Term (logic)

In analogy to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact, in mathematical logic, a term denotes a mathematical object and a formula denotes a mathematical fact. In particular, terms appear as components of a formula.

A first-order term is recursively constructed from constant symbols, variables 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, is a term built from the constant 1, the variable x, and the binary function symbols and ; it is part of the atomic formula which evaluates to true for each real-numbered value of x.

Besides in logic, terms play important roles in universal algebra, and rewriting systems.

Elementary mathematics

In the context of polynomials, sometimes term is used for a monomial with a coefficient: to 'collect like terms' in a polynomial is the operation of making it a linear combination of distinct monomials. Terms, in this sense, are things that are added or subtracted. A series is often represented as the sum of a sequence of terms. Individual factors in an expression representing a product are multiplicative terms. For example, in 6 + 3x − 2, 6, 3x, and −2 are all terms.

Term logic

In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for the way of doing logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late nineteenth century. This entry is an introduction to the term logic needed to understand philosophy texts written before predicate logic came to be seen as the only formal logic of interest. Readers lacking a grasp of the basic terminology and ideas of term logic can have difficulty understanding such texts, because their authors typically assumed an acquaintance with term logic.

Aristotle's system

Aristotle's logical work is collected in the six texts that are collectively known as the Organon. Two of these texts in particular, namely the Prior Analytics and De Interpretatione, contain the heart of Aristotle's treatment of judgements and formal inference, and it is principally this part of Aristotle's works that is about term logic. Modern work on Aristotle's logic builds on the tradition started in 1951 with the establishment by Jan Lukasiewicz of a revolutionary paradigm. The Jan Lukasiewicz approach was reinvigorated in the early 1970s by John Corcoran and Timothy Smiley - which informs modern translations of Prior Analytics by Robin Smith in 1989 and Gisela Striker in 2009.