Term (logic) in the context of Predicate (logic)

In mathematical logic, a tautology (from Ancient Greek: ταυτολογία) is a formula that is true regardless of the interpretation of its component terms, with only the logical constants having a fixed meaning. It is a logical truth. For example, a formula that states "the ball is green or the ball is not green" is always true, regardless of what a ball is and regardless of its colour. Tautology is usually, though not always, used to refer to valid formulas of propositional logic.

The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement. In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. In other words, it cannot be false.

In mathematics, a law is a formula that is always true within a given context. Laws describe a relationship, between two or more expressions or terms (which may contain variables), usually using equality or inequality, or between formulas themselves, for instance, in mathematical logic. For example, the formula ${\displaystyle a^{2}\geq 0}$ is true for all real numbers a, and is therefore a law. Laws over an equality are called identities. For example, ${\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}}$ and ${\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1}$ are identities. Mathematical laws are distinguished from scientific laws which are based on observations, and try to describe or predict a range of natural phenomena. The more significant laws are often called theorems.

Intensional logic is an approach to predicate logic that extends first-order logic, which has quantifiers that range over the individuals of a universe (extensions), by additional quantifiers that range over terms that may have such individuals as their value (intensions). The distinction between intensional and extensional entities is parallel to the distinction between sense and reference.

In formal systems particularly mathematical logic, a function symbol is a non-logical symbol which represents a function or mapping on the domain of discourse, though, formally, does not need to represent anything at all. Function symbols are a basic component in formal languages to form terms. Specifically, if the symbol ${\displaystyle F}$ is a function symbol, then given any constant symbol ${\displaystyle X}$ representing an object in the language, ${\displaystyle F(X)}$ also represents an object in the language. Similarly, if ${\displaystyle T}$ is some term in the language, ${\displaystyle F(T)}$ is also a term. As such, the interpretation of a function symbol must be defined over the whole domain of discourse. Function symbols are a primitive notion, and are therefore not defined in terms of other, more basic concepts.

In typed logic, F is a functional symbol with domain type T and codomain type U if, given any symbol X representing an object of type T, F(X) is a symbol representing an object of type U.One can similarly define function symbols of more than one variable, analogous to functions of more than one variable; a function symbol in zero variables is simply a constant symbol.

In lambda calculus, the Church–Rosser theorem states that, when applying reduction rules to terms, the ordering in which the reductions are chosen does not make a difference to the eventual result.

More precisely, if there are two distinct reductions or sequences of reductions that can be applied to the same term, then there exists a term that is reachable from both results, by applying (possibly empty) sequences of additional reductions. The theorem was proved in 1936 by Alonzo Church and J. Barkley Rosser, after whom it is named.