The autoepistemic logic is a
formal logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
for the representation and reasoning of knowledge about knowledge. While
propositional logic
The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
can only express facts, autoepistemic logic can express knowledge and lack of knowledge about facts.
The
stable model semantics, which is used to give a semantics to
logic programming
Logic programming is a programming, database and knowledge representation paradigm based on formal logic. A logic program is a set of sentences in logical form, representing knowledge about some problem domain. Computation is performed by applyin ...
with
negation as failure, can be seen as a simplified form of autoepistemic logic.
Syntax
The
syntax
In linguistics, syntax ( ) is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituenc ...
of autoepistemic logic extends that of propositional logic by a
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 se ...
[To clarify, the modal operator is a medium white square; this is not a browser rendering issue] indicating knowledge: if
is a formula,
indicates that
is known. As a result,
indicates that
is known and
indicates that
is not known.
This syntax is used for allowing reasoning based on knowledge of facts. For example,
means that
is assumed false if it is not known to be true. This is a form of
negation as failure.
Semantics
The semantics of autoepistemic logic is based on the ''expansions'' of a theory, which have a role similar to
model
A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , .
Models can be divided in ...
s in
propositional logic
The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
. While a propositional model specifies which atomic propositions are true or false, an expansion specifies which formulae
are true and which ones are false. In particular, the expansions of an autoepistemic formula
make this determination for every subformula
contained in
. This determination allows
to be treated as a
propositional formula, as all its subformulae containing
are either true or false. In particular, checking whether
entails in this condition can be done using the rules of the propositional calculus. In order for a specification to be an expansion, it must be that a subformula
is entailed if and only if
has been assigned the value true.
In terms of
possible world semantics, an expansion of
consists of an
S5 model of
in which the possible worlds consist only of worlds where
is true. [The possible worlds need not contain all such consistent worlds; this corresponds to the fact that modal propositions are assigned truth values before checking derivability of the ordinary propositions.] Thus, autoepistemic logic extends S5; the extension is proper, since
and
are tautologies of autoepistemic logic, but not of S5.
For example, in the formula
, there is only a single “boxed subformula”, which is
. Therefore, there are only two candidate expansions, assuming
is true or false, respectively. The check for them being actual expansions is as follows.
is false : with this assumption,
becomes tautological, as
is equivalent to
, and
is assumed true; therefore,
is not entailed. This result confirms the assumption implicit in
being false, that is, that
is not currently known. Therefore, the assumption that
is false is an expansion.
is true : together with this assumption,
entails
; therefore, the initial assumption that is implicit in
being true, i.e., that
is known to be true, is satisfied. As a result, this is another expansion.
The formula
has therefore two expansions, one in which
is not known and one in which
is known. The second one has been regarded as unintuitive, as the initial assumption that
is true is the only reason why
is true, which confirms the assumption. In other words, this is a self-supporting assumption. A logic allowing such a self-support of beliefs is called ''not strongly grounded'' to differentiate them from ''strongly grounded'' logics, in which self-support is not possible. Strongly grounded variants of autoepistemic logic exist.
Generalizations
In
uncertain inference, the known/unknown duality of truth values is replaced by a degree of certainty of a fact or deduction; certainty may vary from 0 (completely uncertain/unknown) to 1 (certain/known). In
probabilistic logic networks, truth values are also given a probabilistic interpretation (''i.e.'' truth values may be uncertain, and, even if almost certain, they may still be "probably" true (or false).)
See also
*
Non-monotonic logic
*
Modal logic
Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields
it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
Notes
References
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{{refend
Logic programming
Epistemic logic