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Epistemic modal logic is a subfield of
modal logic Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
that is concerned with reasoning about
knowledge Knowledge can be defined as awareness of facts or as practical skills, and may also refer to familiarity with objects or situations. Knowledge of facts, also called propositional knowledge, is often defined as true belief that is distinc ...
. While
epistemology Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics. Episte ...
has a long philosophical tradition dating back to
Ancient Greece Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cult ...
, epistemic logic is a much more recent development with applications in many fields, including
philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
,
theoretical computer science Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumsc ...
,
artificial intelligence Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech re ...
,
economics Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and intera ...
and
linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Linguis ...
. While philosophers since
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
have discussed modal logic, and Medieval philosophers such as
Avicenna Ibn Sina ( fa, ابن سینا; 980 – June 1037 CE), commonly known in the West as Avicenna (), was a Persian polymath who is regarded as one of the most significant physicians, astronomers, philosophers, and writers of the Islamic G ...
, Ockham, and
Duns Scotus John Duns Scotus ( – 8 November 1308), commonly called Duns Scotus ( ; ; "Duns the Scot"), was a Scottish Catholic priest and Franciscan friar, university professor, philosopher, and theologian. He is one of the four most important ...
developed many of their observations, it was
C. I. Lewis Clarence Irving Lewis (April 12, 1883 – February 3, 1964), usually cited as C. I. Lewis, was an American academic philosopher. He is considered the progenitor of modern modal logic and the founder of conceptual pragmatism. First a noted logic ...
who created the first symbolic and systematic approach to the topic, in 1912. It continued to mature as a field, reaching its modern form in 1963 with the work of Kripke.


Historical development

Many papers were written in the 1950s that spoke of a logic of knowledge in passing, but the Finnish philosopher
G. H. von Wright Georg Henrik von Wright (; 14 June 1916 – 16 June 2003) was a Finns, Finnish philosopher. Biography G. H. von Wright was born in Helsinki on 14 June 1916 to Tor von Wright and his wife Ragni Elisabeth Alfthan. On the retirement of Ludwig Wit ...
's 1951 paper titled ''An Essay in Modal Logic'' is seen as a founding document. It was not until 1962 that another Finn,
Hintikka Kaarlo Jaakko Juhani Hintikka (12 January 1929 – 12 August 2015) was a Finnish philosopher and logician. Life and career Hintikka was born in Helsingin maalaiskunta (now Vantaa). In 1953, he received his doctorate from the University of Hels ...
, would write ''Knowledge and Belief'', the first book-length work to suggest using modalities to capture the semantics of knowledge rather than the alethic statements typically discussed in modal logic. This work laid much of the groundwork for the subject, but a great deal of research has taken place since that time. For example, epistemic logic has been combined recently with some ideas from dynamic logic to create dynamic epistemic logic, which can be used to specify and reason about information change and exchange of information in
multi-agent systems A multi-agent system (MAS or "self-organized system") is a computerized system composed of multiple interacting intelligent agents.Hu, J.; Bhowmick, P.; Jang, I.; Arvin, F.; Lanzon, A.,A Decentralized Cluster Formation Containment Framework fo ...
. The seminal works in this field are by Plaza, Van Benthem, and Baltag, Moss, and Solecki.


Standard possible worlds model

Most attempts at modeling knowledge have been based on the
possible world A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional logic, intensional and mod ...
s model. In order to do this, we must divide the set of possible worlds between those that are compatible with an agent's knowledge, and those that are not. This generally conforms with common usage. If I know that it is either Friday or Saturday, then I know for sure that it is not Thursday. There is no possible world compatible with my knowledge where it is Thursday, since in all these worlds it is either Friday or Saturday. While we will primarily be discussing the logic-based approach to accomplishing this task, it is worthwhile to mention here the other primary method in use, the
event Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of e ...
-based approach. In this particular usage, events are sets of possible worlds, and knowledge is an operator on events. Though the strategies are closely related, there are two important distinctions to be made between them: * The underlying mathematical model of the logic-based approach are
Kripke semantics Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Jo ...
, while the event-based approach employs the related Aumann structures based on
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
. * In the event-based approach logical formulas are done away with completely, while the logic-based approach uses the system of modal logic. Typically, the logic-based approach has been used in fields such as philosophy, logic and AI, while the event-based approach is more often used in fields such as
game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
and
mathematical economics Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference an ...
. In the logic-based approach, a syntax and semantics have been built using the language of modal logic, which we will now describe.


Syntax

The basic
modal operator A modal connective (or modal operator) is a logical connective for modal logic. It is an operator which forms propositions from propositions. In general, a modal operator has the "formal" property of being non-truth-functional in the following sen ...
of epistemic logic, usually written ''K'', can be read as "it is known that," "it is epistemically necessary that," or "it is inconsistent with what is known that not." If there is more than one agent whose knowledge is to be represented, subscripts can be attached to the operator (\mathit_1, \mathit_2, etc.) to indicate which agent one is talking about. So \mathit_a\varphi can be read as "Agent a knows that \varphi." Thus, epistemic logic can be an example of
multimodal logic A multimodal logic is a modal logic that has more than one primitive modal operator. They find substantial applications in theoretical computer science. Overview A modal logic with ''n'' primitive unary modal operators \Box_i, i\in \ is called an ...
applied for
knowledge representation Knowledge representation and reasoning (KRR, KR&R, KR²) is the field of artificial intelligence (AI) dedicated to representing information about the world in a form that a computer system can use to solve complex tasks such as diagnosing a medic ...
. The dual of ''K'', which would be in the same relationship to ''K'' as \Diamond is to \Box, has no specific symbol, but can be represented by \neg K_a \neg \varphi, which can be read as "a does not know that not \varphi" or "It is consistent with a's knowledge that \varphi is possible". The statement "a does not know whether or not \varphi" can be expressed as \neg K_a\varphi \land \neg K_a\neg\varphi. In order to accommodate notions of
common knowledge Common knowledge is knowledge that is publicly known by everyone or nearly everyone, usually with reference to the community in which the knowledge is referenced. Common knowledge can be about a broad range of subjects, such as science, literat ...
and
distributed knowledge In multi-agent system research, distributed knowledge is all the knowledge that a community of agents possesses and might apply in solving a problem. Distributed knowledge is approximately what "a wise man knows" or what someone who has complete ...
, three other modal operators can be added to the language. These are \mathit_\mathit, which reads "every agent in group G knows" (
mutual knowledge Mutual knowledge in game theory is information known by all participatory agents. However, unlike common knowledge, a related topic, mutual knowledge does not require that all agents are aware that this knowledge is mutual. All common knowledge is ...
); \mathit_\mathit, which reads "it is common knowledge to every agent in G"; and \mathit_\mathit, which reads "it is distributed knowledge to the whole group G." If \varphi is a formula of our language, then so are \mathit_G \varphi, \mathit_G \varphi, and \mathit_G \varphi. Just as the subscript after \mathit can be omitted when there is only one agent, the subscript after the modal operators \mathit, \mathit, and \mathit can be omitted when the group is the set of all agents.


Semantics

As we mentioned above, the logic-based approach is built upon the possible worlds model, the semantics of which are often given definite form in Kripke structures, also known as Kripke models. A Kripke structure ''M'' for ''n'' agents over \Phi is an (''n'' + 2)-tuple (S, \pi, \mathcal_1, ..., \mathcal_n), where S is a nonempty set of ''states'' or ''possible worlds'', \pi is an ''interpretation'', which associates with each state in S a truth assignment to the primitive propositions in \Phi (the set of all primitive propositions), and \mathcal_1, ..., \mathcal_n are
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
s on S for ''n'' numbers of agents. It is important here not to confuse K_i, our modal operator, and \mathcal_i, our accessibility relation. The truth assignment tells us whether or not a proposition ''p'' is true or false in a certain state. So \pi (s)(p) tells us whether ''p'' is true in state ''s'' in model \mathcal. Truth depends not only on the structure, but on the current world as well. Just because something is true in one world does not mean it is true in another. To state that a formula \varphi is true at a certain world, one writes (M,s) \models \varphi, normally read as "\varphi is true at (M,s)," or "(M,s) satisfies \varphi". It is useful to think of our binary relation \mathcal_i as a ''possibility'' relation, because it is meant to capture what worlds or states agent ''i'' considers to be possible; In other words, w\mathcal_i v if and only if \forall \varphi w\models K_i\varphi) \implies (v \models \varphi)/math>, and such v's are called epistemic alternatives for agent ''i''. In idealized accounts of knowledge (e.g., describing the epistemic status of perfect reasoners with infinite memory capacity), it makes sense for \mathcal_i to be an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
, since this is the strongest form and is the most appropriate for the greatest number of applications. An equivalence relation is a binary relation that is reflexive,
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
, and transitive. The accessibility relation does not have to have these qualities; there are certainly other choices possible, such as those used when modeling belief rather than knowledge.


The properties of knowledge

Assuming that \mathcal_i is an equivalence relation, and that the agents are perfect reasoners, a few properties of knowledge can be derived. The properties listed here are often known as the "S5 Properties," for reasons described in the Axiom Systems section below.


The distribution axiom

This axiom is traditionally known as K. In epistemic terms, it states that if an agent knows \varphi and knows that \varphi \implies \psi, then the agent must also know \,\psi. So, : (K_i\varphi \land K_i(\varphi \implies \psi)) \implies K_i\psi This axiom is valid on any frame in relational semantics. This axiom logically establishes
modus ponens In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. ...
as a
rule of inference In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of in ...
for every epistemically possible world.


The knowledge generalization rule

Another property we can derive is that if \phi is valid (i.e. a tautology), then K_i\phi. This does not mean that if \phi is true, then agent i knows \phi. What it means is that if \phi is true in every world that an agent considers to be a possible world, then the agent must know \phi at every possible world. This principle is traditionally called N (Necessitation rule). : \text\models \varphi\textM \models K_i \varphi.\, This rule always preserves truth in relational semantics.


The knowledge or truth axiom

This axiom is also known as T. It says that if an agent knows facts, the facts must be true. This has often been taken as the major distinguishing feature between knowledge and belief. We can believe a statement to be true when it is false, but it would be impossible to ''know'' a false statement. : K_i \varphi \implies \varphi This axiom can also be expressed in its
contraposition In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as Proof by contrapositive, proof by contraposition. The cont ...
as agents cannot ''know'' a false statement: : \varphi \implies \neg K_i \neg \varphi This axiom is valid on any reflexive
frame A frame is often a structural system that supports other components of a physical construction and/or steel frame that limits the construction's extent. Frame and FRAME may also refer to: Physical objects In building construction *Framing (con ...
.


The positive introspection axiom

This property and the next state that an agent has introspection about its own knowledge, and are traditionally known as 4 and 5, respectively. The Positive Introspection Axiom, also known as the KK Axiom, says specifically that agents ''know that they know what they know''. This axiom may seem less obvious than the ones listed previously, and
Timothy Williamson Timothy Williamson (born 1955) is a British philosopher whose main research interests are in philosophical logic, philosophy of language, epistemology and metaphysics. He is the Wykeham Professor of Logic at the University of Oxford, and fe ...
has argued against its inclusion forcefully in his book, ''Knowledge and Its Limits''. : K_i \varphi \implies K_i K_i \varphi Equivalently, this modal axiom 4 says that agents ''do not know'' ''what they do not know that they know'' : \neg K_i K_i \varphi \implies \neg K_i \varphi This axiom is valid on any transitive
frame A frame is often a structural system that supports other components of a physical construction and/or steel frame that limits the construction's extent. Frame and FRAME may also refer to: Physical objects In building construction *Framing (con ...
.


The negative introspection axiom

The Negative Introspection Axiom says that agents ''know that they do not know what they do not know''. : \neg K_i \varphi \implies K_i \neg K_i \varphi Or, equivalently, this modal axiom 5 says that agents ''know'' ''what they do not know that they do not know'' : \neg K_i \neg K_i \varphi \implies K_i \varphi This axiom is valid on any Euclidean
frame A frame is often a structural system that supports other components of a physical construction and/or steel frame that limits the construction's extent. Frame and FRAME may also refer to: Physical objects In building construction *Framing (con ...
.


Axiom systems

Different modal logics can be derived from taking different subsets of these axioms, and these logics are normally named after the important axioms being employed. However, this is not always the case. KT45, the modal logic that results from the combining of K, T, 4, 5, and the Knowledge Generalization Rule, is primarily known as S5. This is why the properties of knowledge described above are often called the S5 Properties. However, it can be proven that modal axiom B is a theorem in S5 (viz. S5 \vdash \mathbf ), which says that ''what an agent does not know that they do not know'' is true: \neg K_i \neg K_i \varphi \implies \varphi. The modal axiom B is true on any symmetric frame, but is very counterintuitive in epistemic logic: How can ''the ignorance on one's own ignorance'' imply truth? It is therefore debatable whether S4 describes epistemic logic better, rather than S5. Epistemic logic also deals with belief, not just knowledge. The basic modal operator is usually written ''B'' instead of ''K''. In this case, though, the knowledge axiom no longer seems right—agents only sometimes believe the truth—so it is usually replaced with the Consistency Axiom, traditionally called D: : \neg B_i \bot which states that the agent does not believe a contradiction, or that which is false. When D replaces T in S5, the resulting system is known as KD45. This results in different properties for \mathcal_i as well. For example, in a system where an agent "believes" something to be true, but it is not actually true, the accessibility relation would be non-reflexive. The logic of belief is called
doxastic logic Doxastic logic is a type of logic concerned with reasoning about beliefs. The term ' derives from the Ancient Greek (''doxa'', "opinion, belief"), from which the English term ''doxa'' ("popular opinion or belief") is also borrowed. Typically, a d ...
.


Multi-agent systems

When there are multiple agents in the
domain of discourse In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range. Overview The domain ...
where each agent ''i'' corresponds to a separate epistemic modal operator K_i, in addition to the axiom schemata for each individual agent listed above to describe the rationality of each agent, it is usually also assumed that the rationality of each agent is
common knowledge Common knowledge is knowledge that is publicly known by everyone or nearly everyone, usually with reference to the community in which the knowledge is referenced. Common knowledge can be about a broad range of subjects, such as science, literat ...
.


Problems with the possible world model and modal model of knowledge

If we take the possible worlds approach to knowledge, it follows that our epistemic agent ''a'' knows all the
logical consequence Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is on ...
s of their beliefs (known as logical omniscience). If Q is a logical consequence of P, then there is no possible world where P is true but Q is not. So if ''a'' knows that P is true, it follows that all of the logical consequences of P are true of all of the possible worlds compatible with ''as beliefs. Therefore, ''a'' knows Q. It is not epistemically possible for ''a'' that not-Q given his knowledge that P. This consideration was a part of what led
Robert Stalnaker Robert Culp Stalnaker (born 1940) is an American philosopher who is Laurance S. Rockefeller Professor Emeritus of Philosophy at the Massachusetts Institute of Technology. He is a Fellow of the American Academy of Arts and Sciences and a Correspond ...
to develop
two-dimensionalism Two-dimensionalism is an approach to semantics in analytic philosophy. It is a theory of how to determine the sense and reference of a word and the truth-value of a sentence. It is intended to resolve the puzzle: How is it possible to discover em ...
, which can arguably explain how we might not know all the logical consequences of our beliefs even if there are no worlds where the propositions we know come out true but their consequences false. Even when we ignore possible world semantics and stick to axiomatic systems, this peculiar feature holds. With K and N (the Distribution Rule and the Knowledge Generalization Rule, respectively), which are axioms that are minimally true of all normal modal logics, we can prove that we know all the logical consequences of our beliefs. If Q is a logical consequence of P (i.e. we have the tautology \models (P \rightarrow Q)), then we can derive K_a (P \rightarrow Q) with N, and using a
conditional proof A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent. Overview The assumed antecedent of a conditional proof is called the conditio ...
with the axiom K, we can then derive K_a P \rightarrow K_a Q with K. When we translate this into epistemic terms, this says that if Q is a logical consequence of P, then ''a'' knows that it is, and if ''a'' knows P, ''a'' knows Q. That is to say, ''a'' knows all the logical consequences of every proposition. This is necessarily true of all classical modal logics. But then, for example, if ''a'' knows that prime numbers are divisible only by themselves and the number one, then ''a'' knows that 8683317618811886495518194401279999999 is prime (since this number is only divisible by itself and the number one). That is to say, under the modal interpretation of knowledge, when ''a'' knows the definition of a prime number, ''a'' knows that this number is prime. This generalizes to any provable theorem in any axiomatic theory (i.e. if ''a'' knows all the axioms in a theory, then ''a'' knows all the provable theorems in that theory). It should be clear at this point that ''a'' is not human (otherwise there would not be any unsolved conjectures in mathematics, like
P versus NP problem The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved. The informal term ''quickly'', used abov ...
or
Goldbach's conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold ...
). This shows that epistemic modal logic is an idealized account of knowledge, and explains objective, rather than subjective knowledge (if anything).See Ted Sider'
Logic for Philosophy
Currently page 230 but subject to change following updates.


Epistemic fallacy (masked-man fallacy)

In
philosophical logic Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical ...
, the
masked-man fallacy In philosophical logic, the masked-man fallacy (also known as the intensional fallacy or epistemic fallacy) is committed when one makes an illicit use of Leibniz's law in an argument. Leibniz's law states that if A and B are the same object, the ...
(also known as the intensional fallacy or epistemic fallacy) is committed when one makes an illicit use of Leibniz's law in an argument. The fallacy is "epistemic" because it posits an immediate identity between a subject's knowledge of an object with the object itself, failing to recognize that Leibniz's Law is not capable of accounting for intensional contexts.


Examples

The name of the fallacy comes from the example: * ''Premise 1'': I know who Bob is. * ''Premise 2'': I do not know who the masked man is * ''Conclusion'': Therefore, Bob is not the masked man. The
premises Premises are land and buildings together considered as a property. This usage arose from property owners finding the word in their title deeds, where it originally correctly meant "the aforementioned; what this document is about", from Latin ''pra ...
may be true and the conclusion false if Bob is the masked man and the speaker does not know that. Thus the argument is a fallacious one. In symbolic form, the above arguments are * ''Premise 1:'' I know who X is. * ''Premise 2:'' I do not know who Y is. * ''Conclusion:'' Therefore, X is not Y. Note, however, that this
syllogism A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. ...
happens in the reasoning by the speaker "I"; Therefore, in the formal
modal logic Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
form, it'll be * ''Premise 1:'' The speaker believes he knows who X is. * ''Premise 2:'' The speaker believes he does not know who Y is. * ''Conclusion:'' Therefore, the speaker believes X is not Y. ''Premise 1'' \mathcal\forall t (t=X\rightarrow K_s(t=X)) is a very strong one, as it is
logically equivalent Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
to \mathcal\forall t (\neg K_s(t=X)\rightarrow t\not=X). It is very likely that this is a
false belief In psychology, theory of mind refers to the capacity to understand other people by ascribing mental states to them (that is, surmising what is happening in their mind). This includes the knowledge that others' mental states may be different fro ...
: \forall t (\neg K_s(t=X)\rightarrow t\not=X) is likely a false proposition, as the ignorance on the proposition t=X does not imply the negation of it is true. Another example: * ''Premise 1:'' Lois Lane thinks Superman can fly. * ''Premise 2:'' Lois Lane thinks Clark Kent cannot fly. * ''Conclusion:'' Therefore Superman and Clark Kent are not the same person. Expressed in
doxastic logic Doxastic logic is a type of logic concerned with reasoning about beliefs. The term ' derives from the Ancient Greek (''doxa'', "opinion, belief"), from which the English term ''doxa'' ("popular opinion or belief") is also borrowed. Typically, a d ...
, the above syllogism is: * ''Premise 1:'' \mathcal_Fly_ * ''Premise 2:'' \mathcal_\neg Fly_ * ''Conclusion:'' Superman\neq Clark The above reasoning is invalid (not truth-preserving). The valid conclusion to be drawn is \mathcal_(Superman\neq Clark).


See also

*
Epistemic closure Epistemic closure is a property of some belief systems. It is the principle that if a subject S knows p, and S knows that p entails q, then S can thereby come to know q. Most epistemological theories involve a closure principle and many skepti ...
*
Epistemology Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics. Episte ...
*
Logic in computer science Logic in computer science covers the overlap between the field of logic and that of computer science. The topic can essentially be divided into three main areas: * Theoretical foundations and analysis * Use of computer technology to aid logicians ...
* ''
Philosophical Explanations ''Philosophical Explanations'' is a 1981 metaphysical, epistemological, and ethical treatise by the philosopher Robert Nozick. The book received positive reviews. Commentators have compared ''Philosophical Explanations'' to the philosopher Richa ...
''


Notes


References

*Anderson, A. and N. D. Belnap. ''Entailment: The Logic of Relevance and Necessity.'' Princeton:
Princeton University Press Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial su ...
, 1975. ASIN B001NNPJL8. * Brown, Benjamin, Thoughts and Ways of Thinking: Source Theory and Its Applications. London:
Ubiquity Press Founded in 2008 by Brian Hole, Ubiquity Press is an academic publisher focusing on open access, peer-reviewed scholarship. Ubiquity Press is a part oUbiquity which also provides full publishing infrastructure and services to university presses, a ...
, 2017

*van Ditmarsch Hans, Halpern Joseph Y., van der Hoek Wiebe and Kooi Barteld (eds.), ''Handbook of Epistemic Logic'', London: College Publications, 2015. * . A classic reference. * Ronald Fagin, Joseph Halpern, Moshe Vardi.
"A nonstandard approach to the logical omniscience problem."
''Artificial'' ''Intelligence'', Volume 79, Number 2, 1995, p. 203-40. *Hendricks, V.F. ''Mainstream and Formal Epistemology.'' New York:
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...
, 2007. * . *Meyer, J-J C., 2001, "Epistemic Logic," in Goble, Lou, ed., ''The Blackwell Guide to Philosophical Logic''.
Blackwell Blackwell may refer to: Places ;Canada * Blackwell, Ontario ;United Kingdom * Blackwell, County Durham, England * Blackwell, Carlisle, Cumbria, England * Blackwell (historic house), South Lakeland, Cumbria, England * Blackwell, Bolsover, Alfre ...
. * Montague, R. "Universal Grammar". ''Theoretica'', Volume 36, 1970, p. 373-398. * . * . See Chapters 13 and 14
downloadable free online


External links

* * * * * * *

—Ho Ngoc Duc. {{Authority control Modal logic Artificial intelligence Formal epistemology