A Quantum Computational Semantics for Epistemic Logical Operators. Part II: Semantics

Transcription

1 Int J heor Phys DOI 0.007/s y A Quantum Computational Semantics for Epistemic Logical Operators. Part II: Semantics Enrico Beltrametti Maria Luisa Dalla Chiara Roberto Giuntini Roberto Leporini Giuseppe Sergioli Received: 5 March 203 / Accepted: 8 June 203 SpringerScience+BusinessMediaNewYork203 Abstract By using the abstract structures investigated in the first Part of this article, we develop a semantics for an epistemic language, which expresses sentences like Alice knows that Bob does not understand that π is irrational. One is dealing with a holistic form of quantum computational semantics, where entanglement plays a fundamental role; thus, the meaning of a global expression determines the contextual meanings of its parts, but generally not the other way around. he epistemic situations represented in this semantics seem to reflect some characteristic limitations of the real processes of acquiring information. Since knowledge is not generally closed under logical consequence, the unpleasant phenomenon of logical omniscience is here avoided. Keywords Contextual meaning Holistic model Epistemic semantic E. Beltrametti Dipartimento di Fisica, Università di Genova, Via Dodecaneso, 33, 646 Genova, Italy enrico.beltrametti@ge.infn.it M.L. Dalla Chiara Dipartimento di Filosofia, Università di Firenze, Via Bolognese 52, 5039 Firenze, Italy dallachiara@unifi.it R. Giuntini G. Sergioli ) Dipartimento di Filosofia e eoria delle Scienze Umane, Università di Cagliari, Via Is Mirrionis, 0923 Cagliari, Italy giuseppe.sergioli@gmail.com R. Giuntini giuntini@unica.it R. Leporini Dipartimento di Ingegneria, Università di Bergamo, viale Marconi 5, Dalmine BG, Italy roberto.leporini@unibg.it

2 Int J heor Phys he Epistemic Quantum Computational Syntax he structures, investigated in the first Part of this article, provide the mathematical basis for the development of our epistemic semantics. he basic intuitive idea can be sketched as follows: pieces of quantum information qumixes) can be denoted by the sentences of a formal language, whose logical connectives correspond to some quantum gates, while the two fundamental epistemic operators to understand and to know) are interpreted as epistemic operations living in semantic models based on convenient epistemic quantum computational structures. Accordingly, sentences like At time t Alice knows that Bob does not understand that π is irrational turn out to denote particular examples of qumixes, representing possible states of quantum objects. Let us first introduce the epistemic language that will be used. his language, indicated by L EpQC,containsatomicsentencessay, thespin-valueinthe x-direction is up ), including two privileged sentences t and f that represent the truth-values ruth and Falsity,respectively.We will use q, r,... as metavariables for atomic sentences, and α, β,... as metavariables for sentences. he quantum computational connectives of L EpQC are: the negation which corresponds to the gate Negation), the square root of the identity id which corresponds to Hadamard-gate), ernary connective which corresponds to the offoli-gate), the exclusive disjunction which corresponds to XOR). he epistemic sub-language of L EpQC contains: a finite set of names for epistemic agents a, b,...); a set of names t, t 2,...)fortheelementsofagiventime-sequence;theepistemic operators U to understand) and K to know). For any sentences α, β, γ,theexpressions α, id α, α, β, γ ), α β are sentences. For any sentence α,foranyagent-namea say, Alice), for any time-name t,theexpressions U α at time t agent a understands the sentence α) andk α at time t agent a knows the sentence α) aresentences.noticethatnestedepistemicoperatorsareallowed:anexpression like K Ub t α is a well-formed sentence. he connectives, id,, are called gate-connectives. AnysubexpressionU or K of an epistemic sentence will be called an epistemic connective. We recall that, for any truth-perspective, the offoli-gate permits one to define a reversible conjunction AND for any ρ DH m) ) and for any σ DH n) )): AND ρ, σ ) := D m,n,) ρ σ P ) ) 0. Accordingly, from a syntactical point of view, it is reasonable to define metalinguistically) the logical conjunction as follows for any sentences α and β): α β := α, β, f). We will now introduce some syntactical notions that will be used in our semantics. Definition. α is called a gate-sentence iff either α is atomic or the principal connective of α is a gate-connective. α is called an epistemic sentence iff α has the form K β. Definition.2 he Atomic Complexity of a Sentence) he atomic complexity Atα) of a sentence α is the number of occurrences of atomic sentences in α. See [, 3, 5 7].

3 Int J heor Phys For instance, At q, q, f)) = 3. We will also indicate by α n) asentencewhoseatomic complexity is n. he notion of atomic complexity plays an important semantic role.as we will see, the meaning of any sentence whose atomic complexity is n shall live in the domain DH n) ).Forthisreason,H Atα)) briefly indicated by H α )willbealsocalledthesemantic space of α. Any sentence α can be naturally decomposed into its parts, giving rise to a special configuration called the syntactical tree of α indicated by Sree α ). Roughly, Sree α can be represented as a finite sequence of levels: where: Level k α). Level α), each Level i α) with i k)isasequenceβ,...,β m ) of subformulas of α; the bottom level Level α) is α; the top level Level k α) is the sequence q,...,q r ),whereq,...,q r are the atomic occurrences in α; for any i with i<k), Level i+ α) is the sequence obtained by dropping the principal gate-connective in all molecular gate-sentences occurring at Level i α), by dropping the epistemic connectives U, K )inallepistemicsentencesoccurringatlevel i α) and by repeating all the atomic sentences that occur at Level i α). ByHeight of α indicated by Heightα))wemeanthenumberoflevelsofthesyntactical tree of α. As an example, consider the following sentence: α = K q q) = K q, q, f) ) say, At time t Alice knows the non-contradiction principle, instantiated by means of the atomic sentence q). he syntactical tree of α is the following sequence of levels: Level 5 α) = q, q, f) Level 4 α) = q, q; f) Level 3 α) = q, q, f) ) Level 2 α) = q, q, f) ) Level α) = K q, q, f) )). Clearly, Height q, q, f)) = 5. More precisely, the syntactical tree of a sentence whose atomic complexity is r) is defined as follows. Definition.3 he Syntactical ree of α) he syntactical tree of α is the following sequence of sentence-sequences: Sree α = Level α),..., Level k α) ),

4 Int J heor Phys where: Level α) = α); Level i+ is defined as follows for any i such that i<k.hepossiblecasesare: ) Level i α) does not contain any connective. Hence, Level i α) = q,...,q r ) and Heightα) = i; 2) Level i α) = β,...,β m ), and for at least one j, β j has a principal) connective. Consider the following sequence of sentence-sequences:,..., m, where with { h = β h ), if β h is atomic; h, otherwise, δ), if β h = δ or β h = id δ; h = γ, δ, θ), if β h = γ, δ, θ); γ, δ), if β h = γ δ; δ), if β h = U δ or β h = K δ. hen, Level i+ α) = m, where represents the sequence-composition. 2 he Epistemic Quantum Computational Semantics We will now give the basic definitions of our semantics. We will apply to epistemic situations a holistic version of quantum computational semantics which has been naturally inspired by the characteristic holistic features of the quantum theoretic formalism). 2 In this semantics any model assigns to any sentence a global meaning that determines the contextual meanings of all its parts from the whole to the parts!). It may happen that one and the same sentence receives different meanings in different contexts. Before defining holistic models, we will first introduce the weaker notion of quasi-model of the language L EpQC. Definition 2. Quasi-model) A quasi-model of the language L EpQC is a system where: M q =, Ag, EpSit, den) 2 See [4]and[2].

5 Int J heor Phys ), Ag, EpSit) is an epistemic quantum computational structure; 3 2) den is a function that interprets the individual names of the language. By simplicity, we put: dena) = a; dent) = t. Apparently, quasi-models represent partial interpretations of the language: while names of times and of agents receive an interpretation in the framework of a given epistemic situation, meanings of sentences are not determined. In the first Part of this article we have seen that knowledge operations cannot be generally represented as qumix gates. At the same time, once fixed an epistemic quantum computational structure S =, Ag, EpSit), one can naturally define the following notion of pseudo-gate with respect to S. Definition 2.2 Pseudo-gate) Let S =, Ag, EpSit) be an epistemic quantum computational structure. A pseudo-gate of S is an operator-product X n ) X nm) m, where any X n i ) i with i m) iseitheraqumixgate D G n i ) with respect to ruthperspective or an epistemic operation U n i ) or K n i ) )ofs. One can show that for any choice of ruth-perspective and of a quasi-model M q =, Ag, EpSit, den), thesyntacticaltreeofasentenceα uniquely determines a sequence of pseudo-gates, that will be called the, M q )-pseudo-gate tree of α. As an example, consider again the sentence α = K q q) = K q, q, f) ) and its syntactical tree. Apparently, Level 4 α) is obtained from Level 5 α) by repeating the first occurrence of q, by negating the second occurrence of q and by repeating f. Hencethepseudo-gatethat corresponds to Level 4 α) will be D I ) D NO ) D I ). Level 3 α) is obtained from Level 4 α) by applying to the three sentences occurring at Level 4 α) the connective.hence the pseudo-gate that corresponds to Level 3 α) will be D,,). Level 2 α) is obtained from Level 3 α) by applying to the sentence occurring at Level 3 α) the connective. Hencethe pseudo-gate that corresponds to Level 2 α) will be D NO 3).Finally,Level α) is obtained from Level 2 α) by applying to the sentence occurring at Level 2 α) the epistemic connective K.Hencethepseudo-gatethatcorrespondstoLevel α) will be K 3). On this basis, the, M q )-pseudo-gate tree of the sentence α = K q q) = K q, q, f) ) can be identified with the following sequence consisting of four pseudo-gates: DI ) D NO ) D I ), D,,), D NO 3), K3) ). Notice that the truth-perspectives and at may be different. he general definition of, M q )-pseudo-gate tree is the following: 3 See Sect. 3 of the first Part of this article.

6 Int J heor Phys Definition 2.3, M q )-pseudo-gate tree) Let α be a sentence such that Heightα) = k. he, M q )-pseudo-gate tree of α is the sequence of pseudo-gates that is defined as follows. Suppose that where <i k). We put: Psree α = α O k ),..., α O ) ), Level i α) = β r ) ),...,β m rm), where any α X r j ) α O i ) = α X r ) α X rm), is a pseudo-gate defined on Hr j ) such that: D I r j ), if β r j ) j is atomic; D NO r j ), if βr j ) j = δ; D I r j ) α X r, if βr j ) j = id δ; j ) = D u,v,w), if β r j ) j = γ u),δ v),θ w) ); D XOR u,v), if βr j ) j = γ u) δ v) ; U r j ), if β r j ) j = U δ; K r j ), if β r j ) j = K δ. Consider now a sentence α and let α O k ),..., α O ) ) be the, Mq )-pseudo-gate tree of α.anychoiceofaqumixρ in H α determines a sequence ρ k,...,ρ ) of qumixes of H α, where: ρ k = ρ ρ k = α O k ) ρ k ). ρ = α O ) ρ 2). he qumix ρ k can be regarded as a possible input-information concerning the atomic parts of α, whileρ represents the output-information about α, giventheinput-informationρ k. Each ρ i corresponds to the information about Level i α), giventheinput-informationρ k. How to determine an information about the parts of α under a given input? It is natural to apply the reduced state function that determines for any state ρ of a composite system S = S + +S n the state Red i,...,i m ρ) of any subsystem S i + +S im where i n,..., i m n). Consider the syntactical tree of α and suppose that: Level i α) = β i,...,β ir ).

7 Int J heor Phys We know that the, M q )-pseudo-gate tree of α and the choice of an input ρ k in H α ) determine a sequence of qumixes: ρ k Level k α) = q,...,q t ). ρ i Level i α) = β i,...,β ir ). ρ Level α) = α). We can consider Red j ρ i ),thereduced information of ρ i with respect to the j-th part. From a semantic point of view, this object can be regarded as a contextual information about β ij the subformula of α occurring at the j-th position at Level i α)) undertheinputρ k. We can now define the notion of holistic model,whichassignsmeaningstoallsentences of the language, for any choice of ruth-perspective. Definition 2.4 Holistic Model) A holistic model of the language L EpQC is a system where: M =,Ag,EpSit, den, M Hol ) ), Ag, EpSit, den) is a quasi-model M q of the language. 2) M Hol is a map that associates to any truth-perspective amap M Hol representing a holistic interpretation of the sentences of the language. he following conditions are required. 2.) For any sentence α, theinterpretation M Hol associates to each level of the syntactical tree of α a meaning,representedbyaqumixlivinginh α the semantic space of α). 2.2) Let α O k ),..., α O ) ) be the, Mq )-pseudo-gate tree of sentence α and let i<heightα).hen, M Hol Leveli α) ) = α O i) M Hol Leveli+ α) )). In other words, the global meaning of each level different from the top level) is obtained by applying the corresponding pseudo-gate to the meaning of the level that occurs immediately above. 2.3) Let Level i α) = β,...,β r ). hen: β j = f Red j M Hol Level i α))) = P ) 0 ; β j = t Red j M Hol Level i α))) = P ),foranyj j r).in other words, the contextual meanings of f and of t are always the -Falsity and the -ruth,respectively. On this basis, we put: M Hol α) := M Hol Level α) ), for any sentence α.

8 Int J heor Phys As an example, consider again the sentence α = K q q) = K q, q, f) ). As we have seen, any, M q )-pseudo-gate-tree of α will have the following form: ake a model such that: DI ) D NO ) D I ), D,,), D NO 3), K3) ). M =,Ag,EpSit, den, M Hol ) M Hol I Level Heightα) α)) = M Hol I q, q, f)) = P ψ,where ψ = ) ) 0. EpSit assigns to agent the epistemic situation at,epd at, U at, K at ), where EpD at = D and K at ρ) = ρ, foranyρ D. Inotherwords, has a maximal epistemic capacity. 4 We obtain: M Hol I K q q)) = P ϕ, where: ϕ =K 3) NO 3) I,,) I I ) NO ) I I ) ) ) ) 0 ) = 0,, + 0, 0, +,, 0 + 2, 0, ). Hence,p I M Hol I K q q))) = 3. his example clearly shows how 4 even an agent with a maximal epistemic capacity does not necessarily know a very simple instance of the non-contradiction principle! Unlike standard compositional semantics, any M Hol α) represents a kind of autonomous semantic context that is not necessarily correlated to the meanings of other sentences. At the same time, given a sentence γ, M Hol determines the contextual meaning, with respect to the context M Hol γ ), ofanyoccurrence of a subformula β in the syntactical tree of γ. Definition 2.5 Contextual Meaning) Consider a sentence γ such that Level i γ ) = β i,...,β ir ). he contextual meaning of the occurrence β ij defined as follows: with respect to the context M Hol γ ) is M Hol γ β i j ) := Red j M Hol Leveli γ ) )). Hence, in particular, we have for any sentence γ M Hol γ γ ) = M Hol Level γ ) ) = M Hol γ ). 4 See Sect. 3 of the first Part of this article.

9 Int J heor Phys Generally, different occurrences β ij and β hk of one and the same subformula β in the syntactical tree of γ may receive different contextual meanings. In other words, we may have: M Hol γ β i j ) M Hol γ β h k ). When this is not the case, we will say that one is dealing with a normal model. Definition 2.6 Normal Holistic Model) A normal holistic model of the language L EpQC is a holistic model M such that for any truth-perspective and for any sentence γ,the interpretation M Hol determines for any occurrence β ij of a subformula β of γ in the syntactical tree of γ the same contextual meaning, which will be uniformly indicated by M Hol γ β). In the following we will always refer to normal holistic models. Suppose that β is a subformula of two different formulas γ and δ. Generally,wehave: M Hol γ β) M Hol δ β). In other words, sentences may receive different contextual meanings in different contexts also in the case of the normal holistic semantics. o what extent do contextual meanings and gates associated to the logical connectives) commute? An answer to this question is given by the following theorem. heorem 2. Consider a holistic model M =, Ag, EpSit, den, M Hol) and ruthperspective.. Let α be a subformula of γ. Suppose that α = β ij the formula occurring at the j-th position of the i-th level in the syntactical tree of γ ), while α = β i+)k. We have: M Hol γ α) = Redj M Hol Leveli γ ) )) = D NO Atα)) Red k M Hol Leveli+ γ ) ))) = D NO Atα)) M Hol γ α)). 2. Let id α be a subformula of γ. Suppose that id α = β ij, while α = β i+)k. We have: M Hol γ id α) = Red j M Hol Leveli γ ) )) = D I Atα)) Red k M Hol Leveli+ γ ) ))) = D I Atα)) M Hol γ α)). 3. Let α,α 2,α 3 ) be a subformula of γ. Suppose that in the syntactical tree of γ : α,α 2,α 3 ) = β ij, while α = β i+)k, α 2 = β i+)k2, α 3 = β i+)k3. We have: M Hol γ α,α 2,α 3 ) ) = Red j M Hol Leveli γ ) )) = D Atα ),Atα 2 ),Atα 3 )) Red k,k 2,k 3 M Hol Leveli+ γ ) ))).

10 Int J heor Phys 4. Let α α 2 be a subformula of γ. Suppose that in the syntactical tree of γ : α α 2 = β ij, while α = β i+)k, α 2 = β i+)k2. We have: M Hol γ α α 2 ) = Red j M Hol Leveli γ ) )) = D XOR Atα ),Atα 2 )) Red k,k 2 M Hol Leveli+ γ ) ))). 5. Let U at α be a subformula of γ. Suppose that U at α = β ij, while α = β i+)k. We have: M Hol γ U α) = Red j M Hol Leveli γ ) )) = U Atα)) Red k M Hol Leveli+ γ ) ))) = U Atα)) M Hol γ α)). 6. Let K at α be a subformula of γ. Suppose that K at α = β ij, while α = β i+)k. We have: M Hol γ K α) = Red j M Hol Leveli γ ) )) = K Atα)) Red k M Hol Leveli+ γ ) ))) = K Atα)) M Hol γ α)). Proof By definition of syntactical tree, of pseudo-gate tree, of normal holistic model and of contextual meaning. Notice that, generally, the contextual meaning of a conjunction is not the conjunction of the contextual meanings of the two members. As a counterexample, consider the following contradictory sentence: whose syntactical tree is: Consider a model M such that: γ = q q = q, q, f), Level 3 γ ) = q, q, f) Level 2 γ ) = q, q, f) Level γ ) = q, q, f). M Hol I Level3 γ ) ) = P 2 0,,0 +,0,0 ) which is a maximally entangled quregister with respect to the first and to the second part.) 5 Hence: M Hol I Level γ ) ) = M Hol I γ ) = M Hol γ Iγ ) = P 2 0,0,0 +,, ) 5 See Sect. 2 of the first Part of this article.

11 Int J heor Phys which is a maximally entangled quregister). At the same time, we have: which is a proper mixture. Consequently: M Hol γ Iq) = M Hol γ I q) = 2 P ) P ), D,,)M Hol γ Iq) M Hol γ I q) M Hol γ If) ) M Hol γ ) I q, q, f). Notice that: p I M Hol γ Iq q)) = 2 ;while:p I D,,,) M Hol γ Iq) M Hol γ I q) M Hol γ If))) = 4. Definition 2.7 Harmonic and Sound Models) Let M =, Ag, EpSit, den, M Hol) be a model. M is called harmonic iff the epistemic structure of M is harmonic, i.e. all agents of the structure share the same truth-perspective.furthermore,theinterpretation-function M Hol is only defined for the truth-perspective. M is called sound iff all agents of M have a sound epistemic capacity i.e. assign the right probability-values to the truth-values of their truth-perspectives). 6 By harmonic epistemic quantum computational semantics sound epistemic quantum computational semantics) wewillmeanthesemanticsthatonlyreferstoharmonicmodels sound models). We can now define the notions of truth, validity and logical consequence. Definition 2.8 Contextual ruth) Let α be a subformula of γ. γ,m,) α the sentence α is true with respect to the context γ, the model M and the truth-perspective ) iff p M Hol γ α)) =. Definition 2.9 ruth) M,) α the sentence α is true with respect to the model M and the truth-perspective )iff α,m,) α. Hence, the concept of truth turns out to be a special case of the concept of contextual truth. Definition 2.0 Contextual Validity) Let α be a subformula of γ. γ,) α the sentence α is valid with respect to the context γ and the truth-perspective )iffforanymodelm, γ,m,) α. γ α the sentence α is valid with respect to the context γ )iffforanytruth-perspective, γ,) α. Definition 2. Validity) α the sentence α is valid with respect to the truth-perspective )iff α,) α. α the sentence α is valid) iff α α. 6 See Sect. 3 of the first Part of this article.

12 Int J heor Phys Definition 2.2 Consequence with Respect to a Quasi-model M q )Letγ be a sentence such that α and β are subformulas of γ and let be ruth-perspective. α γ,m q,) β the sentence β is a consequence of the sentence α with respect to the context γ, the quasi-model M q and the truth-perspective ) iffforanymodelm based on M q : γ,m,) α γ,m,) β. α γ,m q ) β the sentence β is a consequence of the sentence α with respect to the context γ and the quasi-model M q )iffforanytruth-perspective, α γ,m q,) β. Definition 2.3 Logical Consequence) Let γ be a sentence such that α and β are subformulas of γ and let be ruth-perspective. α γ,) β β is a logical consequence of α with respect to the context γ and the truthperspective )iffforanyquasi-modelm q, α γ,m q,) β. α γ β β is a logical consequence of α with respect to the context γ )iffforanytruthperspective, α γ,) β. α β β is a logical consequence of α iff for any context γ such that α and β are subformulas of γ,α γ β. he concepts of consequence and of logical consequence, defined above,correspond to weak concepts, being defined in terms of -ruth, and not in terms of the preorder relation as one could expect). From an intuitive point of view, however, such weak notions turn out to be more interesting in the case of epistemic situations described in the framework of a holistic semantics. Notice that only the contextual notion of logical consequence turns out to satisfy transitivity α γ β and β γ δ α γ δ). Full transitivity α β and β δ α δ)isnaturally violated in the holistic semantics. As expected, in the particular case of the harmonic epistemic semantics where all agents share the same truth-perspective) the definitions of truth, validity and logical consequence can be simplified, since the reference to is no longer necessary. Accordingly, in such a case we will write: Harm γ,m) α; Harm M α harmonic truth); Harm γ α; Harm α harmonic validity); α Harm γ,m q ) β; α Harm γ β; α Harm β harmonic logical consequence). 3 Some Epistemic Situations We will now illustrate some significant examples of epistemic situations that arise in this semantics. We will always refer to models such that dena) = a; dent) = t. M =,Ag,EpSit, den, M Hol )

13 Int J heor Phys ) K α Harm α.intheharmonicsemantics,sentencesthatareknownbyagivenagentat agiventimearetrue. ) is an immediate consequence of the definition of logical consequence and of heorem 2.. 2) As a particular case of ) we obtain: K αk α Harm K α. Knowing of knowing implies knowing. But not the other way around! 3) In the non-harmonic semantics only the two following conditions which are weaker than ) and 2)) hold for any quasi-model M q and any agent of M q : 3.) K α M q, at ) α; 3.2) K αk α M q, at ) K α. 4) K Kb t α Harm α.intheharmonicsemantics,knowingthatanotheragentknowsagiven sentence implies that the sentence in question holds. At the same time, we will have: K Kb t α Harm K α. Alice might know that Bob knows a given sentence, without knowing herself the sentence in question! and 5) In the harmonic sound semantics where for any agent, K at at P ) = P ) K at at P ) 0 = P ) 0 )wehave: Harm K t; Harm K f. Hence, there are sentences that every agent knows. 6) K α β) K α; K α β) K β.knowingaconjunctiondoesnotgenerally imply knowing its members. 7) γ,m) K α and γ,m) K β γ,m) K α β). Knowledge is not generally closed under conjunction. 8) Let M be a model and let be an agent of M. M,at ) K α α). Contradictions are never known. In order to prove 8), suppose, by contradiction, that there exists a model M and an agent such that: M,at ) K α α). hen, p at M Hol at K α α))) =. By definition of model and by heorem 2. we have: M Hol at Kat α α) ) = K at M Hol at α α) ). Consequently, by hypothesis, p at K at M Hol at α α))) =. hus, by the properties of knowledge operations: p at M Hol at α α)) =, which is impossible, owing to the following Lemma of the holistic semantics). Lemma 3. For any sentence α, for any truth-perspective and for any holistic model M, p M Hol α α) ).

14 Int J heor Phys 9) In the non-harmonic semantics the following situation is possible: M,at ) K Kb t f. In other words, according to the truth-perspective of Alice it is true that Alice at time t) knows that Bob at time t)knowsthefalsity of Alice s truth-perspective. As an example, consider a non-harmonic) model M with two agents and b t satisfying the following conditions: a) the epistemic distance between the truth-perspectives of and of b t is greater than or equal to 2.7 In such a case we have: at P ) at bt P ) 0 according to Bob s truth-perspective, Alice s ruth precedes Alice s Falsity); b) K at bt P ) 0 = P ) the information according to which Bob knows Alice s Falsity is true with respect to Alice s truth-perspective); c) K at at P ) = P ) Alice at time t has a sound epistemic capacity). Consider the syntactical tree of K Kb t f: Level 3 K Kb t f) = f) Level 2 K Kb t f) = Kb t f) Level K Kb t f) = K Kb t f). he qumixes assigned by M Hol at to the levels of this tree are: by definition of model); M Hol at Level3 K Kb t f) ) = P ) 0 M Hol at Level2 K Kb t f) ) = K bt M Hol at Level3 K Kb t f) )) = P ) by definition of model and by b)); M Hol at Level K Kb t f) ) = K bt M Hol at Level2 K Kb t f) )) = P ) by definition of model and by c)). Hence, p at M Hol at K Kb t f)) = and M,at ) K Kb t f.noticethat M,at ) K Kb t f M,b t ) Kb t f. In other words, the following situation is possible: According to Alice s truth-perspective, it is true that Alice knows that Bob knows the Falsity. However, according to Bob s truth-perspective it is not true that Bob knows the Falsity. 7 he concept of epistemic distance has been defined in Sect. 2 of the first Part of this article.

15 Int J heor Phys Roughly, we might say: Alice knows that Bob is wrong. However, Bob is not aware of being wrong! he epistemic situations illustrated above seem to reflect pretty well some characteristic limitations of the real processes of acquiring information and knowledge. Owing to the limits of epistemic domains, understanding and knowing are not generally closed under logical consequence. Hence, the unpleasant phenomenon of logical omniscience is here avoided. We have, in particular, that knowledge is not generally closed under logical conjunction, as in fact happens in the case of concrete memories both of human and of artificial intelligence. It is also admitted that an agent can understand or know) a conjunction, without being able to understand to know) its members. Such situation, which might appear prima facie somewhat irrational, seems to be instead deeply in agreement with our use of natural languages, where sometimes agents show to use correctly and to understand some global expressions without being able to understand their meaningful) parts. References. Beltrametti, E., Dalla Chiara, M.L., Giuntini, R., Leporini, R., Sergioli, G.: Epistemic quantum computational structures in a Hilbert-space environment. Fundam. Inform. 5, 4202). doi:0.3233/ FI Beltrametti, E., Dalla Chiara, M.L., Giuntini, R., Sergioli, G.: Quantum teleportation and quantum epistemic semantics. Math. Slovaca 62, 24202) 3. Chiribella, G., D Ariano, G.M., Perinotti, P.: ransforming quantum operations: quantum supermaps. ALett.J.Explor.Front.Phys.83, ) 4. Dalla Chiara, M.L., Freytes, H., Giuntini, R., Ledda, A., Leporini, R., Sergioli, G.: Entanglement as a semantic resource. Found. Phys. 40, ) 5. Fan, H.-y., Hu, L.-y.: Infinite-dimensional Kraus operators for describing amplitudue-damping channel and laser process. Opt. Commun. 282, ) 6. Kraus, K.: States, Effects and Operations. Springer, Berlin 983) 7. Leporini, R., Sergioli, G.: Quantum channels and knowledge operations. Submitted