Search results
Results from the WOW.Com Content Network
Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations, and any other mathematical objects and assembling them into expressions and formulas. Mathematical notation is widely used in mathematics , science , and engineering for representing complex concepts and properties in a concise ...
A formula in first-order logic with no free variable occurrences is called a first-order sentence. These are the formulas that will have well-defined truth values under an interpretation. For example, whether a formula such as Phil(x) is true must depend on what x represents.
Corner quotes, also called “Quine quotes”; for quasi-quotation, i.e. quoting specific context of unspecified (“variable”) expressions; [4] also used for denoting Gödel number; [5] for example “⌜G⌝” denotes the Gödel number of G. (Typographical note: although the quotes appears as a “pair” in unicode (231C and 231D), they ...
A mathematical markup language is a computer notation for representing mathematical formulae, based on mathematical notation. Specialized markup languages are necessary because computers normally deal with linear text and more limited character sets (although increasing support for Unicode is obsoleting very simple uses). A formally ...
A closed formula, also ground formula or sentence, is a formula in which there are no free occurrences of any variable. If A is a formula of a first-order language in which the variables v 1, …, v n have free occurrences, then A preceded by ∀v 1 ⋯ ∀v n is a universal closure of A.
For example, Hamilton uses two symbols = and ≠ when he defines the notion of a valuation v of any well-formed formulas (wffs) A and B in his "formal statement calculus" L. A valuation v is a function from the wffs of his system L to the range (output) { T, F }, given that each variable p 1 , p 2 , p 3 in a wff is assigned an arbitrary truth ...
Ground formulas may be defined by syntactic recursion as follows: A ground atom is a ground formula. If φ {\displaystyle \varphi } and ψ {\displaystyle \psi } are ground formulas, then ¬ φ {\displaystyle \lnot \varphi } , φ ∨ ψ {\displaystyle \varphi \lor \psi } , and φ ∧ ψ {\displaystyle \varphi \land \psi } are ground formulas.
According to model theory, a logical system may be given interpretations which describe whether a given structure - the mapping of formulas to a particular meaning - satisfies a well-formed formula. A structure that satisfies all the axioms of the formal system is known as a model of the logical system.