. Recall that in the intuitionistic setting, J is not definable in terms of unary □ . Instead, it can be viewed as sitting between □ ( p → q ) and □ p → □ q . Indeed, the basic “flat” system 1 HLC ♭ prov es [ 26 ]: ( bl ) □ ( p → q ) → ( p J q ) ( lb ) ( p J q ) → ( □ p → □ q ) The original motiv ation of the Utrecht school to study such a connective came from research on schematic logics of theories ov er intuitionistic arithmetic HA , more specifically from the study of Σ 0 1 -pr eservativity [ 18 , 19 , 38 , 39 , 41 ], which over P A can be seen as the contraposed variant of both Π 0 1 -conservatively and arithmetic interpr etability [ 1 , 3 , 20 , 35 , 40 ]. Subsequently , many other application and interpretations were put forward, see e.g. [ 10 , 26 ]. In particular, in the presence of an additional axiom str (see Section 5 ) the resulting calculus turns out to be the Curry-Howard counterpart (i.e. the inhabitation logic) of (Hask ell) arr ows in functional programming [ 15 , 16 ]. Some what underdeveloped philosophical applications include a generalization of intuitionistic epistemic logic IEL [ 2 ] to intuitionistic logic of entailments IELE [ 10 , Section 2.4] or a fine-grained analysis of the collapse of Lewis’ original 1918 system of strict implication caused by in voluti ve negation. 2 The flat calculus HLC ♭ arises from extending intuitionistic logic with a binary operator J that is normal in it second argument, transitiv e, and satisfies implication necessitation , i.e. deri v ability of ϕ → ψ implies deriv ability of ϕ J ψ . From this, we can obtain the sharp calculus HLC ♯ by adding the axiom: ( di ) ( p J r ) ∧ ( q J r ) → (( p ∨ q ) J r ) This sharp version of the logic can con veniently be interpreted in Kripke-style relational semantics. This per- spectiv e has resulted in numerous correspondence, completeness and finite model property results for this “sharp” semantics [ 17 – 19 , 26 , 47 ], with a recent work showing ho w to use the natural G ¨ odel-McKinsey-T arski translation to transfer metatheory of bimodal classical logics [ 10 ]. In the flat setting, so far one has had to turn to algebraic semantics or a suitable adaptation of Chellas-W eiss semantics for ICK [ 7 , 12 , 42 ], Routley-Meyer semantics for substructural logics [ 5 , 30 , 32 – 34 ], or (generalised) V eltman semantics [ 21 – 23 , 37 ], because a simple Kripke-style semantics for HLC ♭ appeared elusiv e. 1 Naming underwent se veral evolutions. Early references in the Utrecht school [ 17 – 19 , 47 ] denoted the base “flat” system as iP − and the base sharp system as iP , Litak and V isser [ 26 ] replaced iP with iA , and in a subsequent paper the same authors [ 27 ] finally settled for the present notation. 2 It is worth noting here that in later years, having become aware of nascent study of non-classical calculi, Lewis not only followed closely the development of early multi-valued logics, but also on at least one occasion spoke favourably of Brouwer’ s rejection of excluded middle. More information and detailed discussion can be found in Litak and V isser [ 26 ]. 1 In this paper we fill this gap by providing a Kripk ean interpretation for HLC ♭ . This semantics is inspired by recent work on semantics of CK [ 9 ], and crucially relies on using a preor der ⪯ instead of a partial order to interpret the intuitionistic implication. Since the semantic clause for J directly enforces upward persistence (Definition 3.1 ), the most general version of the new semantics (Definition 3.1 ) does not impose any interaction conditions between R and ⪯ . Howe ver , similarly to the case of intuitionistic □ and unlike the sharp interpretation, our language is obli vious to closing R under post-composing with ⪯ (Proposition 3.10 ), and the resulting upwar d-flat frames (Section 3.2 ) prov e con venient for computing correspondents and obtaining completeness results. Using a canonical model construction we prove completeness and the finite model property for HLC ♭ and sev eral of its extensions (Sections 4 and 5.1 ). Guided by the canonical model construction for CK , we use segments rather than prime theories to ha ve a more fine-grained handle on the modal accessibility relation. Still mirroring CK , we sometimes need to restrict our choice of segments, for example when proving completeness for natural variants of K4 and S4 in our setting (Section 5.2 ). When ⪯ is collapsed to equality , turning our frames into standard Kripke frames, our frames turn lb into bi- implication, rather than bl . This does not mean that our semantics trivialises classically: in the preorder setting, validating e xcluded middle simply requires ⪯ to be symmetric, and such a classical variant of our semantics does not collapse J (Example 3.3 ). This creates the opportunity to use our semantics for completeness results for subsystems of standard interpretability logics such as ILM and ILP . In the CK setting, the segment approach can be used to obtain duality results [ 11 ]. While our paper does discuss duality in depth, we include comments for an interested reader such as Remarks 3.8 and 4.10 illustrating difficulties with more standard approaches. Ho we ver , we discuss a promising application in Section 6 in the context of syntactically moti vated notion of extension stability . W e note the relationship of this notion to what one might call open subfr ame construction, and use our semantics to show that HLC ♯ is not e xtension stable, unlike the flat base calculus. 2 Intuitionistic strict implication, sharply and flatly This section pro vides preliminaries and recapitulates kno wn material. Section 2.1 presents the base flat system HLC ♭ . Section 2.2 discusses the sharp variant HLC ♯ together with its known Kripke semantics. Section 2.3 reca- pitulates the algebraic semantics of both systems. Throughout the paper , we denote by L J the language generated by the grammar ϕ :: = p | ⊤ | ⊥ | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ → ϕ | ϕ J ϕ , where p ranges o ver some arbitrary b ut fixed set Prop of proposition letters. W e abbreviate □ ϕ : = ⊤ J ϕ . 2.1 Syntax and axioms of the base flat system A consecution is an expressions of the form Γ ⇒ ϕ , where Γ ∪ { ϕ } ⊆ L J . 2.1 Definition. Let Ax ♭ be an axiomatisation of intuitionistic logic together with the axioms ( k a ) (( p J q ) ∧ ( p J r )) → ( p J ( q ∧ r )) ( tr ) (( p J q ) ∧ ( q J r )) → ( p J r ) If Ax ⊆ L J , the we denote by I ( Ax ) the collection of substitution instances of formulas in Ax, and define the axiomatic system HLC ♭ ⊕ Ax by: ( Ax ) ϕ ∈ I ( Ax ♭ ) ∪ I ( Ax ) Γ ⇒ ϕ ( El ) ϕ ∈ Γ Γ ⇒ ϕ ( MP ) Γ ⇒ ϕ Γ ⇒ ϕ → ψ Γ ⇒ ψ ( N a ) / 0 ⇒ ϕ → ψ Γ ⇒ ϕ J ψ W e say that Γ ⇒ ϕ is pr ovable in HLC ♭ ⊕ Ax, and write Γ ⊢ Ax ϕ , if there exists a tree of consecutions built using the rules above with Γ ⇒ ϕ as root and adequate applications of rules ( El ) and ( Ax ) as leaves. If Ax = / 0 then we abbreviate ⊢ Ax to ⊢ , and if Ax = { ϕ 1 , . . . , ϕ n } then we write HLC ♭ ⊕ ϕ 1 ⊕ · · · ⊕ ϕ n for HLC ♭ ⊕ Ax. Finally , we write Γ ⊢ Ax ∆ if there exist ψ 1 , . . . , ψ m ∈ ∆ such that Γ ⊢ Ax ψ 1 ∨ · · · ∨ ψ m . 2.2 Proposition. F or Ax ⊆ L J and uniform substition σ , the following rules ar e admissible in HLC ♭ ⊕ Ax : Γ ⇒ ϕ Γ , Γ ′ ⇒ ϕ { Γ ⇒ δ | δ ∈ ∆ } ∆ ⇒ ϕ Γ ⇒ ϕ Γ ⇒ ϕ Γ σ ⇒ ϕ σ Γ , ϕ ⇒ ψ Γ ⇒ ϕ → ψ Pr oof. By induction on the height of a deriv ation for the premiss(es). 2 The first three rules sho w that HLC ♭ ⊕ Ax is a monotone, compositional and structural relation, respecti vely . Furthermore, since proof trees are finite we have Γ ⊢ Ax ϕ if and only if there is a finite Γ ′ ⊆ Γ such that Γ ′ ⊢ Ax ϕ . Therefore HLC ♭ ⊕ Ax is a finitary logic [ 24 , Definition 1.4.1]. Where ver possible, we blur the distinction between a logic and the set of its theorems (identified with deriv able consecutions with an empty premise). 2.2 The sharpening 2.3 Definition. Let di be the axiom ( p J r ) ∧ ( q J r ) → (( p ∨ q ) J r , and define the sharp He yting-Lewis calculus by HLC ♯ : = HLC ♭ ⊕ di . The sharp systems is known to allow a simple Kripke-style semantics, with soundness, completeness and the finite model property results for many of its e xtensions [ 10 , 17 – 19 , 26 , 47 ]: 2.4 Definition. A sharp fr ame is a tuple F = ( W , ≤ , R ) consisting of a set W , a partial order ≤ on W , and a relation R on W such that w ≤ vRu implies wRu for all w , v , u ∈ W . A sharp model is formed by adding a valuation that interprets proposition letters as upsets. The interpretation of L J -formulas at a world w in a sharp model M is defined recursiv ely by intutionistic clauses for intuitionistic connectiv es and J interpreted as M , w ⊩ s ϕ J ψ iff for all v ( if wRv and M , v ⊩ s ϕ then M , v ⊩ s ψ ) 2.3 Algebraic semantics While Kripke completeness has been so far only av ailable for the sharp calculus and its reasonably well-behav ed extensions, algebra pro vides an obvious route to wards a generic completeness result. 2.5 Definition. A flat Le wisian Heyting Algebr a Expansion , or L - H A E 3 , is a tuple A : = ⟨ A , ∧ , ∨ , J , → , ⊥ , ⊤⟩ such that ⟨ A , ∧ , ∨ , → , ⊤ , ⊥⟩ is a Heyting algebra and the follo wing laws are satisfied: ( CK ) ( a J b ) ∧ ( a J c ) = a J ( b ∧ c ) , ( CT ) ( a J b ) ∧ ( b J c ) ≤ a J c , ( CI ) a J a = ⊤ . If A additionally satisfies ( CD ) ( a J c ) ∧ ( b J c ) = ( a ∨ b ) J c . then it is a sharp Lewisian He yting Algebra ( L - H AO ). A L - H A E is a L - H AO if and only if its strict r educt , i.e. the reduct without → , is a weak Heyting algebra [ 6 ]. W e note that CK , CD , CT , and CI are referred to as C1 – C4 in [ 6 ]. A valuation v in A , as usual, maps propositional atoms to elements of A and is inducti vely extended to ˆ v defined on all formulas in the obvious way . W e write A , v ⊩ ϕ if ˆ v ( ϕ ) = ⊤ and A ⊩ ϕ if A , v ⊩ ϕ for e very valuation v . For Ax ⊆ L J , we write L - H A E ( Ax ) for the class of L - H A E -algebras A such that A ⊩ ϕ for all ϕ ∈ Ax. Furthermore, we write L - H A E ( Ax ) ⊩ Γ ⇒ ϕ if there exists a finite Γ ′ ⊆ Γ such that all algebras in L - H A E ( Ax ) validate ( V Γ ′ ) → ϕ . Then the usual Lindenbaum-T arski construction giv es: 2.6 Theorem. Let Ax ⊆ L J be a set of axioms and Γ ⇒ ϕ a consecution. Then Γ ⊢ Ax ϕ if and only if L - H A E ( Ax ) ⊩ Γ ⇒ ϕ . 3 Relational semantics f or flat Heyting-Lewis logic W e introduce relational semantics for HLC ♭ , first in the most general version (Section 3.1 ), then in the “upward- flat” v ariant (Section 3.2 ) simplifying calculations of correspondents and completeness proofs and in Section 3.3 compare it to the sharp semantics from Definition 2.4 . 3 W e write L - H A E s for the class of all Lewisian Heyting Algebra Expansions, following Litak and V isser [ 27 ]. The same authors call the class of sharp algebras Lewisian Heyting Algebras with Oper ators and discuss the reasons behind this terminology , whereas de Groot et al. [ 10 ] call the sharp algebras simply Heyting-Le wis algebr as , a name which would prov e rather confusing in this context. 3 3.1 Flat frames 3.1 Definition. A flat frame is a tuple ( W , ⪯ , R ) consisting of a nonempty set W , a preorder ⪯ on W and a relation R on W . A flat model is a pair M = ( F , V ) consisting of a flat frame F = ( W , ⪯ , R ) and a valuation V : Prop → up ( W , ⪯ ) that assigns to each proposition letter p an upset V ( p ) of ( W , ≤ ) . The interpretation of L J -formulas at a world w ∈ W extends the intuitionistic semantics with M , w ⊩ ϕ J ψ iff for all w ′ ⪰ w , if M , v ⊩ ϕ for all v ∈ W such that w ′ Rv then M , v ⊩ ψ for all v ∈ W such that w ′ Rv The truth set of ϕ is giv en by J ϕ K M = { w ∈ W | M , w ⊩ ϕ } . Let Γ ∪ { ϕ } ⊆ L J and let M be a flat model. W e write M , w | = Γ if w satisfies all ψ ∈ Γ , and we say that M validates Γ ⇒ ϕ if M , w | = Γ implies M , w | = ϕ for all worlds w in M . A flat frame F validates Γ ⇒ ϕ if every model of the form ( F , V ) validates the consecution, and it validates a formula ϕ if it validates the consecution / 0 ⇒ ϕ . If Ax ⊆ L J is a set of axioms, then we write Γ ⊩ Ax ϕ , and say that Γ semantically entails ϕ on the class of flat frames for HLC ♭ ⊕ Ax, if e v ery conditional frame that validates all formulas in Ax also v alidates the consecution Γ ⇒ ϕ . Using truth-set notation, we have M , w ⊩ ϕ J ψ iff R [ w ′ ] ⊆ J ϕ K M implies R [ w ′ ] ⊆ J ψ K M , for all w ′ ⪰ w . T o illustrate the subtleties of this semantics (e ven in the classical setting), we gi ve two examples sho wing that ( di ) and the rev erse of ( lb ) are not valid. 3.2 Example. Consider the flat model ( W , ⪯ , R ) where W = { w , v , u } , the intuitionistic acces- sibility relation ⪯ is the refle xi ve closure of the three worlds, and R is given by wRv and wRu : Let V be a valuation such that V ( p ) = { v } , V ( q ) = { u } and V ( r ) = / 0. Then w ⊩ p J r because w has no intuitionistic successor x such that R [ x ] ⊆ V ( p ) , so the truth condition for p J r is vacuously true. Similarly , w ⊩ q J r . But w  ⊩ ( p ∨ q ) J r , because every R -successor of w satisfies p ∨ q , b ut not e very successor satisfies r . This sho ws that di is not v alid on flat frames. w v u 3.3 Example. Consider the flat model depicted on the right with V ( p ) = { s } and V ( q ) = / 0. Then w , v  ⊩ □ p because s  ⊩ p , and hence w ⊩ □ p → □ q . On the other hand, the fact that R [ v ] is contained in V ( p ) but not in V ( q ) shows that w  ⊩ p J q . Therefore w  ⊩ ( □ p → □ q ) → ( p J q ) , i.e. the rev erse direction of ( lb ) is false. w v u s A routine induction on the structure of ϕ allo ws us to pro ve: 3.4 Lemma (Intuitionistic heredity) . Let M = ( W , ⪯ , R , V ) be a flat model. Then for all ϕ ∈ L J and all w , v ∈ W , if M , w ⊩ ϕ and w ⪯ v then M , v ⊩ ϕ . Every flat frame gi ves rise to a HLC ♭ -algebra via its complex algebra. 3.5 Definition. The complex algebra of a flat frame F = ( W , ⪯ , R ) is F + ♭ : = ⟨ up ( W , ⪯ ) , ∩ , ∪ , → , J ♭ , W , / 0 ⟩ , where a J ♭ b : = { w ∈ W | for all v ⪰ w , R [ v ] ⊆ a implies R [ v ] ⊆ b } . 3.6 Lemma. If F = ( W , ⪯ , R ) is a flat frame then F + ♭ is a HLC ♭ -algebra, and F and F + ♭ validate precisely the same consecutions. Pr oof. W e kno w that the upsets with the gi ven operations form a Heyting algebra. A routine verification sho ws that J ♭ satisfies ( CK ), ( CT ) and ( CI ). The second part of the lemma follo ws from the fact that v aluations for F correspond bijectiv ely with assignments in F + ♭ , and that the interpretation of connectives in F corresponds to that in F + ♭ . Combining Lemma 3.6 and algebraic soundness prov es: 3.7 Proposition. F or any Ax ⊆ L J and any Γ ∪ { ϕ } ⊆ L J , we have that Γ ⊢ Ax ϕ implies Γ ⊩ Ax ϕ . 3.8 Remark. One might expect that, at least in the finite setting, turning such a complex algebra back into a flat frame should be straightforward, with join-prime elements pro viding the carrier set of the frame. Example 3.3 illustrates that this is not the case: the Heyting reduct of the dual algebra is the Boolean algebra with three atoms (join-primes). Collapsing the { v , w } cluster would change the equational theory . The right approach to duality , similar to the one pursued in [ 9 , 43 ], uses a suitable algebraic translation of the notion of the notion a se gment introduced in Section 4 , potentially blowing up the number of states (cf. estimates in the proof of Lemma 4.7 ). While many se gments can often be eliminated (cf. Remark 4.9 and Section 5.2 ), care is needed. 4 3.2 Upward-flat frames When only using the modal relation to interpret J , we can make a simplification to our frames and assume that R [ w ] is an upset for each w . 3.9 Definition. A upward-flat fr ame is a flat frame ( W , ⪯ , R ) such that for all w , v , u ∈ W , if wRv ⪯ u then wRu . A upwar d-flat model is a flat model whose underlying frame is upward-flat. The coherence condition on the relation can be read as ( R ◦ ⪯ ) = R . While not strictly required, b ut it simplifies the correspondence results for some of the additional axioms we consider in Section 5 . 3.10 Proposition. Let F = ( W , ⪯ , R ) be a flat frame. Define R ⪯ : = R ◦ ⪯ , i.e. wR ⪯ u if ther e exists a v such that wRv ⪯ u, and let F ⪯ = ( W , ⪯ , R ⪯ ) . Then F and F ⪯ have the same complex alg ebra. Pr oof. Let a be an upset of ( W , ⪯ ) and w ∈ W . Then R [ w ] ⊆ a if and only if R ⪯ [ w ] ⊆ a . This entails that the change from F to F ⪯ leav es the definition of J ♭ unchanged, so that F + ♭ = ( F ⪯ ) + ♭ . It is often easier to find and depict frame correspondence results for upw ard-flat frames than for arbitrary ones. The definition and proposition abov e sho w that these can always be transformed into arbitrary frame conditions: simply replace every occurrence of R with ( R ◦ ⪯ ) . T o illustrate the dif ference, consider the axiom 4 a : ϕ J ( ⊤ J ϕ ) (proof in the appendix). 3.11 Fact. • A flat frame walidates 4 a if and only if for all x , y , z , w such that xRy ⪯ zRw, ther e exists v such that xRv ⪯ w. • An upward-flat fr ame validates 4 a iff R is tr ansitive. 3.3 Relation to sharp semantics The sharp semantics for HLC ♯ can be embedded into flat semantics in a truth preserving way . This gi v es rise to a completeness result for HLC ♯ with respect to flat semantics. W e start with a simple suf ficient condition for a flat frame to validate di . 3.12 Lemma. If a flat frame F = ( W , ⪯ , R ) is pointwise downwar d dir ected, then it validates di Next, we turn a sharp model into a flat one. Intuitiv ely , for each w we create a cluster such that each element of the cluster can modally access precisely one of the worlds in R [ w ] . 3.13 Definition. Let F = ( W , ≤ , R , V ) be a sharp frame. Let F ♭ = ( W ♭ , ⪯ , R , V ♭ ) , where W ♭ : = { ( w , v ) | w ∈ W and wRv } ∪ { ( w , • ) | w ∈ W and R [ w ] = / 0 } ( w , v ) ⪯ ( w ′ , v ′ ) iff w ≤ w ′ V ♭ ( p ) = { ( w , v ) ∈ W ♭ | w ∈ V ( p ) } ( w , v ) R ( w ′ , v ′ ) iff v ≤ w ′ Note that the definition of R ensures that M ♭ is and upw ard-flat model. Moreov er , for each ( w , v ) ∈ W ♭ we hav e that R [( w , v )] is the upward closure (under ⪯ ) of ( v , u ) , for any u ∈ W ∪ {•} such that ( v , u ) ∈ W ♭ . This implies that M is pointwise do wnward directed. 3.14 Proposition. Let M = ( W , ≤ , R , V ) be a sharp model, and M ♭ = ( W ♭ , ⪯ , R , V ♭ ) the corr esponding flat model. Then for all ( w , v ) ∈ X ♭ and all formulas ϕ , we have M , w ⊩ s ϕ if and only if M ♭ , ( w , v ) ⊩ ϕ . Combining the known completeness result for HLC ♯ with respect to sharp frames, the lemma and proposition abov e, and the fact that M ♭ is a pointwise downw ard directed upward-flat model, gi ves: 3.15 Proposition. The logic HLC ♯ is sound and complete with r espect to the class of flat frames such that R [ w ] is a cluster for every w. 4 Canonical models and completeness W e provide a canonical model construction relative to some set Σ that is closed under subformulas. This will give us, at once, the finite model property and strong completeness of the logic. W e use a modification of the canonical model construction for CK, using so-called segments. The idea behind a segment is that it encodes both a world of the frame (a prime theory) as well as its successors. W e start by defining prime Σ -theories. Throughout this subsection, we let Ax be a consistent set of formulas, and Σ denote a set of formulas that contains ⊤ and is closed under subformulas. 5 4.1 Definition. A prime ( Ax , Σ ) -theory is a subset Γ ⊆ Σ that is deductively closed (i.e. if ϕ ∈ Σ and Γ ⊢ Ax ϕ then ϕ ∈ Γ ), consistent (i.e. Γ ⊢ Ax ⊥ ), and Σ -prime (i.e. if ϕ 1 , . . . , ϕ n ∈ Σ and Γ ⊢ Ax ϕ 1 ∨ · · · ∨ ϕ n then ϕ i ∈ Γ for some i ∈ { 1 , . . . , n } ). Write Th Ax , Σ for the set of prime ( Ax , Σ ) -theories. If Σ = L J then we omit reference to Σ and simply write prime Ax -theory instead of prime ( Ax , Σ ) -theory . The Lindenbaum lemma can be proved as usual. W e can use it to obtain prime ( Ax , Σ ) -theories by taking the intersection of the resulting prime Ax-theory with Σ . 4.2 Lemma (Lindenbaum lemma) . Let Γ ∪ ∆ ⊆ L J and suppose Γ ⊢ Ax ∆ . Then ther e exists a prime theory Γ ′ such that Γ ⊆ Γ ′ and Γ ′ ∩ ∆ = / 0 . 4.3 Lemma. If Γ is a prime Ax -theory , then Γ ∩ Σ is a prime ( Ax , Σ ) -theory . Segments comprise of a prime ( Ax , Σ ) -theory together with a suitable set of such theories that encodes the successors of the segment. 4.4 Definition. An ( Ax , Σ ) -se gment is a pair ( Γ , U ) where { Γ } ∪ U ⊆ Th Ax , Σ such that (S1) if ∆ ∈ U and ∆ ⊆ ∆ ′ ∈ Th Ax , Σ then ∆ ′ ∈ U ; (S2) for all ϕ , ψ ∈ Σ , if Γ ⊢ Ax ϕ J ψ and ϕ ∈ ∆ for all ∆ ∈ U , then ψ ∈ ∆ for all ∆ ∈ U . Let SEG Ax , Σ be the set of Σ -segments and define relations by setting ( Γ , U ) ⊂ ∼ ( Γ ′ , U ′ ) if f Γ ⊆ Γ ′ , and ( Γ , U ) R ( Γ ′ , U ′ ) iff Γ ′ ∈ U . Define the (canonical) valuation by V Ax , Σ ( p ) = { ( Γ , U ) ∈ SEG Ax , Σ | p ∈ Γ } . Then F Ax , Σ = ( SEG Ax , Σ , ⊆ , R ) and M Ax , Σ = ( SEG Ax , Σ , ⊂ ∼ , R , V Ax , Σ ) are an upward-flat frame and model, called the full canonical frame and model (with respect to Ax and Σ ) . If Σ = L J then we abbreviate SEG Ax : = SEG Ax , L J and F Ax : = F Ax , L J . Lemmas 4.2 and 4.3 provide a way to construct prime ( Ax , Σ ) -theories, gi ven suitable sets of formulas. The following lemma allo ws us to extend this to a segment: 4.5 Lemma. Let Γ be a prime ( Ax , Σ ) -theory and ϕ ∈ Σ , and define U Γ , ϕ : = { ∆ ∈ Th Ax , Σ | if ψ ∈ Σ and Γ ⊢ Ax ϕ J ψ then ψ ∈ ∆ } . 1. ( Γ , U Γ , ϕ ) is an ( Ax , Σ ) -se gment 2. ϕ ∈ ∆ for all ∆ ∈ U Γ , ϕ 3. If θ ∈ Σ is such that Γ ⊢ Ax ϕ J θ , then ther e exists ∆ ∈ U Γ , ϕ such that θ / ∈ ∆ . 4.6 Lemma. F or all ϕ ∈ Σ and ( Γ , U ) ∈ SEG Ax , Σ we have M Ax , Σ , ( Γ , U ) ⊩ ϕ if f ϕ ∈ Γ . Depending on our choice of Ax and Σ , the canonical model construction giv es rise to a finite model property and a strong completeness result. 4.7 Lemma. Let Ax be a set of axioms. 1. Suppose that for every finite consecution Γ ⇒ ϕ ther e e xists a finite subformula-closed set Σ that contains Γ and ϕ suc h that F Ax , Σ validates Ax . Then HLC ♭ ⊕ Ax has the finite model property . 2. Suppose F Ax validates Ax . Then HLC ♭ ⊕ Ax is str ongly complete with r espect to the class of (upwar d-)flat frames validating Ax . Pr oof. (1) Let Γ ∪ { ϕ } be a finite set of formulas and suppose Γ ⊢ Ax ϕ . Let Σ be as described. Then we can use Lemmas 4.2 and 4.3 to construct a prime Σ -theory Γ ′ extending Γ that does not contain ϕ . Let ( Γ ′ , U ) be a segment in SEG Ax , Σ . It then follows from Lemma 4.6 that that M Σ , ( Γ ′ , U ) ⊩ ψ for all ψ ∈ Γ and M Σ , ( Γ ′ , U )  ⊩ ϕ . So M Σ  ⊩ Γ ⇒ ϕ , hence F Ax , Σ is a flat frame that does not validate Γ ⇒ ϕ . By assumption Σ is finite. A prime Σ -theory is a subset of Σ , so we ha v e at most 2 | Σ | many prime theories, and hence at most 2 | Σ | × 2 2 | Σ | Σ -segments, where | Σ | denotes the size of Σ . Hence F Σ is finite. (2) The proof is identical to the first paragraph of item (1) with Σ = L J . T aking Ax = / 0 and Σ the closure of Γ ∪ { ϕ } under subformulas yields: 6 4.8 Theorem. The logic HLC ♭ has the finite model property and is strongly complete with respect to the class of (upwar d-)flat fr ames. 4.9 Remark. While taking the collection of all ( Ax , Σ ) -segments in Definition 4.4 provides a canonical choice of segments, it is not strictly necessary . Analogous to [ 9 ], we can restrict the shape of segments we use while main- taining the truth lemma and completeness result. This can help create a canonical model that satisfies additional constraints. W e will see example of a restriction in Section 5.2 , where we use this strate gy to ensure that the modal accessibility relation is transitiv e when having 4 a as an axiom. 4.10 Remark. A dif ferent method for obtaining completeness results, employed for instance for HLC ♯ [ 10 ] and intuitionistic modal logic with a □ [ 45 , 46 ], is via a G ¨ odel-McKinsey-T arski translation into classical bimodal logic with an S4-box □ i and a normal box □ m . Our case seems amenable to this treatment: flat frames corresponds precisely to the semantics of S4 ⊕ K , and the interpretation of ϕ J ψ is gi ven by □ i ( □ m ϕ → □ m ψ ) . Howe v er , there is a mismatch between the descriptiv e frames of both logics: a duality for HLC ♭ would resemble that for CK [ 11 ] and use segments. As a consequence it does not seem to be the case that the two types of descripti ve frames line up. This frustrates the transfer of e.g. completeness. 5 Completeness and the fmp f or axiomatic extensions W e inv estigate the e xtension of HLC ♭ with the axioms listed in T able 1 . (The gi ven correspondence conditions are prov en in the appendix.) W e start by using Lemma 4.7 to obtain completeness and the finite model property for certain extensions of HLC ♭ with the listed axioms. In Section 5.2 we modify this canonical model construction to obtain completeness for extensions that include 4 a , and the finite model property for HLC ♭ ⊕ t □ ⊕ 4 a . Axiom Formula Upward-flat correspondent em p ∨ ¬ p ⪯ is symmetric t □ ( ⊤ J p ) → p ( ⪯ ◦ R ) is reflexi ve 4 a p J ( ⊤ J p ) R is transitiv e str ( p → q ) → ( p J q ) wRv implies w ⪯ v p a ( p J q ) → ( ⊤ J ( p J q )) if wRvRs then there exists u ⪰ w such that uRs and R [ u ] ⊆ R [ v ] T able 1: Fi ve axioms and their correspondents for upward-flat frames F = ( W , ⪯ , R ) . 5.1 Remark. Both 4 a and p a often occur in arithmetical contexts. It is w orth noting that while 4 a is the “flat” correspondent of transiti vity , p a is the “sharp” one [ 27 ]. While str is a rather degenerate axiom classically (cf. Remark 5.5 ), intuitionistically it plays an important role, occurring in the logics of Haskell arro ws [ 16 ], guarded (co)recursion, and entailments [ 10 , Section 2.4], and e ven allows a non-trivial arithmetical interpretation as com- pleteness principle . 5.1 Reusing the full canonical model W e begin by focussing on em , str and p a . T ow ards proving completeness and the finite model property for some extensions of HLC ♭ with the axioms from Lemma B.1 , we prov e under what conditions on Σ , the canonical frame F Ax , Σ satisfies the correspondence conditions deri v ed in Lemma B.1 . T o this end, we use the following definition of single negations: if ϕ is a formula then its single negation ∼ ϕ is defined as ∼ ϕ = ψ if ϕ = ¬ ψ for some ψ ∈ L J , and ∼ ϕ = ¬ ϕ otherwise. W e say that a set Σ is closed under single negations if ϕ ∈ Σ implies ∼ ϕ ∈ Σ . 5.2 Lemma. Let Ax be a set of axioms, Σ ⊆ L J a set of formulas that is closed under subformulas, and F Ax , Σ = ( SEG Ax , Σ , ⊂ ∼ , R ) the canonical frame gener ated by Ax and Σ . 1. If Σ is closed under single ne gations and em ∈ Ax , then ⊂ ∼ is symmetric. 2. If t □ ∈ Ax then for all ( Γ , U ) ∈ SEG Ax , Σ ther e e xists ( ∆ , D ) such that ( Γ , U ) ⊂ ∼ ( ∆ , D ) R ( Γ , U ) . 3. If str ∈ Ax then ( Γ , U ) R ( Γ ′ , U ′ ) implies ( Γ , U ) ⊂ ∼ ( Γ ′ , U ′ ) 4. If Σ = L J and p a ∈ Ax then F Ax satisfies the corr espondence condition for p a . 7 5.3 Theor em. Let Ax ⊆ { em , t □ , str , p a } . Then HLC ♭ ⊕ Ax is sound and str ongly complete with respect to the class of (upwar d-)flat fr ames on which the y are valid. Pr oof. Combine Lemma 4.7 ( 2 ) and Lemma 5.2 . 5.4 Theorem. Let Ax ⊆ { em , t □ , str } . Then HLC ♭ ⊕ Ax has the finite model property . Pr oof. Use Lemma 4.7 ( 1 ), taking Σ to be the closure under subformulas and under single negations of Γ ∪ { ϕ } . This is finite when Γ is finite. Lemma 5.2 shows that F Ax , Σ validates the required axiom(s). Remark 5.1 indicates that each of the axioms taken in separation and ev en se veral surprising combinations thereof (for example em ⊕ p a ) are of independent interest. In the presence of str , howe ver , certain careless combi- nations may degenerate. Still, such proofs of degenerac y may also illustrate con veience of our semantics. 5.5 Remark. W e note that HLC ♭ ⊕ em ⊕ str is rather de generate, reducing not only to its o wn □ -fragment, but in fact further still to the classical propositional calculus enriched with a single constant: One can sho w that p J q is equiv alent to ( p → q ) ∨ □ ⊥ . While the algebraic proof is very simple, our semantics allows an e ven more perspicuous argument: In upward-flat frames for this system, ⪯ is an equiv alence relation and R ⊆ ⪯ . In those clusters where R is non-empty , p J q is the same as p → q , and otherwise it reduces to ⊤ J ⊥ , which in such degenerate clusters is equi valent to ⊤ (and else where to ⊥ ). 5.6 Remark. In the logic HLC ♭ ⊕ t □ ⊕ str strict implication collapses to → . T o see this, note that str already gi ves ( p → q ) → ( p J q ) . Combining the correspondence conditions for str and t □ giv es: R ⊆ ⪯ and for every w ∈ W there exists some w ′ in the same ⪯ -cluster (i.e. w ⪯ w ′ ⪯ w ) such that R [ w ′ ] = ↑ ⪯ w . Let us verify that this entails ( p J q ) → ( p → q ) . Let w be a world in an upward-flat model such that w ⊩ p J q and let v ⪰ w be a world that satisfies p . Then we can find some v ′ in the same cluster as v such that R [ v ′ ] = ↑ ⪯ v . By assumption and intuitionistic heredity we then get R [ v ′ ] ⊆ V ( p ) , and since w ⪯ v ′ and w ⊩ p J q this implies R [ v ′ ] ⊆ V ( q ) . In particular, this gi ves v ⊩ q , so it follows that w ⊩ p → q . 5.2 Modifying the full canonical model W e turn our attention to e xtensions of HLC ♭ with sets of axioms that include 4 a . Recall that on upw ard-flat frames, 4 a corresponds to transitivity of the modal accessibility relation. The follo wing example illustrates that we cannot use the full canonical model construction from Section 4 . 5.7 Example. Let Ax = { 4 a } and consider Σ = {⊤ , q } . Then we hav e two prime Σ -theories, {⊤} and {⊤ , q } . Let { Γ } ∪ U ⊆ {{⊤} , {⊤ , q }} and suppose U is upwards closed under inclusion. In order for ( Γ , U ) to be an ( Ax , Σ ) - segment, we need to sho w that for all ϕ , ψ ∈ {⊤ , q } , if Γ ⊢ Ax ϕ J ψ and ϕ ∈ ∆ for all ∆ ∈ U , then ψ ∈ ∆ for all ∆ ∈ U . This gi ves four cases, ⊤ J q , q J ⊤ , q J q and ⊤ J ⊤ . The desired condition is clearly satisfied for the latter three, and a simple countermodel shows that Γ ⊢ Ax ⊤ J q for either choice of Γ . Therefore ( Γ , U ) is an ( Ax , Σ ) -segment for an y choice of Γ and U . In particular , this shows that for Γ : = {⊤} and ∆ : = {⊤ , q } we hav e ( Γ , { ∆ } ) R ( ∆ , { Γ , ∆ } ) R ( Γ , / 0 ) while ( Γ , / 0 ) is not modally accessible from ( Γ , { ∆ } ) . So the modal accessibility relation R of the full canonical frame F Ax , Σ is not transitiv e, hence F Ax , Σ does not validate 4 a . In order to prove completeness for e xtensions of HLC ♭ with 4 a , we used a trimmed v ersion of the canonical model construction from Section 4 . This is obtained by restricting the set SEG Ax , Σ . 5.8 Definition. Let Ax be a consistent set of axioms and Σ a set of formulas that is closed under subformulas and contains ⊤ . W e call an ( Ax , Σ ) -segment ( Γ , U ) pointed if there exists a formula ϕ ∈ Σ such that U = U Γ , γ : = { ∆ ∈ Th Ax , Σ | if ψ ∈ Σ and Γ ⊢ Ax γ J ψ then ψ ∈ ∆ } . By Lemma 4.5 , ev ery prime ( Ax , Σ ) -theory can be extended to a pointed ( Ax , Σ ) -segment. Write SEG p Ax , Σ for the set of pointed ( Ax , Σ ) -segments, and F p Ax , Σ : = ( SEG p Ax , Σ , ⊂ ∼ , R ) and M p Ax , Σ : = ( F p Ax , Σ , V Ax , Σ ) for the pointed canonical frame and model . If Σ = L J we abbreviate F p Ax : = F p Ax , L J . Using precisely the same proof as Lemma 4.6 , we get 5.9 Lemma. Let Ax ⊆ L J be a set of axioms and Σ ⊆ L J as set of formulas that contains ⊤ and is closed under subformulas. Then for all ϕ ∈ Σ and ( Γ , U ) ∈ SEG Ax , Σ we have M Ax , Σ , ( Γ , U ) ⊩ ϕ if f ϕ ∈ Γ . 8 5.10 Theorem. Let Ax ⊆ { em , t □ , str , 4 a } . Then HLC ♭ ⊕ Ax is sound and str ongly complete with r espect to the class of (upwar d-)flat fr ames on which Ax is valid. Pr oof. It suf fices to show that F p Ax validates each of the axioms in Ax. Using the same proof as in Lemma 5.2 shows that if em , t □ or str is in Ax, then F p Ax validates it, so we are left to consider 4 a . So suppose 4 a ∈ Ax. W e need to show that R is transitiv e. T o this end, let ( Γ , U Γ , γ ) R ( ∆ , U ∆ , δ ) R ( Π , U Π , π ) in F p Ax . Suppose γ J ψ ∈ Γ . By 4 a we also have ψ J ( ⊤ J ψ ) ∈ Γ , so tr giv es γ J ( ⊤ J ψ ) ∈ Γ . This entails ⊤ J ψ ∈ ∆ , which by the definition of a segment gi ves ψ ∈ Π . This pro ves that Π ∈ U Γ , γ , so that ( Γ , U Γ , γ ) R ( Π , U Π , π ) , as desired. Finally , using the same kind of canonical model we deriv e the finite model property for the logic HLC ♭ ⊕ t □ ⊕ 4 a . The ke y insight to wards this is that ⊤ J ϕ is equiv alent to ⊤ J ( ⊤ J ϕ ) in this logic, so that it suf fices to close Σ under “single boxes. ” 5.11 Lemma. W e have ⊢ t □ , 4 a ( ⊤ J ϕ ) ↔ ( ⊤ J ( ⊤ J ϕ )) . 5.12 Definition. For ϕ ∈ L J we define ⊠ ϕ : = ( ϕ if ϕ = ⊤ J ψ for some ψ ∈ L J ⊤ J ϕ otherwise A set Σ ⊆ L J is said to be closed under single boxes if ϕ ∈ Σ implies ⊠ ϕ ∈ Σ . Closing a finite set Σ under single boxes at most doubles its size, hence it stays finite. This allows us to construct a finite model with a transitiv e modal relation. 5.13 Theorem. The logic HLC ♭ ⊕ t □ ⊕ 4 a has the finite model pr operty . Pr oof. Let Γ ⇒ ϕ be a finite consecution such that Γ ⊢ t □ , 4 a ϕ . Let Σ be the set of subformulas of Γ ∪ {⊤ , ϕ } closed under single boxes. Then Σ is finite, and we can use Lemmas 4.2 and 4.3 to e xtend Γ to a prime ( Ax , Σ ) -theory Γ ′ containing Γ but not ϕ . Lemma 4.5 then yields an ( Ax , Σ ) -segment ( Γ ′ , U Γ ′ , ⊤ ) which by Lemma 5.9 , under the canonical v aluation, in v alidates Γ ⇒ ϕ . Therefore F p Ax , Σ = ( SEG p Ax , Σ , ⊂ ∼ , R ) inv alidates Γ ⇒ ϕ . T o establish the finite model property , we now ar gue that F Ax , Σ validates t □ and 4 a . Using the same proof as Lemma 5.2 ( 2 ) sho ws that F validates t □ . For 4 a , let ( Γ , U Γ , γ ) , ( ∆ , U ∆ , δ ) and ( Π , U Π , π ) be three ( Ax , Σ ) -segments and suppose ( Γ , U Γ , γ ) R ( ∆ , U ∆ , δ ) R ( Π , U Π , π ) . Let γ , ψ ∈ Σ and suppose Γ ⊢ Ax γ J ψ . By assumption we hav e Γ ⊢ Ax ψ J ( ⊤ J ψ ) , hence by tr we find Γ ⊢ Ax γ J ( ⊤ J ψ ) . This entails Γ ⊢ Ax γ J ⊠ ψ , and since ψ ∈ Σ we have ⊠ ψ ∈ Σ . Therefore we must hav e ⊠ ψ ∈ ∆ , hence ∆ ⊢ Ax ⊤ J ψ . Finally , the definition of a segment and the fact that ( ∆ , U ∆ , δ ) R ( Π , U Π , π ) entails ψ ∈ Π . Thus, we have shown that for any ψ ∈ Σ , Γ ⊢ Ax γ J ψ implies ψ ∈ Π , so that Π ∈ U Γ , γ hence ( Γ , U Γ , γ ) R ( Π , U Π , π ) . Therefore R is transiti ve, so F p Ax , Σ ⊩ 4 a . 6 Open subframes and extension stability Litak and V isser [ 27 ] note a direct connection between the syntactic notion of extension stability , motiv ated by arithmetical interpretations of J , and a special type of nuclei on flat algebras, more specifically open nuclei [ 14 , 28 ]. Recall that nuclei provide an algebraic perspectiv e on subframes in modal logic [ 4 , 13 , 44 ]. In particular , quotienting an algebra by an open nucleus generated by a chosen element a produces an algebra (isomorphic to one) whose Heyting reduct is (isomorphic to) the ideal of elements below a , with suitably restricted J . In the classical setting with unary box, applying this construction to dual algebras of Kripke frames produces the dual algebra of the (not necessarily modally generated!) subframe induced by a ; that is, a Kripke frame whose carrier and modal accessibility relation are restricted to a . In the Heyting setting, the f act that a is an element of the upset algebra means that the carrier set of the corresponding subframe is ⪯ -generated, i.e. an upset. When it comes to R , Proposition 3.10 indicates a certain subtlety: unlike the classical case, the dual algebras of our frames might fail to notice the presence/absence of certain R -edges. Let us reconsider the example of 4 a from F act 3.11 : the corresponding class of arbitrary flat frames does not appear closed with respect to the open subframe construction. Howe ver , over upward-flat frames, the situation changes: transitivity is well-kno wn to be persistent with respect to subframes. T ogether with difficulties in presenting duality for flat subframes noted abo ve (Remarks 3.8 and 4.10 ), this means that some care is needed. Given the space constraints of the present paper , we do not attempt a full discussion here. Nev ertheless, it is illustrati ve to pro vide a semantic discussion of the f ailure of extension stability for HLC ♯ . 9 6.1 Example. Consider the flat model ( W , ⪯ , R ) where W = { w , v , u , z } , the intuitionistic accessibility relation ⪯ is the reflexi ve closure of the four worlds together with z ⪯ v and z ⪯ u , and R is gi ven by wRv , wRz and wRu : w v u z This frame is clearly upward-flat. Moreover , it satisfies the sufficient condition of Lemma 3.12 to validate di . Howe ver , the open subframe obtained by removing z is precisely the one used in Example 3.2 to illustrate the failure of di . In order to turn this counterexample into a formal proof, let us recall the syntactic characterization of extension stability [ 27 ]. Gi ven a formula ϕ and a fresh propositional variable p , define the translation ϕ ⌈ p ⌉ inductiv ely as commuting with the propositional variables and the connecti ves of IPC , with the J clause being • ( ψ J χ ) ⌈ p ⌉ : = (( p → ψ ⌈ p ⌉ ) J ( p → χ ⌈ p ⌉ )) . As □ ϕ is ⊤ J ϕ , we get HLC ♭ ⊢ ( □ ϕ ) ⌈ p ⌉ iff □ ( p → ϕ ⌈ p ⌉ ) . Note that for any logic Λ and any ϕ , if Λ ⊢ ϕ ⌈ p ⌉ , then Λ ⊢ ϕ . A logic Λ is extension stable if, whene ver Λ ⊢ ϕ and p not in ϕ , we ha ve Λ ⊢ p → ϕ ⌈ p ⌉ . 6.2 Theorem. The frame fr om Example 6.1 r efutes s → di ⌈ s ⌉ , i.e., s → ((( s → p ) J ( s → r )) ∧ (( s → q ) J ( s → r )) → (( s → ( p ∨ q )) J ( s → r ))) . Thus, HLC ♯ is not extension stable , and neither is any of its extensions validated by this fr ame . Pr oof. Define V ( s ) to be the complement of z and follow Example 3.2 for other atoms, i.e., V ( p ) = { v } , V ( q ) = { u } and V ( r ) = / 0. One can then follow the reasoning from Example 3.2 , with s in the antecedent used to relati vize reasoning to the three-state open subframe. For contrast, consider 4 a . One can easily see that 4 a ⌈ s ⌉ is equiv alent to a substitution instance of 4 a itself, and hence s → 4 a ⌈ s ⌉ is a theorem of HLC ♭ ⊕ 4 a . This shows that the closure of the corresponding upward-flat frames under open subframes is more important than the apparent failure of such closure in the broader class. In other words, narro wing do wn the class of frames might be essential for gi ving an appropriate duality account. 7 Conclusions and future w ork W e believ e we have demonstrated the potential of the flat semantics. 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URL https://msclogic.illc.uva.nl/theses/archive/ . 11 A Omitted proofs Pr oof of F act 3.11 . • Suppose W validates 4 a , and xRy ≤ zRw for some worlds x , y , z , w . Let V be a valuation of p with V ( p ) = ↑ R [ x ] . Then all worlds in R [ x ] satisfy p , so by assumption they also all satisfy ⊤ J p . In particular , y ⊩ ⊤ J p , and since y ⪯ z and (tri vially) all worlds in R [ z ] satisfy ⊤ , we must hav e w ⊩ p . By definition, this means that w lies abov e some R -successor v of x , as desired. Con versely , suppose W satisfies the frame condition. Let x be an y world. T o sho w that it satisfies 4 a , let x ⪯ x ′ and suppose all worlds in R [ x ′ ] satisfy p . Then we need that all worlds in R [ x ′ ] satisfy ⊤ J p . Let y be such a world. Since ⊤ is always true, we need to prove that y ⪯ zRw implies that w ⊩ p . This follows from the frame condition. So 4 a is valid. • This is a straightforward simplification of the above condition. For readers’ con venience, we provide a direct proof. Suppose R is transiti ve and let w be a world such that R [ w ] ⊆ V ( p ) . Then by assumption R [ R [ w ]] ⊆ J p K , and since wRv ⪯ uRs implies wRuRs we have R [ u ] ⊆ J p K for every u ∈ R [ w ] , so that R [ w ] ⊆ J ⊤ J p K . Con versely , suppose wRvRu . Let V be a v aluation such that V ( p ) = R [ w ] . Then R [ w ] ⊆ V ( p ) , so we must hav e R [ w ] ⊆ J ⊤ J ϕ K . This forces R [ R [ w ]] ⊆ V ( p ) = R [ w ] . In particular, we hav e u ∈ R [ R [ w ]] ⊆ R [ w ] so wRu . Therefore R is transitiv e. Pr oof of Lemma 3.12 . Suppose F is a flat frame that is pointwise do wnward directed, M = ( F , V ) is a flat model based on F and w ∈ W satisfies p J r and q J r . Suppose w ′ ⪰ w and R [ w ′ ] ⊆ J p ∨ q K = V ( p ) ∪ V ( q ) . W e claim that either R [ w ′ ] ⊆ V ( p ) or R [ w ′ ] ⊆ V ( q ) . If this is not the case, then we can find v , u ∈ R [ w ′ ] such that v / ∈ V ( p ) and u / ∈ V ( q ) . By assumption there exists some s ∈ R [ w ′ ] such that s ⪯ v and s ⪯ u . But then s / ∈ V ( p ) ∪ V ( q ) , a contradiction. So we must hav e R [ w ′ ] ⊆ V ( p ) or R [ w ′ ] ⊆ V ( q ) . In either case, using the assumption yields R [ w ′ ] ⊆ V ( r ) . This pro ves w ⊩ ( p ∨ q ) J r , and hence di is v alid on F . Pr oof of Proposition 3.14 . W e use induction on the ϕ , showcasing only the induction step for ϕ = ψ J χ . Suppose M , w ⊩ s ψ J χ . Suppose ( w , v ) ⪯ ( w ′ , v ′ ) and R [( w ′ , v ′ )] ⊆ J ψ K M ♭ . Then w ≤ w ′ and w ′ Rv ′ , so wRv ′ because M is a sharp model. Also R [( w ′ , v ′ )] = { ( u , s ) ∈ W ♭ | v ′ ≤ u } , so by the induction hypothesis we ha ve M , v ′ ⊩ s ψ . The assumption that M , w ⊩ s ψ J χ then gi ves M , v ′ ⊩ s χ , and intuitionistic heredity entails M , u ⊩ s χ for all u ≥ v ′ . Using induction again, this implies R [( w ′ , v ′ )] ⊆ J χ K M ♭ , and hence M ♭ , ( w , v ) ⊩ ψ J χ . Con versely , suppose M ♭ , ( w , v ) ⊩ ψ J χ and suppose wRu and M , u ⊩ s ψ . Then ( w , v ) ⪯ ( w , u ) and by induction R [( w , u )] ⊆ J ψ K M ♭ . This implies R [( w , u )] ⊆ J χ K M ♭ , and hence M , u ⊩ s χ . Therefore M , w ⊩ s ψ J χ . Pr oof of Lemma 4.5 . (1) It follo ws immediately from the definition that ( Γ , U Γ , ϕ ) satisfies ( S1 ), so we focus on proving ( S2 ). Suppose χ , ξ ∈ Σ and Γ ⊢ Ax χ J ξ and χ ∈ ∆ for all ∆ ∈ U Γ , ϕ . Then we must have { ψ ∈ Σ | Γ ⊢ Ax ϕ J ψ } ⊢ χ , because otherwise we could use the Lindenbaum lemma to find some prime Σ -theory in U Γ , ϕ that does not contain χ . (W e can first use the usual Lindenbaum lemma to find a prime theory containing the LHS but not ϕ , and then take its intersection with Σ .) By compactness, we can find ψ 1 , . . . , ψ n ∈ Σ such that Γ ⊢ Ax ϕ J ψ i for all i ∈ { 1 , . . . , n } and ψ 1 , . . . , ψ n ⊢ Ax χ . This implies ϕ J ψ 1 , . . . , ϕ J ψ n ⊢ Ax ϕ J χ , hence using transitivity ϕ J ψ 1 , . . . , ϕ J ψ n , χ J ξ ⊢ Ax ϕ J ξ , . Since Γ deriv es everything on the LHS, we also get Γ ⊢ Ax ϕ J ξ , hence by definition of U Γ , ϕ we have ξ ∈ ∆ for all ∆ ∈ U Γ , ϕ . (2) This follo ws from the f act ( N a ) entails ⊢ ϕ J ϕ for an y ϕ ∈ L J . Therefore Γ ⊢ Ax ϕ J ϕ and hence ϕ ∈ ∆ for all ∆ ∈ U Γ , ϕ by definition. (3) W e claim that { ψ ∈ Σ | Γ ⊢ Ax ϕ J ψ } ⊢ Ax θ . Suppose tow ards a contradiction that this is not the case. Then by compactness we can find ψ 1 , . . . , ψ n ∈ Σ such that ψ 1 , . . . , ψ n ⊢ Ax θ (1) and Γ ⊢ Ax ϕ J ψ i for each i ∈ { 1 , . . . , n } . This implies Γ ⊢ Ax ϕ J ( ψ 1 ∧ · · · ∧ ψ n ) . Furthermore, ( 1 ) entails ⊢ Ax ( ψ 1 ∧ · · · ∧ ψ n ) → θ , so by ( N a ) we get ⊢ Ax ( ψ 1 ∧ · · · ∧ ψ n ) J θ . In particular , this gi ves Γ ⊢ Ax ( ψ 1 ∧ · · · ∧ ψ n ) J θ , so that tr entails Γ ⊢ Ax ϕ J θ , a contradiction. So we hav e { ψ ∈ Σ | Γ ⊢ Ax ϕ J ψ } ⊢ Ax θ . Then Lemma 4.2 giv es a prime Ax-theory ∆ containing ψ for ev ery ψ ∈ Σ such that Γ ⊢ Ax ϕ J ψ , but not θ . By definition ∆ ∩ Σ ∈ U Γ , ϕ , so it is the desired witness. 12 Pr oof of Lemma 5.2 . (1) Suppose ( Γ , U ) ⊂ ∼ ( Γ ′ , U ′ ) . Then ϕ ∈ Γ ′ implies ∼ ϕ / ∈ Γ ′ . Since Γ ⊆ Γ ′ this gives ∼ ϕ / ∈ Γ , hence ϕ ∈ Γ . (2) W e can take ( ∆ , D ) = ( Γ , U Γ , ⊤ ) . Then ( Γ , U ) ⊂ ∼ ( Γ , U Γ , ⊤ ) , and it follo ws from t □ that Γ ∈ U Γ , ⊤ . (3) Suppose ( Γ , U ) R ( Γ ′ , U ′ ) . Then ϕ ∈ Γ implies Γ ⊢ Ax ⊤ → ϕ , hence using str we get Γ ⊢ Ax ⊤ J ϕ . By definition of an ( Ax , Σ ) -segment, this implies that ϕ ∈ ∆ for all ∆ ∈ U . It follows that Γ ⊆ ∆ for all ∆ ∈ U . In particular , this implies Γ ⊆ Γ ′ , hence ( Γ , U ) ⊂ ∼ ( Γ ′ , U ′ ) . (4) Suppose ( Γ , U ) R ( ∆ , D ) . Then ϕ J ψ ∈ Γ implies ⊤ J ( ϕ J ψ ) ∈ Γ , so that ϕ J ψ ∈ ∆ . It follows that ( Γ , D ) is a segment. This implies the correspondence condition, because for any s is the correspondence condition we can take u = ( Γ , D ) . Pr oof of Lemma 4.6 . W e use induction on the structure of ϕ . If ϕ is ⊤ , ⊥ or a proposition letter, the statement is immediate. The cases for ∧ and ∨ follo w using induction and the fact that Γ is prime. Case ϕ = ψ → χ . Suppose ψ → χ ∈ Γ . Let ( Γ , U ) ⊂ ∼ ( Γ ′ , U ′ ) and suppose ( Γ ′ , U ′ ) ⊩ ψ . Then by definition of ⊂ ∼ , deductiv e closure of prime ( Ax , Σ ) -theories, and the induction hypothesis we find ψ ∈ Γ ′ and ψ → χ ∈ Γ ′ . This implies χ ∈ Γ ′ , hence by induction ( Γ ′ , U ′ ) ⊩ χ . This proves ( Γ , U ) ⊩ ψ → χ . Con versely , suppose ψ → χ / ∈ Γ . Then Γ ⊢ Ax ψ → χ , so Γ , ψ ⊢ Ax χ and we can find a prime theory Γ ′ containing Γ , ψ but not χ . Then Γ ′ ∩ Σ is a prime Σ -theory and we can e xtend it to an ( Ax , Σ ) -segment (for e xample by using Lemma 4.5 with ϕ = ⊤ ) which (using induction) satisfies ψ but not χ . Therefore ( Γ , U )  ⊩ ψ → χ . Case ϕ = ψ J χ . If ψ J χ ∈ Γ then we get ( Γ , U ) ⊩ ψ J χ immediately by definition of a segment. So suppose ψ J χ / ∈ Γ . By Lemma 4.5 ( Γ , U Γ , ψ ) is a ( Ax , Σ ) -segment such that ψ ∈ ∆ for all ∆ ∈ U Γ , ψ while χ / ∈ ∆ for some ∆ ∈ U Γ , ψ . Since each ∆ can be extended to a se gment, this prov es ( Γ , U )  ⊩ ψ J χ . Pr oof of Lemma 5.11 . As a substitution instance of t □ we get ⊢ t □ , 4 a ( ⊤ J ( ⊤ J ϕ )) → ( ⊤ J ϕ ) . Con v ersely , combining ⊤ J ϕ with 4 a and tr yields ⊤ J ( ⊤ J ϕ ) . B Correspondence r esults B.1 Lemma. Let F = ( W , ⪯ , R ) be a upward-flat fr ame . Then 1. F validates em if and only if ⪯ is symmetric; 2. F validates t □ if and only if for all w ther e e xists v such that w ⪯ vRw; 3. F validates str if and only if wRv implies w ⪯ v; 4. F validates p a if and only if for all w , v , s satisfying wRvRs there exists u ⪰ w such that s ∈ R [ u ] and R [ u ] ⊆ R [ v ] ; 5. F validates 4 a if and only if for all R is transitive . Pr oof. ( 1 ) Suppose ⪯ is symmetric. Let V be an y valuation and suppose w ∈ W does not satisfy p . Then for all v ⪰ w we hav e v ⪯ w by symmetry , so v  ⊩ p . This prov es w ⊩ ¬ p . Therefore em is v alid. For the con v erse, suppose the frame condition does not hold, so there e xist v , w such that w ⪯ v and v ⪯ w . Let V be a v aluation such that V ( p ) = ↑ v . Then w  ⊩ p because w / ∈ V ( p ) and w  ⊩ ¬ p because w ⪯ v ⊩ p , so em fails. ( 2 ) Suppose the frame condition holds and let V be any valuation. If w ⊩ ⊤ J p then for all v ⪰ w we hav e R [ v ] ⊆ p . By assumption there e xists such a v such that w ∈ R [ v ] , hence w ⊩ p . Therefore t □ is valid. Con versely , suppose t □ is v alid. Let w be any world. Let V be a v aluation such that V ( p ) = S { R [ v ] | v ⪰ w } . Then w ⊩ ⊤ J p , hence w ⊩ p , so we must hav e w ∈ R [ v ] for some v ⪰ w , as desired. ( 3 ) Suppose R ⊆ ⪯ , let V be any valuation, and w ⊩ p → q . If w ⪯ v and R [ v ] ⊆ V ( p ) then by assumption w ⪯ u for all u ∈ R [ v ] , hence u ⊩ q for all such u , so that R [ v ] ⊆ V ( q ) . This proves w ⊩ p J q . So str is valid. For the con verse, suppose the frame condition does not hold. Then we can find w , v ∈ W such that wRv while w ⪯ v . Let V be a valuation such that V ( p ) = R [ w ] and V ( q ) = ↑ w . (Recall that R [ w ] is upwards closed in upward-flat frames.) Then w trivially satisfies p → q , but w  ⊩ p J q because all modal successors of w satisfy p , but not all of them satisfy q (namely v does not satisfy q ). ( 4 ) Suppose the frame condition holds, and let V be any valuation. Suppose w ⊩ p J q . T o sho w that w ⊩ ⊤ J ( p J q ) , we need to prove that w ⪯ w ′ Rv implies v ⊩ p J q . T o this end, let v ′ ⪰ v and assume R [ v ′ ] ⊆ V ( p ) . Then because the frame is upward-flat we hav e w ′ Rv ′ . Now let s ∈ R [ v ′ ] . Then by assumption there exists some u ⪰ w ′ such that s ∈ R [ u ] ⊆ R [ v ′ ] Since w ⊩ p J q and R [ u ] ⊆ V ( p ) we find s ⊩ q . This entails that R [ v ′ ] ⊆ V ( q ) , so v ⊩ p J q , as desired. 13 Con versely , if the frame condition does not hold then we can find w , v , s such that wRvRs and for all u ⪰ w either R [ u ] ⊆ R [ v ] or s / ∈ R [ u ] . T aking V ( p ) = R [ v ] and V ( q ) = R [ v ] \ ↓ s then gives w ⊩ p J q , because R [ u ] ⊆ V ( p ) for all u ⪰ w , while v  ⊩ p J q , so w  ⊩ ⊤ J ( p J q ) . ( 5 ) Suppose R is transiti ve and let w be a world such that R [ w ] ⊆ V ( p ) . Then by assumption R [ R [ w ]] ⊆ J p K , and since wRv ⪯ uRs implies wRuRs we have R [ u ] ⊆ J p K for every u ∈ R [ w ] , so that R [ w ] ⊆ J ⊤ J p K . Con versely , suppose wRvRu . Let V be a v aluation such that V ( p ) = R [ w ] . Then R [ w ] ⊆ V ( p ) , so we must ha ve R [ w ] ⊆ J ⊤ J ϕ K . This forces R [ R [ w ]] ⊆ V ( p ) = R [ w ] . In particular , we have u ∈ R [ R [ w ]] ⊆ R [ w ] so wRu . Therefore R is transitive. 14