Predicate (logic) in the context of Boolean-valued function

In mathematical logic, a term is an arrangement of dependent/bound symbols that denotes a mathematical object within an expression/formula. In particular, terms appear as components of a formula. This is analogous to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact.

A first-order term is recursively constructed from constant symbols, variable symbols, and function symbols.An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluates to true or false in bivalent logics, given an interpretation.For example, ⁠${\displaystyle (x+1)*(x+1)}$⁠ is a term built from the constant 1, the variable x, and the binary function symbols ⁠${\displaystyle +}$⁠ and ⁠${\displaystyle *}$⁠; it is part of the atomic formula ⁠${\displaystyle (x+1)*(x+1)\geq 0}$⁠ which evaluates to true for each real-numbered value of x.

In mathematical logic, especially model theory, non-logical symbols are elements of a formal language whose interpretation may change depending on the model. In first-order logic, these usually consist of constant symbols, function symbols, and predicates. This is in contrast to logical constants which are required to have the same interpretation under every model, such as logical connectives and quantifiers.

A non-logical symbol only has meaning or semantic content when one is assigned to it by means of an interpretation. Consequently, a sentence containing a non-logical symbol lacks meaning except under an interpretation, so a sentence is said to be true or false under an interpretation. These concepts are defined and discussed in the article on first-order logic, and in particular the section on syntax.

The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities x and y are identical if every predicate possessed by x is also possessed by y and vice versa. It states that no two distinct things (such as snowflakes) can be exactly alike, but this is intended as a metaphysical principle rather than one of natural science. A related principle is the indiscernibility of identicals, discussed below.

A form of the principle is attributed to the German philosopher Gottfried Wilhelm Leibniz. While some think that Leibniz's version of the principle is meant to be only the indiscernibility of identicals, others have interpreted it as the conjunction of the identity of indiscernibles and the indiscernibility of identicals (the converse principle). Because of its association with Leibniz, the indiscernibility of identicals is sometimes known as Leibniz's law. It is considered to be one of his great metaphysical principles, the other being the principle of noncontradiction and the principle of sufficient reason (famously used in his disputes with Newton and Clarke in the Leibniz–Clarke correspondence).