aa r X i v : . [ m a t h . L O ] M a y INTERMEDIATE JUSTIFICATION LOGICS: UNIFIED COMPLETENESS RESULTS
NICHOLAS PISCHKE
Abstract.
We introduce abstract intermediate justification logics by extending arbitrary intermediate propo-sitional logics with a subset of specific axioms of (classical) justification logic. We study these intermediatejustification logics semantically out of various perspectives by combining the well-known semantical accesspoints to intermediate logics through algebraic and Kripke-frame based models with the usual semantic machin-ery used by Mkrtychevs, Fittings or Lehmanns and Studers models for classical justification logics. We proveunified completeness theorems for all intermediate justification logics and their corresponding semantics using arespective propositional completeness theorem of the underlying intermediate logic. We consider especially theparticular instances of intuitionistic, classical and G¨odel justification logics because of their previous presencein the literature. Introduction
Justification logics originated in the 90’s from the studies of Artemov (see [1, 2]) regarding the provabilityinterpretation of the modal logic S { , } (or [0 ,
1] as in the case of G¨odel justification logics) but inarbitrary Heyting algebras. The approach via intuitionistic Kripke frames extends the previous considerationsfor semantics of intuitionistic justification logics by new model classes as well as a wider range of applicable logics.All these considerations culminate in general unified completeness theorems based on a semantical character-ization of the underlying intermediate logic. As the class of intermediate justification logics contains especiallythe well-known cases of intuitionistic, G¨odel and classical justification logics, these completeness results moreovercontain all the completeness theorems based on Mkrtychev, Fitting and subset models for classical justification
Key words and phrases.
Justification Logic, Intermediate Logic, Heyting Algebras, Kripke Frames, Completeness. logics as well as the previous completeness theorems for the G¨odel justification logics with respect to [0 , Intermediate justification logics
Syntax and proof calculi.
We consider the propositional language L : φ ::= ⊥ | p | ( φ ∧ φ ) | ( φ ∨ φ ) | ( φ → φ )where p ∈ V ar := { p i | i ∈ N } . We introduce negation as the abbreviation ¬ φ := φ → ⊥ . We also define n ^ i =1 φ n := φ ∧ · · · ∧ φ n for some φ , . . . , φ n ∈ L . The same applies to ∨ . In order to define intermediate logics and later intermediatejustification logics, we need to briefly review some notions regarding propositional substitutions.A substitution in L is a function σ : V ar → L . This function σ naturally extends to L by commutingwith the connectives ∧ , ∨ , → and ⊥ and we write σ ( φ ) also for the image of this extended function.Using this definition of substitutions, we can now give the following definition of an intermediate justificationlogic. Definition 2.1. A intermediate logic (over L ) is a set I ( L which satisfies:(1) the schemes ( A
1) - ( A
9) are contained in Γ;(2) I is closed under modus ponens , that is φ → ψ, φ ∈ I implies ψ ∈ I ;(3) I is closed under substitution in L .Here, the schemes ( A
1) - ( A
9) are given by:( A : φ → ( ψ → φ );( A : ( φ → ( χ → ψ )) → (( φ → χ ) → ( φ → ψ ));( A : ( φ ∧ ψ ) → φ ;( A : ( φ ∧ ψ ) → ψ ;( A : φ → ( ψ → ( φ ∧ ψ ));( A : φ → ( φ ∨ ψ );( A : ψ → ( φ ∨ ψ );( A : ( φ → ψ ) → (( χ → ψ ) → (( φ ∨ χ ) → ψ ));( A : ⊥ → φ .We denote the smallest intermediate propositional logic, that is the logic given by the axiom schemes ( A
1) -( A
9) in L and closed under modus ponens, by IPC . Given a set of formulae Γ ⊆ L , we writeΓ ⊢ I φ iff ∃ γ , . . . , γ n ∈ Γ n ^ i =1 γ i → φ ∈ I ! . On the side of justification logics, we consider the following set of justification terms Jt : t ::= x | c | [ t + t ] | [ t · t ] | ! t where x ∈ V := { x i | i ∈ N } and c ∈ C := { c i | i ∈ N } and the resulting multi-modal language L J : φ ::= ⊥ | p | ( φ ∧ φ ) | ( φ ∨ φ ) | ( φ → φ ) | t : φ where p ∈ V ar and t ∈ Jt . Naturally, the same abbreviations as for L also apply here. Given a set Γ , ∆ ⊆ L J ,we write Γ ⊕ ∆ for the smallest set containing Γ ∪ ∆ which is closed under modus ponens.In order to formulate intermediate justification logics, we consider especially substitutions in L J . These areagain functions σ : V ar → L J which extend uniquely to L J to commuting with ∧ , ∨ , → , ⊥ and the justificationmodalities t :. We again write σ ( φ ) for the image of a formula φ ∈ L J under this extension. By Γ, we denotethe closure of Γ under substitutions in L J . Definition 2.2.
Let I be an intermediate propositional logic. Given the axiom schemes( J ) : t : ( φ → ψ ) → ( s : φ → [ t · s ] : ψ ),(+) : t : φ → [ t + s ] : φ , t : φ → [ s + t ] : φ ,( F ) : t : φ → φ ,( I ) : t : φ → ! t : t : φ ,we consider the following justification logics based on I :(1) IJ := I ⊕ ( J ) ⊕ (+); NTERMEDIATE JUSTIFICATION LOGICS: UNIFIED COMPLETENESS RESULTS 3 (2)
IJ T := IJ ⊕ ( F );(3) IJ := IJ ⊕ ( I );(4) IJ T := IJ ⊕ ( F ) ⊕ ( I ).As before in the propositional case, given Γ ∪ { φ } ⊆ L J , we writeΓ ⊢ IJ L φ iff ∃ γ , . . . , γ n ∈ Γ n ^ i =1 γ i → φ ∈ IJ L ! . Specific instances of intermediate propositional logics and of the resulting intermediate justification logics whichwe consider explicitly in this paper are G := IPC ⊕ ( LIN ) , CPC := IPC ⊕ ( LEM ) , with the schemes ( LIN ): ( φ → ψ ) ∨ ( ψ → φ ), ( LEM ): φ ∨ ¬ φ ,over L . Definition 2.3.
Let I be an intermediate propositional logic and IJ L ∈ {IJ , IJ T , IJ , IJ T } . A constant specification for IJ L is a set CS of formulae from L J of the form c i n : · · · : c i : φ where n ≥ c i k ∈ CS for all k and φ is an axiom instance of IJ L , that is φ ∈ I or φ is an instance of thejustification axiom schemes ( J ) , (+) , ( F ) , ( I ) (depending on IJ L ).Constant specifications can be used to augment proof systems and increase the amount of justified formulaewhich they can prove. Definition 2.4.
Let I be an intermediate propositional logic and IJ L ∈ {IJ , IJ T , IJ , IJ T } andlet CS be a constant specification for IJ L . We write Γ ⊢ IJ L CS φ for Γ ∪ CS ⊢ IJ L φ with Γ ∪ { φ } ⊆ L J .Note, that the above definition of a constant specification is different from the usual one in the literature.Normally, one works with a specific set of axiom schemes for the propositional base of the justification logic inquestion and allows φ in c i n : · · · : c i : φ only to be an instance of these axioms where here, we allow φ to be an arbitrary theorem of I . This is out ofpure convenience as not introducing the notion of axiomatic systems for intermediate logics makes the followingdefinitions and results regarding constant specifications more clean to state. Note however, that all results canbe appropriately adapted to constant specifications over axiomatic bases.If one follows this line of defining axiomatic bases of intermediate logics however, one can similarly defineaxiomatically appropriate constant specifications as in the classical case and obtain analogues for the liftinglemma and internalization. We omit this here as well (see [3, 4, 25] for these concepts in the classical case).As a straightforward application of classical techniques (see e.g. [25]), one can show directly that all theintermediate justification logics are conservative over their corresponding intermediate logic and L . Lemma 2.5.
Let I be an intermediate logic and let IJ L ∈ {IJ , IJ T , IJ , IJ T } . Let CS be a constantspecification for IJ L . For any φ ∈ L : ⊢ IJ L CS φ iff ⊢ I φ . The proof is a natural generalization from the classical case, see e.g. [25] for a version for
CPCJ T
4. Further,we want to mention the deduction theorem for
IJ L CS . Lemma 2.6.
Let I be an intermediate logic and let IJ L ∈ {IJ , IJ T , IJ , IJ T } . Further, let CS bea constant specification for IJ L . For any Γ ∪ { φ, ψ } ⊆ L J : Γ ∪ { φ } ⊢ IJ L CS ψ iff Γ ⊢ IJ L CS φ → ψ. Extended propositional languages.
In later sections, it will be convenient to consider intermediatelogics over different sets of variables. For this, we consider the language L ( X ) : φ ::= ⊥ | x | ( φ ∧ φ ) | ( φ ∨ φ ) | ( φ → φ )where X is a countably infinite set of variables. The same notational abbreviations as before also apply here.Note also, that naturally L ( V ar ) = L . A particular choice different from V ar for X in the following will bethe set V ar ⋆ := V ar ∪ { φ t | φ ∈ L J , t ∈ Jt } . NICHOLAS PISCHKE
Here, we write L ⋆ := L ( V ar ⋆ ).For the following definition, note that any bijection t : V ar → X can be naturally extended to a bijection t : L → L ( X ) through recursion on L by commuting with ∧ , ∨ , → and ⊥ . Also, such a bijection t : V ar → X always exists as both X and V ar are countably infinite.
Definition 2.7.
Let I be an intermediate logic and let t : V ar → X be some (any) bijection extended to t : L → L ( X ). We write I ( X ) := t [ I ].Note, that here also I ( V ar ) = I . Remark . In the above definition, it is indeed not important which bijection f : V ar → X is fixed as I isclosed under substitutions. Further, naturally I ( X ) is closed under modus ponens and under substitutions ofvariables in X by formulae in L ( X ).Given I ( X ) and Γ ∪ { φ } ⊆ L ( X ), we write Γ ⊢ I ( X ) φ if as before ∃ γ , . . . , γ n ∈ Γ n ^ i =1 γ i → φ ∈ I ( X ) ! . In the following, we will also write I ⋆ for the particular case of I ( V ar ⋆ ).3. Algebraic semantics for intermediate justification logics
We move on to the first main line of semantics for intermediate justification logics studied here, extendingthe model-theoretic approaches of Mkrtychev, Fitting as well as Lehmann and Studer to take values in arbitraryHeyting algebras. The models which we introduce, as well as the techniques used later to prove correspondingcompleteness theorems, are similar to those from [34] where completeness theorems of the particular case ofG¨odel justification logics with respect to models over the particular Heyting algebra [ , ] G (see the last section)were considered.3.1. Heyting algebras and propositional semantics.
We give some preliminaries on Heyting algebras andtheir relevant notions as a primer for the later definitions.
Definition 3.1. A Heyting algebra is structure A = h A, ∧ A , ∨ A , → A , A , A i such that h A, ∧ A , ∨ A , A , A i isa bounded lattice with largest element 1 A and smallest element 0 A and → A is a binary operation with(1) x → A x = 1 A ,(2) x ∧ A ( x → A y ) = x ∧ A y ,(3) y ∧ A ( x → A y ) = y ,(4) x → A ( y ∧ A z ) = ( x → A y ) ∧ A ( x → A z ),where we write a ≤ A b for a ∧ A b = a .Note, that this order ≤ A on A is always a partial order. Given a Heyting algebra A , we write ¬ A x := x → A A . We call a Heyting algebra A linear if x ≤ A y or y ≤ A x for all x, y ∈ A . A is called a Boolean algebra, if x → A y = ¬ A x ∨ A y for all x, y ∈ A .We collect some facts about Heyting algebras which are of use later. Lemma 3.2.
Let A = h A, ∧ A , ∨ A , → A , A , A i be a Heyting algebra. Then, for all x, y, z, w ∈ A :(1) x ∧ A y ≤ A z iff x ≤ A y → A z ;(2) x ≤ A y iff x → A y = 1 A ;(3) → A x = x ;(4) if x ≤ A y , then y → A z ≤ A x → A z ;(5) ( x → A y ) ∧ A ( z → A w ) ≤ A ( x ∧ A z ) → A ( y ∧ A w ) . These properties are quite immediate from the definition of Heyting algebras. For a modern reference onbasic properties of Heyting algebras, see e.g. [33].Another particular property of Heyting algebras important in this note is that of completeness . Definition 3.3.
A Heyting algebra A is complete if every set X ⊆ A has a join and a meet with respect to ≤ A , that is for every X ⊆ A there are s X , i X ∈ A such that: • ∀ x ∈ X : x ≤ A s X and if x ≤ A s for all x ∈ X , then s X ≤ A s ; • ∀ x ∈ X : i X ≤ A x and if i ≤ A x for all x ∈ X , then i ≤ A i X . NTERMEDIATE JUSTIFICATION LOGICS: UNIFIED COMPLETENESS RESULTS 5
We denote these (unique) joins and meets, s X and i X , by W X and V X , respectively. Given a class ofHeyting algebras C , we write C fin for the subclass of all finite algebras and C com for the subclass of all completeHeyting algebras in C . Naturally, every finite Heyting algebra is complete.Given an (extended) propositional language L ( X ), we can give an algebraic interpretation using variousclasses of particular Heyting algebras. Definition 3.4.
Let A be a Heyting algebra. A propositional evaluation of L ( X ) is a function f : L ( X ) → A which satisfies the following equations:(1) f ( ⊥ ) = 0 A ;(2) f ( φ ∧ ψ ) = f ( φ ) ∧ A f ( ψ );(3) f ( φ ∨ ψ ) = f ( φ ) ∨ A f ( ψ );(4) f ( φ → ψ ) = f ( φ ) → A f ( ψ ).We denote the set of all A -valued propositional evaluations of L ( X ) by Ev ( A ; L ( X )). Definition 3.5.
Let C be a class of Heyting algebras and Γ ∪ { φ } ⊆ L ( X ). We write Γ || = C φ if ∀ A ∈ C : ∀ f ∈ Ev ( A ; L ( X )) : f [Γ] ⊆ { A } implies f ( φ ) = 1 A . If in particular C = { A } , we write || = A for the corresponding relation. Definition 3.6.
Let I be an intermediate logic and let X be a set of variables. We say that I ( X ) is (strongly)complete with respect to a class C of Heyting algebras if for any Γ ∪ { φ } ⊆ L ( X ): Γ ⊢ I ( X ) φ iff Γ || = C φ .Although not particularly important for the rest of the paper, every intermediate logic actually has at leastone class of Heyting algebras with respect to which it is strongly complete (namely its variety). We collect thisin the following fact. Fact 3.7.
For every intermediate logic I and any set of variables X , there is a class of Heyting algebras C suchthat I ( X ) is strongly complete with respect to C . For a modern reference of the proof, see again e.g. [33]. Correspondingly, we introduce the following notation.We write C ∈ Alg ( I ( X )), C ∈ Alg com ( I ( X )) or C ∈ Alg fin ( I ( X )) if C is a class of Heyting algebras, of completeHeyting algebras or of finite Heyting algebras with respect to which I ( X ) is strongly complete. Note, that here C ∈ Alg ( I ( X )) iff C ∈ Alg ( I ( Y ))for arbitrary sets of variables X, Y and similarly for
Alg com ( I ( X )) and Alg fin ( I ( X )).3.2. Algebraic Mkrtychev models.
The first class of semantics which we consider are algebraic Mkrtychevmodels. The classical Mkrtychev models were introduced in [30], originally for the logic of proofs, and mark thefirst non-provability semantics. The generalization of the Mkrtychev models to the other classical justificationlogics J , J T , J is due to Kuznets [22]. In some contexts, especially [4, 25], these models are also calledbasic models. The following algebraic models also generalize the work on [0 , Definition 3.8 (Algebraic Mkrtychev model) . Let A be a Heyting algebra. An ( A -valued) algebraic Mkrtychevmodel is a structure M = h A , Vi such that V : L J → A fulfils(1) V ( ⊥ ) = 0 A ,(2) V ( φ ∧ ψ ) = V ( φ ) ∧ A V ( ψ ),(3) V ( φ ∨ ψ ) = V ( φ ) ∨ A V ( ψ ),(4) V ( φ → ψ ) = V ( φ ) → A V ( ψ ),for all φ, ψ ∈ L J and such that it satisfies(i) V ( t : ( φ → ψ )) ∧ A V ( s : φ ) ≤ A V ([ t · s ] : ψ ),(ii) V ( t : φ ) ∨ A V ( s : φ ) ≤ A V ([ t + s ] : φ ),for all t, s ∈ Jt and φ, ψ ∈ L J .We write M | = φ if V ( φ ) = 1 A and M | = Γ if M | = γ for all γ ∈ Γ where Γ ⊆ L J . Definition 3.9.
Let M = h A , Vi be an A -valued algebraic Mkrtychev model. We call M (1) factive if V ( t : φ ) ≤ A V ( φ ), and(2) introspective if V ( t : φ ) ≤ A V (! t : t : φ ). Definition 3.10.
Let C be a class of Heyting algebras. Then:(1) CAMJ denotes the class of all A -valued Mkrtychev models, for all A ∈ C ; NICHOLAS PISCHKE (2)
CAMJT denotes the class of all factive
CAMJ -models;(3)
CAMJ4 denotes the class of all introspective
CAMJ -models;(4)
CAMJT4 denotes the class of all factive and introspective
CAMJ -models.
Definition 3.11.
Let A be a Heyting algebra and let M = h A , Vi be an algebraic Mkrtychev model. Further,let CS be a constant specification (for some proof calculus). We say that M respects CS if V ( c : φ ) = 1 A forall c : φ ∈ CS .If C is a class of algebraic Mkrtychev models, then we denote the subclass of all models from C respecting aconstant specification CS by C CS . Definition 3.12.
Let C be a class of algebraic Mkrtychev models and let Γ ∪ { φ } ⊆ L J . We write:(1) Γ | = C φ if ∀ M = h A , Vi ∈ C (cid:18) ^ A {V ( γ ) | γ ∈ Γ } ≤ A V ( φ ) (cid:19) ;(2) Γ | = C φ if ∀ M = h A , Vi ∈ C (cid:16) M | = Γ ⇒ M | = φ (cid:17) . Lemma 3.13.
Let I be an intermediate logic, IJ L ∈ {IJ , IJ T , IJ , IJ T } , let CS be a constantspecification logic, and let C ∈ Alg ( I ) . For any Γ ∪ { φ } ⊆ L J : Γ ⊢ IJ L CS φ implies Γ | = CAMJL CS φ. Proof.
We only show that ⊢ IJ L φ implies | = CAMJL φ . This already suffices for the strong completeness statementabove by the following argument using the deduction theorem for the respective logics and compactness of theprovability relations:Γ ⊢ IJ L CS φ impl. ∃ Γ ⊆ Γ ∪ CS finite (Γ ⊢ IJ L φ )impl. ∃ Γ ⊆ Γ ∪ CS finite (cid:16) ⊢ IJ L ^ Γ → φ (cid:17) impl. ∃ Γ ⊆ Γ ∪ CS finite (cid:16) | = CAMJL ^ Γ → φ (cid:17) impl. ∃ Γ ⊆ Γ ∪ CS finite ∀ M = h A , Vi ∈
CAMJL (cid:18) ^ A {V ( γ ) | γ ∈ Γ } ≤ A V ( φ ) (cid:19) impl. ∀ M = h A , Vi ∈
CAMJL (cid:18) ^ A {V ( γ ) | γ ∈ Γ ∪ CS } ≤ A V ( φ ) (cid:19) impl. ∀ M = h A , Vi ∈
CAMJL CS (cid:18) ^ A {V ( γ ) | γ ∈ Γ } ≤ A V ( φ ) (cid:19) impl. Γ | = CAMJL CS φ. We show that ⊢ IJ L φ implies | = CAMJL φ as follows: by the definition of IJ L , it suffices to show that | = CAMJL φ for φ ∈ I as well as for φ ∈ ( J ) ∪ (+) or even (depending on the choice of IJ L ) φ ∈ ( F ) ∪ ( I ) and that itis preserved under modus ponens. The latter is immediate. For the former, note that in the case of φ beinga justification axiom, the choice of CAMJL is such that all models satisfy conditions (i) and (ii) of Definition3.8 (validating ( J ) and (+)) and (depending on IJ L ) are factive or introspective given ( F ) or ( I ) and thusvalidate those immediately.If now φ ∈ I , then by definition there is a subsitution σ : V ar → L J and a formula ψ ∈ I such that φ = σ ( ψ ).Let A ∈ C and M = h A , Vi be a CAMJ -model. Then, we may define f : χ
7→ V ( σ ( χ ))for χ ∈ L . By definition of M and properties of σ , we have that f is a well-defined evaluation on A . By thechoice of C , we have that ψ ∈ I implies f ( ψ ) = 1 A and thus V ( σ ( ψ )) = V ( φ ) = 1. As M was arbitrary, we have | = CAMJ φ . (cid:3) Algebraic Fitting models.
The second algebraic semantics which we consider is based on algebraicFitting models, derived from the fundamental possible-world semantics of Fitting [12, 14] which combined theearlier work of Mkrtychev on syntactic evaluations with the usual semantics of non-explicit modal logics basedon modal Kripke models. As a generalization, we allow the accessibility, evidence and evaluation functions totake values in Heyting algebras. We have to restrict to complete Heyting algebras however, as we want certainalgebraic equations to be satisfied which involve infima and suprema. The algebraic Fitting models presentedhere again generalize the previously introduced many-valued Fitting models from [16, 34] from the context ofthe G¨odel justification logics.
Definition 3.14.
Let A be a complete Heyting algebra. An ( A -valued) algebraic Fitting model is a structure M = h A , W , R , E , Vi with NTERMEDIATE JUSTIFICATION LOGICS: UNIFIED COMPLETENESS RESULTS 7 • W 6 = ∅ , • R : W × W → A , • E : W × Jt × L J → A , • V : W × L J → A ,such that it fulfills the conditions(1) V ( w, ⊥ ) = 0 A ,(2) V ( w, φ ∧ φ ) = V ( w, φ ) ∧ A V ( w, ψ ),(3) V ( w, φ ∨ φ ) = V ( w, φ ) ∨ A V ( w, ψ ),(4) V ( w, φ → φ ) = V ( w, φ ) → A V ( w, ψ ),(5) V ( w, t : φ ) = E w ( t, φ ) ∧ A ^ A {R ( w, v ) → A V ( v, φ ) | v ∈ W} ,for all w ∈ W and such that it satisfies(i) E w ( t, φ → ψ ) ∧ A E w ( s, φ ) ≤ A E w ( t · s, ψ ),(ii) E w ( t, φ ) ∨ A E w ( s, φ ) ≤ A E w ( t + s, φ ).for all w ∈ W , all t, s ∈ Jt and all φ, ψ ∈ L J .We write ( M , w ) | = φ for V ( w, φ ) = 1 A and ( M , w ) | = Γ if ( M , w ) | = γ for all γ ∈ Γ. Definition 3.15.
Let M = h A , W , R , E , Vi be an A -valued Fitting model. We call M (i) reflexive if ∀ w ∈ W (cid:0) R ( w, w ) = 1 A (cid:1) ,(ii) transitive if ∀ w, v, u ∈ W (cid:0) R ( w, v ) ∧ A R ( v, u ) ≤ A R ( w, u ) (cid:1) ,(iii) monotone if ∀ v, w ∈ W∀ t ∈ Jt, φ ∈ L J (cid:0) E w ( t, φ ) ∧ A R ( w, v ) ≤ A E v ( t, φ ) (cid:1) ,(iv) introspective if it is transitive, monotone and satisfies E w ( t, φ ) ≤ A E w (! t, t : φ )for all w ∈ W and all t ∈ Jt , φ ∈ L J ,(v) accessibility-crisp if ∀ w, v ∈ W (cid:0) R ( w, v ) ∈ { A , A } (cid:1) . Definition 3.16.
Let C be a class of complete Heyting algebras. Then:(1) CAFJ denotes the class of all A -valued Fitting models, for all A ∈ C ;(2) CAFJT denotes the class of all reflexive
CAFJ -models;(3)
CAFJ4 denotes the class of all introspective
CAFJ -models;(4)
CFJT4 denotes the class of all
CAFJ4 -models which are reflexive.By C c , we denote the class of all accessibility-crisp models in C for some class C of algebraic Fitting models. Definition 3.17.
Let A be a complete Heyting algebra and let M = h A , W , R , E , Vi be a A -valued algebraicFitting model. We say that M respects a constant specification CS (for some proof system) if V ( w, c : φ ) = 1 A for all w ∈ W and all c : φ ∈ CS .Given a class C of algebraic Fitting model, we denote the subclass of all algebraic Fitting models in C respecting a constant specification CS (for some proof system) by C CS . Definition 3.18.
Let C be a class of algebraic Fitting models and Γ ∪ { φ } ⊆ L J . We write:(1) Γ | = C φ if ∀ M = h A , W , R , E , Vi ∈ C ∀ w ∈ W (cid:18) ^ A {V ( w, γ ) | γ ∈ Γ } ≤ A V ( w, φ ) (cid:19) ;(2) Γ | = C φ if ∀ M = h A , W , R , E , Vi ∈ C ∀ w ∈ W (cid:16) ( M , w ) | = Γ ⇒ ( M , w ) | = φ (cid:17) . Lemma 3.19.
Let I be an intermediate logic and IJ L ∈ {IJ , IJ T , IJ , IJ T } . Let CS be a constantspecification for IJ L and let C ∈ Alg com ( I ) . For any Γ ∪ { φ } ⊆ L J : Γ ⊢ IJ L CS φ implies Γ | = CAFJL CS φ. Proof.
As before, we only show ⊢ IJ L φ implies | = CAFJL φ . The argument from the proof of Lemma 3.13 abouthow to obtain strong soundness can be straightforwardly adapted to the case of algebraic Fitting models.To see that ⊢ IJ L φ implies | = CAFJL φ , note that it is again enough to show the claim for φ ∈ I or φ being ajustification axiom (depending on IJ L ).If φ ∈ I , then by the choice of C one may repeat the argument from the proof of Lemma 3.13 locally forevery V ( w, · ) with w ∈ W to obtain | = CAFJ φ .As the algebraic Fitting models are slightly more complex in their evaluation of the justification modalities,we actually show the validity of the justification axiom schemes in their respective model classes. For this, let M = h A , W , R , E , Vi ∈
CAFJ and let w ∈ W . NICHOLAS PISCHKE (1) Consider the axiom scheme ( J ). Then, we have ^ A {R ( w, v ) → A V ( v, φ → ψ ) | v ∈ W} ∧ A ^ A {R ( w, v ) → A V ( v, φ ) | v ∈ W}≤ A ^ A {R ( w, v ) → A V ( v, ψ ) | v ∈ W} Further we have E w ( t, φ → ψ ) ∧ A E w ( s, φ ) ≤ A E w ( t · s, ψ ) by condition (i) of Definition 3.14. Thus: V ( w, t : ( φ → ψ )) ∧ A V ( w, s : φ )= E w ( t, φ → ψ ) ∧ A ^ A {R ( w, v ) → A V ( v, φ → ψ ) | v ∈ W}∧ A E w ( s, φ ) ∧ A ^ A {R ( w, v ) → A V ( v, φ ) | v ∈ W}≤ A E w ( t · s, ψ ) ∧ A ^ A {R ( w, v ) → A V ( v, ψ ) | v ∈ W} = V ( w, [ t · s ] : ψ ) . The claim follows from the above as by residuation, we have V ( w, t : ( φ → ψ )) ≤ A V ( w, s : φ ) → A V ( w, [ t · s ] : ψ ).(2) Consider the axiom scheme (+). We only show V ( w, t : φ ) ≤ A V ( w, [ t + s ] : φ ). The other part followssimilarly. By condition (ii) of Definition 3.14, we have E w ( t, φ ) ≤ A E w ( t, φ ) ∨ A E w ( s, φ ) ≤ A E w ( t + s, φ )and thus V ( w, t : φ ) = E w ( t, φ ) ∧ A ^ A {R ( w, v ) → A V ( v, φ ) | v ∈ W}≤ A E w ( t + s, φ ) ∧ A ^ A {R ( w, v ) → A V ( v, φ ) | v ∈ W} = V ( w, [ t + s ] : φ ) . (3) Consider the axiom scheme ( F ) and assume that M is reflexive. We have R ( w, w ) = 1 A and thus: V ( w, t : φ ) = E w ( t, φ ) ∧ A ^ A {R ( w, v ) → A V ( v, φ ) | v ∈ W}≤ A ^ A {R ( w, v ) → A V ( v, φ ) | v ∈ W}≤ A R ( w, w ) → A V ( w, φ )= V ( w, φ ) . (4) Consider the axiom scheme ( I ). Assume that M is introspective. By the transitivity of R , we have atfirst ^ A {R ( w, u ) → A V ( u, φ ) | u ∈ W} ≤ A R ( w, v ) → A ^ A {R ( v, u ) → A V ( u, φ ) | u ∈ W} . To see this, note that we have R ( w, v ) ∧ A ^ A {R ( w, u ) → A V ( u, φ ) | u ∈ W} ≤ A R ( w, v ) ∧ A (cid:0) R ( w, u ) → A V ( u, φ ) (cid:1) ≤ A R ( w, v ) ∧ A (cid:0)(cid:0) R ( w, v ) ∧ A R ( v, u ) (cid:1) → A V ( u, φ ) (cid:1) = R ( w, v ) ∧ A (cid:0) R ( w, v ) → A ( R ( v, u )) → A V ( u, φ ) (cid:1) ≤ A R ( w, u ) → A V ( u, φ )for all u ∈ W . By taking the infimum over u , we have R ( w, v ) ∧ A ^ A {R ( w, u ) → A V ( u, φ ) | u ∈ W} ≤ A ^ A {R ( v, u ) → A V ( u, φ ) | u ∈ W} . Further, we have by monotonicity that E w ( t, φ ) ≤ A R ( w, v ) → A E v ( t, φ ) . NTERMEDIATE JUSTIFICATION LOGICS: UNIFIED COMPLETENESS RESULTS 9
Therefore, we have V ( w, t : φ ) = E w ( t, φ ) ∧ A ^ A {R ( w, u ) → A V ( u, φ ) | u ∈ W}≤ A (cid:0) R ( w, v ) → A E v ( t, φ ) (cid:1) ∧ A (cid:18) R ( w, v ) → A ^ A {R ( v, u ) → A V ( u, φ ) | u ∈ W} (cid:19) ≤ A R ( w, v ) → A (cid:18) E v ( t, φ ) ∧ A ^ A {R ( v, u ) → A V ( u, φ ) | u ∈ W} (cid:19) = R ( w, v ) → A V ( v, t : φ ) . By taking the infimum, we have E w ( t, φ ) ∧ A ^ A {R ( w, u ) → A V ( u, φ ) | u ∈ W} ≤ A ^ A {R ( w, v ) → A V ( v, t : φ ) | v ∈ W} . Thus, we have for any v ∈ W by introspectivity: V ( w, t : φ ) ≤ A E w (! t, t : φ ) ∧ A ^ A {R ( w, v ) → A V ( v, t : φ ) | v ∈ W} = V ( w, ! t : t : φ ) . (cid:3) Algebraic subset models.
The last algebraic semantics which we consider is based on algebraic gener-alizations of the subset models for classical justification logic by Lehmann and Studer [26]. Similar as with theprevious algebraic Fitting models, we allow all involved functions to take arbitrary values in Heyting algebras,again restricting ourselves to complete Heyting algebras to be able to formulate certain regularity conditions.
Definition 3.20 (Algebraic subset model) . Let A be a complete Heyting algebra with domain A . An ( A -valued)algebraic subset model is a structure M = h A , W , W , E , Vi with • W 6 = ∅ , • W ⊆ W , W = ∅ , • E : Jt × W × W → A , • V : W × L J → A ,such that for all w ∈ W , V fulfills the conditions(1) V ( w, ⊥ ) = 0 A ,(2) V ( w, φ ∧ ψ ) = V ( w, φ ) ∧ A V ( w, ψ ),(3) V ( w, φ ∨ ψ ) = V ( w, φ ) ∨ A V ( w, ψ ),(4) V ( w, φ → ψ ) = V ( w, φ ) → A V ( w, ψ ),(5) V ( w, t : φ ) = ^ A {E t ( w, v ) → A V ( v, φ ) | v ∈ W} ,and such that it is regular , that is for all w ∈ W :(i) E s + t ( w, v ) ≤ A E s ( w, v ) ∧ A E t ( w, v ) for all v ∈ W ;(ii) for all v ∈ W : E s · t ( w, v ) ≤ A ^ A { M ws,t ( ψ ) → A V ( v, ψ ) | ψ ∈ L J } with M ws,t ( ψ ) := _ A {V ( w, s : ( φ → ψ )) ∧ A V ( w, t : φ ) | φ ∈ L J } . We write ( M , w ) | = φ for V ( w, φ ) = 1 A and ( M , w ) | = Γ for V ( w, γ ) = 1 A for all γ ∈ Γ.The function E is actually a straightforward A -valued generalization of the E -function from [26] as it is infact nothing more than a different representation of the function E : Jt × W → A W which maps terms and worlds to A -valued subsets of W . Definition 3.21.
Let M = h A , W , W , E , Vi be an A -valued subset model. We call M (i) reflexive if ∀ w ∈ W ∀ t ∈ Jt (cid:0) E t ( w, w ) = 1 A (cid:1) ,(ii) introspective if ∀ w ∈ W ∀ v ∈ W∀ t ∈ Jt (cid:18) E ! t ( w, v ) ≤ A ^ A {V ( w, t : φ ) → A V ( v, t : φ ) | φ ∈ L J } (cid:19) ,(iii) accessibility-crisp if ∀ t ∈ Jt ∀ w, v ∈ W (cid:0) E t ( w, v ) ∈ { A , A } (cid:1) . Definition 3.22.
Let C be a class of complete Heyting algebras. Then: (1) CASJ denotes the class of all A -valued subset models, for all A ∈ C ;(2) CASJT denotes the class of all A -valued reflexive subset models, for all A ∈ C ;(3) CASJ4 denotes the class of all A -valued introspective subset models, for all A ∈ C ;(4) CASJT4 denotes the class of all A -valued reflexive and introspective subset models, for all A ∈ C .Given a class C of algebraic subset models, we denote the class of all accessibility-crisp models in C by C c . Definition 3.23.
Let A be a complete Heyting algebra and let M = h A , W , W , E , Vi be a A -valued algebraicsubset model. Further, let CS be a constant specification (for some proof calculus). We say that M respects CS if V ( w, c : φ ) = 1 A for all c : φ ∈ CS and all w ∈ W .Given a class C of algebraic subset models, we write C CS for the class of all models from C which respect CS . As before, there are two natural consequence relations to consider here. Definition 3.24.
Let Γ ∪ { φ } ⊆ L J and C be a class of algebraic subset models. We write(1) Γ | = C φ if ∀ M = h A , W , W , E , Vi ∈ C ∀ w ∈ W (cid:18) ^ A {V ( w, γ ) | γ ∈ Γ } ≤ A V ( w, φ ) (cid:19) ;(2) Γ | = C φ if ∀ M = h A , W , W , E , Vi ∈ C ∀ w ∈ W (cid:16) ( M , w ) | = Γ ⇒ ( M , w ) | = φ (cid:17) .We write Γ | = A JL φ or Γ | = A JL φ for Γ | = { A } JL φ or Γ | = { A } JL φ , respectively. Lemma 3.25.
Let I be an intermediate logic and IJ L ∈ {IJ , IJ T , IJ , IJ T } . Further, let CS be aconstant specification for IJ L and let C ∈ Alg com ( I ) . For any Γ ∪ { φ } ⊆ L J , we have: Γ ⊢ IJ L CS φ implies Γ | = CASJL CS φ. Proof.
By the same reasoning as in Lemmas 3.13 and 3.13, we only show ⊢ IJ L φ implies | = CASJL φ . Similarly,it suffices to show the claim for φ ∈ I as well as the justifications axioms (based on IJ L ).We may repeat the argument from the previous soundness proofs that φ ∈ I implies | = CASJ φ by constructinga similar propositional evaluation f locally for every V ( w, · ) over every w ∈ W .We thus only show the validity of (1) ( J ), (2) (+), (3) ( F ) and (4) (4) in their respective model classes. Forthis, let M = h A, W , W , E , Vi be a CASJL CS -model and let w ∈ W .(1) We show( † ) V ( w, t : ( φ → ψ )) ∧ A V ( w, s : φ ) ≤ A V ( w, [ t · s ] : ψ ) . For this, let v ∈ W . We then have V ( w, t : ( φ → ψ )) ∧ A V ( w, s : φ ) ≤ A _ A {V ( w, t : ( φ → ψ )) ∧ A V ( w, s : φ ) | φ ∈ L J }≤ A E t · s ( w, v ) → A V ( v, ψ )through condition (ii) of Definition 3.20. Therefore, we have V ( w, t : ( φ → ψ )) ∧ A V ( w, s : φ ) ≤ A ^ A {E t · s ( w, v ) → A V ( v, ψ ) | v ∈ W} = V ( w, [ t · s ] : ψ )as v was arbitrary, which is ( † ).(2) Let v ∈ W . We have V ( w, t : φ ) = ^ A {E t ( w, u ) → A V ( u, φ ) | u ∈ W}≤ A E t ( w, v ) → A V ( v, φ ) ≤ A E t + s ( w, v ) → A V ( v, φ )through condition (i) in Definition 3.20. As v was arbitrary, we have V ( w, t : φ ) ≤ A ^ A {E t + s ( w, v ) → A V ( v, φ ) | v ∈ W} = V ( w, [ t + s ] : φ ) . One shows similarly that V ( w, s : φ ) ≤ A V ( w, [ t + s ] : φ ).(3) M is reflexive by assumption. Therefore, we have E t ( w, w ) = 1 A as w ∈ W and thus V ( w, t : φ ) = ^ A {E t ( w, v ) → A V ( v, φ ) | v ∈ W}≤ A E t ( w, w ) → A V ( w, φ ) ≤ A V ( w, φ ) . Therefore, V ( w, t : φ ) → A V ( w, φ ) = 1. NTERMEDIATE JUSTIFICATION LOGICS: UNIFIED COMPLETENESS RESULTS 11 (4) M is introspective by assumption. Thus, we have V ( w, t : φ ) ≤ A E ! t ( w, v ) → A V ( v, t : φ )for any v ∈ W by the introspectivity and we have therefore: V ( w, t : φ ) ≤ A ^ A {E ! t ( w, v ) → A V ( v, t : φ ) | v ∈ W} = V ( w, ! t : t : φ ) . (cid:3) Completeness for algebraic semantics
To approach completeness, we translate the language L J to L ⋆ by introducing the translation ⋆ : L J → L ⋆ using recursion on L J with the following clauses: • ⊥ ⋆ := ⊥ ; • p ⋆ := p ; • ( φ ◦ ψ ) ⋆ := φ ⋆ ◦ ψ ⋆ with ◦ ∈ {∧ , ∨ , →} ; • ( t : φ ) ⋆ := φ t .Using the above translation, we can convert formulae containing justification modalities into formulae of L ⋆ and use semantic results for the intermediate logic in question over L ⋆ to derive results for the correspondingintermediate justification logic. This approach, especially in the context of algebra-valued modal logics, goesback to Caicedo and Rodriguez work [6] (see also [39]) and was previously also applied in the context of many-valued justification logics (see [34]).For this, the following lemma provides a way to interpret modal systems in extended propositional systems.For this, given a proof calculus S over a language L , we write T h S := { φ ∈ L | ⊢ S φ } . Lemma 4.1.
Let I be an intermediate logic and IJ L ∈ {IJ , IJ T , IJ , IJ T } and CS be a constantspecification for IJ L . For any Γ ∪ { φ } ⊆ L J : Γ ⊢ IJ L CS φ iff Γ ⋆ ∪ ( T h
IJ L CS ) ⋆ ⊢ I ⋆ φ ⋆ . Proof.
We prove both directions separately. In any way, recall that ⋆ is a bijection between L J and L ⋆ .For the direction from left to right, notice that it suffices to show Γ ⋆ ∪ ( T h
IJ L CS ) ⋆ ⊢ I ⋆ φ ⋆ for(i) φ ∈ Γ, or(ii) φ ∈ CS , or(iii) φ ∈ IJ L ,and that it is preserved under modus ponens. The latter is obvious by definition of ⋆ . For (i) of the former, wehave φ ⋆ ∈ Γ ⋆ and thus Γ ⋆ ∪ ( T h
IJ L CS ) ⋆ ⊢ I ⋆ φ ⋆ . For (ii) and (iii), we have ⊢ IJ L CS φ , and thus φ ⋆ ∈ ( T h
IJ L CS ) ⋆ .This gives again Γ ⋆ ∪ ( T h
IJ L CS ) ⋆ ⊢ I ⋆ φ ⋆ .For the direction from right to left, note that also here it suffices to show Γ ⊢ IJ L CS φ for(a) φ ⋆ ∈ Γ ⋆ , or(b) φ ⋆ ∈ ( T h
IJ L CS ) ⋆ , or(c) φ ⋆ ∈ I ⋆ ,and that also here, it is preserved under modus ponens. The latter is again immediate. For (a) of the former,we have φ ∈ Γ which gives Γ ⊢ IJ L CS φ directly. For (b), we have ⊢ IJ L CS φ by definition. For (c), we have φ ∈ I . To see this, note that by the definition of I ⋆ = I ( L ⋆ ), we have φ ⋆ = σ ( ψ ) for some ψ ∈ I and somebijection t : V ar → V ar ⋆ . Now, the function σ t : p ( q if t ( p ) = qs : φ if t ( p ) = φ s is a substitution from V ar to L J and we have φ = σ t ( ψ ) . Thus, we have φ ∈ I and thus φ ∈ IJ L , i.e. Γ ⊢ IJ L CS φ . (cid:3) The rest of this section is devoted countermodel constructions, converting algebraic evaluations of L ⋆ intocorresponding algebraic Mkrtychev, Fitting or subset models and deriving corresponding completeness resultsfor the intermediate justification logics from this. Completeness w.r.t. algebraic Mkrtychev models.Definition 4.2.
Let I be an intermediate logic and let IJ L ∈ {IJ , IJ T , IJ , IJ T } where CS isa constant specification for IJ L . Let A be a Heyting algebra and v ∈ Ev ( A ; L ⋆ ). The canonical algebraicMkrtychev model w.r.t. A and v is the structure M c,M A ,v ( IJ L CS ) = h A , V c i defined by: V c ( φ ) := v ( φ ⋆ ) . Lemma 4.3.
For any Heyting algebra A , any v ∈ Ev ( A ; L ⋆ ) with v [( T h
IJ L CS ) ⋆ ] ⊆ { A } and any choice of IJ L CS , M c,M A ,v ( IJ L CS ) is a well-defined A -valued algebraic Fitting model. Further:(a) if ( F ) is an axiom scheme of IJ L CS , then M c,M A ,v ( IJ L CS ) is factive;(b) if ( I ) is an axiom scheme of IJ L CS , then M c,M A ,v ( IJ L CS ) is introspective.Proof. As v ∈ Ev ( A ; L ⋆ ), we have items (1) - (4) from Definition 3.8. Then, as additionally v [( T h
IJ L CS ) ⋆ ] ⊆{ A } , we have V c ( t : ( φ → ψ )) ∧ A V c ( s : φ ) = v (( φ → ψ ) t ) ∧ A v ( φ s ) ≤ A v ( ψ [ t · s ] )= V c ([ t · s ] : ψ )and V c ( t : φ ) ∨ A V c ( s : φ ) = v ( φ t ) ∨ A v ( φ s ) ≤ A v ( φ [ t + s ] )= V c ([ t + s ] : φ )regarding items (i) and (ii) of Definition 3.8. Now, regarding (a), if ( F ) is an axiom scheme of IJ L CS , wenaturally have V c ( t : φ ) = v ( φ t ) ≤ v ( φ ⋆ ) = V c ( φ ) . As for item (b), if ( I ) is an axiom scheme of IJ L CS , we have V c ( t : φ ) = v ( φ t ) ≤ A v (( t : φ ) ! t ) = V c (! t : t : φ ) . (cid:3) Theorem 4.4.
Let I be an intermediate logic and let IJ L ∈ {IJ , IJ T , IJ , IJ T } where CS is aconstant specification for IJ L . Further, let C ∈ Alg ( I ) .For any Γ ∪ { φ } ⊆ L J , the following are equivalent:(1) Γ ⊢ IJ L CS φ ;(2) Γ | = CAMJL CS φ ;(3) Γ | = CAMJL CS φ .Proof. (1) implies (2) comes from Lemma 3.13 and (2) implies (3) is natural. For (3) implies (1), supposeΓ IJ L CS φ . Then, by Lemma 4.1, we haveΓ ⋆ ∪ ( T h
IJ L CS ) ⋆ I ⋆ φ ⋆ which gives that there exists a A ∈ C and v ∈ Ev ( A ; L ⋆ ) such that v [Γ ⋆ ] ⊆ { A } , v [( T h
IJ L CS ) ⋆ ] ⊆ { A } and v ( φ ) < A by assumption on C . By Lemma 4.3, we have that M c,M A , v ( IJ L CS ) is a well-defined CAMJL -model and bydefinition, it follows that: M | = Γ and M = φ. Also, M c,M A ,v ( IJ L CS ) respects CS . As we have ⊢ IJ L CS c : φ for c : φ ∈ CS , we have φ c ∈ ( T h
IJ L CS ) ⋆ andthus v ( φ c ) = 1 A . By definition, we have V c ( c : φ ) = v ( φ c ) = 1 A . Thus, we have Γ = CAMJL CS φ . (cid:3) NTERMEDIATE JUSTIFICATION LOGICS: UNIFIED COMPLETENESS RESULTS 13
Completeness w.r.t. algebraic Fitting models.Definition 4.5.
Let I be an intermediate propositional logic and let IJ L ∈ {IJ , IJ T , IJ , IJ T } where CS is a constant specification for IJ L . Let A be a complete Heyting algebra. The canonical algebraicFitting model w.r.t. A is the structure M c,F A ( IJ L CS ) = h A , W c , W c , E c , V c i defined as follows: • W c := { v ∈ Ev ( A ; L ⋆ ) | v [( T h
IJ L CS ) ⋆ ] ⊆ { A }} ; • R c ( v, w ); = ( A if ∀ t ∈ Jt ∀ φ ∈ L J (cid:0) v ( φ t ) ≤ A w ( φ ⋆ ) (cid:1) ;0 A otherwise; • E cv ( t, φ ) := v ( φ t ); • V c ( v, φ ) := v ( φ ⋆ ). Lemma 4.6.
For any complete Heyting algebra A and any choice of IJ L CS , M c,F A ( IJ L CS ) is a well-defined A -valued algebraic Fitting model. Further:(a) if ( F ) is an axiom scheme of IJ L CS , then M c,F A ( IJ L CS ) is reflexive;(b) if ( I ) is an axiom scheme of IJ L CS , then M c,F A ( IJ L CS ) is introspective.Proof. Condition (1) - (4) from Definition 3.14 follow immediately for any v ∈ W c as v ∈ Ev ( A ; L ⋆ ) and by thedefinition of ⋆ . For item (5), we have v ( φ t ) ≤ A w ( φ ⋆ )for any w ∈ W c with R c ( v, w ) = 1 A . Thus, we have v ( φ t ) ≤ A ^ A { w ( φ ⋆ ) | w ∈ W c , R c ( v, w ) = 1 A } = ^ A {R c ( v, w ) → A w ( φ ⋆ ) | w ∈ W c } . Therefore E cv ( t, φ ) ∧ A ^ A {R c ( v, w ) → A w ( φ ⋆ ) | w ∈ W c } = v ( φ t ) . For item (i), note that E cv ( t, φ → ψ ) ∧ A E cv ( s, φ ) = v (( φ → ψ ) t ) ∧ A v ( φ s ) ≤ A v ( ψ [ t · s ] )= E cv ( t · s, ψ )where the inequality follows using the axiom scheme ( J ) as v [( T h
IJ L CS ) ⋆ ] ⊆ { A } and v ∈ Ev ( A ; L ⋆ ).For item (ii), note that E cv ( t, φ ) = v ( φ t ) ≤ A v ( φ [ t + s ] ) = E cv ( t + s, φ )and similarly for s through the axiom scheme (+) as again v [( T h
IJ L CS ) ⋆ ] ⊆ { A } and v ∈ Ev ( A ; L ⋆ ). Thus,we have E cv ( t, φ ) ∨ A E cv ( s, φ ) ≤ A E cv ( t + s, φ ) . On to item (a), if ( F ) is an axiom scheme of IJ L CS , then we naturally have v ( φ t ) ≤ A v ( φ ⋆ )for any φ ∈ L J and any t ∈ Jt . Thus, especially we have R c ( v, v ) = 1 A by definition and thus R c is reflexive.For item (b), note at first that by the axiom scheme ( I ), we have E cv ( t, φ ) = v ( φ t ) ≤ A v (( t : φ ) ! t ) = E cv (! t, t : φ )for any φ ∈ L J and any t ∈ Jt by definition of the canonical model. Further, we have that R c is transitive. Forthis, let R c ( v, w ) = R c ( w, u ) = 1 A . Then, we have for any φ ∈ L J and any t ∈ Jt : v ( φ t ) ≤ A v (( t : φ ) ! t ) ≤ A w ( φ t ) ≤ A u ( φ ⋆ )and thus R c ( v, u ) = 1 A . For the property of monotonicity, suppose R c ( v, w ) = 1 A . Then, we have E cv ( t, φ ) = v ( φ t ) ≤ A v (( t : φ ) ! t ) ≤ A w ( φ t ) = E cw ( t, φ )which is monotonicity. (cid:3) Theorem 4.7.
Let I be an intermediate logic and let IJ L ∈ {IJ , IJ T , IJ , IJ T } where CS is aconstant specification for IJ L . Further, let C ∈ Alg com ( I ) .For any Γ ∪ { φ } ⊆ L J , the following are equivalent:(1) Γ ⊢ IJ L CS φ ;(2) Γ | = CAFJL CS φ ;(3) Γ | = CAFJL CS φ ;(4) Γ | = CAFJL c CS φ . Proof.
Again, (1) implies (2) follows from Lemma 3.19 and (2) implies (3) as well as (3) implies (4) are obvious.Thus, suppose Γ IJ L CS φ . Then, by Lemma 4.1, we have thatΓ ⋆ ∪ ( T h
IJ L CS ) ⋆ G ⋆ φ ⋆ . By the assumption on C , we have that there exists a A ∈ C and a v ∈ Ev ( A , L ⋆ ) such that v [Γ ⋆ ] ⊆ { A } , v [( T h
IJ L CS ) ⋆ ] ⊆ { A } but v ( φ ⋆ ) = 1 A . Thus, by definition of M c,F A ( IJ L CS ), we have v ∈ W c . Further, by definition we have V c ( v, γ ) = 1 A for all γ ∈ Γbut V c ( v, φ ) = 1 A . Using Lemma 4.6, M c,F A ( IJ L CS ) is again a well-defined CAFJL -model and as before, itrespects CS . Thus, we have Γ = CAFJL c CS φ as M c,F A ( IJ L CS ) is accessibility-crisp. (cid:3) Completeness w.r.t. algebraic subset models.Definition 4.8.
Let I be an intermediate propositional logic and let IJ L ∈ {IJ , IJ T , IJ , IJ T } where CS is a constant specification for IJ L . Let A be a complete Heyting algebra. The canonical algebraicsubset model w.r.t. A is the structure M c,S A ( IJ L CS ) = h A , W c , W c , E c , V c i defined as follows: • W c := A L J ; • W c := { v ∈ Ev ( A ; L ⋆ ) | v [( T h
IJ L CS ) ⋆ ] ⊆ { A }} ; • E ct ( v, w ) := ( A if ∀ φ ∈ L J (cid:0) v ( φ t ) ≤ A w ( φ ⋆ ) (cid:1) ;0 A otherwise; • V c ( v, φ ) := v ( φ ⋆ ). Lemma 4.9.
For any complete Heyting algebra A and any choice of IJ L CS , M c,S A ( IJ L CS ) is well-defined A -valued algebraic subset model. Further:(a) if ( F ) is an axiom scheme of IJ L CS , then M c,S A ( IJ L CS ) is reflexive;(b) if ( I ) is an axiom scheme of IJ L CS , then M c,S A ( IJ L CS ) is introspective.Proof. To show that M c A ( IJ L CS ) is well-defined, we have to verify the conditions (1) - (5) and (i), (ii) fromDefinition 3.20. For this, let v ∈ W c . We only show (5) from the former, as (1) - (4) follows naturally from v ∈ Ev ( A ; L ⋆ ).For (5), we show the equality in two steps. At first, note that ^ A {E ct ( v, w ) → A V c ( w, φ ) | w ∈ W c } = ^ A {E ct ( v, w ) → A w ( φ ⋆ ) | w ∈ W c } = ^ A { w ( φ ⋆ ) | w ∈ W c , E ct ( v, w ) = 1 A } . Now, by definition we have V c ( v, t : φ ) = v ( φ t ) ≤ A w ( φ ⋆ )for any w ∈ W c . Thus, we naturally have V c ( v, t : φ ) ≤ A ^ A { w ( φ ⋆ ) | w ∈ W c , E ct ( v, w ) = 1 A } . For the other direction, consider v t : L J → A, ψ ⋆ ψ t . Then, we have that v t ∈ W c and further v ( ψ t ) ≤ A v t ( φ ⋆ )by definition. Thus E ct ( v, v t ) = 1 A and therefore ^ A { w ( φ ⋆ ) | w ∈ W c , E ct ( v, w ) = 1 A } ≤ A v t ( φ ⋆ ) = v ( φ t ) . Let further w ∈ W c .(i) Suppose E ct + s ( v, w ) = 1 A . Then, we have (as v ∈ Ev ( A ; L ⋆ ) and v [( T h
IJ L CS ) ⋆ ] ⊆ { A } ) v ( φ t ) ≤ A v ( φ [ t + s ] ) ≤ A w ( φ ⋆ )through axiom scheme (+) for any φ ∈ L J and similarly for v ( φ s ). Thus, we have E cs ( v, w ) = E ct ( v, w ) =1 A . NTERMEDIATE JUSTIFICATION LOGICS: UNIFIED COMPLETENESS RESULTS 15 (ii) Suppose E ct · s ( v, w ) = 1 A . We write ( M c ) vt,s as a shorthand for ( M c,S A ( IJ L CS )) vt,s . Then to show( M c ) vt,s ( ψ ) ≤ A w ( ψ ⋆ )for every ψ ∈ L J , it suffices to show (for an arbitrary ψ ∈ L J ):( † ) ( M c ) vt,s ( ψ ) ≤ A v ( ψ t · s ) . ( † ) however follows from( M c ) vt,s ( ψ ) = _ A {V c ( v, t : ( φ → ψ )) ∧ A V c ( v, s : φ ) | φ ∈ L J } = _ A { v (( φ → ψ ) t ) ∧ A v ( φ s ) | φ ∈ L J }≤ A v ( ψ t · s ) . It remains to show items (a) and (b).(a) Assume that ( F ) is an axiom scheme of IJ L CS . Let v ∈ W c and let t ∈ Jt . We then have naturallythat v ( φ ⋆ → ψ ⋆ ) ∧ A v ( φ ⋆ ) ≤ A v ( ψ ⋆ )for any φ, ψ ∈ L J as v ∈ Ev ( A ; L ⋆ ). Now, using that ( F ) is an axiom scheme of IJ L CS , we have forany φ ∈ L J that v ( φ t → φ ⋆ ) = 1 A through v [( T h
IJ L CS ) ⋆ ] ⊆ { A } and thus v ( φ t ) ≤ A v ( φ ⋆ )as v ∈ Ev ( A ; L ⋆ ) again. This gives E ct ( v, v ) = 1 A .(b) Assume that ( I ) is an axiom scheme of IJ L CS and let again v ∈ W c , w ∈ W c and t ∈ Jt . Assume E ! t ( v, w ) = 1 A and let φ ∈ L J be arbitrary. We have, as v ( φ t ) ≤ A v (( t : φ ) ! t )through v ∈ W c , that V ( v, t : φ ) = v ( φ t ) ≤ A v (( t : φ ) ! t ) ≤ A w ( φ t )= V ( w, t : φ )where the last inequality follows from E ! t ( v, w ) = 1 A . (cid:3) Theorem 4.10.
Let I be an intermediate logic and let IJ L ∈ {IJ , IJ T , IJ , IJ T } where CS is aconstant specification for IJ L . Let further C ∈ Alg com ( I ) For any Γ ∪ { φ } ⊆ L J , the following are equivalent:(1) Γ ⊢ IJ L CS φ ;(2) Γ | = CASJL CS φ ;(3) Γ | = CASJL CS φ ;(4) Γ | = CASJL c CS φ .Proof. (1) implies (2) comes from Lemma 3.25. (2) implies (3) and (3) implies (4) is obvious. So, assume thatΓ IJ L CS φ . Through Lemma 4.1, we first haveΓ ⋆ ∪ ( T h
IJ L CS ) ⋆ I ⋆ φ ⋆ By the choice of C , there is a complete Heyting algebra A ∈ C with an evaluation v ∈ Ev ( A ; L ⋆ ) such that( † ) v [Γ ⋆ ] ∪ v [( T h
IJ L CS ) ⋆ ] ⊆ { A } but v ( φ ⋆ ) = 1 A . By Lemma 4.9, we have that M c,S A ( IJ L CS ) ∈ CASJL through ( † ). Also M c,S A ( IJ L CS ) naturally respects CS and is accessibility-crisp. Further, through ( † ) and the definition of M c,S A ( IJ L CS ), we have V c ( v, γ ) = 1 A for all γ ∈ Γ but V c ( v, φ ) = 1 A and thus again per definition Γ = CASJL c CS φ . (cid:3) Frame semantics for intermediate justification logics
As a second semantic approach, we extend not Heyting algebras but intuitionistic Kripke frames for inter-mediate logics with the semantic machinery of the models of Mkrtychev, Fitting or of Lehmann and Studer.This extends the work on intuitionistic Mkrtychev and Fitting models (under different terminology) fromMarti and Studer in [27] to wider classes of logics. The intuitionistic subset models based on Kripke framesintroduced later are completely new in the literature.5.1.
Kripke frames and propositional semantics.
We review some concepts from Kripke frames for propo-sitional intermediate logics (see e.g. [15, 31]). For this, we need some terminology from the context of the theoryof partial orders first.
Definition 5.1.
We call a partial order, that is a structure h F, ≤i such that ≤ is a binary relation on thenon-empty set F which satisfies the conditions(1) x ≤ x (reflexivity),(2) x ≤ y and y ≤ x implies x = y (antisymmetry),(3) x ≤ y and y ≤ z implies x ≤ z (transitivity),for all x, y, z ∈ F , a Kripke frame .A set X ⊆ F is called a cone (or upset ), if ∀ x ∈ X ∀ y ∈ F ( x ≤ y ⇒ y ∈ X ) . We denote the smallest cone containing a set X of a partial order h F, ≤i by ↑ X . A cone X is called principal if X = ↑ { x } for some element x . It is straightforward that ↑ { x } = { y ∈ F | y ≥ x } and that ↑ X = [ x ∈ X ↑ { x } . A Kripke frame G = h G, ≤ ′ i is an (induced) subframe of a Kripke frame F = h F, ≤i , if G ⊆ F and ≤ ′ = ≤∩ ( G × G ). In this case, we also write G = F ↾ G . A Kripke frame is called principal if its domain is principal. Definition 5.2.
Let F = h F, ≤i be a Kripke frame. A ( L ( X ) -)Kripke model based on F is a structure M = h F , (cid:13) i with (cid:13) ⊆ F × X which satisfies x ≤ y and x (cid:13) p implies y (cid:13) p for all p ∈ X .A Kripke model N = h G , (cid:13) ′ i is called an (induced) submodel of a Kripke model M = h F , (cid:13) i if G is an inducedsubframe of F and for all p ∈ X : { x ∈ G | x (cid:13) ′ p } = { x ∈ F | x (cid:13) p } ∩ G. We write N = M ↾ G in this case.Given a Kripke model M = h F , (cid:13) i , we introduce the satisfaction relation | = for formulae from L ( X ) asfollows. Given a x ∈ F , we define recursively: • ( M , x ) = ⊥ ; • ( M , x ) | = p if x (cid:13) p ; • ( M , x ) | = φ ∧ ψ if ( M , x ) | = φ and ( M , x ) | = ψ ; • ( M , x ) | = φ ∨ ψ if ( M , x ) | = φ or ( M , x ) | = ψ ; • ( M , x ) | = φ → ψ if ∀ y ∈ F ( x ≤ y ⇒ ( M , x ) = φ or ( M , x ) | = ψ ).We write M | = φ if ( M , x ) | = φ for any x ∈ F , ( M , x ) | = Γ if ( M , x ) | = γ for all γ ∈ Γ and M | = Γ if ( M , x ) | = Γfor all x ∈ F .A fundamental property of Kripke models is that the monotonicity of propositional variables extends to allformulae. More precisely, we have the following: Lemma 5.3.
Let M = h F , (cid:13) i be a L ( X ) Kripke model. Then, for all φ ∈ L ( X ) and all x, y ∈ F : x ≤ y and ( M , x ) | = φ implies ( M , y ) | = φ. The proof is an easy induction on the structure of L ( X ). Given a class of Kripke frames C , we write M ( C ; L ( X )) for the class of all Kripke models over L ( X ) with underlying Kripke frames from C . Given asingle frame F , we also write M ( F ; L ( X )) for M ( { F } ; L ( X )).Using these definitions, there are now two definitions of consequence to consider. NTERMEDIATE JUSTIFICATION LOGICS: UNIFIED COMPLETENESS RESULTS 17
Definition 5.4.
Let Γ ∪ { φ } ⊆ L ( X ) and C be a class of Kripke models. Then, we write:(1) Γ | = C φ if ∀ M ∈ C ∀ x ∈ D ( M ) (cid:16) ( M , x ) | = Γ ⇒ ( M , x ) | = φ (cid:17) ;(2) Γ | = g C φ if ∀ M ∈ C (cid:16) M | = Γ ⇒ M | = φ (cid:17) .Further, if C is now a class of Kripke frames, we write:(3) Γ | = C φ if Γ | = M ( C ; L ( X )) φ ;(4) Γ | = g C φ if Γ | = g M ( C ; L ( X )) φ . Definition 5.5.
Let I be an intermediate logic, X a countably infinite set of variables and C be a class ofKripke frames.(1) We say that I ( X ) is strongly complete w.r.t. C if Γ ⊢ I ( X ) φ iff Γ | = C φ .(2) We say that I ( X ) is strongly globally complete w.r.t. C if Γ ⊢ I ( X ) φ iff Γ | = g C φ .Given a class of Kripke frames C , we write C ∈ KFr ( I ) or C ∈ KFr g ( I ) if I is strongly (locally) complete orstrongly globally complete w.r.t. C , respectively. We also write C ∈ KFr ( I ) ∩ KFr g ( I ) for C ∈ KFr ( I ) and C ∈ KFr g ( I ).The global version will later prove to be important in the completeness considerations. Two things shallbe noted in this context. First, it is well known that there are Kripke incomplete intermediate logics, that isintermediate logics where there is no class of Kripke frames for which the logic is (even weakly) complete. Thisis (well-known to be) connected with the corresponding problem of Kripke incomplete modal logics and thefirst such logic was constructed in [37]. All following considerations involving propositional completeness w.r.t.classes of Kripke frames thus implicitly assume that such a class exists.Further, if an intermediate logic is characterized by a class of Kripke frames locally , there is a simple extendedclass of frames which characterizes the logic globally . More precisely, we have the following: Lemma 5.6.
Let C be a class of Kripke frames and let C be the closure of C under principal subframes. Let Γ ∪ { φ } ⊆ L ( X ) . Then, we have:(1) Γ | = C φ iff Γ | = C φ ;(2) Γ | = C φ iff Γ | = g C φ .Proof. For (1), we have naturally the direction from right to left. For the converse, note that for all φ ∈ L , all M over frames from C and all x ∈ D ( M ), we have( M , x ) | = φ iff ( M ↾ ( ↑ { x } ) , x ) | = φ. Thus, the claim follows from the fact that for every F ∈ C , we have F ∈ C or F = G ↾ ( ↑ { x } ) for some G ∈ C and some x ∈ D ( G ).For (2), we naturally have the direction from left to right. For the converse, consider Γ | = g C φ , that is ∀ N ∈ M ( C ; L ) (cid:16) N | = Γ ⇒ N | = φ (cid:17) . Let F ∈ C and M ∈ M ( F ; L ) as well as x ∈ F and suppose ( M , x ) | = Γ. Consider M ′ := M ↾ ( ↑ { x } ) . Then, we have M ′ | = Γ by Lemma 5.3 and as M ′ ∈ M ( C , L ) as C is closed under principal subframes. Wethus have M ′ | = φ by Γ | = g C φ , i.e. especially ( M , x ) | = φ . Thus, we have Γ | = C φ . (cid:3) Intuitionistic Mkrtychev models.
We continue our semantical investigations into intermediate justifi-cation logics by extending the approach of Mkrtychevs syntactic models by intuitionistic Kripke frames. Theseintuitionistic Mkrtychev models are akin to the previously considered models from [27] for
IPCJ T
Definition 5.7.
Let F = h F, ≤i be a Kripke frame. An intuitionistic Mkrtychev model based on F is a structure M = h F , E , (cid:13) i such that (cid:13) ⊆ F × V ar and E : Jt × F → L J satisfy(1) x ≤ y and x (cid:13) p implies y (cid:13) p for all p ∈ V ar ,(2) x ≤ y and φ ∈ E t ( x ) implies φ ∈ E t ( x ) for all φ ∈ L J and all t ∈ Jt ,for all x, y ∈ F as well as(i) E t ( x ) ⊐ E s ( x ) ⊆ E [ t · s ] ( x ) for all x ∈ F and all t, s ∈ Jt ,(ii) E t ( x ) ∪ E s ( x ) ⊆ E [ t + s ] ( x ) for all x ∈ F and all t, s ∈ Jt ,where Γ ⊐ ∆ := { φ ∈ L J | ψ → φ ∈ Γ , ψ ∈ ∆ for some ψ ∈ L J } for Γ , ∆ ⊆ L J . Given an intuitionistic Mkrtychev model M over a Kripke frame F = h F, ≤i , we also write D ( M ) := F andcall F the domain of M . Note that we use F to denote the domain of a model but also ( F ) to denote the axiomscheme of factivity for the intermediate justification logics.Over an intuitionistic Mkrtychev model M = h F , E , (cid:13) i , we introduce the following local satisfaction relationby recursion: • ( M , x ) = ⊥ ; • ( M , x ) | = p if x (cid:13) p ; • ( M , x ) | = φ ∧ ψ if ( M , x ) | = φ and ( M , x ) | = ψ ; • ( M , x ) | = φ ∨ ψ if ( M , x ) | = φ or ( M , x ) | = ψ ; • ( M , x ) | = φ → ψ if ∀ y ∈ F ( x ≤ y ⇒ ( M , x ) = φ or ( M , x ) | = ψ ); • ( M , x ) | = t : φ if φ ∈ E t ( x ).We write ( M , x ) | = Γ if ( M , x ) | = γ for all γ ∈ Γ. Further, we have the following immediate lemma.
Lemma 5.8.
Let F = h F, ≤i be a Kripke frame and let M be a intuitionistic Mkrtychev model over F . For any φ ∈ L J and all x, y ∈ F : x ≤ y and ( M , x ) | = φ implies ( M , y ) | = φ. Definition 5.9.
Let F = h F, ≤i be a Kripke frame and M = h F , E , (cid:13) i be an intuitionistic Mkrtychev model.We call M (1) factive if φ ∈ E t ( x ) implies ( M , x ) | = φ , and(2) introspective if t : E t ( x ) ⊆ E ! t ( x ) where t : Γ = { t : γ | γ ∈ Γ } . Definition 5.10.
Let C be a class of Kripke frames. Then, we write:(1) CKMJ for the class of all intuitionistic Mkrtychev models over frames from C ;(2) CKMJT for the class of all factive intuitionistic Mkrtychev models over frames from C ;(3) CKMJ4 for the class of all introspective intuitionistic Mkrtychev models over frames from C ;(4) CKMJT4 for the class of all factive and introspective intuitionistic Mkrtychev models over C . Definition 5.11.
Let M = h F , E , (cid:13) i be a intuitionistic Mkrtychev model and let CS be a constant specification(for some proof calculus). We say that M respects CS if for all x ∈ F and all c : φ ∈ CS : φ ∈ E c ( x ).Given a class C of intuitionistic Mkrtychev models, we denote the class of all intuitionistic Mkrtychev modelsrespecting a constant specification CS by C CS . Definition 5.12.
Let C be a class of intuitionistic Mkrtychev models. We write Γ | = C φ if for all M ∈ C andall x ∈ D ( M ): ( M , x ) | = Γ implies ( M , x ) | = φ . Lemma 5.13.
Let I be an intermediate logic and IJ L ∈ {IJ , IJ T , IJ , IJ T } . Let CS be a constantspecification for IJ L and let C ∈ KFr ( I ) . For any Γ ∪ { φ } ⊆ L J : Γ ⊢ IJ L CS φ implies Γ | = CKMJL CS φ. Proof.
By an argument similar to the one of Lemma 3.13, we may reduce strong to weak soundness:Γ ⊢ IJ L CS φ impl. ∃ Γ ⊆ Γ ∪ CS finite (cid:16) ⊢ IJ L ^ Γ → φ (cid:17) impl. ∃ Γ ⊆ Γ ∪ CS finite (cid:16) | = CKMJL ^ Γ → φ (cid:17) impl. ∃ Γ ⊆ Γ ∪ CS finite ∀ M ∈ CKMJL ∀ x ∈ D ( M ) ∀ y ≥ x (( M , y ) | = Γ ⇒ ( M , y ) | = φ )impl. ∃ Γ ⊆ Γ ∪ CS finite ∀ M ∈ CKMJL ∀ x ∈ D ( M ) (( M , x ) | = Γ ⇒ ( M , x ) | = φ )impl. ∀ M ∈ CKMJL ∀ x ∈ D ( M ) (( M , x ) | = Γ ∪ CS ⇒ ( M , x ) | = φ )impl. ∀ M ∈ CKMJL CS ∀ x ∈ D ( M ) (( M , x ) | = Γ ⇒ ( M , x ) | = φ ) . Thus, we only show that ⊢ IJ L φ implies | = CKMJL φ . As before, by definition of IJ L , it suffices to show | = CKMJL φ for φ ∈ I or φ being an instance of the justification axioms (depending on IJ L ). For both, let M = h F , E , (cid:13) i ∈ CKMJL as well as x ∈ D ( M ).If φ ∈ I , then there is a substitution σ : V ar → L J such that φ = σ ( ψ ) for some ψ ∈ I . By the choice of C ,we have that ( N , y ) | = ψ for any N = h F , (cid:13) ′ i and any y ∈ D ( F ). Define a particular (cid:13) ′ by y (cid:13) ′ p iff ( M , y ) | = σ ( p )for any p ∈ V ar and any y ∈ F and define N := h F , (cid:13) ′ i . Then, it is straightforward to see that ( M , y ) | = σ ( χ )iff ( N , y ) | = χ for any χ ∈ L and thus especially, we have ( M , x ) | = φ . This gives | = CKMJL φ . NTERMEDIATE JUSTIFICATION LOGICS: UNIFIED COMPLETENESS RESULTS 19 If φ is an instance of ( J ) or (+), then the conditions (i) and (ii) of Definition 5.7, respectively, give thevalidity of φ immediately.Similarly if φ is an instance of ( F ) and M is factive or φ is an instance of ( I ) and M is introspective, therespective validity of φ follows immediately by the definition of factive or introspective intuitionistic Mkrtychevmodels, that is (1) or (2) of Definition 5.9 (cid:3) Intuitionistic Fitting models.
We continue with intuitionistic Fitting models, combining various streamsof semantics in non-classical modal logics by extending the approach using intuitionistic modal Kripke modelsof [32] for intuitionistic modal logics by the machinery of evidence functions for explicit modalities in the senseof Fitting (or conversely extending Fittings models with the machinery of intuitionistic Kripke frames). Inany way, the models which we introduce are akin to a model class from [27] for
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Definition 5.14.
Let F = h F, ≤i be a Kripke frame. An intuitionistic Fitting model based on F is a structure M = h F , R , E , (cid:13) i such that (cid:13) ⊆ F × V ar , R ⊆ F × F and E : Jt × F → L J satisfy(1) x ≤ y and x (cid:13) p imply y (cid:13) p for all p ∈ V ar ,(2) x ≤ y and φ ∈ E t ( x ) imply φ ∈ E t ( y ) for all φ ∈ L J and all t ∈ Jt ,(3) x ≤ y implies R [ y ] ⊆ R [ x ].for all x, y ∈ F as well as(i) E t ( x ) ⊐ E s ( x ) ⊆ E [ t · s ] ( x ) for all x ∈ F and all t, s ∈ Jt ,(ii) E t ( x ) ∪ E s ( x ) ⊆ E [ t + s ] ( x ) for all x ∈ F and all t, s ∈ Jt .Over an intuitionistic Fitting model M = h F , R , E , (cid:13) i , we introduce the following local satisfaction relationby recursion: • ( M , x ) = ⊥ ; • ( M , x ) | = p if x (cid:13) p ; • ( M , x ) | = φ ∧ ψ if ( M , x ) | = φ and ( M , x ) | = ψ ; • ( M , x ) | = φ ∨ ψ if ( M , x ) | = φ or ( M , x ) | = ψ ; • ( M , x ) | = φ → ψ if ∀ y ∈ F ( x ≤ y ⇒ ( M , x ) = φ or ( M , x ) | = ψ ); • ( M , x ) | = t : φ if φ ∈ E t ( x ) and ∀ y ∈ R [ x ] ( M , y ) | = φ .We write ( M , x ) | = Γ if ( M , x ) | = γ for all γ ∈ Γ. Also, given an intuitionistic Fitting model M over a Kripkeframe F = h F, ≤ , V i , we write again D ( M ) = F . Lemma 5.15.
Let M be an intuitionistic Fitting model over a Kripke frame F = h F, ≤i . For any φ ∈ L J andall x, y ∈ F : x ≤ y and ( M , x ) | = φ imply ( M , y ) | = φ. Definition 5.16.
Let F = h F, ≤i be a Kripke frame and M = h F , R , E , (cid:13) i be an intuitionistic Fitting model.We call M (1) reflexive if R is reflexive,(2) transitive if R is transitive,(3) monotone if E t ( x ) ⊆ E t ( y ) for y ∈ R [ x ],(4) introspective if it is transitive, monotone and t : E t ( x ) ⊆ E ! t ( x ) for all t ∈ Jt and x ∈ F . Definition 5.17.
Let C be a class of intuitionistic Fitting models. We write Γ | = C φ if for all M ∈ C and all x ∈ D ( M ): ( M , x ) | = Γ implies ( M , x ) | = φ . Definition 5.18.
Let C be a class of Kripke frames. Then, we write:(1) CKFJ for the class of all intuitionistic Fitting models over frames from C ;(2) CKFJT for the class of all reflexive intuitionistic Fitting models over frames from C ;(3) CKFJ4 for the class of all introspective intuitionistic Fitting models over frames from C ;(4) CKFJT4 for the class of all reflexive and introspective intuitionistic Fitting models over frames from C . Lemma 5.19.
Let I be an intermediate logic and IJ L ∈ {IJ , IJ T , IJ , IJ T } . Let CS be a constantspecification for IJ L and let C ∈ KFr ( I ) . Then, for any Γ ∪ { φ } ⊆ L J : Γ ⊢ IJ L CS φ implies Γ | = CKFJL CS φ. Proof.
As in Lemma 5.13, we may restrict ourselves to weak soundness only. Here, it again suffices to onlyverify ( M , x ) | = φ for any φ ∈ I or φ being a instance of a justification axiom (depending on IJ L ) as well asany M ∈ CKFJL and any x ∈ D ( M ). The case for φ ∈ I can be handled similarly as in Lemma 5.13. We thus only show the validity of (1) ( J ),(2) (+) as well as (3) ( F ) and (4) ( I ) in their respective model classes. For this, let M = h F , R , E , (cid:13) i be anintuitionistic Fitting model with F ∈ C .(1) We show ( M , x ) | = t : ( φ → ψ ) impl. ( M , x ) | = ( s : φ → [ t · s ] : ψ )for any φ, ψ ∈ L J , any t, s ∈ Jt and any x ∈ F . For this, suppose ( M , x ) | = t : ( φ → ψ ), that is bydefinition φ → ψ ∈ E t ( x ) as well as ∀ y ∈ R [ x ] ( M , y ) | = φ → ψ. Let y ≥ x and suppose ( M , y ) | = s : φ , that is φ ∈ E s ( y ) and ∀ z ∈ R [ y ] ( M , z ) | = φ. By condition (2) of Definition 5.14, we have that φ → ψ ∈ E t ( y ). By condition (i), we have thus that ψ ∈ E [ t · s ] ( y ). Now, let z ∈ R [ y ]. As above, we have ( M , z ) | = φ and by condition (3) of Definition 5.14,we have that z ∈ R [ x ] and thus ( M , z ) | = φ → ψ . Thus, we have especially ( M , z ) | = ψ .Therefore, we have ∀ z ∈ R [ y ] ( M , z ) | = ψ and in combination with ψ ∈ E [ t · s ] ( y ), we have ( M , y ) | =[ t · s ] : ψ . As y was arbitrary, we have ( M , x ) | = s : φ → [ t · s ] : ψ .As x ∈ F was arbitrary, we have ( M , x ) | = t : ( φ → ψ ) → ( s : φ → [ t · s ] : ψ ).(2) Let x ∈ F be arbitrary. Suppose ( M , x ) | = t : φ , that is φ ∈ E t ( x ) and ∀ y ∈ R [ x ] ( M , x ) | = φ. The former gives φ ∈ E [ t + s ] ( x ) by condition (ii) of Definition 5.14 and this combined with the lattergives ( M , x ) | = [ t + s ] : φ. As x ∈ F war arbitrary, we have ( M , x ) | = t : φ → [ t + s ] : φ . One shows ( M , x ) | = s : φ → [ t + s ] : φ ina similar way.(3) Suppose M is reflexive and let x ∈ F . Suppose( M , x ) | = t : φ that is especially we have ∀ y ∈ R [ x ] ( M , y ) | = φ . As R is reflexive, we have x ∈ R [ x ] and thus( M , x ) | = φ . As x was arbitrary, we have( M , x ) | = t : φ → φ. (4) Let M be introspective and let x ∈ F . Suppose that ( M , x ) | = t : φ , that is φ ∈ E t ( x ) and ∀ y ∈ R [ x ] ( M , x ) | = φ. The former gives at first t : φ ∈ E ! t ( x ) by introspectivity. Now, let y ∈ R [ x ] be arbitrary. By themonotonicity aspect of introspectivity, we have φ ∈ E t ( y ) as φ ∈ E t ( x ). Now, let z ∈ R [ y ]. Bytransitivity of R , we have z ∈ R [ x ] and thus ( M , z ) | = φ by assumption. Summarized, we have φ ∈ E t ( y ) and ∀ z ∈ R [ y ] ( M , z ) | = φ, that is ( M , y ) | = t : φ for all y ∈ R [ x ] and this combined with t : φ ∈ E ! t ( x ) gives ( M , x ) | =! t : t : φ . As x was arbitrary, we have ( M , x ) | = t : φ → ! t : t : φ . (cid:3) Intuitionistic subset models.
The last semantics which we introduce, based on intuitionistic Kripkeframes, extends the considerations of Lehmann and Studer from [26] about their subset models to these inter-mediate cases. This semantics seems to have not appeared in the literature before.
Definition 5.20.
Let F = h F , ≤i be a Kripke frame. An intuitionistic subset model over F is a structure M = h F , F, E , (cid:13) i with F ⊇ F E : Jt → F × F and (cid:13) ⊆ F × L J and which satisfies(1) x ≤ y and x (cid:13) p imply y (cid:13) p for all p ∈ V ar ,(2) x ≤ y implies E t [ y ] ⊆ E t [ x ] for all t ∈ Jt ,for all x, y ∈ F as well as(i) x (cid:13) ⊥ ,(ii) x (cid:13) φ ∧ ψ iff x (cid:13) φ and x (cid:13) ψ ,(iii) x (cid:13) φ ∨ ψ iff x (cid:13) φ or x (cid:13) ψ ,(iv) x (cid:13) φ → ψ iff ∀ y ≥ x : y (cid:13) φ or y (cid:13) ψ ,(v) x (cid:13) t : φ iff ∀ y ∈ E t [ x ] : y (cid:13) φ ,for any x ∈ F and such that it satisfies: NTERMEDIATE JUSTIFICATION LOGICS: UNIFIED COMPLETENESS RESULTS 21 (a) E [ t + s ] [ x ] ⊆ E t [ x ] ∩ E s [ x ];(b) E [ t · s ] [ x ] ⊆ { y ∈ F | ∀ φ ∈ ( M ) xt,s ( y (cid:13) φ ) } where we define( M ) vt,s := { φ ∈ L J | ∃ ψ ∈ L J ∀ y ∈ F ( y ∈ E t [ x ] ⇒ y (cid:13) ψ → φ and y ∈ E s [ x ] ⇒ y (cid:13) ψ ) } . We write D ( M ) for F and D ( M ) for F . Also, given x ∈ D ( M ), we write ( M , x ) | = φ if x (cid:13) φ and ( M , x ) | = Γif ( M , x ) | = γ for all γ ∈ Γ, given Γ ∪ { φ } ⊆ L J . We write M | = φ if for all x ∈ D ( M ): ( M , x ) | = φ and similarlyfor sets Γ. Note the emphasis on D ( M ), not D ( M ). Lemma 5.21.
Let F = h F , ≤i be a Kripke frame and M = h F , F, E , (cid:13) i be an intuitionistic subset model over F . Then, for all φ ∈ L J and all x, y ∈ F : x ≤ y and x (cid:13) φ imply y (cid:13) φ. Definition 5.22.
Let F = h F , ≤i be a Kripke frame and M = h F , F, E , (cid:13) i be an intuitionistic subset model.We call M (1) reflexive if x ∈ E t [ x ] for all x ∈ F and all t ∈ Jt ,(2) introspective if E ! t [ x ] ⊆ { y ∈ F | ∀ φ ∈ L J ( x (cid:13) t : φ ⇒ y (cid:13) t : φ ) } . Definition 5.23.
Let C be a class of intuitionistic subset models. We write Γ | = C φ if for all M ∈ C and all x ∈ D ( M ): ( M , x ) | = Γ implies ( M , x ) | = φ . Definition 5.24.
Let C be a class of Kripke frames. Then, we write:(1) CKSJ for the class of all intuitionistic subset models over frames from C ;(2) CKSJT for the class of all reflexive intuitionistic subset models over frames from C ;(3) CKSJ4 for the class of all introspective intuitionistic subset models over frames from C ;(4) CKSJT4 for the class of all reflexive and introspective intuitionistic subset models over frames from C . Definition 5.25.
Let F = h F , ≤i be a Kripke frame and M = h F , F, E , (cid:13) i be an intuitionistic subset modelover F . Let CS be a constant specification (for some proof system). We say that M respects CS if x (cid:13) c : φ for all c : φ ∈ CS and all x ∈ F .Given a class C of intuitionistic subset models, we write C CS for the subclass of all models respecting aconstant specification CS . Lemma 5.26.
Let I be an intermediate logic and IJ L ∈ {IJ , IJ T , IJ , IJ T } . Let CS be a constantspecification for IJ L . Let C ∈ KFr ( I ) . For any Γ ∪ { φ } ⊆ L J : Γ ⊢ IJ L CS φ implies Γ | = CkSJL CS φ. Proof.
Reasoning as in Lemmas 5.13 and 5.19, we restrict the argument and only show the validity of ( J ) , (+) , ( F )and ( I ) in their respective model classes. For this, let M = h F , F, E , (cid:13) i be a intuitionistic subset model over aKripke frame F and let x ∈ F .(1) For ( J ), suppose x (cid:13) t : ( φ → ψ ). Now, we want to show x (cid:13) s : φ → [ t · s ] : ψ . For this, let y ≥ x .Note that then y ∈ F as ≤ is only a relation on F . Suppose y (cid:13) s : φ , that is( † ) ∀ z ∈ E s [ y ] z (cid:13) φ. Further, x (cid:13) t : ( φ → ψ ) implies y (cid:13) t : ( φ → ψ ) by Lemma 5.21, that is( ‡ ) ∀ z ∈ E t [ y ] z (cid:13) φ → ψ. Let z ∈ E [ t · s ] [ y ]. Then, by property (b) of Definition 5.20, we have z ∈ { w ∈ F | ∀ χ ∈ ( M ) yt,s w (cid:13) χ } and thus it suffices to show ψ ∈ ( M ) yt,s . But this is immediate by ( † ) and ( ‡ ). Thus ∀ z ∈ E [ t · s ] [ y ] z (cid:13) ψ which is y (cid:13) [ t · s ] : ψ and thus x (cid:13) s : φ → [ t · s ] : ψ . As x was arbitrary, we have x (cid:13) t : ( φ → ψ ) → ( s : φ → [ t · s ] : ψ ) for all x ∈ F .(2) Suppose x (cid:13) t : φ . That is, we have ∀ y ∈ E t [ x ] y (cid:13) φ. Then, by condition (a) of Definition 5.20, we have ∀ y ∈ E [ t + s ] [ x ] ⊆ E t [ x ] y (cid:13) φ which is x (cid:13) [ t + s ] : φ . As x was arbitrary, we have x (cid:13) t : φ → [ t + s ] : φ for any x ∈ F . Similarly,one shows x (cid:13) s : φ → [ t + s ] : φ for any x ∈ F . (3) Let M be reflexive with x (cid:13) t : φ . Then, we have ∀ y ∈ E t [ x ] y (cid:13) φ, that is as M is reflexive x ∈ E t [ x ] and thus x (cid:13) φ . As x was arbitrary, we have x (cid:13) t : φ → φ for any x ∈ F .(4) Let M be introspective x (cid:13) t : φ . Then, we have ∀ y ∈ E ! t [ x ] y (cid:13) t : φ by definition of introspectivity but this is exactly x (cid:13) ! t : t : φ . Again we have x (cid:13) t : φ → ! t : t : φ forany x ∈ F as x was arbitrary. (cid:3) Completeness for frame semantics
In this section, we prove the corresponding completeness theorems for the intermediate justification logicstogether with their previously introduced semantics based on Mkrtychev, Fitting or subset models over intu-itionistic Kripke frames. The permissible classes of frames for the completeness theorems derive, similarly asthe permissible classes of Heyting algebras from the completeness theorems for the algebraic models, from theunderlying intermediate logic where we especially rely on the global completeness statement introduced earlier.6.1.
Completeness w.r.t. intuitionistic Mkrtychev models.Definition 6.1.
Let F = h F, ≤i be a Kripke frame and let N ∈ M ( F ; L ⋆ ). We define the canonical intuitionisticMkrtychev model over N as the structure M c,M N = h F , E c , (cid:13) c i by setting:(1) x (cid:13) c p iff x (cid:13) ∗ p ;(2) E ct ( x ) := { φ ∈ L J | x (cid:13) ∗ φ t } . Lemma 6.2.
Let F = h F, ≤i be a Kripke frame, let N ∈ M ( F ; L ⋆ ) and let M c,M N = h F , E c , (cid:13) c i as above. Thenfor all φ ∈ L J and all x ∈ F : ( M c,M N , x ) | = φ iff ( N , x ) | = φ ⋆ . Proof.
We prove it by induction on φ . The claim is immediate for p ∈ V ar by definition. Suppose the claim istrue for φ, ψ . Then, we have at first( M c,M N , x ) | = φ ∧ ψ iff ( M c,M N , x ) | = φ and ( M c,M N , x ) | = ψ iff ( N , x ) | = φ ⋆ and ( N , x ) | = ψ ⋆ iff ( N , x ) | = ( φ ∧ ψ ) ⋆ and similarly for ∨ . For → , we have( M c,M N , x ) | = φ → ψ iff ∀ y ≥ x : ( M c,M N , y ) | = φ implies ( M c,M N , y ) | = ψ iff ∀ y ≥ x : ( N , y ) | = φ ⋆ implies ( N , y ) | = ψ ⋆ iff ( N , x ) | = φ ⋆ → ψ ⋆ . Lastly, we have ( M c,M N , x ) | = t : φ iff φ ∈ E ct ( x )iff x (cid:13) ∗ φ t iff ( N , x ) | = φ t . (cid:3) Lemma 6.3.
Let F = h F, ≤i be a Kripke frame and let N ∈ M ( F ; L ⋆ ) such that additionally N | = ( T h
IJ L CS ) ⋆ .Then M c,M N is a well-defined intuitionistic Mkrtychev model. Further:(a) if ( F ) is an axiom scheme of IJ L , then M c,M N is factive;(b) if ( I ) is an axiom scheme of IJ L , then M c,M N is introspective.Proof. We first show properties (1) and (2) of Definition 5.7. For this, let x, y ∈ F with x ≤ y . For (1), we have x (cid:13) c p ⇒ x (cid:13) ∗ p ⇒ y (cid:13) ∗ p ⇒ y (cid:13) c p and for (2), we have φ ∈ E t ( x ) ⇒ x (cid:13) ∗ φ t ⇒ y (cid:13) ∗ φ t ⇒ φ ∈ E t ( y ) . Both follow from Lemma 5.3 applied to N . NTERMEDIATE JUSTIFICATION LOGICS: UNIFIED COMPLETENESS RESULTS 23
For properties (i) and (ii), let x ∈ F and t, s ∈ Jt . Then, at first for (i), let φ ∈ E ct ( x ) ⊐ E cs ( x ), that is bydefinition ∃ ψ ∈ L J : ψ → φ ∈ E ct ( x ) and ψ ∈ E cs ( x ) . Untangling the definition of E c , we have x (cid:13) ∗ ( ψ → φ ) t and x (cid:13) ∗ ψ s . As we have N | = ( T h
IJ L CS ) ⋆ , we have by the ⋆ -translation of the axiom scheme ( J ) that ∀ y ≥ x : y (cid:13) ∗ ( ψ → φ ) t and y (cid:13) ∗ ψ s implies y (cid:13) ∗ φ [ t · s ] and thus especially, as x ≥ x , we have x (cid:13) ∗ φ [ t · s ] and thus φ ∈ E c [ t · s ] ( x ).For (ii), let φ ∈ E ct ( x ) ∪ E cs ( x ). Then, we have x (cid:13) ∗ φ t or x (cid:13) ∗ φ s . By the ⋆ -translation of the axiom scheme(+) and N | = ( T h
IJ L CS ) ⋆ , we have in either case as before x (cid:13) ∗ φ [ t + s ] .Now, for (a), if ( F ) is an axiom scheme of IJ L , then we have by N | = ( T h
IJ L CS ) ⋆ again, that x (cid:13) ∗ φ t implies ( N , x ) | = φ ⋆ . By the definition of E c and Lemma 6.2, we have thus φ ∈ E ct ( x ) implies ( M c,M N , x ) | = φ and thus M c,M N is factive.For (b), if ( I ) is an axiom scheme of IJ L , then we have x (cid:13) ∗ φ t implies x (cid:13) ∗ ( t : φ ) ! t and by definition that is φ ∈ E ct ( x ) implies t : φ ∈ E c ! t ( x )which is t : E ct ( x ) ⊆ E c ! t ( x ) and thus M c,M N is introspective. (cid:3) Theorem 6.4.
Let I be an intermediate logic, IJ L ∈ {IJ , IJ T , IJ , IJ T } and let CS be a constantspecification for IJ L . Let C ∈ KFr ( I ) ∩ KFr g ( I ) . For any Γ ∪ { φ } ⊆ L J , we have: Γ ⊢ IJ L CS φ iff Γ | = CKMJL CS φ. Proof.
The direction from left to right follows from Lemma 5.13. For the converse, suppose Γ IJ L CS φ . ByLemma 4.1, we have Γ ⋆ ∪ ( T h
IJ L CS ) ⋆ I ⋆ φ ⋆ and by assumption on the global strong completeness of I w.r.t. C , there is a N = h F , (cid:13) ∗ i ∈ M ( C ; L ⋆ ) suchthat N | = Γ ⋆ ∪ ( T h
IJ L CS ) ⋆ but N = φ ⋆ . By Lemma 6.3, we have M c,M N ∈ CKMJL for the corresponding canonical intuitionistic Mkrtychev model. ByLemma 6.2, we have M c,M N | = CS and thus M c,M N ∈ CKMJL CS as well as M c,M N | = Γ but M c,M N = φ. Thus, we have Γ = CKMJL CS φ . (cid:3) Completeness w.r.t. intuitionistic Fitting models.Definition 6.5.
Let F = h F, ≤i be a Kripke frame and let N ∈ M ( F ; L ⋆ ). We define the canonical intuitionisticFitting model over N as the structure M c,F N = h F , R c , E c , (cid:13) c i by setting:(1) x (cid:13) c p iff x (cid:13) ∗ p ;(2) E ct ( x ) := { φ ∈ L J | x (cid:13) ∗ φ t } ;(3) ( x, y ) ∈ R c iff ∀ t ∈ Jt ∀ φ ∈ L J ( x (cid:13) ∗ φ t ⇒ ( N , y ) | = φ ⋆ ). Lemma 6.6.
Let F = h F, ≤i be a Kripke frame, let N ∈ M ( F ; L ⋆ ) and define M c,F N as above. For any φ ∈ L J and all x ∈ F : ( M c,F N , x ) | = φ iff ( N , x ) | = φ ⋆ . Proof.
The claim is again proved by induction on the structure of the formula. We only consider the modalcase. Suppose the claim holds for all x ∈ F and some φ ∈ L J .At first, suppose ( N , x ) | = φ t , i.e. x (cid:13) ∗ φ t . Then, naturally φ ∈ E ct ( x ) by definition. Let further y ∈ R c [ x ].Then, as x (cid:13) ∗ φ t , we have ( N , y ) | = φ ⋆ by definition and thus ( M c,F N , x ) | = φ by induction hypothesis. Thus, wehave φ ∈ E ct ( x ) and ∀ y ∈ R c [ x ] ( M c,F N , x ) | = φ and thus ( M c,F N , x ) | = t : φ .Conversely, suppose ( N , x ) = φ t , that is x (cid:13) ∗ φ t . Thus, by definition φ
6∈ E ct ( x ) and thus( M c,F N , x ) = t : φ immediately by definition. (cid:3) Lemma 6.7.
Let F = h F, ≤i be a Kripke frame and let N ∈ M ( F ; L ⋆ ) such that additionally N | = ( T h
IJ L CS ) ⋆ .Then M c,F N is a well-defined intuitionistic Fitting model. Further:(a) if ( F ) is an axiom scheme of IJ L , then M c,F N is reflexive;(b) if ( I ) is an axiom scheme of IJ L , then M c,F N is introspective.Proof. For properties (1) - (3) of Definition 5.14, let x, y ∈ F with x ≤ y . For (1) and (2), we have as before x (cid:13) c p ⇒ x (cid:13) ∗ p ⇒ y (cid:13) ∗ p ⇒ y (cid:13) c p and φ ∈ E t ( x ) ⇒ x (cid:13) ∗ φ t ⇒ y (cid:13) ∗ φ t ⇒ φ ∈ E t ( y )by Lemma 5.3 for N . For (3), let z ∈ R [ y ], that is we have ∀ t ∈ Jt ∀ φ ∈ L J ( y (cid:13) ∗ φ t ⇒ ( N , z ) | = φ ⋆ ) . Then, for any t ∈ Jt and any φ ∈ L J we have, if x (cid:13) ∗ φ t that y (cid:13) ∗ φ t by Lemma 5.3 and thus by the above( N , z ) | = φ ⋆ . Thus z ∈ R [ x ] and thus R [ y ] ⊆ R [ x ].We have properties (1) and (2) of Definition 5.14 in the same way as in the proof of Lemma 6.3. For (3), let z ∈ R c [ y ] for x ≤ y , that is ∀ t ∈ Jt ∀ φ ∈ L J ( y (cid:13) ∗ φ t ⇒ ( N , z ) | = φ ⋆ ) . If x (cid:13) ∗ φ t , then y (cid:13) ∗ φ t by Lemma 5.3 and thus by the above ( N , z ) | = φ ⋆ . Thus z ∈ R c [ y ].For property (i), let φ ∈ E ct ( x ) ⊐ E cs ( x ), i.e. ∃ ψ ∈ L J ( ψ → φ ∈ E ct ( x ) and ψ ∈ E cs ( x )) . Thus, by definition, we have x (cid:13) ∗ ( ψ → φ ) t and x (cid:13) ∗ ψ s and thus, as N | = ( T h
IJ L CS ) ⋆ , we have x (cid:13) ∗ φ [ t · s ] , that is φ ∈ E ct · s ( x ).For property (ii), note that x (cid:13) ∗ φ t implies x (cid:13) ∗ φ [ t + s ] again by N | = ( T h
IJ L CS ) ⋆ and thus φ ∈ E ct ( x ) implies φ E ct + s ( x ) and similarly for φ ∈ E cs ( x ).Suppose that ( F ) is an axiom scheme of IJ L . Then, we have ∀ t ∈ Jt ∀ φ ∈ L J ( x (cid:13) ∗ φ t ⇒ ( N , x ) | = φ ⋆ )as N | = ( T h
IJ L CS ) ⋆ and this is exactly ( x, x ) ∈ R c .Suppose that ( I ) is an axiom scheme of IJ L . As in the case of intuitionistic Mkrtychev models, one shows t : E ct ( x ) ⊆ E c ! t ( x ) . For the transitivity of R c , let ( x, y ) , ( y, z ) ∈ R c . that is, we have ∀ t ∈ Jt ∀ φ ∈ L J ( x (cid:13) ∗ φ t ⇒ ( N , y ) | = φ ⋆ )as well as ∀ t ∈ Jt ∀ φ ∈ L J ( y (cid:13) ∗ φ t ⇒ ( N , z ) | = φ ⋆ ) . As ( I ) is an axiom scheme and N | = ( T h
IJ L CS ) ⋆ , we have w (cid:13) ∗ φ t ⇒ w (cid:13) ∗ ( t : φ ) ! t NTERMEDIATE JUSTIFICATION LOGICS: UNIFIED COMPLETENESS RESULTS 25 for any w ∈ F . Thus, especially we have x (cid:13) ∗ φ t ⇒ x (cid:13) ∗ ( t : φ ) ! t ⇒ y (cid:13) ∗ φ t ⇒ ( N , z ) | = φ ⋆ using Lemma 6.6. Thus, by definition we have ( x, z ) ∈ R c .For the monotonicity, let y ∈ R c [ x ] and let φ ∈ E t ( x ). The former gives ∀ t ∈ Jt ∀ φ ∈ L J ( x (cid:13) ∗ φ t ⇒ ( N , y ) | = φ ⋆ )and the latter gives x (cid:13) ∗ φ t . As N | = ( T h
IJ L CS ) ⋆ , we have especially x (cid:13) ∗ ( t : φ ! t ). By the above, we have( N , y ) | = φ t , that is y (cid:13) ∗ φ t and thus φ ∈ E ct ( y ). Thus, M c,F N is monotone and thus M c,F N is introspective. (cid:3) Theorem 6.8.
Let I be an intermediate logic, IJ L ∈ {IJ , IJ T , IJ , IJ T } and let CS be a constantspecification for IJ L . Let C ∈ KFr ( I ) ∩ KFr g ( I ) . For any Γ ∪ { φ } ⊆ L J , we have: Γ ⊢ IJ L CS φ iff Γ | = CKFJL CS φ. Proof.
The direction from left to right now comes from Lemma 5.19. For the converse, we again use Lemma4.1 and the assumptions on C to obtain a N = h F , (cid:13) ∗ i ∈ M ( C ; L ⋆ ) with N | = Γ ⋆ ∪ ( T h
IJ L CS ) ⋆ but N = φ ⋆ . Using Lemmas 6.7 and 6.6, we obtain as before that M c,F N ∈ CKFJL CS as well as M c,F N | = Γ but M c,F N = φ, that is Γ = CKFJL CS φ . (cid:3) Completeness w.r.t. intuitionistic subset models.Definition 6.9.
Let F = h F , ≤i be a Kripke frame and let N ∈ M ( F ; L ⋆ ). We define the canonical intuition-istic subset model over N as the structure M c,S N = h F , F c , E c , (cid:13) c i by setting:(1) F c = F ∪ S x ∈ F { x t | t ∈ Jt } ;(2) ( x, y ) ∈ E ct iff ∀ φ ∈ L J ( x (cid:13) ∗ φ t ⇒ y (cid:13) c φ ) for all x, y ∈ F c ;(3) for x ∈ F and t ∈ Jt :(a) x (cid:13) c φ iff ( N , x ) | = φ ⋆ ;(b) x t (cid:13) c φ iff ( N , x ) | = φ t . Lemma 6.10.
Let F = h F , ≤i be a Kripke frame, let N ∈ M ( F ; L ⋆ ) and define M c,S N as above. For any φ ∈ L J and any x ∈ F : ( M c,S N , x ) | = φ iff ( N , x ) | = φ ⋆ . Proof.
This is immediate by definition as we have ( M c,S N , x ) | = φ iff x (cid:13) c φ iff ( N , x ) | = φ ⋆ , given a x ∈ F . (cid:3) The simplicity of the above lemma is in contrast to the truth lemmas for the previous canonical modelsover Kripke frames. In the context of intuitionistic subset models, the relation (cid:13) completely encodes the truthvalues of formulae to be able to cope with ”irregular” worlds. This comes with the expense of conditions ofwell-definedness for (cid:13) and thus, the previous complexity of showing an equivalence like the one of the abovelemma is shifted into the following result.
Lemma 6.11.
Let F = h F , ≤i be a Kripke frame and let N ∈ M ( F ; L ⋆ ) such that additionally N | =( T h
IJ L CS ) ⋆ . Then M c,S N is a well-defined intuitionistic subset model. Further:(I) if ( F ) is an axiom scheme of IJ L , then M c,S N is reflexive;(II) if ( I ) is an axiom scheme of IJ L , then M c,S N is introspective.Proof. We begin with properties (i) - (v) from Definition 5.20. For this, let x ∈ F . The properties (i) - (iii) areimmediate by using the respective properties of (cid:13) ∗ and the fact that ⋆ commutes with ⊥ , ∧ , ∨ .For (iv), note that we have x (cid:13) c φ → ψ iff ( N , x ) | = φ ⋆ → ψ ⋆ iff ∀ y ≥ x (( N , y ) = φ ⋆ or ( N , y ) | = ψ ⋆ )iff ∀ y ≥ x ( y (cid:13) c φ or y (cid:13) c ψ )where it is instrumental that ≤ is a relation on F only.For (v), we have for one by definition that ∀ y ∈ E ct [ x ] ∀ φ ∈ L J (( N , x ) | = φ t ⇒ ( N , y ) | = φ ⋆ ) , that is we have x (cid:13) c t : φ ⇒ x (cid:13) ∗ φ t ⇒ ∀ y ∈ E ct [ x ] (( N , y ) | = φ ⋆ ) ⇔ ∀ y ∈ E ct [ x ] ( y (cid:13) c φ ) . For another, we have x t ∈ E t [ x ] as we have x t (cid:13) c φ iff x (cid:13) ∗ φ t by definition. Thus, if x (cid:13) c t : φ , then x (cid:13) ∗ φ t and thus x t (cid:13) c φ . Therefore x (cid:13) c t : φ ⇒ ∃ y ∈ E t [ x ] ( y (cid:13) c φ ) . Concluding, we have x (cid:13) c t : φ iff ∀ y ∈ E ct [ x ] ( y (cid:13) c φ ).Regarding properties (1) and (2) of Definition 5.20, let x ≤ y for x, y ∈ F . Property (1) follows as in theproof of Lemma 6.7 by (3).(a) of Definition 6.9. For property (2), let z ∈ E ct [ y ]. Thus, we have ∀ φ ∈ L J ( y (cid:13) c t : φ ⇒ z (cid:13) c φ )and if x (cid:13) c t : φ , then as x, y ∈ F , we have x (cid:13) ∗ φ t and thus y (cid:13) ∗ φ t which is y (cid:13) c t : φ . By the above, wehave z (cid:13) c φ and thus z ∈ E ct [ x ].Now, on to properties (a), (b) of Definition 5.20. For (a), let y ∈ E c [ t + s ] [ x ], that is we have ∀ φ ∈ L J ( x (cid:13) ∗ φ [ t + s ] ⇒ ( N , y ) | = φ ⋆ )by definition. Now, by assumption as N | = ( T h
IJ L CS ) ⋆ we have x (cid:13) ∗ φ t implies x (cid:13) ∗ φ [ t + s ] and x (cid:13) ∗ φ s implies x (cid:13) ∗ φ [ t + s ] . Therefore, we have ∀ φ ∈ L J ( x (cid:13) ∗ φ t ⇒ x (cid:13) ∗ φ [ t + s ] ⇒ ( N , y ) | = φ ⋆ )and ∀ φ ∈ L J ( x (cid:13) ∗ φ s ⇒ x (cid:13) ∗ φ [ t + s ] ⇒ ( N , y ) | = φ ⋆ )which is y ∈ E ct [ x ] ∩ E cs [ x ].For (b), let y ∈ E c [ t · s ] [ x ], that is( † ) ∀ φ ∈ L J ( x (cid:13) ∗ φ [ t · s ] ⇒ ( N , y ) | = φ ⋆ ) . Let φ ∈ ( M c,S N ) xt,s , that is there is a ψ ∈ L J such that ∀ z ∈ F c ( z ∈ E ct [ x ] ⇒ z (cid:13) c ψ → φ and z ∈ E cs [ x ] ⇒ z (cid:13) c ψ ) . By property (v), we have that x (cid:13) c t : ( ψ → φ ) and x (cid:13) c s : ψ , i.e. by definition as x ∈ F :( N , x ) | = ( ψ → φ ) t and ( N , x ) | = ψ s and by N | = ( T h
IJ L CS ) ⋆ and axiom ( J ), we have( N , x ) | = φ [ t · s ] . Thus, by ( † ), we have ( N , y ) | = φ ⋆ and thus by definition y (cid:13) c φ .Assume second to last that ( F ) is an axiom scheme of IJ L . Then, we have x (cid:13) ∗ φ t ⇒ ( N , x ) | = φ ⋆ for all x ∈ F and all φ ∈ L J , t ∈ Jt as N | = ( T h
IJ L CS ) ⋆ and thus, by definition we have x ∈ E ct [ x ] for all t ∈ Jt .Assume last that ( I ) is an axiom scheme of IJ L . Let y ∈ E ! t [ x ], that is( ‡ ) ∀ φ ∈ L J ( x (cid:13) ∗ φ ! t ⇒ ( N , y ) | = φ ⋆ ) . Let φ ∈ L J and assume x (cid:13) c t : φ , that is x (cid:13) ∗ φ t . Then, as N | = ( T h
IJ L CS ) ⋆ , we have x (cid:13) ∗ ( t : φ ) ! t . By ( ‡ )we have ( N , y ) | = ( t : φ ) ⋆ , that is ( N , y ) | = φ t and thus by definition y (cid:13) c t : φ . (cid:3) Theorem 6.12.
Let I be an intermediate logic, IJ L ∈ {IJ , IJ T , IJ , IJ T } and let CS be a constantspecification for IJ L . Let C ∈ KFr ( I ) ∩ KFr g ( I ) . Then, for any Γ ∪ { φ } ⊆ L J , we have: Γ ⊢ IJ L CS φ iff Γ | = CKSJL CS φ. NTERMEDIATE JUSTIFICATION LOGICS: UNIFIED COMPLETENESS RESULTS 27
Proof.
Lemma 5.26 gives the direction from left to right. As before, Lemma 4.1 as well as the assumptions on C give a L ⋆ -model N = h F , (cid:13) ∗ i ∈ M ( C ; L ⋆ ) with N | = ( T h
IJ L CS ) ⋆ , N | = Γ ⋆ but N = φ ⋆ . The first part give that M c,S N is a well-defined CKSJL -model by Lemma 6.11 and Lemma 6.10 gives M c,S N | = Γ but M c,S N = φ. as well as M c,S N | = CS , i.e. M c,S N ∈ CKSJL CS . Therefore, by definition Γ = CKSJL CS φ . (cid:3) Conclusion
The completeness theorems proved in this paper show that behind any previous completeness result in theliterature stands a unified completeness theorem lifting classes of algebras or classes of Kripke frames completefor some intermediate logic to a complete model class for the corresponding justification logic. Key to this is ofcourse the strong completeness assumption of the underlying propositional logic and it remains open whetherthere are similar liftings of weak propositional completeness to weak completeness on the justification side.We want to acknowledge that the algebraic results can be generalized in an immediate way. E.g. with analgebraic Fitting model M = h A , W , R , E , Vi over a complete Heyting algebra A , A can be generalized to notbe complete but only card( W ) + -complete, similarly as in the case of the Kripke-models taking values in Heytingalgebras for intuitionistic modal logics from Ono [32]. This of course also applies to the algebraic subset models.This study of the general class of intermediate justification logics was initiated, originally, to study exten-sions of the G¨odel-McKinsey-Tarski translation and modal companions (see e.g. [19, 29] and also [7, 11]) tothe language of justification logics and to relate the intermediate justification logics with hybrid justificationlogics (in the sense of [5, 13]). A relationship between hybrid justification logics and intermediate justificationlogics could prove to be a similarly fruitful connection as between intermediate logics and modal logics over S The special cases of
IPC , G and CPC . As special instances of the completeness results, we in particularhave the following algebraic and frame-based completeness theorems for
IPC , G and CPC . For this, considerfirst the following collection of algebraic completeness results:We use H to denote the class of all Heyting algebras. An important instance of a linear Heyting algebra isthe standard G¨odel algebra [ , ] G given by[ , ] G := h [0 , , min , max , ⇒ , , i where x ⇒ y := ( x ≤ yy otherwisefor x, y ∈ [0 , { , } B := h{ , } , min , max , ⇒ , , i with the above function ⇒ restricted to { , } . Theorem 7.1.
We have the following algebraic completeness results:(1)
IPC is strongly complete with respect to H fin ;(2) G is strongly complete with respect to [ , ] G ;(3) CPC is strongly complete with respect to { , } B . All items are folklore by now. For example, item (2) was proven by Dummett in [10]. Based on Theorems4.4, 4.7 and 4.10, we obtain the following particular corollaries.
Corollary . Let
IPCJ L ∈ {IPCJ , IPCJ T , IPCJ , IPCJ T } where CS is a constant specificationfor IPCJ L . For any Γ ∪ { φ } ⊆ L J , the following are equivalent:(1) Γ ⊢ IPCJ L CS φ ;(2) Γ | = H fin AMJL CS φ ;(3) Γ | = H fin AFJL c CS φ ;(4) Γ | = H fin ASJL c CS φ . Corollary . Let
GJ L ∈ {GJ , GJ T , GJ , GJ T } where CS is a constant specification for GJ L . Forany Γ ∪ { φ } ⊆ L J , the following are equivalent:(1) Γ ⊢ GJ L CS φ ;(2) Γ | = , ] G AMJL CS φ ;(3) Γ | = , ] G AFJL c CS φ ;(4) Γ | = , ] G ASJL c CS φ . Corollary . Let
CPCJ L ∈ {CPCJ , CPCJ T , CPCJ , CPCJ T } where CS is a constant specificationfor CPCJ L . For any Γ ∪ { φ } ⊆ L J , the following are equivalent:(1) Γ ⊢ CPCJ L CS φ ;(2) Γ | = { , } B AMJL CS φ ;(3) Γ | = { , } B AFJL c CS φ ;(4) Γ | = { , } B ASJL c CS φ .These theorems already contain several well-known results from the literature on semantics for justificationlogics. At first, the equivalence between (1) and (5), (6) and (7) in Corollary 7.4 are the known completenesstheorems of Mkrtychev [30], Fitting [12, 14] as well as Lehmann and Studer [26], respectively. Further, theequivalence between (1) and (5), (6) in Corollary 7.3 are among the completeness results previously obtainedfor G¨odel justification logic in [34].The previous work on semantics of intuitionistic justification logic in the sense of the present paper is mainly[27], where the authors considered models for IPCJ T
IPCJ T
IPCJ L , GJ L and
CPCJ L using the semantic characterization of their base logics based on Kripke frames. Let IF be the class ofall intuitionistic Kripke frames and let LIF , SIF be the class of all linear intuitionistic Kripke frames and of allsingle-point intuitionistic Kripke frames, respectively. As a well-known result, we have:
Theorem 7.5.
For any Γ ∪ { φ } ⊆ L J :(1) Γ ⊢ IPC φ iff Γ | = IK φ ;(2) Γ ⊢ G φ iff Γ | = LIK φ ;(3) Γ ⊢ CPC φ iff Γ | = SIK φ . Item (1) goes back to Kripkes work [21]. Item (2) and (3) are not that easily traceable ((3) is quite immediate),but for more modern references see e.g. [8]. Combining this with the fact that IF as well as LIF and
SIF areclosed under principal subframes, we have by Lemma 5.6:
Corollary . For any Γ ∪ { φ } ⊆ L J :(1)’ Γ ⊢ IPC φ iff Γ | = g IF φ ;(2)’ Γ ⊢ G φ iff Γ | = g LIF φ ;(3)’ Γ ⊢ CPC φ iff Γ | = g SIF φ .Thus, the completeness theorems based on Kripke frames apply and we obtain the following completenesstheorems. Corollary . Let
IPCJ L ∈ {IPCJ , IPCJ T , IPCJ , IPCJ T } where CS is a constant specificationfor IPCJ L . For any Γ ∪ { φ } ⊆ L J , the following are equivalent:(1) Γ ⊢ IPCJ L CS φ ; NTERMEDIATE JUSTIFICATION LOGICS: UNIFIED COMPLETENESS RESULTS 29 (2) Γ | = IFKMJL CS φ ;(3) Γ | = IFKFJL CS φ ;(4) Γ | = IFKSJL CS φ . Corollary . Let
GJ L ∈ {GJ , GJ T , GJ , GJ T } where CS is a constant specification for GJ L . Forany Γ ∪ { φ } ⊆ L J , the following are equivalent:(1) Γ ⊢ GJ L CS φ ;(2) Γ | = LIFKMJL CS φ ;(3) Γ | = LIFKFJL CS φ ;(4) Γ | = LIFKSJL CS φ . Corollary . Let
CPCJ L ∈ {CPCJ , CPCJ T , CPCJ , CPCJ T } where CS is a constant specificationfor CPCJ L . For any Γ ∪ { φ } ⊆ L J , the following are equivalent:(1) Γ ⊢ CPCJ L CS φ ;(2) Γ | = SIFKMJL CS φ ;(3) Γ | = SIFKFJL CS φ ;(4) Γ | = SIFKSJL CS φ .In particular, the equivalences between (1), (2) and (3) in Corollary 7.7, for IPC , are the completenesstheorems obtained by Marti and Studer in [27].7.2.
Some less-well-known intermediate justification logics.
Finite-valued G¨odel logics.
Prominent strengthenings of the infinite valued G¨odel logic G (or G¨odel-Dummet logic) are the finite valued G¨odel logics G n . These actually predate G in the sense that this sequence ofintermediate logics is the one used by G¨odel in [18] for his investigations about intuitionistic logic. G was laterdefined by Dummet in [10]. Axiomatically, we can give the following description of G n . Consider the axiomscheme( BC ) n : W ni =0 V j
Corollary . Let G n J L ∈ {G n J , G n J T , G n J , G n J T } and let CS be a constant specification for G n J L . For any Γ ∪ { φ } ⊆ L J , the following are equivalent:(1) Γ ⊢ G n J L CS φ ;(2) Γ | = V ( n ) G AMJL CS φ ;(3) Γ | = V ( n ) G AFJL c CS φ ;(4) Γ | = V ( n ) G ASJL c CS φ .7.2.2. The logic of the weak law of the excluded middle.
Consider the axiom scheme ( W LEM ): ¬¬ φ ∨ ¬ φ and the corresponding logic of the weak law of the excluded middle, also known as Jankov’s logic (introducedin [20]), given by KC := IPC ⊕ ( W LEM ) . A classical result of Gabbai [15] (see also Smorynski’s [38]) is the completeness result in terms of special Kripkeframes. For this consider the following definition:
Definition 7.12.
A Kripke frame h F, ≤i has topwidth k , if there are k maximal nodes x , . . . , x k such that forevery y ∈ F , there is a i ∈ { , . . . , k } such that y ≤ x i .Let TIF k be the class of all intuitionistic Kripke frames with topwidth k . Then, one obtains the followingsemantical characterization. Theorem 7.13 (Gabbai [15]) . KC is strongly complete w.r.t. TIF . Consindering that the model class in question is closed under principal subframes, we have the followingcorollary based on Lemma 5.6.
Corollary . KC is strongly globally complete w.r.t. TIF .This results in the following completeness theorem for justification logics based on Jankov’s logic as a corollaryof Theorems 6.4, 6.8 and 6.12. Corollary . Let
KCJ L ∈ {KCJ , KCJ T , KCJ , KCJ T } where CS is a constant specification for KCJ L . For any Γ ∪ { φ } ⊆ L J , the following are equivalent:(1) Γ ⊢ KCJ L CS φ ;(2) Γ | = TIF KMJL CS φ ;(3) Γ | = TIF KFJL CS φ ;(4) Γ | = TIF KSJL CS φ . References [1] S. Artemov. Operational Modal Logic. Technical Report MSI 95-29, Cornell University, 1995. Ithaca, NY.[2] S. Artemov. Explicit Provability and Constructive Semantics.
The Bulleting of Symbolic Logic , 7(1):1–36, 2001.[3] S. Artemov. The logic of justification.
The Review of Symbolic Logic , 1(4):477–513, 2008.[4] S. Artemov and M. Fitting.
Justification Logic: Reasoning with Reasons , volume 216 of
Cambridge Tracts in Mathematics .Cambridge University Press, 2019.[5] S. Artemov and E. Nogina. Introducing justification into epistemic logic.
Journal of Logic and Computation , 15(6):1059–1073,2005.[6] X. Caicedo and R. O. Rodriguez. Standard G¨odel Modal Logics.
Studia Logica , 94:189–214, 2010.[7] A. Chagrov and M. Zakharyashchev. Modal companions of intermediate propositional logics.
Studia Logica , 51:49–82, 1992.[8] A. Ciabattoni and M. Ferrari. Hypersequent Calculi for some Intermediate Logics with Bounded Kripke Models.
Journal ofLogic and Computation , 11(2):283–294, 2001.[9] A. Ciabattoni, N. Galatos, and K. Terui. From Axioms to Analytic Rules in Nonclassical Logics. In , pages 229–240, 2008.[10] M. Dummett. A propositional calculus with denumerable matrix.
Journal of Symbolic Logic , 24(2):97–106, 1959.[11] M. Dummett and E. Lemmon. Modal logics between S4 and S5.
Zeitschrift f¨ur mathematische Logik und Grundlagen derMathematik , 5:250–264, 1959.[12] M. Fitting. A Semantics for the Logic of Proofs. Technical Report TR-2003012, City University of New York, 2003. PhDProgram in Computer Science.[13] M. Fitting. Semantics and tableaus for LPS4. Technical Report TR2004016, CUNY Ph.D. Program in Computer Science,2004.[14] M. Fitting. The logic of proofs, semantically.
Annals of Pure and Applied Logic , 132(1):1–25, 2005.[15] D. Gabbay.
Semantical Investigations in Heyting’s Intuitionistic Logic , volume 148 of
Synthese Library . Springer Netherlands,1981.[16] M. Ghari. Justification Logics in a Fuzzy Setting.
ArXiv e-prints , 2014. arXiv, math.LO, 1407.4647.[17] M. Ghari. Pavelka-style fuzzy justification logics.
Logic Journal of the IGPL , 24(5):743–773, 2016.[18] K. G¨odel. Zum intuitionistischen Aussagenkalk¨ul.
Anzeiger der Akademie der Wissenschaften in Wien , 69:65–66, 1932.[19] K. G¨odel. Eine Interpretation des intuitionistischen Aussagenkalk¨uls.
Ergebnisse eines mathematischen Kolloquiums , 4:39–40,1933.[20] A. Jankov. Calculus of the weak law of the excluded middle.
Izv. Akad. Nauk SSSR, Ser. Mat. , 32:1044–1051, 1968.[21] S. Kripke. Semantical Analysis of Intuitionistic Logic I. In
Formal Systems and Recursive Functions , volume 40 of
Studies inLogic and the Foundations of Mathematics , pages 92–130. 1965.[22] R. Kuznets. On the complexity of explicit modal logics. In
International Workshop on Computer Science Logic. Proceedings ,volume 1862 of
Lecture Notes in Computer Science , pages 371–383. Springer, 2000.[23] R. Kuznets.
Complexity Issues in Justification Logic . PhD thesis, City University of New York, 2008.[24] R. Kuznets, S. Marin, and L. Straburger. Justification logic for constructive modal logic. In , pages 371–383. 2017.[25] R. Kuznets and T. Studer.
Logics of Proofs and Justifications , volume 80 of
Mathematical Logic and Foundations . CollegePublications, 2019.[26] E. Lehmann and T. Studer. Subset models for justification logic.
Logic, Language, Information and Computation (WoLLIC) ,page 433449, 2019.[27] M. Marti and T. Studer. Intuitionistic Modal Logic made Explicit.
IfCoLog Journa of Logics and their Applications , 3(5):877–901, 2016.[28] M. Marti and T. Studer. The Internalized Disjunction Property for Intuitionistic Justification Logic. In
Advances in ModalLogic , 2018.
NTERMEDIATE JUSTIFICATION LOGICS: UNIFIED COMPLETENESS RESULTS 31 [29] J. McKinsey and A. Tarski. Some theorems about the sentential calculi of Lewis and Heyting.
The Journal of Symbolic Logic ,13:1–15, 1948.[30] A. Mkrtychev. Models for the logic of proofs. In
Proceedings of Logical Foundations of Computer Science LFCS’97 , volume1234 of
Lecture Notes in Computer Science , pages 266–275. Springer, 1997.[31] H. Ono. Kripke models and intermediate logics.
Publications of the Research Institute for Mathematical Sciences, KyotoUniversity , 6:461–476, 1971.[32] H. Ono. On Some Intuitionistic Modal Logics.
Publications of the Research Institute for Mathematical Sciences, Kyoto Uni-versity , 13:687–722, 1977.[33] H. Ono.
Proof Theory and Algebra in Logic , volume 2 of
Short Textbooks in Logic . Springer Singapore, 2019.[34] N. Pischke. A note on strong axiomatization of G¨odel Justification Logic.
Studia Logica , 2019. https://doi.org/10.1007/s11225-019-09871-4 .[35] N. Pischke. Standard G¨odel modal logics are not realized by G¨odel justification logics.
ArXiv e-prints , 2019. arXiv, math.LO,1907.04583.[36] N. Preining. G¨odel Logics - A Survey. In
Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2010. Proceed-ings , volume 6397 of
Lecture Notes in Computer Science , pages 30–51. Springer, 2010.[37] V. Shetman. On Incomplete Propositional Logics.
Soviet Mathematics Doklady , 18:985–989, 1977.[38] C. Smorynski.
Investigations of Intuitionistic Formal Systems by Means of Kripke Models . PhD thesis, University of Illinoisat Chicago, 1973.[39] A. Vidal.
On modal expansions of t-norm based logics with rational constants . PhD thesis, Artificial Intelligence ResearchInstitute (IIIA - CSIC) and Universitat de Barcelona, 2015.
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