Our investigation is inspired by recent approaches to a formal concept of intuitionistic knowledge, i.e. formalizations of knowledge that are in accordance with intuitionistic or constructive reasoning. In particular, we consider Intuitionistic Epistemic Logic introduced by Artemov and Protopopescu  where intuitionistic knowledge is explained as the product of verification. Some principles of that approach are adopted by Lewitzka  and incorporated into a family of modal systems – originally introduced in . The resulting epistemic logics are systems for the reasoning about intuitionistic truth, i.e. proof, and a kind of intuitionistic knowledge based on an informal notion of justification (cf. ). In the present paper, we extend modal logic with the purpose of formalizing a new concept of constructive knowledge which, in our multi-agent setting, relies on the intuition that agent knows if has gained access to a proof of . This paradigma also admits a concept of common knowledge that we adopt from  and interpret here constructively. The basic motivation behind this access-based concept is the idea that to know , in a constructive sense, means something like to understand, to become aware of, to effect, … a proof of proposition (or, if one prefers, a solution to problem ), and that the agent possibly has to spent some effort and ressources to execute this activity. We feel that the intuition of finding access to a proof of captures those ideas in some abstract way. In the following, we shortly discuss the above mentioned concepts of intuitionistic knowledge found in the literature and then present the notion of access-based knowledge. In the subsequent sections, we shall see that all three concepts can be formalized and studied within a unifying framework of algebraic (and relational) semantics.
1.1 Verification-based knowledge: Artemov and Protopopescu 2016
Artemov and Protopopescu  propose an intuitionistic concept of knowledge which is in accordance with the proof-reading semantics of intuitionistic propositional logic , i.e. with well-known Brouwer-Heyting-Kolmogorov (BHK) interpretation. Knowledge is viewed as the product of a verification. The intuitive notion of verification generalizes proof as intuitionistic truth in the sense that a proof is ‘the strictest kind of a verification’. means that it is verified that proposition holds intuitionistically, i.e. there is evidence that has a proof (even if a concrete proof is not delivered nor specified in the process of verification). Under this interpretation, the following principles hold and represent an axiomatization of in the language of augmented with knowledge operator :
(i) all schemes of theorems of
(ii) (distribution of knowledge)
(iv) (intuitionistic reflection)
Note that (iv) reads ‘Known (i.e. verified) propositions cannot be proved to be false’. Since the process of verification, in general, does not deliver a concrete proof, the classical reflection principle (factivity of knowledge) is not valid. Modus Ponens is the unique inference rule of the resulting deductive system. It is shown in  that is sound and complete w.r.t. a possible-worlds semantics. Although the notion of verification is only intuitively given in , it is shown by Protopopescu  that also an arithmetical interpretation can be provided.
1.2 Adopting a justification-based view: Lewitzka 2017, 2019
Logic was introduced in  together with a hierarchy of classical Lewis-style modal logics for the reasoning about intuitionistic truth, i.e. proof. A formula reads ‘ is proved (i.e. has an actual proof)’. Semantics is given by a class of Heyting algebras where intuitionistic truth is represented by the top element of the Heyting lattice, and classical truth is modeled by a designated ultrafilter. Formulas of the form
are theorems and express that is a predicate for intuitionistic truth: is classically true iff holds intuitionistically (i.e. denotes the top element of the underlying Heyting algebra). An essential feature is the definability of an identity connective by such that the identity axioms of R. Suszko’s basic non-Fregean logic, the Sentential Calculus with Identity (cf. ), are satisfied:111In , the identity connective is a primitive symbol of the object language.
(iii) .222 is the result of substituting for any occurrence of variable in .
reads ‘ and have the same meaning (denotation, Bedeutung)’. We refer to the axioms (i)–(iii) as the axioms of propositional identity, and particularly to (iii) as the Substitution Principle (SP). Since these axioms are theorems of our modal systems, we are dealing with specific classical non-Fregean logics which essentially means that the ‘Fregean Axiom’ does not hold, i.e. formulas with the same truth value may have different meanings. That is, the denotation 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 are specific non-Fregean theories with the property that for any formulas : is a theorem iff is intuitionistically valid, i.e. valid in standard BHK semantics extended by proof-interpretation clauses for additional operators. Thus, in any model, intuitionistically equivalent formulas denote the same proposition whereas formulas such as and have, in general, different meanings. This determines, in a sense, the ‘degree of intensionality’ of our logics. The highest degree of this kind of intensionality is achieved in Suszko’s where for all formulas , it holds that is a theorem iff .
A further feature of our modal systems is that they contain a copy of and thus combine with classical propositional logic in the following precise sense. If is a set of formulas in the propositional language of , then
where . In particular, for any propositional formula , is a theorem of iff is a theorem of system . That is, is a ‘translation’, actually an embedding, of into classical modal logic , cf.  ( can be replaced here with any member of the hierarchy ‘epistemic extensions’). Obviously, this embedding of into classical modal systems is simpler than the well-known standard translation of into modal logic due to Gödel. We argued in  that the -style system is an adequate system for reasoning about proof showing that it is complete w.r.t. extended BHK semantics, i.e. w.r.t. intuitionistic reasoning. This semi-formal result is formally confirmed by soundness and completeness of w.r.t. a relational semantics based on intuitionistic general frames, cf. . For this reason, we consider here as the basis of our epistemic extensions. In , we extended to the epistemic logic taking into account principles coming from . is further studied in  where its algebraic semantics is complemented by relational semantics. can be axiomatized in the following way:
(INT) All formulas which have the form of an -tautology
(vi) (intuitionistic reflection)
(viii) (weak co-reflection)333Replacing this scheme with results in a deductively equivalent system.
(TND) (tertium non datur)
The reference rules are Modus Ponens (MP) and Intuitionistic Axiom Necessitation (AN): ‘If is an intuitionistically acceptable axiom, i.e. any axiom distinct from (TND), then infer .’ Actually, we argued in  that all schemes (i)–(viii) above are intuitionistically acceptable, i.e. sound w.r.t. BHK semantics extended by constructive interpretations of the modal and epistemic operators, respectively. Logic is axiomatized by (INT), (i)–(v) and (TND) along with the same inference rules where, again, (AN) applies to all axioms but (TND).
Notice that in the more expressive modal language, we are able to weaken the original axiom of co-reflection from . Of course, the resulting formalization of knowledge then no longer captures the notion of verification as axiomatized in . Instead, we proposed in  to consider an informal notion of justification or reason to motivate the new formalization.444There is a family of sophisticated Justification Logics found in the literature (see, e.g.,  for an overview) where justifications along with operations on them are explicitly formalized. These aspects are not contained in logic . Instead, the notion of justification is understood in a primitive and completely informal and unspecified way. Accordingly, we suppose that is known by the agent if he has an epistemic justification, reason for . What the agent recognizes or accepts as an epistemic justification depends essentially from its internal conditions, reasoning capabilities, convictions, etc. Contrary to the more objective and agent-invariant concept of verification, the notion of justification is agent-dependent. We postulate that the agent recognizes at least all actual proofs, i.e. all effected constructions, as epistemic justifications. This ensures the validity 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 his knowledge. Full co-reflection in its original form must be rejected under this justification-based view.
Of course, a justification does not constitute a proof: classical reflection (factivity of knowledge), , must be rejected. Nevertheless, if the agent has an epistemic justification of proposition , then cannot be proved to be false, i.e. holds intuitionistically. Therefore, intuitionistic reflection (vi) from is adopted. We also assume that if the agent has justifications for and for , respectively, then he obtains a justification for .555This is an established principle in Justification Logics with a precise formalization, cf. . Thus, we adopt distribution of knowledge, axiom (vii) above, too.
1.3 Access-based knowledge
We propose here a concept of constructive knowledge which relies on the intuition that an agent knows a proposition if he has found an access to a proof of . In some specific context, ‘to find an access to a proof’ may be interpreted as ‘to understand a proof’, ‘to become aware of a proof’, etc. We consider a multi-agent scenario based on the following ontological assumptions (see also ):
We are given a universe of possible proofs, i.e. a universe of potential constructions, mathematical possibilities. The creative subject666This term was used by Brouwer and we adopt it here four our short, informal explanation. establishes the intuitionistic truth of propositions by effecting constructions. These effected constructions are the actual proofs among the possible proofs, i.e., the established intuitionistic truths. The universe of possible proofs exists objectively and can be explored by reasoning subjects.777Since we are reasoning about proof in classical logic, i.e. from a classical point of view, we adopt a platonist perspective which we combine with the constructive approach. Notice that the BHK interpretation of implication implicitly contains a universal quantification: ‘A proof of consists in a construction such that for all proofs : if is a proof of , then is a proof of ’. The range of that universal quantifier is the given universe of possible proofs. A possible proof may be a hypothetical, potential construction, not necessarily effected by the creative subject. It can be conceived as a set of conditions on a construction rather than the construction itself (cf. [3, 4]). We expect that these conditions are not in conflict with effected constructions, i.e. they are ‘consistent’ with the actual proofs. There is a set of agents distinct from the creative subject. Each agent can obtain knowledge by accessing possible proofs, where ‘accessing a proof’ is a constructive procedure or activity that any agent is able to carry out, possibly by spending some effort and resources. A (possible) proof of the proposition “agent knows ” is given by a (possible) proof of along with an access to that proof found by . Actual proofs, i.e. the constructions effected by the creative subject, are immediately available and thus trivially accessible. That is, each agent’s knowledge comprises at least intuitionistic truth established by the creative subject. Finally, there is a designated subset of possible proofs that determines the facts, i.e. the ‘classical truths’.
By the proof predicate on the object language, we may explicitly distinguish between actual proofs and non-effected, possible proofs. As before, reads classically ‘ has an actual proof (i.e. is proved)’, and reads ‘ has a possible proof’.888Of course, is a theorem of the Lewis-style systems , cf. . In , we extended standard BHK interpretation by the following clause for the modal operator:
A proof of consists in presenting an actual proof of .999We assume that the presentation of an actual proof of involves some proof-checking procedure which depends only from the given actual proof itself and from .
Since actual proofs are effected, available constructions, every agent has the same immediate, trivial access to them. We denote this unique, trivial access by . It might be regarded as an access created by the ‘empty action’ (no effort must be spent). On the other hand, if some proof is accessed via , then must be an actual proof. That is, we postulate the following:
The proofs accessed via (by any agent) are exactly the actual proofs.
We establish the following proof-interpretation clause for the knowledge operator:
A proof of is a tuple , where is an access, found by agent , to a proof of proposition .
If is a proof of , then we write also instead of in order to emphasize the involved agent. Note that for any and any proposition , is a proof of iff is an actual proof of . Then the principles and are intuitionistically acceptable. In fact, given the presentation of an actual proof of , that actual proof must be of the form , where is an access to the actual proof of (thus is the trivial access). The construction that maps to yields a proof of the former principle. This also shows that any actual proof of a formula is of the form , where is an actual proof of . Now, one recognizes that the construction that for any actual proof of returns the tuple gives rise to a proof of the latter principle. Consequently, and are sound w.r.t. extended BHK semantics, i.e. . Of course, and denote generally different propositions.
It is clear that from a (possible) proof of , the (possible) proof of can be extracted. This procedure yields a proof of . Hence, classical reflection (factivity of knowledge) is intuitionistically acceptable. On the other hand, intuitionistic reflection , an axiom of verification-based knowledge, must be rejected (for similar reasons as it is rejected in the justification-based approach discussed above). In fact, given a (possible) proof of , we cannot expect that agent has gained any access to , there is no logical evidence for such an access. The access-based approach validates the following disjunction property of knowledge: . A BHK proof derives immediately from the clauses for and disjunction. We postulate the following two Combination Principles:
(C1) If is an access to proof , and is an access to proof , and is a construction converting into the proof , then any agent which has gained both accesses and is able to create a combined access to proof . We assume that , for any access and the trivial access .
(C2) If is an access to proof , and is an access to proof , then a composed access to proof can be found. That is, if is a proof and is a proof, then is a proof. We assume that , for any access .
(C1) warrants intuitionistic validity of . A proof is given by the construction that for any proof of returns the function mapping any proof of to the proof of . Principle (C2) warrants the following intuitive epistemic law: ‘If knows that knows , then knows ’. That is, is intuitionistically acceptable.
As usual, the fact that everyone in group knows is expressed by the formula . Recall that is recursively defined by and , for . Also recall that knowledge distributes over conjunction. The concept of ‘ is common knowledge among the agents of group ’, notation: , is often informally defined as follows:
That is, is true iff the infinitely many formulas , , , … are true. However, standard formalizations found in the literature (see, e.g., [7, 13]) involve additionally properties that go beyond that basic intuition. In fact, standard possible worlds semantics of epistemic logic with common knowledge validates also the following introspection principle as a theorem of standard axiomatizations:
But if we take (3) seriously and understand common knowledge as such an infinite conjunction, then principle (4) does not necessarily follow. Of course, in many ‘natural’ situations, such as the popular example of the moody children (cf. [7, 13]), common knowledge arises at once after some finite amount of communication 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 an infinite process of communication steps. In , p. 416, for instance, an unrealistic version of the well-known coordinated-attack problem is discussed. If the messenger between the two generals is able to double his speed every time around, and his first journey takes one hour, then it follows that after exactly two hours he has visited 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 able to 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 two generals knows that , for every natural (where is the delivered message). However, we cannot conclude that the generals do know the infinite collection of facts as a single proposition . In fact, new knowledge is attained after each finite number of communication steps between the two agents, but there is no further communication beyond the limit step. This example shows that if is attained (possibly by an infinite number of steps), we cannot expect in general that also holds for . Thus, principle (4) is not valid. However, under the assumption that in all known natural situations where common knowledge arises, it arises in a similar way as in the example of the moody children, we may accept (4) as an additional axiom. Since our modeling deviates from the possible worlds approach, we are able to treat both versions of common knowledge: 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 introspection principle (4). The axiomatization and semantic modeling of the basic version of common knowledge is adopted from  where it was originally developed in a general, classical non-Fregean setting. We add here principle (4) and provide a constructive, access-based interpretation which proves to be sound w.r.t. our extended 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 standard possible worlds semantics. Instead, we propose a semantic characterization by means of intended models, a solution that we shall discuss in some detail in the last section.
Let be a group and let be a (possible) proof. We call an access to a common access in , or a -common access, if the following hold:
(a) all agents of have gained the same access to
(b) is self-referential in , i.e. for any and any proof , if is a proof, then so is .
The next result shows that the particular choice of proof in Definition 1.1 is not relevant.
Let be a -common access to . If some has access to some proof , then is also a -common access to proof .
If has the access to proof , then is a proof. Since is a -common access, item (b) of Definition 1.1 implies that is a proof, for any . By composition principle (C2) above, is a proof for any . Also by (C2), . Thus, is a proof, for all . That is, is a -common access to . ∎
For any group , the trivial access is a -common access (to any actual proof).
Recall that the proofs accessed via (by any agent) are exactly the actual proofs. Thus, all agents have access to any actual proof. If is a proof, for some proof , then and must be actual proofs. Thus, can be accessed via (by any agent). By Definition 1.1, is a -common access, for any . ∎
A proof-interpretation clause for must take into account the respective version of common knowledge. Let us first consider the basic version of common knowledge given by the infinite conjunction expressed in (3) above. We consider two proposals:
A ‘proof’ of consists in an infinite sequence of proofs such that is a proof of .
A ‘proof’ of consists in a proof of together with a construction that for a given proof of , , returns a proof of .
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 an inductive definition of a construction that possibly needs an infinite amount of time to produce all the different proofs of the infinitely many formulas , . It seems that any approach to the basic intuition (3) of common knowledge (without introspection) involves some form of infinity that makes a constructive treatment hard or impossible. Therefore, we will focus on the stronger, introspective version of common knowledge which can be constructively described by the following simple and finitary clause:
A proof of is a tuple , where is a proof of and is a -common access to .
We consider the introspective version of common knowledge. Imagine a math lecture. The lecturer writes a proof of a theorem on the blackboard. It is clear that at the end of the lecture, there is common knowledge of in the group of students who listened the lecture. We interpret the situation constructively in the following way. Let be the lecture and let be the proof of written on the blackboard. Then all students of group share the same access to . Hence, condition (a) of Definition 1.1 is satisfied. During the lecture, the students can see each other listening the lecture. Thus, every student has access via to the proof of , for any . This yields proofs of , for any , and so on … . Of course, the same arguments apply to any other statement with proof presented in lecture . Then is self-referential in in the sense of Definition 1.1, i.e. condition (b) holds true. Thus, is a -common access to , and is a proof of in the sense of the clause for above.
Next, we present some principles of common knowledge which are sound w.r.t. extended BHK interpretation.
. Every agent has the trivial access to an actual proof of . We already saw that is self-referential in the group of all agents . Consequently, the function that maps any actual proof of to the actual proof of gives rise to an actual proof of .
, . Suppose is a proof of . Then, in particular, is a proof of . Since is self-referential in , is a proof, for every . Thus, is a proof of . Then the mapping is an effected construction, i.e. actual proof, for .
. Let be a proof of . Then is a -common access to proof of . In particular, is self-referential in . Thus, for some (for any) , is a proof. Then by Lemma 1.2, is a -common access to proof . By definition, then is a proof of . Thus, the mapping represents an actual proof of .
. Let be an actual proof of . Let be a proof of . Then converts into a proof of . Each has the trivial access to , since is an actual proof. And each has the access to proof . By combination principle (C1), each gains the access to proof . By Lemma 1.2, then is also a -common access to . Thus, is a proof of . Of course, the function is an effected construction, i.e. an actual proof. Then the construction that for any actual proof of returns a presentation (including proof-checking) of function , constitutes an actual proof of .
Finally, we show that and have exactly the same actual proofs, independently of and . In fact, is an actual proof of iff is an actual proof of and iff is an actual proof of and the trivial access is a -common access to iff is an actual proof of . This shows in particular that the actual proofs (not all possible proofs) of , and , respectively, can be converted into each other, i.e. holds for all and all groups .101010Cf. Lemma 2.3(b) below. However, , and will denote, in general, pairwise distinct propositions.
2 The logics of access-based knowledge and
We extend, in the following, system by axioms for knowledge and common knowledge in an augmented epistemic object language. As before, is a fixed finite set of agents, and groups of agents are always non-empty subsets .
The object language is defined over the following set of symbols: an infinite set of propositional variables , logical connectives , , , , , modal operator and epistemic operators , for , and , for every group of agents. Then the set of formulas is the smallest set that contains and is closed under the following conditions: , , , , , where , , , .
We use the following abbreviations: , , , (propositional identity), .
We consider the following axiom schemes:
(INT) any scheme which has the form of an -tautology111111It would be sufficient to fix here a finite set of schemes that axiomatize .
(vi) (reflection, factivity of knowledge)
(x) (only for introspective common knowledge)
(xii) , for any
(xiii) , for any
(xiv) , for any non-empty
(xv) (for introspective common knowledge)
Except of (TND), all schemes above are intuitionistically acceptable in the sense that they are sound w.r.t. extended BHK semantics considering the access-based interpretation of epistemic operators. For most of the epistemic axioms, this is shown in the last section. In , we saw that the modal axioms, in particular (iv) and (v), are sound w.r.t. extended BHK semantics. For the convenience of the reader, we recall here the argumentation. Before, we show that
is intuitionistically acceptable.121212Actually, is a theorem of , cf. Theorem 3.7(vii) in . Of course, either there is an actual proof of or there is no such proof. Since an actual proof is immediately available, it can be decided which one of the two alternatives is the case. In the former case, that actual proof is available and can be presented (proof-checked). This yields an actual proof of . In the latter case, we conclude that has no possible proof at all. In fact, any (possible) proof of would, by the BHK clause, involve an actual proof of which, by hypothesis, does not exist. Thus, the identity function on proofs, as an effected construction, constitutes an actual proof of , i.e. of . We have shown that for any proposition , either we can present an actual proof of or we can present an actual proof of , and we are able to indicate which one of the two alternatives is the case. Thus, (5) is intuitionistically valid.
Soundness of (iv) . Suppose we are given a proof of . By definition, consists in the presentation of an actual proof of . The presentation (including proof-checking) depends only from the actual proof and from and no further hypotheses. Thus, is itself an effected construction, an actual proof. The presentation of as an actual proof of yields an actual proof of . Thus, the construction that converts into is an actual proof of .131313This shows in particular that any possible proof of must be an actual proof of which is in accordance with the fact that is a theorem of , cf. Theorem 3.7(v) in .
Soundness of (v) . Suppose is a proof of . Then (i.e. ) must have an actual proof for otherwise, by (5) above, would have an actual proof contradicting that has proof . But then we may present a witness of an actual proof of , namely the identity function on proofs which is, trivially, an effected construction. Its presentation (including proof-checking) results in an actual proof of . We have presented a construction that for any possible proof of returns a proof of .
Recall that our basic logic for the reasoning about proof is given by the axiom schemes (INT), (i)–(v) and (TND) plus the inference rules of Modus Ponens (MP) and Intuitionistic Axiom Necessitation (AN): ‘If is an intuitionistically acceptable axiom, i.e. any axiom distinct from (TND), then infer .’ We define as the multi-agent logic of access-based knowledge and introspective common knowledge with agents.141414Letter ‘A’ refers to ‘access-based knowledge’ while ‘C’ stands for ‘common knowledge’. is given by + (vi)–(xv). That is, is axiomatized by the complete list of axioms above along with the rules of (MP) and (AN). The logic is given in the same way as but without the schemes (x) and (xv). is intended to formalize access-based common knowledge as an infinite conjunction according to (3) without introspection. Obviously, both and are super-logics of . As usual, we define a derivation of from a set as a finite sequence of formulas such that each member of the sequence is an axiom, an element of or the result of an application of the rules of (MP) or (AN) to formulas occurring at preceding positions. Recall that (AN) only applies to axioms of the underlying system that are different from tertium non datur.
For any formulas , , the following hold in all systems extending :
(a) If is a theorem derivable without (TND), then is a theorem.
(b) The Deduction Theorem holds.
(c) The Substitution Principle (SP) holds: .
The following are theorems:
(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 congruence relation on . (SP) then follows by induction on . This is shown for the logical connectives, the modal operator and the knowledge operator in [9, 10, 11]. We consider here only the new operator of common knowledge. We must show that is a theorem scheme. By axioms (xi), (ix), (ii) and propositional calculus, we get . By item (a), distribution and axiom (ii), we obtain the assertion.
(d): The first part of (d) is originally shown in  for sublogic . We present here a simpler derivation: 1. ; 2. , by distribution and (MP); 3. , by (AN); 4. , by (MP); 5. , by Deduction Theorem; 6. , by (AN); 7. , by distribution and (MP); 8. , by (AN); 9. , by distribution and (MP); 10. , by 7. and 9.; 12. , by 5. and 9. This shows the first part of (d). The second part now follows by item (a).
(e): Consider the intuitionistic tautologies and , apply rule (AN), distribution, intuitionistic propositional calculus. The other way round, consider the intuitionistic tautology , apply (AN), distribution and intuitionistic propositional calculus. Finally, apply item (a). The second equation follows similarly using propositional calculus and axiom (i).
(f): This result is originally proved in , Theorem 3.7(vii).
(g): Use (f), i.e. , and propositional calculus. Actually, by (a), it is enough to show that and derive without (TND).
(h): Using (f) and axiom (i), one derives . Then (d) along with propositional caluclus yields .
(i): From and the contraposition of theorem we derive without using (TND). Now, apply item (a). The second assertion is clear by scheme (v) and item (a).
(j): By (e), is a theorem. Observe that is a theorem since is an intuitionistic tautology. By the Substitution Principle (SP), we may replace by in every context. Hence, is a theorem and so is its contrapositive . Then by the second assertion of (g), we derive , i.e. . Note that (TND) does not occur in the derivations. Thus, we may apply item (a) and obtain (j). ∎
The following are theorems of and of :
Moreover, axiom scheme (xiii) is redundant in , i.e. it is derivable from the remaining axioms.
(a): is an instance of scheme (xi). Now, consider (ix) and (iii) along with applications of rules (AN) and (MP). This yields the second assertion of (a). Using (xi) and (xii), one derives . Thus, is a theorem. The first assertion of (a) now follows in a similar way as the second one.
(b): The derivations of and are straightforward. Now, (b) follows by Lemma 2.2 (a).
(c): We show the second assertion. is a theorem by scheme (viii). is an intuitionistic tautology, thus is a theorem. Now, one easily derives . Then, by distribution of knowledge, is a theorem. Applying Lemma 2.2 (a) yields the second assertion of (c). The proof of the first assertion of (c) is straightforward.
(d): is an instance of scheme (vi). By (AN), is a theorem. Then the first part of (a), together with (MP), yields .
Finally, we prove the last assertion. By (a), is a theorem. By scheme (xii), (AN) and (MP), is a theorem. This, thogether with scheme (xv), yields scheme (xiii) . By Lemma 2.2 (a), we may apply (AN) to that formula. This shows that without scheme (xiii) is equivalent to .151515Notice that the argument does not work in where scheme (xv) is not available. ∎
3 Algebraic semantics
It is well-known that the class of all Heyting algebras constitutes a semantics for .161616It is enough to consider Heyting algebras with the Disjunction Property as in Definition 3.2. A propositional formula evaluates to the top element of any given Heyting algebra , under any assignment of elements of to propositional variables, if and only if is a theorem of . In this sense, the greatest element of any given Heyting algebra represents intuitionistic truth, and we have strong completeness: if and only if for any Heyting algebra and any assignment , if is intuitionistically true in under , then so is .
Recall that a Heyting algebra is a bounded lattice such that for all elements , the subset has a greatest element , called the relative pseudo-complement of with respect to , where is the infimum (meet) operation and is the lattice ordering. For a Heyting algebra , we use the notation , where is the universe and , , , are the usual operations for join, meet, least element and relative pseudo-complement (implication), respectively. The greatest element is given by , and the pseudo-complement (negation) of is defined by . A subset of the universe is called a filter if the following conditions are satisfied: ; and for any : if and , then (cf. ). A filter is a proper filter if . A prime filter is a proper filter such that implies or , for any . Finally, an ultrafilter is a maximal proper filter. Every ultrafilter satisfies for all elements : or . It follows that mirrors the classical behaviour of logical connectives and represents, in this sense, classical truth. In particular, every ultrafilter is prime. Also recall that in any Heyting algebra, for any elements , the equivalence holds true.
Furthermore, the following facts will be useful:
Let be a Heyting algebra with universe . Then the following hold.
(i) Any proper filter is the intersection of all prime filters containing it.
(ii) Let be a filter, and . Then iff for all prime filters : implies .
(iii) If the smallest filter is prime, then for all : iff for all prime filters : implies .
(i): Let be a proper filter of . For every , there is a prime filter containing such that . In fact, by Zorn’s Lemma, there is an ultrafilter with that property. Then .
(ii): Let be a prime filter, . The left-to-right implication of the assertion is clear by definition of a filter. Suppose . Consider . We claim that is a filter. Obviously, . Suppose and , for . Then and . Since is an intuitionistic tautology, we conclude that , whence and is a filter. Let . Of course, . Since is the greatest element such that , it follows that . Thus, . That is, . We have shown: . Obviously, and, by hypothesis, . By (i), it follows that there is a prime filter extending such that and . We have . By contraposition, the right-to-left implication of assertion (ii) follows.
(iii): Suppose is a prime filter. The equivalence is a well-known property of Heyting algebras. The assertion now follows from (ii). ∎
A model is given by a Heyting algebra expansion
with universe whose elements are called propositions, a designated ultrafilter which is the set of classically true propositions, and additionally unary operations , such that the following truth conditions are satisfied:
(i) has the Disjunction Property: for all , implies or . That is, the smallest filter is prime.
(ii) For all :
(iii) For every prime filter , and for all and all groups , the following conditions (a)–(e) are fulfilled:
(a) The set is a filter.
(b) The set is a filter.
(c) For every ultrafilter : ; in particular, is a proper filter and .
(d) , whenever .
(e) For any : if then , whenever .
(f) , whenever .
Notice that the definition involves a relational structure given by the set of prime filters which can be viewed as ‘worlds’ ordered by set-theoretical inclusion. Actually, this yields a relational semantics based on intuitionistic general frames (cf. ) with some additional structure regarding the epistemic ingredients. This kind of relational semantics was explicitly defined and studied for the logics , and in  where also its equivalence to algebraic semantics is shown. Considering Definition 3.2 above and following the constructions presented in , that frame-based semantics extends straightforwardly to a semantics with common knowledge equivalent to the algebraic conditions given in Definition 3.2. For space reasons, we skip here the details. Intuitively, is the set of propositions known by agent at ‘world’ , and is the set of propositions that are common knowledge in at ‘world’ . Intuitionistic truth is represented by ‘world’ , the smallest prime filter; and classical truth is determined by a designated ‘maximal world’ . Observe that is true at ‘world’ (i.e. ) iff is true at the ‘root world’ iff is true at all ‘worlds’ (i.e. is contained in all prime filters). Thus, regarding the modal operator, we actually have a -style Kripke model combined with the properties of an intuitionistic Kripke model for constructive reasoning.
Let be a model. We have , for all and all . Moreover, the operations and are monotonic on , i.e. implies and .
By truth condition (i), is a prime filter. Now, consider and in truth conditions (iii)(a) and (iii)(b). Then the first assertion of the Lemma follows. Suppose and . By Lemma 3.1(iii), it is enough to show: implies , for all prime filters . Let be a prime filter. Then implies implies implies . The assertion regarding the operators follows similarly. ∎
Let be a model. In the following, we consider the truth conditions given in Definition 3.2.
is an -model if, instead of (iii)(c), the following stronger condition (c)* is satisfied: For every prime filter and every , is a prime filter and .
is an -model if condition (c)* holds, is a prime filter, for every prime filter , and the following additional truth condition (g) is fulfilled for every prime filter , every group and every :
(g) If , then .
The Heyting algebra reduct of with ultrafilter and operators and (i.e. , single-agent case) is called an -model. Only the conditions (i), (ii), and (iii)(a) and (c) are relevant.
The Heyting algebra reduct of with ultrafilter and operator is called an -model. Of course, only the conditions (i) and (ii) are relevant.
The Heyting algebra reduct of with operator (single-agent case: ) is said to be an algebraic -model if the following additional truth condition of intuitionistic co-reflection (IntCo) is satisfied:
(IntCo) , for every prime filter , where .
Besides that condition, only (i), (iii)(a) and (iii)(c) are relevant.
Algebraic semantics for and is originally presented in  and [11, 12], respectively, in essentially the way as formulated in the next Theorem 3.5. Algebraic semantics of , in the form as presented in , is also described in Theorem 3.5 below.
A Heyting algebra expansion
with ingredients as before is a model in the sense of Definition 3.2 if and only if the following conditions are fulfilled for all , all and all groups :
(A) has the Disjunction Property