Propositional formulaIn propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional formula may also be called a propositional expression, a sentence, or a sentential formula. A propositional formula is constructed from simple propositions, such as "five is greater than three" or propositional variables such as p and q, using connectives or logical operators such as NOT, AND, OR, or IMPLIES; for example: (p AND NOT q) IMPLIES (p OR q).
Sheffer strokeIn Boolean functions and propositional calculus, the Sheffer stroke denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both". It is also called non-conjunction, or alternative denial since it says in effect that at least one of its operands is false, or NAND ("not and"). In digital electronics, it corresponds to the NAND gate. It is named after Henry Maurice Sheffer and written as or as or as or as in Polish notation by Łukasiewicz (but not as ||, often used to represent disjunction).
Recursive definitionIn mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set (Aczel 1977:740ff). Some examples of recursively-definable objects include factorials, natural numbers, Fibonacci numbers, and the Cantor ternary set. A recursive definition of a function defines values of the function for some inputs in terms of the values of the same function for other (usually smaller) inputs.
Type–token distinctionThe type–token distinction is the difference between naming a class (type) of objects and naming the individual instances (tokens) of that class. Since each type may be exemplified by multiple tokens, there are generally more tokens than types of an object. For example, the sentence "A rose is a rose is a rose" contains three word types: three word tokens of the type a, three word tokens of the type rose and two word tokens of the type is. The distinction is important in disciplines such as logic, linguistics, metalogic, typography and computer programming.
Signature (logic)In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. They are rarely made explicit in more philosophical treatments of logic.
MetalogicMetalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths. The basic objects of metalogical study are formal languages, formal systems, and their interpretations.
Axiom schemaIn mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which may or may not be required to satisfy certain conditions. Often, such conditions require that certain variables be free, or that certain variables not appear in the subformula or term.
Predicate variableIn mathematical logic, a predicate variable is a predicate letter which functions as a "placeholder" for a relation (between terms), but which has not been specifically assigned any particular relation (or meaning). Common symbols for denoting predicate variables include capital roman letters such as , and , or lower case roman letters, e.g., . In first-order logic, they can be more properly called metalinguistic variables.
Substitution (logic)A substitution is a syntactic transformation on formal expressions. To apply a substitution to an expression means to consistently replace its variable, or placeholder, symbols with other expressions. The resulting expression is called a substitution instance, or instance for short, of the original expression. Where ψ and φ represent formulas of propositional logic, ψ is a substitution instance of φ if and only if ψ may be obtained from φ by substituting formulas for symbols in φ, replacing each occurrence of the same symbol by an occurrence of the same formula.
