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Accessibility Relation
An accessibility relation is a relation which plays a key role in assigning truth values to sentences in the relational semantics for modal logic. In relational semantics, a modal formula's truth value at a ''possible world'' w can depend on what's true at another possible world v, but only if the accessibility relation R relates w to v. For instance, if P holds at some world v such that wRv, the formula \Diamond P will be true at w. The fact wRv is crucial. If R did not relate w to v, then \Diamond P would be false at w unless P also held at some other world u such that wRu. Accessibility relations are motivated conceptually by the fact that natural language modal statements depend on some, but not all alternative scenarios. For instance, the sentence "It might be raining" is not generally judged true simply because one can imagine a scenario where it was raining. Rather, its truth depends on whether such a scenario is ruled out by available information. This fact can be for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Kripke Model
Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * Simple (album), ''Simple'' (album), by Andy Yorke, 2008, and its title track * Simple (Florida Georgia Line song), "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnny Mathis from the 1984 album ''A Special Part of Me'' * "Simple", a song by Collective Soul from the 1995 album ''Collective Soul (1995 album), Collective Soul'' * "Simple", a song by Katy Perry from the 2005 soundtrack to ''The Sisterhood of the Traveling Pants (film), The Sisterhood of the Traveling Pants'' * "Simple", a song by Khalil from the 2017 album ''Prove It All'' * "Simple", a song by Kreesha Turner from the 2008 album ''Passion (Kreesha Turner album), Passion'' * "Simple", a song by Ty Dolla Sign from the 2017 album ''Beach House 3'' deluxe version * Simple (video game series), ''Simple'' (video game series), budget-priced console games Businesses and organisations * Simple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relation (math)
In mathematics, a relation on a set may, or may not, hold between two given set members. For example, ''"is less than"'' is a relation on the set of natural numbers; it holds e.g. between 1 and 3 (denoted as 1 is an asymmetric relation, but ≥ is not. Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation defined by is neither symmetric nor antisymmetric, let alone asymmetric. ; : for all , if and then . A transitive relation is irreflexive if and only if it is asymmetric. For example, "is ancestor of" is a transitive relation, while "is parent of" is not. ; : for all , if then or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : for all , or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : every nonempty subset of contains a minimal element with respect to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Joyal. It was first conceived for modal logics, and later adapted to intuitionistic logic and other non-classical systems. The development of Kripke semantics was a breakthrough in the theory of non-classical logics, because the model theory of such logics was almost non-existent before Kripke (algebraic semantics existed, but were considered 'syntax in disguise'). Semantics of modal logic The language of propositional modal logic consists of a countable set, countably infinite set of propositional variables, a set of truth-functional Logical connective, connectives (in this article \to and \neg), and the modal operator \Box ("necessarily"). The modal operator \Diamond ("possibly") is (classically) the duality (mathematics)#Duality in log ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 systems by adding unary operators \Diamond and \Box, representing possibility and necessity respectively. For instance the modal formula \Diamond P can be read as "possibly P" while \Box P can be read as "necessarily P". Modal logics can be used to represent different phenomena depending on what kind of necessity and possibility is under consideration. When \Box is used to represent epistemic necessity, \Box P states that P is epistemically necessary, or in other words that it is known. When \Box is used to represent deontic necessity, \Box P states that P is a moral or legal obligation. In the standard relational semantics for modal logic, formulas are assigned truth values relative to a ''possible world''. A formula's truth value at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 modal logic. Their metaphysics, metaphysical status has been a subject of controversy in philosophy, with Modal realism, modal realists such as David Lewis (philosopher), David Lewis arguing that they are literally existing alternate realities, and others such as Robert Stalnaker arguing that they are not. Logic Possible worlds are one of the foundational concepts in modal logic, modal and intensional logics. Formulas in these logics are used to represent statements about what ''might'' be true, what ''should'' be true, what one ''believes'' to be true and so forth. To give these statements a formal interpretation, logicians use structures containing possible worlds. For instance, in the relational semantics for classical propositional mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Language
In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages can take different forms, such as speech or signing. They are distinguished from constructed and formal languages such as those used to program computers or to study logic. Defining natural language Natural language can be broadly defined as different from * artificial and constructed languages, e.g. computer programming languages * constructed international auxiliary languages * non-human communication systems in nature such as whale and other marine mammal vocalizations or honey bees' waggle dance. All varieties of world languages are natural languages, including those that are associated with linguistic prescriptivism or language regulation. ( Nonstandard dialects can be viewed as a wild type in comparison with standard l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linguistic Modality
In linguistics and philosophy, modality refers to the ways language can express various relationships to reality or truth. For instance, a modal expression may convey that something is likely, desirable, or permissible. Quintessential modal expressions include modal auxiliaries such as "could", "should", or "must"; modal adverbs such as "possibly" or "necessarily"; and modal adjectives such as "conceivable" or "probable". However, modal components have been identified in the meanings of countless natural language expressions, including counterfactuals, propositional attitudes, evidentials, habituals, and generics. Modality has been intensely studied from a variety of perspectives. Within linguistics, typological studies have traced crosslinguistic variation in the strategies used to mark modality, with a particular focus on its interaction with tense–aspect–mood marking. Theoretical linguists have sought to analyze both the propositional content and discourse effects ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deontic Logic
Deontic logic is the field of philosophical logic that is concerned with obligation, permission, and related concepts. Alternatively, a deontic logic is a formal system that attempts to capture the essential logical features of these concepts. It can be used to formalize imperative logic, or directive modality in natural languages. Typically, a deontic logic uses ''OA'' to mean ''it is obligatory that A'' (or ''it ought to be (the case) that A''), and ''PA'' to mean ''it is permitted (or permissible) that A'', which is defined as PA\equiv \neg O\neg A. Note that in natural language, the statement "You may go to the zoo OR the park" should be understood as Pz\land Pp instead of Pz\lor Pp, as both options are permitted by the statement; See Hans Kamp's paradox of free choice for more details. When there are multiple agents involved in the domain of discourse, the deontic modal operator can be specified to each agent to express their individual obligations and permissions. For e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 systems by adding unary operators \Diamond and \Box, representing possibility and necessity respectively. For instance the modal formula \Diamond P can be read as "possibly P" while \Box P can be read as "necessarily P". Modal logics can be used to represent different phenomena depending on what kind of necessity and possibility is under consideration. When \Box is used to represent epistemic necessity, \Box P states that P is epistemically necessary, or in other words that it is known. When \Box is used to represent deontic necessity, \Box P states that P is a moral or legal obligation. In the standard relational semantics for modal logic, formulas are assigned truth values relative to a ''possible world''. A formula's truth value at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Possible Worlds
Possible Worlds may refer to: * Possible worlds, concept in philosophy * ''Possible Worlds'' (play), 1990 play by John Mighton ** ''Possible Worlds'' (film), 2000 film by Robert Lepage, based on the play * Possible Worlds (studio) * ''Possible Worlds'', poetry book by Peter Porter * ''Possible Worlds'', book by J. B. S. Haldane * ''Possible Worlds'', 1995 album by Markus Stockhausen Markus Stockhausen (born May 2, 1957) is a German trumpeter and composer. His recordings and performances have typically alternated between jazz and chamber or opera music, the latter often in collaboration with his father, composer Karlheinz Sto ... See also * * * Possible (other) * World (other) {{dab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Attitude
A propositional attitude is a mental state held by an agent toward a proposition. Linguistically, propositional attitudes are denoted by a verb (e.g. "believed") governing an embedded "that" clause, for example, 'Sally believed that she had won'. Propositional attitudes are often assumed to be the fundamental units of thought and their contents, being propositions, are true or false from the perspective of the person. An agent can have different propositional attitudes toward the same proposition (e.g., "S believes that her ice-cream is cold," and "S fears that her ice-cream is cold"). Propositional attitudes have directions of fit: some are meant to reflect the world, others to influence it. One topic of central concern is the relation between the modalities of assertion and belief, perhaps with intention thrown in for good measure. For example, we frequently find ourselves faced with the question of whether or not a person's assertions conform to his or her beliefs. Disc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modal Depth
In modal logic, the modal depth of a formula is the deepest nesting of modal operators (commonly \Box and \Diamond). Modal formulas without modal operators have a modal depth of zero. Definition Modal depth can be defined as follows. Let MD(\phi) be a Function (mathematics), function that computes the modal depth for a modal formula \phi: :MD(p) = 0, where p is an atomic formula. :MD(\top) = 0 :MD(\bot) = 0 :MD(\neg \varphi) = MD(\varphi) :MD(\varphi \wedge \psi) = max(MD(\varphi), MD(\psi)) :MD(\varphi \vee \psi) = max(MD(\varphi), MD(\psi)) :MD(\varphi \rightarrow \psi) = max(MD(\varphi), MD(\psi)) :MD(\Box \varphi) = 1 + MD(\varphi) :MD(\Diamond \varphi) = 1 + MD(\varphi) Example The following computation gives the modal depth of \Box ( \Box p \rightarrow p ): :MD(\Box ( \Box p \rightarrow p )) = :1 + MD( \Box p \rightarrow p) = :1 + max(MD(\Box p), MD(p)) = :1 + max(1 + MD(p), 0) = :1 + max(1 + 0, 0) = :1 + 1 =:2 Modal depth and semantics The modal depth of a formu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
