aa r X i v : . [ c s . L O ] J un Access-based Intuitionistic Knowledge
Steffen Lewitzka * June 30, 2020
Abstract
We introduce the concept of access-based intuitionistic knowledge whichrelies on the intuition that agent i knows ϕ if i has found access to a proof of ϕ . Basic principles are distribution and factivity of knowledge as well as (cid:3) ϕ → K i ϕ and K i ( ϕ ∨ ψ ) → ( K i ϕ ∨ K i ψ ) , where (cid:3) ϕ reads ‘ ϕ is proved’.The formalization extends a family of classical modal logics designed in [Le-witzka 2015, 2017, 2019] as combinations of IPC and
CPC and as systemsfor the reasoning about proof, i.e. intuitionistic truth. We adopt a formal-ization of common knowledge from [Lewitzka 2011] and interpret it here as access-based common knowledge. We compare our proposal with recent ap-proaches to intuitionistic knowledge [Artemov and Protopopescu 2016; Le-witzka 2017, 2019] and bring together these different concepts in a unifyingsemantic framework based on Heyting algebra expansions.
Our investigation is inspired by recent approaches to a formal concept of intu-itionistic knowledge , i.e. formalizations of knowledge that are in accordance withintuitionistic or constructive reasoning. In particular, we consider IntuitionisticEpistemic Logic
IEL introduced by Artemov and Protopopescu [2] where intu-itionistic knowledge is explained as the product of verification . Some principlesof that approach are adopted by Lewitzka [11] and incorporated into a family ofmodal systems L – L originally introduced in [10]. The resulting epistemic log-ics are systems for the reasoning about intuitionistic truth, i.e. proof, and a kindof intuitionistic knowledge based on an informal notion of justification (cf. [12]).In the present paper, we extend modal logic L with the purpose of formalizinga new concept of constructive knowledge which, in our multi-agent setting, relies * Universidade Federal da Bahia – UFBA, Instituto de Matem´atica e Estatistica, Departamento deCiˆencia da Computac¸ ˜ao, 40170-110 Salvador BA, Brazil, [email protected]
1n the intuition that agent i knows ϕ if i has gained access to a proof of ϕ . Thisparadigma also admits a concept of common knowledge that we adopt from [8]and interpret here constructively. The basic motivation behind this access-basedconcept is the idea that to know ϕ , in a constructive sense, means something like to understand, to become aware of, to effect , ... a proof of proposition ϕ (or, if oneprefers, a solution to problem ϕ ), and that the agent possibly has to spent someeffort and ressources to execute this activity. We feel that the intuition of findingaccess to a proof of ϕ captures those ideas in some abstract way. In the follow-ing, we shortly discuss the above mentioned concepts of intuitionistic knowledgefound in the literature and then present the notion of access-based knowledge. Inthe subsequent sections, we shall see that all three concepts can be formalized andstudied within a unifying framework of algebraic (and relational) semantics. Artemov and Protopopescu [2] propose an intuitionistic concept of knowledgewhich is in accordance with the proof-reading semantics of intuitionistic propo-sitional logic
IPC , i.e. with well-known Brouwer-Heyting-Kolmogorov (BHK)interpretation. Knowledge is viewed as the product of a verification . The intu-itive notion of verification generalizes proof as intuitionistic truth in the sense thata proof is ‘the strictest kind of a verification’. Kϕ means that it is verified thatproposition ϕ holds intuitionistically, i.e. there is evidence that ϕ has a proof (evenif a concrete proof is not delivered nor specified in the process of verification).Under this interpretation, the following principles hold and represent an axiomati-zation of IEL in the language of
IPC augmented with knowledge operator K :(i) all schemes of theorems of IPC (ii) K ( ϕ → ψ ) → ( Kϕ → Kψ ) (distribution of knowledge)(iii) ϕ → Kϕ (co-reflection)(iv) Kϕ → ¬¬ ϕ (intuitionistic reflection)Note that (iv) reads ‘Known (i.e. verified) propositions cannot be proved tobe false’. Since the process of verification, in general, does not deliver a concreteproof, the classical reflection principle (factivity of knowledge) Kϕ → ϕ is notvalid. Modus Ponens is the unique inference rule of the resulting deductive system.It is shown in [2] that IEL is sound and complete w.r.t. a possible-worlds semantics.Although the notion of verification is only intuitively given in
IEL , it is shown byProtopopescu [14] that also an arithmetical interpretation can be provided.2 .2 Adopting a justification-based view: Lewitzka 2017, 2019
Logic L was introduced in [10] together with a hierarchy L ( L ( L ( L ofclassical Lewis-style modal logics for the reasoning about intuitionistic truth, i.e.proof. A formula (cid:3) ϕ reads ‘ ϕ is proved (i.e. ϕ has an actual proof)’. Semantics isgiven by a class of Heyting algebras where intuitionistic truth is represented by thetop element of the Heyting lattice, and classical truth is modeled by a designatedultrafilter. Formulas of the form(1) (cid:3) ϕ ↔ ( ϕ ≡ ⊤ ) are theorems and express that (cid:3) is a predicate for intuitionistic truth: (cid:3) ϕ is clas-sically true iff ϕ holds intuitionistically (i.e. ϕ denotes the top element of theunderlying Heyting algebra). An essential feature is the definability of an identityconnective by ϕ ≡ ψ := (cid:3) ( ϕ ↔ ψ ) such that the identity axioms of R. Suszko’sbasic non-Fregean logic, the Sentential Calculus with Identity SCI (cf. [5]), aresatisfied: (i) ϕ ≡ ϕ (ii) ( ϕ ≡ ψ ) → ( ϕ ↔ ψ ) (iii) ϕ ≡ ψ → χ [ x := ϕ ] ≡ χ [ x := ψ ] . ϕ ≡ ψ reads ‘ ϕ and ψ have the same meaning (denotation, Bedeutung )’. Werefer to the axioms (i)–(iii) as the axioms of propositional identity, and particularlyto (iii) as the Substitution Principle (SP). Since these axioms are theorems of ourmodal systems, we are dealing with specific classical non-Fregean logics whichessentially means that the ‘Fregean Axiom’ ( ϕ ↔ ψ ) → ( ϕ ≡ ψ ) does not hold,i.e. formulas with the same truth value may have different meanings. That is, thedenotation of a formula is generally more than a truth value: it is a proposition,i.e. an element of a given model. Actually, all our logics extending L are specificnon-Fregean theories with the property that for any formulas ϕ, ψ : ϕ ≡ ψ is atheorem iff ϕ ↔ ψ is intuitionistically valid, i.e. valid in standard BHK semanticsextended by proof-interpretation clauses for additional operators. Thus, in anymodel, intuitionistically equivalent formulas denote the same proposition whereasformulas such as ϕ and ¬¬ ϕ have, in general, different meanings. This determines,in a sense, the ‘degree of intensionality’ of our logics. The highest degree of thiskind of intensionality is achieved in Suszko’s SCI where for all formulas ϕ , ψ itholds that ϕ ≡ ψ is a theorem iff ϕ = ψ . In SCI , the identity connective is a primitive symbol of the object language. ϕ [ x := ψ ] is the result of substituting ψ for any occurrence of variable x in ϕ . IPC andthus combine
IPC with classical propositional logic
CPC in the following precisesense. If Φ ∪ { ϕ } is a set of formulas in the propositional language of IPC , then(2) Φ ⊢ IP C ϕ ⇔ (cid:3) Φ ⊢ L (cid:3) ϕ, where (cid:3) Φ := { (cid:3) ψ | ψ ∈ Φ } . In particular, for any propositional formula ϕ , ϕ is a theorem of IPC iff (cid:3) ϕ is a theorem of system L . That is, ϕ (cid:3) ϕ is a‘translation’, actually an embedding, of IPC into classical modal logic L , cf. [10]( L can be replaced here with any member of the hierarchy L ⊆ L ⊆ L ⊆ L ⊆ ‘epistemic extensions’). Obviously, this embedding of IPC into classical modalsystems is simpler than the well-known standard translation of
IPC into modallogic S due to G ¨odel. We argued in [12] that the S -style system L is an ade-quate system for reasoning about proof showing that it is complete w.r.t. extendedBHK semantics, i.e. w.r.t. intuitionistic reasoning. This semi-formal result is for-mally confirmed by soundness and completeness of L w.r.t. a relational semanticsbased on intuitionistic general frames, cf. [12]. For this reason, we consider here L as the basis of our epistemic extensions. In [11], we extended L to the epis-temic logic EL taking into account principles coming from IEL . EL is furtherstudied in [12] where its algebraic semantics is complemented by relational seman-tics. EL can be axiomatized in the following way:(INT) All formulas which have the form of an IPC -tautology(i) (cid:3) ( ϕ ∨ ψ ) → ( (cid:3) ϕ ∨ (cid:3) ψ ) (ii) (cid:3) ϕ → ϕ (iii) (cid:3) ( ϕ → ψ ) → ( (cid:3) ϕ → (cid:3) ψ ) (iv) (cid:3) ϕ → (cid:3)(cid:3) ϕ (v) ¬ (cid:3) ϕ → (cid:3) ¬ (cid:3) ϕ (vi) Kϕ → ¬¬ ϕ (intuitionistic reflection)(vii) K ( ϕ → ψ ) → ( Kϕ → Kψ ) (viii) (cid:3) ϕ → (cid:3) Kϕ (weak co-reflection) (TND) ϕ ∨ ¬ ϕ ( tertium non datur )The reference rules are Modus Ponens (MP) and Intuitionistic Axiom Necessi-tation (AN): ‘If ϕ is an intuitionistically acceptable axiom, i.e. any axiom distinctfrom (TND), then infer (cid:3) ϕ .’ Actually, we argued in [12] that all schemes (i)–(viii)above are intuitionistically acceptable, i.e. sound w.r.t. BHK semantics extendedby constructive interpretations of the modal and epistemic operators, respectively. Replacing this scheme with (cid:3) ϕ → Kϕ results in a deductively equivalent system. L is axiomatized by (INT), (i)–(v) and (TND) along with the same infer-ence rules where, again, (AN) applies to all axioms but (TND).Notice that in the more expressive modal language, we are able to weaken theoriginal axiom of co-reflection from IEL . Of course, the resulting formalizationof knowledge then no longer captures the notion of verification as axiomatized in
IEL . Instead, we proposed in [12] to consider an informal notion of justification or reason to motivate the new formalization. Accordingly, we suppose that ϕ isknown by the agent if he has an epistemic justification, reason for ϕ . What theagent recognizes or accepts as an epistemic justification depends essentially fromits internal conditions, reasoning capabilities, convictions, etc. Contrary to themore objective and agent-invariant concept of verification, the notion of justifica-tion is agent-dependent. We postulate that the agent recognizes at least all actualproofs , i.e. all effected constructions, as epistemic justifications. This ensures thevalidity of weak co-reflection (viii). However, a possible proof as a potential, non-effected construction is, in general, not accepted by the agent as a reason for hisknowledge. Full co-reflection in its original form ϕ → Kϕ must be rejected underthis justification-based view.Of course, a justification does not constitute a proof: classical reflection (fac-tivity of knowledge), Kϕ → ϕ , must be rejected. Nevertheless, if the agent hasan epistemic justification of proposition ϕ , then ϕ cannot be proved to be false, i.e. ¬¬ ϕ holds intuitionistically. Therefore, intuitionistic reflection (vi) from IEL isadopted. We also assume that if the agent has justifications for ϕ → ψ and for ϕ ,respectively, then he obtains a justification for ψ . Thus, we adopt distribution ofknowledge, axiom (vii) above, too.
We propose here a concept of constructive knowledge which relies on the intuitionthat an agent knows a proposition ϕ if he has found an access to a proof of ϕ .In some specific context, ‘to find an access to a proof’ may be interpreted as ‘tounderstand a proof’, ‘to become aware of a proof’, etc. We consider a multi-agentscenario based on the following ontological assumptions (see also [12]):We are given a universe of possible proofs , i.e. a universe of potential construc-tions, mathematical possibilities. The creative subject establishes the intuitionis- There is a family of sophisticated Justification Logics found in the literature (see, e.g., [1] for anoverview) where justifications along with operations on them are explicitly formalized. These aspectsare not contained in logic EL . Instead, the notion of justification is understood in a primitive andcompletely informal and unspecified way. This is an established principle in Justification Logics with a precise formalization, cf. [1]. This term was used by Brouwer and we adopt it here four our short, informal explanation. actual proofs among the possible proofs, i.e., the established intuitionistictruths. The universe of possible proofs exists objectively and can be explored byreasoning subjects. A possible proof may be a hypothetical, potential construc-tion, not necessarily effected by the creative subject. It can be conceived as a setof conditions on a construction rather than the construction itself (cf. [3, 4]). Weexpect that these conditions are not in conflict with effected constructions, i.e. theyare ‘consistent’ with the actual proofs. There is a set I = { , ..., N } of N ≥ agents distinct from the creative subject. Each agent can obtain knowledge by ac-cessing possible proofs, where ‘accessing a proof’ is a constructive procedure oractivity that any agent is able to carry out, possibly by spending some effort andresources. A (possible) proof of the proposition “agent i knows ϕ ” is given by a(possible) proof of ϕ along with an access to that proof found by i . Actual proofs,i.e. the constructions effected by the creative subject, are immediately availableand thus trivially accessible. That is, each agent’s knowledge comprises at leastintuitionistic truth established by the creative subject. Finally, there is a designatedsubset of possible proofs that determines the facts, i.e. the ‘classical truths’.By the proof predicate on the object language, we may explicitly distinguishbetween actual proofs and non-effected, possible proofs. As before, (cid:3) ϕ readsclassically ‘ ϕ has an actual proof (i.e. ϕ is proved)’, and ♦ ϕ := ¬ (cid:3) ¬ ϕ reads ‘ ϕ has a possible proof’. In [12], we extended standard BHK interpretation by thefollowing clause for the modal operator:• A proof of (cid:3) ϕ consists in presenting an actual proof of ϕ . Since actual proofs are effected, available constructions, every agent i ∈ I hasthe same immediate, trivial access to them. We denote this unique, trivial accessby s . It might be regarded as an access created by the ‘empty action’ (no effortmust be spent). On the other hand, if some proof t is accessed via s , then t mustbe an actual proof. That is, we postulate the following:• The proofs accessed via s (by any agent) are exactly the actual proofs. Since we are reasoning about proof in classical logic, i.e. from a classical point of view, weadopt a platonist perspective which we combine with the constructive approach. Notice that theBHK interpretation of implication implicitly contains a universal quantification: ‘A proof of ϕ → ψ consists in a construction u such that for all proofs t : if t is a proof of ϕ , then u ( t ) is a proof of ψ ’.The range of that universal quantifier is the given universe of possible proofs. Of course, (cid:3) ϕ → ♦ ϕ is a theorem of the Lewis-style systems L ⊆ L ⊆ L ⊆ L , cf. [12]. We assume that the presentation of an actual proof of ϕ involves some proof-checking procedurewhich depends only from the given actual proof itself and from ϕ .
6e establish the following proof-interpretation clause for the knowledge oper-ator:• A proof of K i ϕ is a tuple ( s, t ) , where s is an access, found by agent i , to aproof t of proposition ϕ .If ( s, t ) is a proof of K i ϕ , then we write also ( si, t ) instead of ( s, t ) in order toemphasize the involved agent. Note that for any i ∈ I and any proposition ϕ , ( s , t ) is a proof of K i ϕ iff t is an actual proof of ϕ . Then the principles (cid:3) K i ϕ → (cid:3) ϕ and (cid:3) ϕ → (cid:3) K i ϕ are intuitionistically acceptable. In fact, given the presentation of anactual proof of (cid:3) K i ϕ , that actual proof must be of the form ( s, t ) , where s is anaccess to the actual proof t of ϕ (thus s = s is the trivial access). The constructionthat maps ( s, t ) to t yields a proof of the former principle. This also shows that anyactual proof of a formula K i ϕ is of the form ( s , t ) , where t is an actual proofof ϕ . Now, one recognizes that the construction that for any actual proof t of ϕ returns the tuple ( s , t ) gives rise to a proof of the latter principle. Consequently, (cid:3) ( (cid:3) K i ϕ → (cid:3) ϕ ) and (cid:3) ( (cid:3) ϕ → (cid:3) K i ϕ ) are sound w.r.t. extended BHK semantics,i.e. (cid:3) K i ϕ ≡ (cid:3) ϕ . Of course, K i ϕ and ϕ denote generally different propositions.It is clear that from a (possible) proof ( si, t ) of K i ϕ , the (possible) proof t of ϕ can be extracted. This procedure yields a proof of K i ϕ → ϕ . Hence, classical re-flection (factivity of knowledge) is intuitionistically acceptable. On the other hand, intuitionistic reflection ϕ → K i ϕ , an axiom of verification-based knowledge, mustbe rejected (for similar reasons as it is rejected in the justification-based approachdiscussed above). In fact, given a (possible) proof t of ϕ , we cannot expect thatagent i has gained any access to t , there is no logical evidence for such an access.The access-based approach validates the following disjunction property of knowl-edge: K i ( ϕ ∨ ψ ) → ( K i ϕ ∨ K i ψ ) . A BHK proof derives immediately from theclauses for K i and disjunction. We postulate the following two Combination Prin-ciples :(C1) If s is an access to proof t , and s ′ is an access to proof u , and t is a con-struction converting u into the proof t ( u ) , then any agent which has gained bothaccesses s and s ′ is able to create a combined access s + s ′ to proof t ( u ) . Weassume that s + s = s = s + s , for any access s and the trivial access s .(C2) If t is an access to proof u , and s is an access to proof ( t, u ) , then acomposed access s ◦ t to proof u can be found. That is, if ( tj, u ) is a proof and ( si, ( tj, u )) is a proof, then (( s ◦ t ) i, u ) is a proof. We assume that s ◦ s = s , forany access s . 7C1) warrants intuitionistic validity of K i ( ϕ → ψ ) → ( K i ϕ → K i ψ ) . Aproof is given by the construction that for any proof ( s, t ) of K i ( ϕ → ψ ) returnsthe function mapping any proof ( s ′ , u ) of K i ϕ to the proof ( s + s ′ , t ( u )) of K i ψ .Principle (C2) warrants the following intuitive epistemic law: ‘If i knows that j knows ϕ , then i knows ϕ ’. That is, K i K j ϕ → K i ϕ is intuitionistically acceptable.As usual, the fact that everyone in group G = { i , ..., i k } knows ϕ is expressedby the formula E G ϕ := K i ϕ ∧ ... ∧ K i k ϕ . Recall that E nG ϕ is recursively definedby E G ϕ := ϕ and E k +1 G ϕ := E G E kG ϕ , for k ≥ . Also recall that knowledgedistributes over conjunction. The concept of ‘ ϕ is common knowledge among theagents of group G ’, notation: C G ϕ , is often informally defined as follows:(3) C G ϕ ⇔ ^ n ∈ N E nG ϕ That is, C G ϕ is true iff the infinitely many formulas ϕ , E G ϕ , E G ϕ , ... are true.However, standard formalizations found in the literature (see, e.g., [7, 13]) involveadditionally properties that go beyond that basic intuition. In fact, standard possibleworlds semantics of epistemic logic with common knowledge validates also thefollowing introspection principle as a theorem of standard axiomatizations:(4) C G ϕ → C G C G ϕ (introspection of common knowledge)But if we take (3) seriously and understand common knowledge as such an in-finite conjunction, then principle (4) does not necessarily follow. Of course, inmany ‘natural’ situations, such as the popular example of the moody children (cf.[7, 13]), common knowledge arises at once after some finite amount of communica-tion steps, and one may regard (4) as an evident principle in those cases. However,one may construct examples where common knowledge is actually attained in aninfinite process of communication steps. In [7], p. 416, for instance, an unrealisticversion of the well-known coordinated-attack problem is discussed. If the messen-ger between the two generals is able to double his speed every time around, andhis first journey takes one hour, then it follows that after exactly two hours he hasvisited both camps an infinite number of times delivering each time the message“attack at down” sent from the other general, and the generals will finally be ableto carry out a coordinated attack because they have attained common knowledge.We may state that after the two hours of infinitely many journeys, each of the twogenerals knows that E nG ϕ , for every natural n (where ϕ is the delivered message).However, we cannot conclude that the generals do know the infinite collection offacts { E nG ϕ | n ∈ N } as a single proposition V n ∈ N E nG ϕ . In fact, new knowl-edge is attained after each finite number of communication steps between the two8gents, but there is no further communication beyond the limit step. This exampleshows that if C G ϕ is attained (possibly by an infinite number of steps), we cannotexpect in general that also K i C G ϕ holds for i ∈ G . Thus, principle (4) is not valid.However, under the assumption that in all known natural situations where com-mon knowledge arises, it arises in a similar way as in the example of the moodychildren , we may accept (4) as an additional axiom. Since our modeling deviatesfrom the possible worlds approach, we are able to treat both versions of commonknowledge: the basic one which is given by an infinite conjunction in the form of(3), and the stronger version which extends the basic version by the introspectionprinciple (4). The axiomatization and semantic modeling of the basic version ofcommon knowledge is adopted from [8] where it was originally developed in ageneral, classical non-Fregean setting. We add here principle (4) and provide aconstructive, access-based interpretation which proves to be sound w.r.t. our ex-tended BHK semantics. We are not able to represent the infinite conjunction of (3)in our object language by a fixed-point axiom or similar solutions working in stan-dard possible worlds semantics. Instead, we propose a semantic characterizationby means of intended models , a solution that we shall discuss in some detail in thelast section. Definition 1.1.
Let G be a group and let t be a (possible) proof. We call an access s to t a common access in G , or a G -common access, if the following hold:(a) all agents of G have gained the same access s to t (b) s is self-referential in G , i.e. for any i, j ∈ G and any proof u , if ( si, u ) is aproof, then so is ( sj, ( si, u )) . The next result shows that the particular choice of proof t in Definition 1.1 isnot relevant. Lemma 1.2.
Let s be a G -common access to t . If some i ∈ G has access s to someproof u , then s is also a G -common access to proof u .Proof. If i ∈ G has the access s to proof u , then ( si, u ) is a proof. Since s is a G -common access, item (b) of Definition 1.1 implies that ( sj, ( si, u )) is a proof,for any j ∈ G . By composition principle (C2) above, (( s ◦ s ) j, u ) is a proof forany j ∈ G . Also by (C2), s ◦ s = s . Thus, ( sj, u ) is a proof, for all j ∈ G . That is, s is a G -common access to u . Lemma 1.3.
For any group G , the trivial access s is a G -common access (to anyactual proof).Proof. Recall that the proofs accessed via s (by any agent) are exactly the actualproofs. Thus, all agents have access s to any actual proof. If ( s , t ) is a proof, for9ome proof t , then t and ( s , t ) must be actual proofs. Thus, ( s , t ) can be accessedvia s (by any agent). By Definition 1.1, s is a G -common access, for any G .A proof-interpretation clause for C G ϕ must take into account the respectiveversion of common knowledge. Let us first consider the basic version of commonknowledge given by the infinite conjunction expressed in (3) above. We considertwo proposals:• A ‘proof’ of C G ϕ consists in an infinite sequence of proofs ( t n ) n ∈ N suchthat t n is a proof of E nG ϕ .• A ‘proof’ of C G ϕ consists in a proof t of ϕ together with a construction thatfor a given proof of E nG ϕ , n ≥ , returns a proof of E G E nG ϕ = E n +1 G ϕ .Unfortunately, both clauses are problematic from a constructivist point of view.The first one describes a proof as an infinite object. The second one gives aninductive definition of a construction that possibly needs an infinite amount of timeto produce all the different proofs of the infinitely many formulas E nG ϕ , n ≥ . Itseems that any approach to the basic intuition (3) of common knowledge (withoutintrospection) involves some form of infinity that makes a constructive treatmenthard or impossible. Therefore, we will focus on the stronger, introspective versionof common knowledge which can be constructively described by the followingsimple and finitary clause:• A proof of C G ϕ is a tuple ( s, t ) , where t is a proof of ϕ and s is a G -commonaccess to t . Example 1.4.
We consider the introspective version of common knowledge. Imag-ine a math lecture. The lecturer writes a proof of a theorem ϕ on the blackboard. Itis clear that at the end of the lecture, there is common knowledge of ϕ in the group G of students who listened the lecture. We interpret the situation constructively inthe following way. Let s be the lecture and let t be the proof of ϕ written on theblackboard. Then all students of group G share the same access s to t . Hence,condition (a) of Definition 1.1 is satisfied. During the lecture, the students can seeeach other listening the lecture. Thus, every student j ∈ G has access via s to theproof ( si, t ) of K i ϕ , for any i ∈ G . This yields proofs ( sj, ( si, t )) of K j K i ϕ , forany j, i ∈ G , and so on ... . Of course, the same arguments apply to any otherstatement ψ with proof u presented in lecture s . Then s is self-referential in G inthe sense of Definition 1.1, i.e. condition (b) holds true. Thus, s is a G -commonaccess to t , and ( s, t ) is a proof of C G ϕ in the sense of the clause for C G ϕ above. (cid:3) ϕ → (cid:3) C G ϕ . Every agent has the trivial access s to an actual proof t of ϕ . Wealready saw that s is self-referential in the group of all agents I . Consequently,the function that maps any actual proof t of ϕ to the actual proof ( s , t ) of C G ϕ gives rise to an actual proof of (cid:3) ϕ → (cid:3) C G ϕ . C G ϕ → C G K i ϕ , i ∈ G . Suppose ( s, t ) is a proof of C G ϕ . Then, in partic-ular, ( si, t ) is a proof of K i ϕ . Since s is self-referential in G , ( sj, ( si, t )) isa proof, for every j ∈ G . Thus, ( s, ( si, t )) is a proof of C G K i ϕ . Then themapping ( s, t ) ( s, ( si, t )) is an effected construction, i.e. actual proof, for C G ϕ → C G K i ϕ . C G ϕ → C G C G ϕ . Let ( s, t ) be a proof of C G ϕ . Then s is a G -common accessto proof t of ϕ . In particular, s is self-referential in G . Thus, for some (for any) j ∈ G , ( sj, ( s, t )) is a proof. Then by Lemma 1.2, s is a G -common access toproof ( s, t ) . By definition, ( s ( s, t )) then is a proof of C G C G ϕ . Thus, the mapping ( s, t ) ( s ( s, t )) represents an actual proof of C G ϕ → C G C G ϕ . (cid:3) ( ϕ → ψ ) → (cid:3) ( C G ϕ → C G ψ ) . Let t be an actual proof of ϕ → ψ . Let ( s, u ) be a proof of C G ϕ . Then t converts u into a proof t ( u ) of ψ . Each i ∈ G has thetrivial access s to t , since t is an actual proof. And each i ∈ G has the access s toproof u . By combination principle (C1), each i ∈ G gains the access s + s = s to proof t ( u ) . By Lemma 1.2, s then is also a G -common access to t ( u ) . Thus, ( s, t ( u )) is a proof of C G ψ . Of course, the function f t : ( s, u ) ( s, t ( u )) is aneffected construction, i.e. an actual proof. Then the construction that for any actualproof t of ϕ → ψ returns a presentation (including proof-checking) of function f t ,constitutes an actual proof of (cid:3) ( ϕ → ψ ) → (cid:3) ( C G ϕ → C G ψ ) .Finally, we show that K i ϕ and C G ϕ have exactly the same actual proofs, indepen-dently of i and G . In fact, ( s, t ) is an actual proof of K i ϕ iff t is an actual proofof ϕ and s = s iff t is an actual proof of ϕ and the trivial access s = s is a G -common access to t iff ( s, t ) is an actual proof of C G ϕ . This shows in particularthat the actual proofs (not all possible proofs) of ϕ , K i ϕ and C G ϕ , respectively,can be converted into each other, i.e. (cid:3) ϕ ≡ (cid:3) K i ϕ ≡ (cid:3) C G ϕ holds for all i ∈ I and all groups G . However, ϕ , K i ϕ and C G ϕ will denote, in general, pairwisedistinct propositions. L AC − N and L ACN
We extend, in the following, system L by axioms for knowledge and commonknowledge in an augmented epistemic object language. As before, I = { , ..., N } Cf. Lemma 2.3(b) below.
11s a fixed finite set of N ≥ agents, and groups of agents are always non-emptysubsets G ⊆ I . Definition 2.1.
The object language is defined over the following set of symbols:an infinite set of propositional variables V = { x , x , ... } , logical connectives ⊥ , ¬ , ∨ , ∧ , → , modal operator (cid:3) and epistemic operators K i , for i ∈ I , and C G ,for every group G of agents. Then the set of formulas F m is the smallest set thatcontains V ∪ {⊥} and is closed under the following conditions: ϕ, ψ ∈ F m ⇒¬ ϕ , ( ϕ ∗ ψ ) , (cid:3) ϕ , K i ϕ , C G ϕ ∈ F m , where ∗ ∈ {∨ , ∧ , →} , i ∈ I , G ⊆ I , G = ∅ . We use the following abbreviations: ⊤ := ( ⊥ → ⊥ ) , ¬ ϕ := ( ϕ → ⊥ ) , ( ϕ ↔ ψ ) := ( ϕ → ψ ) ∧ ( ψ → ϕ ) , ϕ ≡ ψ := (cid:3) ( ϕ ↔ ψ ) (propositional identity), ♦ ϕ := ¬ (cid:3) ¬ ϕ .We consider the following axiom schemes:(INT) any scheme which has the form of an IPC -tautology (i) (cid:3) ( ϕ ∨ ψ ) → ( (cid:3) ϕ ∨ (cid:3) ψ ) (ii) (cid:3) ϕ → ϕ (iii) (cid:3) ( ϕ → ψ ) → ( (cid:3) ϕ → (cid:3) ψ ) (iv) (cid:3) ϕ → (cid:3)(cid:3) ϕ (v) ¬ (cid:3) ϕ → (cid:3) ¬ (cid:3) ϕ (vi) K i ϕ → ϕ (reflection, factivity of knowledge)(vii) K i ( ϕ → ψ ) → ( K i ϕ → K i ψ ) (viii) K i ( ϕ ∨ ψ ) → ( K i ϕ ∨ K i ψ ) (ix) C G ( ϕ → ψ ) → ( C G ϕ → C G ψ ) (x) C G ( ϕ ∨ ψ ) → ( C G ϕ ∨ C G ψ ) (only for introspective common knowledge)(xi) (cid:3) ϕ → (cid:3) C G ϕ (xii) C G ϕ → K i ϕ , for any i ∈ G (xiii) C G ϕ → C G K i ϕ , for any i ∈ G (xiv) C G ϕ → C G ′ ϕ , for any non-empty G ′ ⊆ G (xv) C G ϕ → C G C G ϕ (for introspective common knowledge)(TND) ϕ ∨ ¬ ϕ Except of (TND), all schemes above are intuitionistically acceptable in thesense that they are sound w.r.t. extended BHK semantics considering the access-based interpretation of epistemic operators. For most of the epistemic axioms, thisis shown in the last section. In [12], we saw that the modal axioms, in particular(iv) and (v), are sound w.r.t. extended BHK semantics. For the convenience of the It would be sufficient to fix here a finite set of schemes that axiomatize
IPC . (cid:3) ϕ ∨ ¬ (cid:3) ϕ is intuitionistically acceptable. Of course, either there is an actual proof of ϕ orthere is no such proof. Since an actual proof is immediately available, it can bedecided which one of the two alternatives is the case. In the former case, that ac-tual proof is available and can be presented (proof-checked). This yields an actualproof of (cid:3) ϕ . In the latter case, we conclude that (cid:3) ϕ has no possible proof at all. Infact, any (possible) proof of (cid:3) ϕ would, by the BHK clause, involve an actual proofof ϕ which, by hypothesis, does not exist. Thus, the identity function on proofs, asan effected construction, constitutes an actual proof of (cid:3) ϕ → ⊥ , i.e. of ¬ (cid:3) ϕ . Wehave shown that for any proposition ϕ , either we can present an actual proof of (cid:3) ϕ or we can present an actual proof of ¬ (cid:3) ϕ , and we are able to indicate which oneof the two alternatives is the case. Thus, (5) is intuitionistically valid.Soundness of (iv) (cid:3) ϕ → (cid:3)(cid:3) ϕ . Suppose we are given a proof s of (cid:3) ϕ . Bydefinition, s consists in the presentation of an actual proof t of ϕ . The presentation(including proof-checking) depends only from the actual proof t and from ϕ andno further hypotheses. Thus, s is itself an effected construction, an actual proof.The presentation of s as an actual proof of (cid:3) ϕ yields an actual proof u of (cid:3)(cid:3) ϕ .Thus, the construction that converts s into u is an actual proof of (cid:3) ϕ → (cid:3)(cid:3) ϕ . Soundness of (v) ¬ (cid:3) ϕ → (cid:3) ¬ (cid:3) ϕ . Suppose s is a proof of ¬ (cid:3) ϕ . Then ¬ (cid:3) ϕ (i.e. (cid:3) ϕ → ⊥ ) must have an actual proof for otherwise, by (5) above, (cid:3) ϕ would have an actual proof contradicting that ¬ (cid:3) ϕ has proof s . But then wemay present a witness of an actual proof of (cid:3) ϕ → ⊥ , namely the identity functionon proofs which is, trivially, an effected construction. Its presentation (includingproof-checking) results in an actual proof t of (cid:3) ¬ (cid:3) ϕ . We have presented a con-struction that for any possible proof s of ¬ (cid:3) ϕ returns a proof t of (cid:3) ¬ (cid:3) ϕ .Recall that our basic logic for the reasoning about proof L is given by the ax-iom schemes (INT), (i)–(v) and (TND) plus the inference rules of Modus Ponens(MP) and Intuitionistic Axiom Necessitation (AN): ‘If ϕ is an intuitionistically ac-ceptable axiom, i.e. any axiom distinct from (TND), then infer (cid:3) ϕ .’ We define L ACN as the multi-agent logic of access-based knowledge and introspective com-mon knowledge with N ≥ agents. L ACN is given by L + (vi)–(xv). That is, Actually, (cid:3) ( (cid:3) ϕ ∨ ¬ (cid:3) ϕ ) is a theorem of L , cf. Theorem 3.7(vii) in [12]. This shows in particular that any possible proof of (cid:3) ϕ must be an actual proof of (cid:3) ϕ which isin accordance with the fact that ♦(cid:3) ϕ → (cid:3)(cid:3) ϕ is a theorem of L , cf. Theorem 3.7(v) in [12]. Letter ‘A’ refers to ‘access-based knowledge’ while ‘C’ stands for ‘common knowledge’. ACN is axiomatized by the complete list of axioms above along with the rules of(MP) and (AN). The logic L AC − N is given in the same way as L ACN but withoutthe schemes (x) and (xv). L AC − N is intended to formalize access-based commonknowledge as an infinite conjunction according to (3) without introspection. Ob-viously, both L ACN and L AC − N are super-logics of L . As usual, we define aderivation of ϕ from a set Φ as a finite sequence of formulas ϕ , ..., ϕ n = ϕ suchthat each member of the sequence is an axiom, an element of Φ or the result of anapplication of the rules of (MP) or (AN) to formulas occurring at preceding posi-tions. Recall that (AN) only applies to axioms of the underlying system that aredifferent from tertium non datur . Lemma 2.2.
For any formulas ϕ , ψ , the following hold in all systems extending L :(a) If ϕ is a theorem derivable without (TND), then (cid:3) ϕ is a theorem.(b) The Deduction Theorem holds.(c) The Substitution Principle (SP) holds: ϕ ≡ ψ → χ [ x := ϕ ] ≡ χ [ x := ψ ] .The following are theorems:(d) (cid:3) ϕ ↔ ( ϕ ≡ ⊤ ) and (cid:3) ϕ ≡ ( ϕ ≡ ⊤ ) (e) (cid:3) ( ϕ ∧ ψ ) ≡ ( (cid:3) ϕ ∧ (cid:3) ψ ) and (cid:3) ( ϕ ∨ ψ ) ≡ ( (cid:3) ϕ ∨ (cid:3) ψ ) (f) (cid:3) ( (cid:3) ϕ ∨ ¬ (cid:3) ϕ ) (g) ¬¬ (cid:3) ϕ ≡ (cid:3) ϕ and ¬ ( (cid:3) ϕ ∧ (cid:3) ψ ) ≡ ( ¬ (cid:3) ϕ ∨ ¬ (cid:3) ψ ) (h) ( (cid:3) ϕ ≡ ⊤ ) ∨ ( (cid:3) ϕ ≡ ⊥ ) (i) (cid:3) ( ϕ → ♦ ϕ ) and (cid:3) ( ♦ ϕ → (cid:3)♦ ϕ ) (j) (cid:3) ( ♦ ( ϕ ∨ ψ ) → ( ♦ ϕ ∨ ♦ ψ )) Proof. (a) and (b) can be shown by induction on the length of derivations.(c): Roughly speaking, it is enough to show that propositional identity is a congru-ence relation on
F m . (SP) then follows by induction on χ . This is shown for thelogical connectives, the modal operator and the knowledge operator in [9, 10, 11].We consider here only the new operator of common knowledge. We must showthat ( ϕ ≡ ψ ) → ( C G ϕ ≡ C G ψ ) is a theorem scheme. By axioms (xi), (ix), (ii)and propositional calculus, we get (cid:3) ( ϕ ↔ ψ ) → ( C G ϕ ↔ C G ψ ) . By item (a),distribution and axiom (ii), we obtain the assertion.(d): The first part of (d) is originally shown in [9] for sublogic L . We present herea simpler derivation: 1. ( ϕ ≡ ⊤ ) ⊢ (cid:3) ( ⊤ → ϕ ) ; 2. ( ϕ ≡ ⊤ ) ⊢ (cid:3) ⊤ → (cid:3) ϕ , bydistribution and (MP); 3. ( ϕ ≡ ⊤ ) ⊢ (cid:3) ⊤ , by (AN); 4. ( ϕ ≡ ⊤ ) ⊢ (cid:3) ϕ , by (MP);5. ⊢ ( ϕ ≡ ⊤ ) → (cid:3) ϕ , by Deduction Theorem; 6. ⊢ (cid:3) ( ϕ → ( ⊤ → ϕ )) , by (AN);7. ⊢ (cid:3) ϕ → (cid:3) ( ⊤ → ϕ ) , by distribution and (MP); 8. ⊢ (cid:3) ( ϕ → ( ϕ → ⊤ )) , by(AN); 9. ⊢ (cid:3) ϕ → (cid:3) ( ϕ → ⊤ ) , by distribution and (MP); 10. ⊢ (cid:3) ϕ → ϕ ≡ ⊤ , by7. and 9.; 12. ⊢ (cid:3) ϕ ↔ ϕ ≡ ⊤ , by 5. and 9. This shows the first part of (d). Thesecond part now follows by item (a). 14e): Consider the intuitionistic tautologies ( ϕ ∧ ψ ) → ϕ and ( ϕ ∧ ψ ) → ψ , applyrule (AN), distribution, intuitionistic propositional calculus. The other way round,consider the intuitionistic tautology ϕ → ( ψ → ( ϕ ∧ ψ )) , apply (AN), distributionand intuitionistic propositional calculus. Finally, apply item (a). The second equa-tion follows similarly using propositional calculus and axiom (i).(f): This result is originally proved in [12], Theorem 3.7(vii).(g): Use (f), i.e. (cid:3) ϕ ∨ ¬ (cid:3) ϕ , and propositional calculus. Actually, by (a), it isenough to show that ¬¬ (cid:3) ϕ → (cid:3) ϕ and ¬ ( (cid:3) ϕ ∧ (cid:3) ψ ) → ( ¬ (cid:3) ϕ ∨ ¬ (cid:3) ψ ) derivewithout (TND).(h): Using (f) and axiom (i), one derives (cid:3)(cid:3) ϕ ∨ (cid:3) ¬ (cid:3) ϕ . Then (d) along withpropositional caluclus yields ( (cid:3) ϕ ≡ ⊤ ) ∨ ( (cid:3) ϕ ≡ ⊥ ) .(i): From ϕ → ¬¬ ϕ and the contraposition of theorem (cid:3) ¬ ϕ → ¬ ϕ we derive ϕ → ♦ ϕ without using (TND). Now, apply item (a). The second assertion is clearby scheme (v) and item (a).(j): By (e), ( (cid:3) ¬ ϕ ∧ (cid:3) ¬ ψ ) → (cid:3) ( ¬ ϕ ∧ ¬ ψ ) is a theorem. Observe that ¬ ( ϕ ∨ ψ ) ≡ ( ¬ ϕ ∧ ¬ ψ ) is a theorem since ¬ ( ϕ ∨ ψ ) ↔ ( ¬ ϕ ∧ ¬ ψ ) is an intuitionistic tautol-ogy. By the Substitution Principle (SP), we may replace ¬ ϕ ∧ ¬ ψ by ¬ ( ϕ ∨ ψ ) in every context. Hence, ( (cid:3) ¬ ϕ ∧ (cid:3) ¬ ψ ) → (cid:3) ¬ ( ϕ ∨ ψ ) is a theorem and so is itscontrapositive ¬ (cid:3) ¬ ( ϕ ∨ ψ ) → ¬ ( (cid:3) ¬ ϕ ∧ (cid:3) ¬ ψ ) . Then by the second assertion of(g), we derive ¬ (cid:3) ¬ ( ϕ ∨ ψ ) → ( ¬ (cid:3) ¬ ϕ ∨ ¬ (cid:3) ¬ ψ ) , i.e. ♦ ( ϕ ∨ ψ ) → ( ♦ ϕ ∨ ♦ ψ ) .Note that (TND) does not occur in the derivations. Thus, we may apply item (a)and obtain (j). Lemma 2.3.
The following are theorems of L ACN and of L AC − N :(a) (cid:3) ( ϕ → ψ ) → (cid:3) ( K i ϕ → K i ψ ) and (cid:3) ( ϕ → ψ ) → (cid:3) ( C G ϕ → C G ψ ) (b) (cid:3) ϕ ≡ (cid:3) K i ϕ and (cid:3) ϕ ≡ (cid:3) C G ϕ (c) K i ( ϕ ∧ ψ ) ≡ ( K i ϕ ∧ K i ψ ) and K i ( ϕ ∨ ψ ) ≡ ( K i ϕ ∨ K i ψ ) (d) (cid:3) ( K i K j ϕ → K i ϕ ) Moreover, axiom scheme (xiii) is redundant in L ACN , i.e. it is derivable from theremaining axioms.Proof. (a): (cid:3) ( ϕ → ψ ) → (cid:3) C G ( ϕ → ψ ) is an instance of scheme (xi). Now,consider (ix) and (iii) along with applications of rules (AN) and (MP). This yieldsthe second assertion of (a). Using (xi) and (xii), one derives (cid:3) ϕ → (cid:3) K i ϕ . Thus, (cid:3) ( ϕ → ψ ) → (cid:3) K i ( ϕ → ψ ) is a theorem. The first assertion of (a) now followsin a similar way as the second one.(b): The derivations of (cid:3) ϕ ↔ (cid:3) K i ϕ and (cid:3) ϕ ↔ (cid:3) C G ϕ are straightforward. Now,(b) follows by Lemma 2.2 (a).(c): We show the second assertion. K i ( ϕ ∨ ψ ) → ( K i ϕ ∨ K i ψ ) is a theorem byscheme (viii). ϕ → ( ϕ ∨ ψ ) is an intuitionistic tautology, thus (cid:3) ( ϕ → ( ϕ ∨ ψ )) is a theorem. Now, one easily derives K i ( ϕ → ( ϕ ∨ ψ )) . Then, by distribution of15nowledge, K i ϕ → K i ( ϕ ∨ ψ ) is a theorem. Applying Lemma 2.2 (a) yields thesecond assertion of (c). The proof of the first assertion of (c) is straightforward.(d): K j ϕ → ϕ is an instance of scheme (vi). By (AN), (cid:3) ( K j ϕ → ϕ ) is a theorem.Then the first part of (a), together with (MP), yields (cid:3) ( K i K j ϕ → K i ϕ ) .Finally, we prove the last assertion. By (a), (cid:3) ( C G ϕ → K i ϕ ) → (cid:3) ( C G C G ϕ → C G K i ϕ ) is a theorem. By scheme (xii), (AN) and (MP), C G C G ϕ → C G K i ϕ is atheorem. This, thogether with scheme (xv), yields scheme (xiii) C G ϕ → C G K i ϕ .By Lemma 2.2 (a), we may apply (AN) to that formula. This shows that L ACN without scheme (xiii) is equivalent to L ACN . It is well-known that the class of all Heyting algebras constitutes a semantics for
IPC . A propositional formula ϕ evaluates to the top element of any given Heyt-ing algebra H , under any assignment of elements of H to propositional variables, ifand only if ϕ is a theorem of IPC . In this sense, the greatest element of any givenHeyting algebra represents intuitionistic truth, and we have strong completeness: Φ ⊢ IPC ϕ if and only if for any Heyting algebra H and any assignment γ ∈ H V , if Φ is intuitionistically true in H under γ , then so is ϕ . Recall that a Heyting algebrais a bounded lattice such that for all elements a, b , the subset { c | f ∧ ( a, c ) ≤ b } has a greatest element f → ( a, b ) , called the relative pseudo-complement of a withrespect to b , where f ∧ is the infimum (meet) operation and ≤ is the lattice ordering.For a Heyting algebra H , we use the notation H = ( M, f ∨ , f ∧ , f ⊥ , f → ) , where M is the universe and f ∨ , f ∧ , f ⊥ , f → are the usual operations for join, meet, leastelement and relative pseudo-complement (implication), respectively. The greatestelement is given by f ⊤ := f → ( f ⊥ , f ⊥ ) , and the pseudo-complement (negation) of m ∈ M is defined by f ¬ ( m ) := f → ( m, f ⊥ ) . A subset F ⊆ M of the universe M is called a filter if the following conditions are satisfied: f ⊤ ∈ F ; and for any m, m ′ ∈ M : if m ∈ F and f → ( m, m ′ ) ∈ F , then m ′ ∈ F (cf. [6]). A filter F is aproper filter if f ⊥ / ∈ F . A prime filter is a proper filter F such that f ∨ ( m, m ′ ) ∈ F implies m ∈ F or m ′ ∈ F , for any m, m ′ ∈ M . Finally, an ultrafilter is a max-imal proper filter. Every ultrafilter satisfies for all elements m ∈ M : m ∈ U or f ¬ ( m ) ∈ U . It follows that U mirrors the classical behaviour of logical connec-tives and represents, in this sense, classical truth. In particular, every ultrafilter isprime. Also recall that in any Heyting algebra, for any elements m, m ′ , the equiv-alence m ≤ m ′ ⇔ f → ( m, m ′ ) = f ⊤ holds true.Furthermore, the following facts will be useful: Notice that the argument does not work in L AC − N where scheme (xv) is not available. It is enough to consider Heyting algebras with the Disjunction Property as in Definition 3.2. emma 3.1. Let H be a Heyting algebra with universe M . Then the followinghold.(i) Any proper filter is the intersection of all prime filters containing it.(ii) Let P be a filter, and a, b ∈ M . Then f → ( a, b ) ∈ P iff for all prime filters P ′ ⊇ P : a ∈ P ′ implies b ∈ P ′ .(iii) If the smallest filter { f ⊤ } is prime, then for all a, b ∈ M : a ≤ b iff for allprime filters P : a ∈ P implies b ∈ P .Proof. (i): Let F be a proper filter of H . For every a ∈ M r F , there is a primefilter P a containing F such that a / ∈ P a . In fact, by Zorn’s Lemma, there is anultrafilter with that property. Then F = T a/ ∈ F P a .(ii): Let P be a prime filter, a, b ∈ M . The left-to-right implication of the assertionis clear by definition of a filter. Suppose f → ( a, b ) / ∈ P . Consider F a,P := { c ∈ M | f → ( a, c ) ∈ P } . We claim that F a,P is a filter. Obviously, f ⊤ ∈ F a,P .Suppose c ∈ F a,P and f → ( c, d ) ∈ F a,P , for c, d ∈ M . Then f → ( a, c ) ∈ P and f → ( a, f → ( c, d )) ∈ P . Since (( x → y ) ∧ ( x → ( y → z )) → ( x → z ) is an intuitionistic tautology, we conclude that f → ( a, d ) ∈ P , whence d ∈ F a,P and F a,P is a filter. Let c ∈ P . Of course, f ∧ ( a, c ) ≤ c . Since f → ( a, c ) is thegreatest element x such that f ∧ ( a, x ) ≤ c , it follows that c ≤ f → ( a, c ) . Thus, f → ( a, c ) ∈ P . That is, c ∈ F a,P . We have shown: P ⊆ F a,P . Obviously, a ∈ F a,P and, by hypothesis, b / ∈ F a,P . By (i), it follows that there is a prime filter P ′ extending F a,P such that a ∈ P ′ and b / ∈ P ′ . We have P ⊆ F a,P ⊆ P ′ . Bycontraposition, the right-to-left implication of assertion (ii) follows.(iii): Suppose { f ⊤ } is a prime filter. The equivalence a ≤ b ⇔ f → ( a, b ) = f ⊤ is awell-known property of Heyting algebras. The assertion now follows from (ii). Definition 3.2.
A model M is given by a Heyting algebra expansion M = ( M, TRUE , f ∨ , f ∧ , f ⊥ , f → , f (cid:3) , ( f K i ) i ∈ I , ( f C G ) ∅ = G ⊆ I ) with universe M whose elements are called propositions, a designated ultrafilter TRUE ⊆ M which is the set of classically true propositions, and additionallyunary operations f (cid:3) , f K i , f C G such that the following truth conditions are satis-fied:(i) M has the Disjunction Property: for all m, m ′ ∈ M , f ∨ ( m, m ′ ) = f ⊤ implies m = f ⊤ or m ′ = f ⊤ . That is, the smallest filter { f ⊤ } is prime.(ii) For all m ∈ M : f (cid:3) ( m ) = ( f ⊤ , if m = f ⊤ f ⊥ , else iii) For every prime filter F ⊆ M , and for all i ∈ I and all groups G , the followingconditions (a)–(e) are fulfilled:(a) The set BEL i ( F ) := { m ∈ M | f K i ( m ) ∈ F } is a filter.(b) The set COMMON G ( F ) := { m ∈ M | f C G ( m ) ∈ F } is a filter.(c) For every ultrafilter U ⊇ F : BEL i ( F ) ⊆ U ; in particular, BEL i ( F ) is aproper filter and BEL i ( TRUE ) ⊆ TRUE .(d)
COMMON G ( F ) ⊆ BEL i ( F ) , whenever i ∈ G .(e) For any m ∈ M : if m ∈ COMMON G ( F ) then f K i ( m ) ∈ COMMON G ( F ) ,whenever i ∈ G .(f) COMMON G ( F ) ⊆ COMMON G ′ ( F ) , whenever G ′ ⊆ G . Notice that the definition involves a relational structure given by the set ofprime filters which can be viewed as ‘worlds’ ordered by set-theoretical inclusion.Actually, this yields a relational semantics based on intuitionistic general frames (cf. [6]) with some additional structure regarding the epistemic ingredients. Thiskind of relational semantics was explicitly defined and studied for the logics L , EL and IEL in [12] where also its equivalence to algebraic semantics is shown.Considering Definition 3.2 above and following the constructions presented in [12],that frame-based semantics extends straightforwardly to a semantics with commonknowledge equivalent to the algebraic conditions given in Definition 3.2. For spacereasons, we skip here the details. Intuitively,
BEL i ( F ) is the set of propositionsknown by agent i at ‘world’ F , and COMMON G ( F ) is the set of propositionsthat are common knowledge in G at ‘world’ F . Intuitionistic truth is representedby ‘world’ { f ⊤ } , the smallest prime filter; and classical truth is determined by adesignated ‘maximal world’ TRUE . Observe that f (cid:3) ( m ) is true at ‘world’ F (i.e. f (cid:3) ( m ) ∈ F ) iff m is true at the ‘root world’ { f ⊤ } iff m is true at all ‘worlds’ (i.e.is contained in all prime filters). Thus, regarding the modal operator, we actuallyhave a S -style Kripke model combined with the properties of an intuitionisticKripke model for constructive reasoning. Lemma 3.3.
Let M be a model. We have f K i ( f ⊤ ) = f ⊤ = f C G ( f ⊤ ) , for all i ∈ I and all ∅ = G ⊆ I . Moreover, the operations f K i and f C G are monotonic on M ,i.e. m ≤ m ′ implies f K i ( m ) ≤ f K i ( m ′ ) and f C G ( m ) ≤ f C G ( m ′ ) .Proof. By truth condition (i), { f ⊤ } is a prime filter. Now, consider F = { f ⊤ } and m = f ⊤ in truth conditions (iii)(a) and (iii)(b). Then the first assertion ofthe Lemma follows. Suppose m, m ′ ∈ M and m ≤ m ′ . By Lemma 3.1(iii),it is enough to show: f K i ( m ) ∈ F implies f K i ( m ′ ) ∈ F , for all prime filters F . Let F be a prime filter. Then f K i ( m ) ∈ F implies m ∈ BEL i ( F ) implies m ′ ∈ BEL i ( F ) implies f K i ( m ′ ) ∈ F . The assertion regarding the operators f C G follows similarly. 18 efinition 3.4. Let M be a model. In the following, we consider the truth condi-tions given in Definition 3.2.• M is an L AC − N -model if, instead of (iii)(c), the following stronger condition(c)* is satisfied: For every prime filter F and every i ∈ I , BEL i ( F ) is aprime filter and BEL i ( F ) ⊆ F .• M is an L ACN -model if condition (c)* holds,
COMMON G ( F ) is a primefilter, for every prime filter F , and the following additional truth condition(g) is fulfilled for every prime filter F , every group G and every m ∈ M :(g) If m ∈ COM M ON G ( F ) , then f C G ( m ) ∈ COM M ON G ( F ) .• The Heyting algebra reduct of M with ultrafilter TRUE and operators f (cid:3) and f K (i.e. I = { } , single-agent case) is called an EL -model. Only theconditions (i), (ii), and (iii)(a) and (c) are relevant.• The Heyting algebra reduct of M with ultrafilter TRUE and operator f (cid:3) iscalled an L -model. Of course, only the conditions (i) and (ii) are relevant.• The Heyting algebra reduct of M with operator f K (single-agent case: I = { } ) is said to be an algebraic IEL -model if the following additionaltruth condition of intuitionistic co-reflection (IntCo) is satisfied:(IntCo) F ⊆ BEL ( F ) , for every prime filter F , where BEL ( F ) := BEL ( F ) .Besides that condition, only (i), (iii)(a) and (iii)(c) are relevant. Algebraic semantics for L and EL is originally presented in [10] and [11,12], respectively, in essentially the way as formulated in the next Theorem 3.5.Algebraic semantics of IEL , in the form as presented in [11], is also described inTheorem 3.5 below.
Theorem 3.5.
A Heyting algebra expansion M = ( M, TRUE , f ∨ , f ∧ , f ⊥ , f → , f (cid:3) , ( f K i ) i ∈ I , ( f C G ) ∅ = G ⊆ I ) with ingredients as before is a model in the sense of Definition 3.2 if and only if thefollowing conditions are fulfilled for all m, m ′ ∈ M , all i ∈ I and all groups G :(A) M has the Disjunction Property(B) f (cid:3) ( m ) = ( f ⊤ , if m = f ⊤ f ⊥ , else(C) f K i ( f → ( m, m ′ )) ≤ f → ( f K i ( m ) , f K i ( m ′ )) (D) f C G ( f → ( m, m ′ )) ≤ f → ( f C G ( m ) , f C G ( m ′ )) E) f C G ( m ) ≤ f K i ( m ) , whenever i ∈ G (F) f C G ( m ) ≤ f C G ( f K i ( m )) , whenever i ∈ G (G) f C G ( m ) ≤ f C G ′ ( m ) , whenever G ′ ⊆ G (H) f C G ( f ⊤ ) = f ⊤ (I) f K i ( m ) ≤ f ¬ ( f ¬ ( m )) .– M is an L AC − N -model if instead of (I) the stronger condition (I)* f K i ( m ) ≤ m holds, and for all m, m ′ ∈ M : f K i ( f ∨ ( m, m ′ )) ≤ f ∨ ( f K i ( m ) , f K i ( m ′ )) .– M is an L ACN -model if it is an L AC − N -model and for all m, m ′ ∈ M and allgroups G , f C G ( f ∨ ( m, m ′ )) ≤ f ∨ ( f C G ( m ) , f C G ( m ′ )) and introspection of com-mon knowledge f C G ( m ) ≤ f C G ( f C G ( m )) are satisfied. – The appropriate reduct of M is an EL -model if we drop common knowledgeand consider the single agent case I = { } and only the conditions (A), (B), (C)and (I), and f K ( f ⊤ ) = f ⊤ instead of (H)– The appropriate reduct of M is an L -model if we exclude all epistemic ingredi-ents and consider only the conditions (A), (B).– The appropriate reduct of M is an IEL -model if we drop common knowledge,consider the single agent case I = { } and the condtions (A), (C), (I), and addi-tionally (IntCo): m ≤ f K ( m ) , for all m ∈ M . Theorem 3.5 is useful for model constructions. It hides the relational struc-ture on prime theories which is often not relevant for the construction of an alge-braic model. The proof of Theorem 3.5 is straightforward and relies essentially onLemma 3.1(iii) and filter properties.
Definition 3.6.
Given a model M , an assignment is a function γ : V → M thatextends in the canonical way to an ‘homomorphism’ γ ∗ : F m → M . We simplifynotation and write γ instead of the uniquely determined γ ∗ . The tuple ( M , γ ) iscalled an interpretation. We consider two kinds of satisfaction relations betweeninterpretations and formulas. If M is an IEL -model, then we define ( M , γ ) (cid:15) IEL ϕ : ⇔ γ ( ϕ ) = f ⊤ , where ϕ belongs here to the sublanguage F m e ⊆ F m , i.e. the language of
IEL .If
L ∈ { L , EL , L AC − N , L ACN } and M is an L -model, then we define ( M , γ ) (cid:15) L ϕ : ⇔ γ ( ϕ ) ∈ TRUE , where ϕ is any formula of the underlying object language of the respective logic.If the context it allows, we omit the index L . Of course, the satisfaction relationsextend to sets of formulas in the usual way. Note that introspection along with (E) and (I)* implies f C G ( m ) = f C G ( f C G ( m )) . In this sense,common knowledge is a fixed point. Also notice that (F) follows from introspection of commonknowledge, (E) and monotonicity of f C G . L is one of the logics IEL , L , EL , L AC − N , L ACN , and Φ ∪ { ϕ } is a set of formulas of the respectiveobject language, then Φ (cid:13) L ϕ : ⇔ for every interpretation ( M , γ ) , where M is an L -model, ( M , γ ) (cid:15) L Φ implies ( M , γ ) (cid:15) L ϕ . We consider the logics
IEL and L ACN and show that they are sound and completew.r.t. their respective classes of algebraic models. Soundness and completeness of L AC − N , EL and L then follows similarly. Theorem 4.1.
For any Φ ∪ { ϕ } ⊆ F m e , Φ ⊢ IEL ϕ implies Φ (cid:13) IEL ϕ .Proof. It is enough to show that all axioms of
IEL are true, i.e. denote the topelement in every algebraic
IEL -model under every assignment. This is clear forformulas having the form of an intuitionistic tautology. The validity of the remain-ing axioms follows from the conditions (C), (I) and (IntCo) of Theorem 3.5.Weak completeness of
IEL w.r.t. algebraic semantics is shown in [11]. Forthe convenience of the reader, we outline here a proof which is based on the al-ternative definition of algebraic
IEL -models given in Definition 3.4. We considerthe Lindenbaum-Tarski algebra of
IEL . Its elements are the equivalence classes ϕ modulo logical equivalence in IEL , for ϕ ∈ F m e . By IPC and epistemic axiomsof
IEL it follows that the operations f ∗ ( ϕ, ψ ) := ϕ ∗ ψ , ∗ ∈ {∨ , ∧ , →} , f ⊥ := ⊥ and f K ( ϕ ) := Kϕ are all well-defined. This yields a Heyting algebra M with op-erator f K and lattice ordering ϕ ≤ ψ ⇔ ⊢ IPC ϕ → ψ . In [2], it is shown that IEL has the Disjunction Property. Thus, M has the Disjunction Property, i.e. f ⊤ is thesmallest prime filter. We show that the conditions (iii)(a) and (iii)(c) of Definition3.2 are satisfied. For every prime filter F , the set BEL ( F ) := { ϕ | f K ( ϕ ) ∈ F } isa filter because of the distribution axiom of IEL and the fact that ⊤ → K ⊤ is a the-orem which ensures that f K ( ⊤ ) = K ⊤ = ⊤ = f ⊤ ∈ F and thus f ⊤ ∈ BEL ( F ) .Hence, (iii)(a) holds. Now suppose F is a prime filter and U is an ultrafilter suchthat F ⊆ U . Since Kϕ → ¬¬ ϕ is a theorem of IEL , we have Kϕ ≤ ¬¬ ϕ .Then ϕ ∈ BEL ( F ) implies f K ( ϕ ) ∈ F implies ¬¬ ϕ ∈ F implies ϕ ∈ U . Hence, BEL ( F ) ⊆ U . Thus, the truth conditions of an algebraic IEL -model as establishedin Definitions 3.4 and 3.2 are satisfied. Let γ ∈ M V be the assignment x x . Byinduction on formulas, one shows γ ( ϕ ) = ϕ for every formula ϕ ∈ F m e . Then ( M , γ ) (cid:15) ϕ iff γ ( ϕ ) = ϕ = f ⊤ = ⊤ iff ⊢ IEL ϕ ↔ ⊤ iff ⊢ IEL ϕ . Corollary 4.2.
For every formula ϕ ∈ F m e , ⊢ IEL ϕ ⇔ (cid:13) IEL ϕ . heorem 4.3. Let L be the logic L , EL , L AC − N or L ACN . For any set Φ ∪ { ϕ } of the respective object language, Φ ⊢ L ϕ implies Φ (cid:13) L ϕ .Proof. It suffices to consider logic L ACN . Let M be an L ACN -model and γ ∈ M V an assignment. We show that all axioms denote classically true propositions,i.e. elements of ultrafilter TRUE . This is clear for (TND). We claim that theremaining axioms denote the top element of the Heyting lattice. Then follows thatalso rule (AN) is sound. Of course, all intuitionistic tautologies and substitution-instances denote f ⊤ . Note that all other axioms are of the form: ϕ → ψ . Since γ ( ϕ → ψ ) = f → ( γ ( ϕ ) , γ ( ψ )) = f ⊤ iff γ ( ϕ ) ≤ γ ( ψ ) , it is enough to show that(*) γ ( ϕ ) ≤ γ ( ψ ) holds true. For this purpose, it might be more comfortable to useTheorem 3.5 instead of the model definitions. Concerning the axioms (i)–(v), (*)follows from condition (B): for any m ∈ M , either f (cid:3) ( m ) = f ⊤ or f (cid:3) ( m ) = f ⊥ .Referring to axiom (xi), (*) follows by truth condition (B) along with the firstassertion of Lemma 3.3: f C G ( f ⊤ ) = f ⊤ . Concerning the remaining axioms, (*)follows from corresponding conditions given in Theorem 3.5. Finally, rule (MP)is sound because TRUE is a filter. The assertion of the Theorem now follows byinduction on derivations.Completeness of the logics L and EL w.r.t. algebraic semantics is shown in[10] and [11], respectively. Following the same strategy, we sketch out a complete-ness proof of L ACN w.r.t. the class of L ACN -models. It is enough to show that everyconsistent set of formulas is satisfied by some interpretation based on an L ACN -model. Let Φ ⊆ F m be consistent. By Zorn’s Lemma, Φ has a maximal consistentextension Ψ . We construct a model for Ψ . Let ≈ Ψ be the relation on formulas de-fined by ϕ ≈ Ψ ψ : ⇔ Ψ ⊢ ϕ ≡ ψ . Using the Substitution Principle (SP), one showsthat ≈ Ψ is a congruence relation on the resulting ‘algebra of formulas’, where theconnectives, modal and epistemic operators are viewed as operations on F m (cf[10, 11]). For ϕ ∈ F m , we denote by ϕ := ϕ Ψ the congruence class of ϕ modulo ≈ Ψ . Then the sets M = { ϕ | ϕ ∈ F m } and TRUE = { ϕ | ϕ ∈ Ψ } along with thefollowing operations on M : f ⊥ := ⊥ , f ⊤ := ⊤ , f (cid:3) ( ϕ ) := (cid:3) ϕ , f K i ( ϕ ) := K i ϕ , f C G ( ϕ ) := C G ϕ , f ∗ ( ϕ, ψ ) := ϕ ∗ ψ , where ∗ ∈ {∨ , ∧ , →} , are all well-defined.We claim that this yields an L ACN -model M . Clearly, M is based on a Heytingalgebra: all IPC -theorems of the form ϕ ↔ ψ are contained in Ψ . Then rule (AN)implies Ψ ⊢ ϕ ≡ ψ , i.e. ϕ = ψ , hence all equations that determine a Heyting al-gebra are satisfied. TRUE ⊆ M is an ultrafilter because Ψ is maximal consistent.By Lemma 2.2(d), for any m ∈ M : f (cid:3) ( m ) ∈ TRUE iff m = f ⊤ . The axioms (iv)and (v) then ensure truth condition (ii) of a model, cf. Definition 3.2. Truth condi-tion (i), the Disjunction Property, now follows by axiom (i). From Lemma 2.3(b) itfollows that K i ⊤ and C G ⊤ are theorems. This, along with the distribution axioms,implies that the sets BEL i ( F ) and COMMON G ( F ) are filters, for any prime filter22 of the Heyting algebra. Also the remaining truth conditions (iii)(c)–(g) of an L ACN -model follow straightforwardly from corresponding axioms. As in similarsituations, it might be more comfortable to use Theorem 3.5 here to verify all theseconditions. Let γ ∈ M V be the assignment defined by x x . Then γ ( ϕ ) = ϕ , forevery ϕ ∈ F m . Thus, ϕ ∈ Ψ ⇔ ϕ ∈ TRUE ⇔ γ ( ϕ ) ∈ TRUE ⇔ ( M , γ ) (cid:15) ϕ .In particular, ( M , γ ) (cid:15) Φ ⊆ Ψ . Hence, every set consistent in L ACN is satisfied byan L ACN -model; and analogously for L , EL and L AC − N . Theorem 4.4 (Completeness) . Let L be the logic L , EL , L AC − N or L ACN . Forany set Φ ∪ { ϕ } of the respective object language, Φ (cid:13) L ϕ implies Φ L ⊢ ϕ . Intuitively, by an intended model we mean a model where common knowledge hasits intended meaning, i.e. C G ϕ is true iff V n ∈ N E nG ϕ is true, for any formula ϕ and any group G . Since infinite conjunctions cannot be expressed in the finitaryobject language, our axiomatization ensures only that truth of C G ϕ implies thetruth of all E nG ϕ , n ∈ N . Thus, a non-intended model is a model where for someformula ϕ , E nG ϕ is true for every n ∈ N , but C G ϕ is false. Both intended as wellas non-intended models exist as we shall see at the end of this section.We would like to point out here that it is not unusual that the intended propertiesof a formalized concept are not completely captured by the axiomatization but areinstead represented by a standard model or by certain intended models . The phe-nomenon is well-known from classical first-order logic. Compactness argumentsgenerally show that a given first-order theory with infinite models has also modelswith counter-intuitive or unexpected properties, non-standard elements, etc. Theexistence of such non-intended (or non-standard) models is unproblematic as longas enough intended and meaningful models exist.In the following, we characterize intended L AC − N - and L ACN -models. For thispurpose, we adopt and apply some notions and results from [8] where commonknowledge is axiomatized and modeled in essentially the same way, although thisis done in a general, classical non-Fregean setting. In this section, by a model wealways mean an L AC − N - or an L ACN -model.
Definition 5.1.
Let M be a model. For every i ∈ I and every group G , we put BEL i := BEL i ( TRUE ) and COMMON G := COMMON G ( TRUE ) , which arethe sets of propositions known by agent i , and the sets of propositions that arecommon knowledge in G , respectively. Of course, this basic intuition also holds for our stronger introspective notion of common knowl-edge which additionally has the property: C G ϕ ↔ C G C G ϕ . efinition 5.2. Let M be a model, G be a group. We say that a set X ⊆ M ofpropositions is closed under G if the following hold:(a) X ⊆ T i ∈ G BEL i , i.e. the propositions of X are known by all agents of G ,(b) if m ∈ X and i ∈ G , then f K i ( m ) ∈ X .By GREATEST G we denote the greatest set closed under G , i.e. the union of allsets which are closed under G . Formally, the set
COMMON G represents common knowledge in G . On theother hand, the set GREATEST G captures the concept of common knowledge in G in an intuitive way . Do these two sets coincide? By the definitions, we have: Lemma 5.3.
Let M be a model. For any group G , COMMON G is closed under G . In particular, COMMON G ⊆ GREATEST G ⊆ T i ∈ G BEL i . Relative to an interpretation ( M , γ ) , common knowledge given as an infiniteconjunction according to (3) is expressed in the following way: ( M , γ ) (cid:15) C G ϕ ⇔ for all r ≥ and for all sequences ( i , ..., i r ) of agents of G , it holds that ( M , γ ) (cid:15) K i K i ...K i r ϕ . If ϕ ∈ F m denotes the proposition m ∈ M , i.e. γ ( ϕ ) = m , then that is equivalentto: f C G ( m ) ∈ TRUE ⇔ for all r ≥ and for all sequences ( i , ..., i r ) of agentsof G , it holds that f K i ( f K i ( ... ( f K ir ( m )) ... )) ∈ TRUE . Definition 5.4.
Suppose M is a model. Let m ∈ M and G be a group. We callthe set X G,m given by all elements f K i ( f K i ( ... ( f K ir ( m )) ... )) , where r ≥ and ( i , i , ..., i r ) is any sequence of agents of G , the closure of m under G or the G -closure of m . Lemma 5.5.
Let M be a model. For any m ∈ M , f C G ( m ) ∈ TRUE implies X G,m ⊆ TRUE .Proof.
Let f C G ( m ) ∈ TRUE , i.e. m ∈ COMMON G . Applying successivelytruth condition (iii)(e) of a model (Definition 3.2), one recognizes that any element f K i ( f K i ( ... ( f K ir ( m )) ... )) belongs to COMMON G , where i , ..., i r ∈ G and r ≥ . Hence, X G,m ⊆ COMMON G ⊆ TRUE , and the assertion follows.The next result, also adopted from [8], gives a sufficient and necessary condi-tion for common knowledge having the intended meaning in a given model (inde-pendently of the fact whether we are dealing with the basic notion or with intro-spective common knowledge). Of course, repetitions of agents are allowed in the sequences. For r = 0 , we define K i K i ...K i r ϕ := ϕ . Again, for r = 0 we let f K i ( f K i ( ... ( f K ir ( m )) ... )) := m . heorem 5.6 ([8]) . Let M be a model, G be a group. The following conditionsare equivalent:(i) COMMON G = GREATEST G (ii) For any m ∈ M , f C G ( m ) ∈ TRUE iff X G,m ⊆ TRUE .Proof.
Let
COMMON G = GREATEST G . By Lemma 5.5, we know that forany m ∈ M , f C G ( m ) ∈ TRUE implies X G,m ⊆ TRUE . Suppose m ∈ M and X G,m ⊆ TRUE . Then the G -closure of m , X G,m , is closed under G in the senseof Definition 5.2. Since GREAT EST G is the greatest set closed under G , wehave m ∈ X G,m ⊆ GREATEST G = COMMON G . Hence, f C G ( m ) ∈ TRUE and (i) ⇒ (ii) holds true. Now, suppose (ii) holds true and m ∈ GREATEST G .Since GREATEST G is closed under G , we have f K i ( f K i ( ... ( f K ir ( m )) ... )) ∈ GREATEST G , for any r ≥ and any sequence ( i , i , ..., i r ) of agents of G .Thus, X G,m ⊆ GREATEST G ⊆ TRUE . By (ii), f C G ( m ) ∈ TRUE , i.e. m ∈ COMMON G . Thus, GREATEST G ⊆ COMMON G and (i) follows. Definition 5.7.
A model is said to be an intended model if for each group G , COMMON G = GREATEST G . Corollary 5.8.
Let M be a model. Suppose that for all m ∈ M and all groups G ,the G -closure of m is finite (and thus its infimum exists) and the following holds: f C G ( m ) ∈ TRUE ⇔ V X G,m ∈ TRUE . Then M is an intended model. Example 5.9.
Simple examples of models are given by linearly ordered Heytingalgebras. Note that such Heyting algebras always have the Disjunction Property;actually, all proper filters are prime. We modify and extend an example from [11].It is based on the Heyting algebra over the closed interval M := [0 , of realswith its usual linear ordering and the unique ultrafilter TRUE = (0 , . To agents i = 1 , , ..., N we assign elements b (1) < b (2) < ... < b ( N ) ∈ (0 , , respec-tively, and consider the prime filters BEL i = { m ∈ M | b ( i ) ≤ m } . Then BEL ) BEL ) ... ) BEL N are the sets of propositions known by agent i ,respectively. We define f K i ( m ) := m if m ∈ BEL i , and f K i ( m ) := 0 other-wise. Then follows that BEL i ( F ) = F ∩ BEL i , for any prime filter F . Thus,each BEL i ( F ) is a filter contained in F , in accordance with the truth conditions(iii)(a) and (iii)(c)* of Definitions 3.2 and 3.4. For each group G and m ∈ M ,we define f C G ( m ) := f K iG ( m ) , where i G ∈ G is the greatest number referringto an agent of G . Then common knowledge in G is given by COMMON G = BEL i G = T { BEL j | j ∈ G } and COMMON G ( F ) = BEL i G ( F ) for any primefilter F . Of course, we put f (cid:3) (1) := 1 and f (cid:3) ( m ) := 0 for ≤ m < . Nowone recognizes that all truth conditions of an L ACN -model are satisfied (use the efinitions 3.2 and 3.4 or/and Theorem 3.5). It is an intended model since for each G , COMMON G is the greatest set closed under G : it is clear that COMMON G is closed under G ; and it is the greatest set with that property because for any m ∈ M r COMMON G , we have m / ∈ BEL i G . Unfortunately, common knowl-edge in G is trivial in the sense that it coincides with ‘everyone in G knows’: E G ϕ is true iff C G ϕ is true. We modify the model in the following way. We onlychange common knowledge in the group G = { , ..., N } of all agents and leaveall other definitions as before. Let c be a real number such that b ( N ) < c < .Then we consider COMMON G := { m ∈ M | m ≥ c } ( BEL N and de-fine f C G ( m ) := m if m ∈ COMMON G , and f C G ( m ) := 0 otherwise. Again,one verifies that the resulting structure is an L ACN -model. Since
COMMON G is a proper subset of BEL N , common knowledge in G is no longer trivial, i.e. itis strictly stronger than ‘everyone knows’. However, the resulting model is notan intended one: COMMON G ( GREATEST G = BEL N . Finally, we con-struct an intended model with non-trivial common knowledge. For i = 1 , ..., N ,the sets BEL i are defined as before. For the singleton group G = { } , we put COMMON { } := BEL , and for any m ∈ M : f G ( m ) := f K ( m ) , with f K defined as below. For all other groups G , we define, with the same real number c as above, COMMON G := P := { m ∈ M | m ≥ c } , and f C G ( m ) := m if m ∈ P , and f C G ( m ) := 0 otherwise. Thus, all groups distinct from the single-ton group { } have exactly the same common knowledge given by the prime filter P = { m ∈ M | m ≥ c } ( BEL N . The f K i , i = 1 , ..., N , are now defined in thefollowing way: f K i ( m ) = b , if m ∈ BEL i r Pm, if m ∈ P , elseNotice that for any agent i = 1 , m / ∈ P implies f K i ( m ) / ∈ BEL i . Thus, forall groups G = { } , P = { m ∈ M | m ≥ c } is the greatest closed set under G . And for G = { } , BEL = COMMON { } is the greatest set closed under G . Hence, the eventual model is an intended model. Of course, f (cid:3) is defined asbefore. Similarly as in the previous example, one checks that all truth conditionsof an L ACN -model are satisfied. Common knowledge in any group distinct fromthe singleton { } is not trivial, i.e. it is stronger than ‘everyone knows’: P isa proper subset of each BEL i . This model can be transformed into an intended L AC − N -model modifying for some G = { } the function f C G in the followingway: Let d be a real such that < d < c . Then define f C G ( m ) := m − d if m ∈ COMMON G = P , and f C G ( m ) := 0 otherwise. Now there are some m ∈ COMMON G such that f C G ( m ) / ∈ COMMON G . Hence, the model cannotbe an L ACN -model. But the truth conditions of an L AC − N -model are still satisfied. eferences [1] S. Artemov and M. Fitting, Justification Logic , In: The Stanford En-cyclopedia of Philosophy (Winter 2016 Edition), Edward N. Zalta (ed.),https://plato.stanford.edu/archives/win2016/entries/logic-justification[2] S. Artemov and T. Protopopescu,
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