The Bernays–Schönfinkel is a decidable fragment of first-order logic formulas, in prenex \forsome^* \forall^* (not contain function symbols).
This class of logic is called effectively propositional (EPR) since it can be effectively translated into propositional logic formulas by a process of grounding or instantiation.
The idea behind what we call class is after we have gone to infinity and beyond that. Let us say there are sets; later, they grow bigger and bigger and become infinite. Then, it comes to a question if the collection of what we have accumulated so far is still a set?.
In Naive Set Theory, one can freely collect all objects which satisfy certain properties into a set. Hence, this can easily lead to a paradox. Famous ones are
A class is a collection of sets (or mathematical objects) that can be unambiguously defined by a property that all its members share. Its precise definition depends on a foundational context. For example, Zermelo-Fraenkel set theory defines the notion informally whereas Von Neumann-Bernays-Godel set theory define it as entities that are not members of another entity.
A class that is not a set is called a proper class. On the other hand, a class that is a set may be called a small class. For example, the class of all ordinal numbers and the class of all sets are proper classes.
In general, the paradoxes of naive set theory explains why not all classes can be sets. For example, Russell’s paradox suggests that the class of all sets which do not contain themselves is proper; and Burali-Forti paradox suggests that the class of all ordinal numbers is proper. Hence, the paradoxes tell us that some collections may not be sets. The notion of proper class tells us that we can still talk about those collections even though they are not sets per se. For instance, we can still talk about the collection of all ordinals or prove that some properties hold, despite it is not a set, e.g.
Naive set theory may give rise to paradoxes. Axiomatic set theory was originally devised to get rid of such paradoxes. Today, when mathematicians talk about set theory as a field, they usually mean axiomatic set theory. Informal applications of set theory in other fields are sometimes referred to as applications of naive set theory, but are understood to be justified in terms of an axiomatic system.
Let us note that naive set theory may refer to several distinct notions, e.g.
In mathematics, an indexed family is a collection of values associated with indices. For example, a family of real numbers, indexed by the integers is a collection of real numbers, where each integer is associated with one of the real numbers.
Formally, an indexed family is the same thing as a mathematical function, i.e. a function with a domain J and codomain X is equivalent to a family of elements X indexed by elements of J. Their differences are only conceptual basis. Indexed families are interpreted as collections instead of as functions.
Let n be the finite set {1, 2, …, n} where n is a positive integer.
Ones may observe that a sequence is a special type of indexed family accompanying the notion of ordering, e.g. x_1 is before x_2. This notion is not presented in the indexed family unless the indexing set has some sort of order relation defined on it.
Monotonicity of entailment is a property of logical systems which states that the hypotheses of any derived fact may be freely extended with additional assumptions, i.e. given a new premise, those logical systems can only entail new fact.
By its nature, Description Logics (DLs) is a monotonic logic because they only infer new fact, if possible. If some contradictions have found during the inference procedure, they will just say that the new knowledge base is not satisfiable (or inconsistent).
Non-monotonic logics are devised to capture and represent defeasible inferences, i.e. a kind of inference in which reasoners draw tentative conclusions, enabling reasoners to retract their conclusions based on further evidence.
A monotonic logic cannot handle various reasoning tasks such as reasoning by default, abductive reasoning, belief revision, and so on.
In Miyatake, there is a small gym in which you can do several kinds of sports, viz. Basketball and Badminton.
Going to the place is quite convenient. It is in walking distance. Normally, I spend around 10 mins (by walk) from JAIST to the gym. Many JAIST students go there to play basketball on every Friday from 5:30 - 8:30pm. The gym also opens on other days, but we need to make sure with the staffs at front first. This is because the gym also provides services for Badminton. According to what I know, the gym also permit to play basketball on Saturday from 4:30 - 6:30pm.