Onerepublic Logic – Start Again

Epistemic logic is the logic of knowledge and belief. It provides insight into the backdrop of individual knowers, has provided a means to model complicated scenarios involving groups of knowers and has improved our understanding of the dynamics of inquiry.

1. The Logic of Individual Knowers

Epistemic logic gets its start with the recognition that expressions like 'knows that' or 'believes that' have systematic backdrop that are amenable to formal study. In improver to its relevance for traditional philosophical bug, epistemic logic has many applications in informatics and economics. Examples range from robotics, network security and cryptography applications to the report of social and coalitional interactions of various kinds.

Modern treatments of the logic of knowledge and belief grow out of the work of a number of philosophers and logicians writing from 1948 through the 1950s. Rudolf Carnap, Jerzy Los, Arthur Prior, Nicholas Rescher, One thousand.H. von Wright and others recognized that our discourse concerning knowledge and belief exhibits systematic features that admit of an evident-deductive treatment. Amid the many of import papers that appeared in the 1950s, von Wright's seminal work (1951) is widely recognized as having initiated the formal study of epistemic logic equally we know it today. Von Wright's insights were extended by Jaakko Hintikka in his book Knowledge and Conventionalities: An Introduction to the Logic of the Two Notions (1962). In the 1980s and 1990s, epistemic logicians focused on the logical properties of systems containing groups of knowers and later nonetheless on the epistemic features of so-chosen "multi-modal" contexts.

While this article deals with modern developments, epistemic logic has a venerable history. In De Sophisiticis Elenchis as well equally in the Prior and Posterior Analytics Aristotle mentions some aspects of the logic of knowledge and belief. In the medieval menses, Buridan, Duns Scotus and Ockham extended Aristotle's insights in a serial of detailed reflections whose themes and bug would be familiar to gimmicky epistemic logicians (Boh 1993, Knuutila 1993).

Contemporary epistemic logic may appear quite technical and removed from traditional epistemological reflections. All the same, it assumes as its starting indicate some features of the logical behavior of epistemic concepts that are completely obvious. For instance, challenge to know p and q implies that you know q. Furthermore, most systems of epistemic logic brainstorm with an supposition like to G.E. Moore'south intuitively obvious observation that one cannot coherently assert "p just I practice non believe (know) p". Additional assumptions that serve as the footing for most epistemic logics include the recognition that cognition implies veracity. If I know p then p must be the instance. Thus, commonsense observations concerning the beliefs of the term "knows that", accept served as the starting point for later technical developments.

For the almost part, epistemic logic focuses on propositional knowledge. Here, an agent or a grouping of agents bears the propositional mental attitude knowing towards some proposition. Then, when one says: "Zoë knows that at that place is a chicken in the yard", one asserts that Zoë is the agent who bears the propositional attitude knowing towards the proffer expressed by "there is a craven in the yard". Beyond straightforward propositional knowledge of this kind, epistemic logic too suggests ways to systematize the logic of questions and answers (Zoë knows why Murphy barked) and provides insight into the relationships betwixt multiple modes of identification (Zoë knows that this man is the main) and also perhaps even into questions of procedural "know-how". Epistemic logicians have found means to formally treat a wide multifariousness of knowledge claims in propositional terms.

Syntactically, the language of propositional epistemic logic is simply a matter of augmenting the linguistic communication of propositional logic with a unary epistemic operator One thousand _c such that

K c A

reads

"Agent c knows A

and similarly for belief

B c A

reads

"Agent c believes A

for some arbitrary proposition A.

Hintikka provided a semantic interpretation of epistemic and doxastic operators which we can present in terms of standard possible earth semantics forth the following lines (Hintikka 1962):

Grand c A:	in all possible worlds compatible with what c knows, it is the case that A
B c A:	in all possible worlds compatible with what c believes, it is the case that A

The basic assumption is that whatsoever ascription of propositional attitudes like noesis and belief, involves dividing the ready of possible worlds in 2: Those worlds uniform with the attitude in question and those that are incompatible with information technology.

The set of worlds attainable to an agent depends on his or her informational resources at that instant. It is possible to capture this dependency by introducing a relation of accessibility, R, on the ready of possible worlds. To express the thought that for agent c, the earth westward′ is compatible with his information state, or attainable from the possible globe west which c is currently in, it is required that R holds betwixt westward and w′. This relation is written Rww′ and reads "world west′ is accessible from w". The world w′ is said to be an epistemic or doxastic alternative to earth due west for agent c, depending on whether cognition or belief is the considered attitude. Given the to a higher place semantical interpretation, if a proposition A is true in all worlds which amanuensis c considers possible then c knows A.

A possible world semantics for a propositional epistemic logic with a single agent c then consists of a frame F which in plough is a pair <W,R _c > such that Westward is a non-empty gear up of possible worlds and R _c is a binary accessibility relation (relative to agent c) over W. A model M for an epistemic organisation consists of a frame and a denotation part φ assigning sets of worlds to atomic propositional formulas. Propositions are taken to be sets of possible worlds; namely the set of possible worlds in which they are true. Let atom be the set up of atomic propositional formulae, then φ :cantlet P(West), where P denotes the powerset functioning. The model M = <Westward,R _c ,φ> is chosen a Kripke-model and the resulting semantics Kripke-semantics (Kripke 1963): An diminutive propositional formula, a, is said to exist truthful in a world west in M (written M,west ⊨ a) iff west is in the set up of possible worlds assigned to a, i.e., Thou,w ⊨ a iff westward ∈ φ(a) for all a ∈ atom . The formula Yard _c A is truthful in an world w (i.e., Thousand,w ⊨ K _c A) iff ∀w′∈W, if R_cww′, so Grand,due west′ ⊨ A. The semantics for the Boolean connectives follow the usual recursive recipe. A modal formula is said to be valid in a frame iff the formula is true for all possible assignments in all worlds in the frame.

Similar semantics may be formulated for the belief operator. Since a belief is not necessarily true but rather probably true, possibly true, or probable to be true, we must alter our arroyo to the semantics of belief appropriately. For instance, conventionalities may exist modeled by assigning a sufficiently high degree of probability to the proposition in question and determining the doxastic alternatives accordingly. The truth-atmospheric condition for the doxastic operator are divers in a way similar to that of the knowledge operator and the model may also exist expanded to accommodate the ii operators simultaneously.

An important feature of possible world semantics is that many common epistemic axioms correspond to algebraic properties of the frame in the following sense: A modal precept is valid in a frame if and only if the accessibility relation satisfies some algebraic condition. For example, the axiom expressing the veridicality holding that if a proposition is known by c, and then A is true,

(one) K _c A → A,

is valid in all frames in which the accessibility relation is reflexive in the sense that ∀w∈Westward : Rww. Every possible world is attainable from itself. Similarly if the accessibility relation satisfies the condition that ∀ w, w′, west″ ∈ Westward : Rww′ ∧ Rw′west″ → Rww″ then the axiom reflecting the idea that the agent knows that he knows A if he does,

(2) K _c A → K _c Yard _c A,

is valid in all transitive frames. Other axioms of epistemic import require still other relational properties to be met in order to be valid in all frames.

A nomenclature due to Lemmon (1977) and later on refined past Bull and Segerberg (1984) is helpful while cataloguing the axioms typically considered interesting for epistemic logic (Table i):

K	K c (A→A′) → (K c A → K c A′)
D	K c A → ¬K c ¬A
T	K c A → A
4	Thou c A → K c Grand c A
5	¬Thou c A → One thousand c ¬K c A
.ii	¬G c ¬Yard c A → K c ¬Chiliad c ¬A
.iii	G c (K c A → Yard c A′) ∨ Thou c (Thou c A′ → Yard c A)
.iv	A → (¬Thousand c ¬K c A → K c A)

Table one: Common Epistemic Axioms

These axioms in proper combinations make up epistemic modal systems of varying forcefulness depending on the modal formulas valid in the respective systems given the algebraic properties assumed for the accessibility relation.

Although the axiom K by itself constitutes the organisation K, the weakest system of epistemic interest is usually considered to be system T. [Note: The reader should have intendance to distinguish the epistemic operator One thousand, the modal axiom K and the system of axioms K in what follows. Similarly, we distinguish the axiom T from the organisation T.] The system T includes the axioms T and G as valid axioms. Boosted modal strength may be obtained past extending T with other axioms fatigued from the higher up pool altering the frame semantics to validate the boosted axioms. By way of case, while K _c A → A is valid in system T, K _c A → A, G _c A → M _c 1000 _c A and ¬K _c A → Thou _c ¬Thou _c A are all valid in S5 but not in T. Arrangement T has a reflexive accessibility relation, S5 has an equivalence relation of accessibility. The arrows in Table ii symbolize that the arrangement to which the arrow is pointing is included in the system from which the pointer originates and hence reflect relative strength. And then S5 is the strongest and S4 the weakest of the ones listed.

Epistemic Systems
KT4	=	S4
KT4 + .2	=	S4.2	↑
KT4 + .3	=	S4.3	↑
KT4 + .4	=	S4.4	↑
KT5	=	S5	↑

Table 2: Relative Forcefulness of Epistemic Systems Between S4 and S5

1 of the important tasks of epistemic logic is to catalogue all sound and consummate systems of such logics in society to allow us to selection the most 'appropriate' ones. The logics range from S4 over the intermediate systems S4.ii–S4.4 to S5. Past mode of instance, Hintikka settled for S4 (1962), Kutschera argued for S4.4 (1976), Lenzen suggested S4.ii (1978), van der Hoek has proposed to strengthen knowledge according to system S4.iii (1996). van Ditmarsch, van der Hoek and Kooi together with Fagin, Halpern, Moses and Vardi (Fagin et al. 1995) and others assume cognition to exist S5 valid. Like abyss cataloguing is bachelor for conventionalities where axiom T is dropped and usually replaced by D (to avoid the condition of truth for belief only retain consistency among beliefs) yieding systems like KD4–KD45 for belief. This likewise paves the way for combining epistemic and doxastic systems and for studying the interplay between knowledge and belief (see Voorbraak 1993). Care should be taken however not to collapse cognition and belief in the combined systems as have been noted by Lenzen (1978) and Stalnaker (1996), among others.

A especially cancerous philosophical problem for epistemic logic is related to closure properties. Axiom K, can nether certain circumstances be generalized to a closure belongings for an agent's knowledge which is implausibly potent — logical omniscience:

Whenever an amanuensis c knows all of the formulas in a set Γ and A follows logically from Γ, then c also knows A.

In detail, c knows all theorems (letting Γ = ø), and he knows all logical consequences of whatsoever formula which he knows (letting Γ consist of a unmarried formula). There are diverse ways of dealing with logical omniscience. Some of the first proposals for solving the problem of logical omniscience introduce semantical entities which explain why the amanuensis appears to exist, but in fact is not really guilty of logical omniscience. These entities were called 'impossible possible worlds' past Hintikka (1975). The basic idea is that an agent may mistakenly count among the worlds consistent with his or her noesis, some worlds containing logical contradictions. The mistake is simply a production of limited resources; the agent may not be in a position to detect the contradiction and may erroneously count them equally 18-carat possibilities. Similar entities called 'seemingly possible' worlds are introduced by Rantala (1975) in his urn-model analysis of logical omniscience. Allowing impossible possible worlds or seemingly possible worlds in which the semantic valuation of the formulas is capricious to a sure extent provides a fashion of making the appearance of logical omniscience less threatening. Afterwards all, on whatever realistic account of epistemic agency, the agent is probable to consider (albeit inadvertantly) worlds in which the laws of logic practice not concord. Since no real epistemic principles agree broadly enough to encompass impossible and seemingly possible worlds, some conditions must exist applied to epistemic models such that they cohere with epistemic principles.

Computer scientists have proposed that what is being modelled in epistemic logic is not cognition simpliciter but a related concept which is immune to logical omniscience. The epistemic operator Thou _c A should be read as 'agent c knows implicitly A', 'A follows from c'southward knowledge', 'A is amanuensis c's possible cognition', etc. Propositional attitudes like these should supersede the usual 'agent c knows A'. While there exists some variation, the locutions all suggest modeling implicit noesis or what is implicitly represented in an agent's information state rather than explicit cognition (Fagin et al. 1995, and others). The agents neither have to compute knowledge nor tin can they be held responsible for answering queries based on their knowledge nether the implicit understanding of noesis. Logical omniscience is an epistemological condition for implicit knowledge, merely the amanuensis may really neglect to realize this status.

2. Groups of Knowers

Single-agent systems may exist extended to groups or multi-agent systems. Following the standard treatment provided by Fagin, Halpern, Moses and Vardi (Fagin et al. 1995) we tin can syntactically broaden the language of propositional logic with due north knowledge operators, ane for each agent involved in the group of agents under consideration. The primary difference betwixt the semantics given for a mono-agent and a multi-amanuensis semantics is roughly that due north accessibility relations are introduced. A modal organisation for north agents is obtained by joining together n modal logics where for simplicity it may exist causeless that the agents are homogenous in the sense that they may all be described by the same logical arrangement. An epistemic logic for n agents consists of n copies of a certain modal logic. In such an extended epistemic logic it is possible to limited that some amanuensis in the group knows a certain fact, that an agent knows that another agent knows a fact etc. It is possible to develop the logic fifty-fifty further: Not only may an agent know that some other agent knows a fact, just they may all know this fact simultaneously. From here it is possible to express that everyone knows that anybody knows that anybody knows, that…. That it is common cognition.

As Lewis noted in his book Convention (1969) a convention requires common knowledge amid the agents that detect it. A variety of norms, social and linguistic practices, amanuensis interactions and games presuppose common noesis (Aumann 1994). A relatively simple way of defining common knowledge is non to division the grouping of agents into subsets with dissimilar common 'knowledges' but only to ascertain common noesis for the entire group of agents. Once multiple agents have been added to the syntax, the linguistic communication is augmented with an additional operator c. CA is then interpreted every bit 'It is common knowledge amongst the agents that A'. Well-formed formulas follow the standard recursive recipe with a few, only obvious, modifications taking into account the multiple agents. An auxiliary operator E is besides introduced such that EA means 'Everyone knows that A'. EA is divers as the conjunction K ₁ A ∧ 1000 ₂ A ∧ … ∧ Thousand _n A.

To semantically translate n cognition operators, binary accessibility relations R _{due north} are defined over the set of possible worlds W. A special accessibility relation, R°, is introduced to interpret the operator of mutual knowledge. The relation must be flexible enough to express the relationship betwixt individual and common knowledge. The idea is to let the accessibility relation for c be the transitive closure of the union of the accessibility relations corresponding to the singular cognition operators. The model G for an epistemic system with n agents and mutual noesis is appropriately a structure One thousand = <W,R ₁,R ₂,…,R _n ,R°,φ>, where West is a not-empty space of possible worlds, R ₁,R ₂,…,R _{due north} ,R° are accessibility relations over W for which R° = (R _ane ∪ R ₂ ∪ … ∪ R _n ) and φ over again is the denotation office assigning worlds to atomic propositional formula φ :atom P(Westward). The semantics for the Boolean connectives remain intact. The formula K _i A is truthful in a world w, i.eastward., Grand,due west ⊨ Chiliad _i A iff ∀w′∈Due west : if R _i ww′, then 1000,w′ ⊨ A. The formula CA is true in a world w, i.east., M,w ⊨ CA iff R°ww′ implies Yard,west′ ⊨ A. Varying the properties of the accessibility relations R _one,R ₂,…,R _n , as described to a higher place results in different epistemic logics. For example system K with common noesis is adamant by all frames, while arrangement S4 with mutual knowledge is determined by all reflexive and transitive frames. Similar results can be obtained for the remaining epistemic logics (Fagin et al. 1995).

3. Active Agenthood

A significant difference between alethic and epistemic logic is the introduction of the agent c to the syntax. But what role does the amanuensis play in epistemic logic? At the early stages in the development of the logic they primarily served as indices on the accessibility relation between possible worlds. Withal, in that location is naught particularly epistemic about being an alphabetize, and epistemic logicians soon began recognizing the central role of the agent much more explicitly. An agent may have knowledge which is S4.3 valid thereby obtaining a certain epistemic forcefulness. An important set of questions seem to be how the agent has to acquit in gild to gain the epistemic forcefulness that he has. To make epistemic logic pertinent to epistemology, computer science, bogus intelligence and cerebral psychology the activeness of agents must be included in our formal considerations. The original symbolic notation of a knowing agent also suggests this: An agent term should exist inside the scope of the noesis operator — not exterior as Hintikka notes (1998). Inquiring agents are agents who read data, change their minds, interact or have mutual noesis, act co-ordinate to strategies and play games, accept memory and act upon information technology, follow diverse methodological rules, expand, contract or revise their knowledge bases, etc. all in the pursuit of knowledge. Inquiring agents are agile agents (Hendricks 2003).

Game theory is about strategies for winning games in the context of other agents. Game theory has therefore played a prominent function in reflections on epistemic agency (the written report of the beliefs of interactive epistemic agents). Aumann, van Benthem, Brandenburger, Fagin, Halpern, Keisler, Moses, Stalnaker, Vardi and others accept demonstrated how logical epistemology uncovers important features of agent rationality showing how game theory adds to the general agreement of notions like knowledge, conventionalities and belief revision.^[1] Baltag, Moss, Solecki combine epistemic logic with belief revision theory to study actions and belief updates in games (Baltag et al. 1999).

Mixing the theory of belief change and epistemic logic furnishes an illustrative example of active agents. The thought dates back to the mid 1990s. Alchourrón, Gärdenfors and Makinson's seminal belief revision theory (AGM) from the 1980s is a theory virtually the rational change of behavior for expansions, contractions and revisions in light of new (possibly alien) evidence (Alchourrón 1985, Gärdenfors 1988). In 1994, de Rijke showed that the AGM-axioms governing expansion and revision may be translated into the object language of dynamic modal logic (de Rijke 1994). At most the same time, Segerberg demonstrated how the unabridged theory of belief revision could be formulated in modal logic.

Segerberg merged the static first generation doxastic logic with the dynamics of belief alter into 'dynamic doxastic logic' (Segerberg 1995). Doxastic operators in the logic of belief like B _c A may be captured by AGM in the sense that 'A is in c'southward conventionalities-set T', or ¬B _c ¬A becomes '¬A is not in c's belief-ready T'. Similarly for other combinations of the conventionalities operator with negation. An immediate difference betwixt the two perspectives is that while AGM can express dynamic operations on belief-sets like expansions ('A is in c's conventionalities-set T expanded by D', i.e., A ∈ T+D), revisions ('A is in c's belief-set T revised by D', i.east., A ∈ T*D), and contractions ('A is in c's belief-set T contracted by D', i.e. A ∈ T−D), no such dynamics are immediately expressible in the standard language of doxastic logic. On the other mitt, action languages include operators like [ν] and <ν> which are prefixed to a well-formed formula A. On Segerberg'due south interpretation, [ν]A (<ν>A) mean that 'afterward every (some) fashion of performing action ν it is the instance that A'. By introducing iii new operators [+], [*], and [−] into the doxastic linguistic communication, the iii dynamic operations on belief-sets may be rendered every bit [+D]B _c A, [*D]B _c A and [−D]B _c A.

Later on revising the original belief revision theory such that changes of beliefs happen in 'hypertheories' or concentric spheres enumerated according to entrencement Segerberg (1999a, 199b) has provided several axiomatizations of the dynamic doxastic logic together with soundness and completeness results. The dynamic doxastic logic paradigm may also be extended to iterated belief revision as studied past Lindstrøm and Rabinowicz (1997) and accommodate various forms of agent introspection. A related approach drawn upwards by van Ditmarsch, van der Hoek and Kooi's new 'dynamic epistemic logic' studies how data changes and how deportment with epistemic impact on agents may be modelled (Hoek et al. 2003, Ditmarsch et al. 2006).

Active agenthood is also realizable straight on the agent level. One may also choose to endow the agents with epistemic capacities facilitating special epistemic behaviors. Fagin, Halpern, Moses and Vardi have for instance considered 'perfect recall' (Fagin et al. 1995): interacting agents' knowledge in the dynamic system may increase as time goes by simply the agents may even so shop quondam data. The agent'south current local land is an encoding of all events that have happened so far in the run. Perfect call back is in turn an epistemic recommendation telling the agent to remember his earlier epistemic states.

At that place are other structural properties of agents being studied in the literature of dynamic epistemic logics. In an epistemic logic suited for modeling various games of imperfect information van Benthem refers to such properties as 'styles of playing' (van Benthem 2000). Properties like 'bounded retentivity', various 'mechanisms for information updates' and 'compatible strategies', infallibility, consistency etc. have been investigated. Agents as explcitly learning mechanims are also integral parts of Kelly's (1996) computational epistemology and a related approach called modal operator epistemology (Hendricks 2001, 2003).

Researchers in artificial intelligence have additionally been trying to draw and specify the behaviour of intelligent/rational agents by extensions of epistemic of logic by augmenting logics of time, action and conventionalities with modalities for desires and intentions (see Meyer 2003), in particular, his discussion of the BDI-framework of Rao and Georgeff in Department five.ii).

4. Multi-Modalities

When epistemic logic was notwithstanding in its infancy, Dana Scott noted:

Hither is what I consider ane of the biggest mistakes of all in modal logic: concentration on a system with just i modal operator. The only manner to accept any philosophically meaning results in deontic logic or epistemic logic is to combine these operators with: Tense operators (otherwise how can you formulate principles of change?); the logical operators (otherwise how can you compare the relative with the absolute?); the operators like historical or concrete necessity (otherwise how can y'all relate the agent to his environment?); and so on and and so on. (Scott 1970, 143)

Nowadays, there are various means in which multi-modalities may exist realized in epistemic logic. One standard manner is to follow Fagin et al. (1995). in reinterpreting agents and possible worlds relative to systems of agents.^[2]

In a system of agents each individual agent is considered to be in some 'local land'. The whole organisation — as the sum of the local agents and other envrionmental features — is accordingly in some `global state'. The dynamics may be modeled by defining what is referred to as a 'run' over the arrangement which is a office from time to global states. The run may exist construed as an account of the behavior of the organization for possible executions. Pairs of runs and times requite ascent to `points'. For every time, the system is in some global state as a office of the particular fourth dimension. The system may exist thought of as a series of runs rather than agents. In this example, what is beingness modeled are the possible behaviors of the system over a collection of executions. A organization of this kind defines a Kripke-construction with an equivalence relation over points. The accessibility relation is specified with respect to possible points such that some point due west′ is accessible from the electric current indicate westward if the agent is in the same local state at w and westward′. Noesis is defined with respect to the agents' local country. Truth of a formula is given with respect to a betoken. If truth is relative to a point then there is a question of when which opens up for the introduction of temporal operators. One may for instance define a universal future-tense operator such that a formula is true relative to the current point and all afterwards points. The mixture of epistemic and temporal operators can handle claims about the temporal development of knowledge in the organisation.

v. Epistemic Logic and Epistemology

More often than not speaking, gimmicky epistemology is organized around two major goals: (1) The long-standing goal of providing a definition of noesis and simultaneously responding to the challenge of skepticism, and (2) The goal of modeling the dynamics of epistemic and doxastic states. The beginning of these goals has, for the most function, been a concern of philosophers who rely on thought experiments, traditional conceptual analysis or intuitions-based methods of various kinds. By contrast, philosophers working with epistemic logic accept pursued the second goal. The apparent divergence of both enterprises can exist reconciled to some extent in one case one recognizes that both goals relate to a 3rd, and possibly more full general trouble, namely (3) the problem of understanding the rationality of inquiry. This problem is of equal importance to both epistemic logicians and traditional epistemologists. Dynamical treatments of epistemic logic and insights into the logic of inquiry from epistemic logicians speak directly to this third, unifying goal. In recent decades, it is precisely the dynamical model of knowledge and inquiry that has concerned philosophically-inclined epistemic logicians. For a systematic handling of the interplay between epistemology and epistemic logic we refer to Hendricks and Symons 2006.

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Source: https://meinong.stanford.edu/entries/logic-epistemic/

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