41.
For All -- from MathWorld
- Weisstein, Eric W.
If a proposition P is true for all B, this is written P\forall B. \forall is one of the two so-called quantifiers, and translates the universal quantifier \forall. In Mathematica version 4.0, the command Experimental`ForAllRealQ[ineqs, vars] can be used to determine if the system of real equations and inequalities ineqs is satisfied for all real values of the variables vars. See also: Almost All, Exists, For Some, Implies, Quantifier, Universal Quantifier, Universal Sentence
42.
For Some -- from MathWorld
- Weisstein, Eric W.
An expression occurring in existential sentences. "For some x" is the same as "\exists x." Unlike in everyday language, it is does not necessarily refer to a plurality of elements, and so might be more clearly represented in colloquial English as "for at least one." See also: Existential Quantifier, For All
43.
Formal Language -- from MathWorld
- Weisstein, Eric W.
In mathematics, a formal language is normally defined by an alphabet and formation rules. The alphabet of a formal language is a set of symbols on which this language is built. Some of the symbols in an alphabet may have a special meaning. The formation rules specify which strings of symbols count as well-formed. The well-formed strings of symbols are also called words, expressions, formulas, or terms. The formation rules are usually recursive. Some rules postulate that such and such expressions...
44.
Formula -- from MathWorld
- Weisstein, Eric W.
In mathematics, a formula is a fact, rule, or principle that is expressed in terms of mathematical symbols. Examples of formulas include equations, equalities, identities, inequalities, and asymptotic expressions. The term "formula" is also commonly used in the theory of logic to mean sentential formula (also called a propositional formula), i.e., a formula in propositional calculus. The correct Latin plural form of formula is "formulae," although the less...
45.
Free Variable -- from MathWorld
- Weisstein, Eric W.
An occurrence of a variable in a logic formula which is not inside the scope of a quantifier. See also: Bound, Quantifier, Sentence
46.
Game of Logic -- from MathWorld
- Weisstein, Eric W.
The Game of Logic, described by Lewis Carroll--author of Alice in Wonderland--in 1887 (Carroll 1972) consists of discussing the meaning of propositions like "Some fresh cakes are sweet," and is an instructive introduction to the concepts of logic. The game takes place in a world divided into four quadrants. In the northwest quadrant, the cakes are fresh and sweet, in the northeast, they are fresh and not-sweet, in the southwest, they are not-fresh and sweet, and in the southeast, they...
47.
General Quantifier -- from MathWorld
- Weisstein, Eric W.
See: Universal Quantifier
48.
Generalized Completeness Theorem -- from MathWorld
- Weisstein, Eric W.
The proposition that every consistent generalized theory has a model. The theorem is true if the axiom of choice is assumed. See also: Axiom of Choice
49.
Goal -- from MathWorld
- Weisstein, Eric W.
A Horn clause without a positive literal. See also: Horn Clause, Positive Literal
50.
Grammar -- from MathWorld
- Weisstein, Eric W.
A grammar defining formal language L is a quadruple (N, T, R, S), where N is a finite set of nonterminals, T is a finite set of terminal symbols, R is a finite set of productions, and S is an element of N. The set T of terminal symbols is L's alphabet. Nonterminals are symbols representing language constructs. The sets N and T should not intersect. S is called the start symbol. Productions are rules of the form: \alpha\to \beta, where both \alpha and \beta are strings of terminals and...
51.
Ground Atom -- from MathWorld
- Weisstein, Eric W.
Consider a clause (disjunction of literals) obtained from those of a first-order predicate calculus formula \Phi in Skolemized form \forall x_1 \dots \forall x_n S. Then an atomic statement obtained from those of S by replacing all variables by elements of the Herbrand universe H of S is called a ground atom. The set of all ground atoms that can be formed from predicate symbols from S and terms from H is called the Herbrand base. See also: Atomic Statement, Ground Clause, Herbrand Base,...
52.
Ground Clause -- from MathWorld
- Weisstein, Eric W.
Consider a clause (disjunction of literals) obtained from those of a first-order predicate calculus sentential formula \Phi in Skolemized form \forall x_1 \dots \forall x_n S, then a clause obtained from those of S by replacing all variables by elements of the Herbrand universe H of S is called a ground clause. See also: Ground Atom, Ground Literal, Herbrand Base, Herbrand Universe
53.
Ground Literal -- from MathWorld
- Weisstein, Eric W.
Consider a clause (disjunction of literals) obtained from those of a first-order predicate calculus formula \Phi in Skolemized form \forall x_1 \dots \forall x_n S. Then a literal obtained from those of S by replacing all variables by elements of the Herbrand universe H of S is called a ground literal. See also: Ground Atom, Ground Clause, Herbrand Base, Herbrand Universe
54.
Herbrand Base -- from MathWorld
- Weisstein, Eric W.
The set of all ground atoms that can be formed from predicate symbols from a clause in Skolemized form S and terms from the Herbrand universe H of S. See also: Ground Atom, Herbrand Universe
55.
Herbrand Universe -- from MathWorld
- Weisstein, Eric W.
Consider a first-order logic formula \Phi in Skolemized form \forall x_1 \dots \forall x_n S. Then the Herbrand universe H of S is defined by the following rules. 1. All constants from S belong to H. If there are no constants in S, then H contains an arbitrary constant c. 2. If t_1 \in H,\dots, t_n \in H, and an n-place function f occurs in S, then f(t_1,\dots,t_n) \in H. The clauses (disjunctions of literals) obtained from those of S by replacing all variables by elements of the Herbrand...
56.
Horn Clause -- from MathWorld
- Weisstein, Eric W.
A clause (i.e., a disjunction of literals) is called a Horn clause if it contains at most one positive literal. Horn clauses are usually written as L_1, \dots, L_n \Rightarrow L (\equiv \lnot L_1 \vee \dots \vee\lnot L_n \vee L) or L_1, \dots, L_n \Rightarrow (\equiv \lnot L_1 \vee \dots \vee \lnot L_n), where n\geq 0 and L is the only positive literal. A definite clause is a Horn clause that has exactly one positive literal. A Horn clause without a positive literal is called a goal. Horn...
57.
Hypothesis -- from MathWorld
- Weisstein, Eric W.
A hypothesis is a proposition that is consistent with known data, but has been neither verified nor shown to be false. In statistics, a hypothesis (sometimes called a statistical hypothesis) refers to a statement on which hypothesis testing will be based. Particularly important statistical hypotheses include the null hypothesis and alternative hypothesis. In symbolic logic, a hypothesis is the first part of an implication (with the second part being known as the predicate). In general...
58.
Implication -- from MathWorld
- Weisstein, Eric W.
See also: Implies
59.
Independence Axiom -- from MathWorld
- Weisstein, Eric W.
...under construction...
60.
Individual -- from MathWorld
- Weisstein, Eric W.
One of the basic objects treated in a given formal language system. The term is sometimes also used as a synonym for urelement. See also: Urelement
