Term logic in the context of "Consistency"

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory ${\displaystyle T}$ is consistent if there is no formula ${\displaystyle \varphi }$ such that both ${\displaystyle \varphi }$ and its negation ${\displaystyle \lnot \varphi }$ are elements of the set of consequences of ${\displaystyle T}$. Let ${\displaystyle A}$ be a set of closed sentences (informally "axioms") and ${\displaystyle \langle A\rangle }$ the set of closed sentences provable from ${\displaystyle A}$ under some (specified, possibly implicitly) formal deductive system. The set of axioms ${\displaystyle A}$ is consistent when there is no formula ${\displaystyle \varphi }$ such that ${\displaystyle \varphi \in \langle A\rangle }$ and ${\displaystyle \lnot \varphi \in \langle A\rangle }$. A trivial theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory is a syntactic notion, whose semantic counterpart is satisfiability. A theory is satisfiable if it has a model, i.e., there exists an interpretation under which all axioms in the theory are true. This is what consistent meant in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead.

In a sound formal system, every satisfiable theory is consistent, but the converse does not hold. If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive logic, the logic is called complete. The completeness of the propositional calculus was proved by Paul Bernays in 1918 and Emil Post in 1921, while the completeness of (first order) predicate calculus was proved by Kurt Gödel in 1930, and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931). Stronger logics, such as second-order logic, are not complete.

In term logic (a branch of philosophical logic), the square of opposition is a diagram representing the relations between the four basic categorical propositions. The origin of the square can be traced back to Aristotle's tractate On Interpretation and its distinction between two oppositions: contradiction and contrariety. However, Aristotle did not draw any diagram; this was done several centuries later.

Many-valued logic (also multi- or multiple-valued logic) is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., true and false) for any proposition. Classical two-valued logic may be extended to n-valued logic for n greater than 2. Those most popular in the literature are three-valued (e.g., Łukasiewicz's and Kleene's, which accept the values true, false, and unknown), four-valued, nine-valued, the finite-valued (finitely-many valued) with more than three values, and the infinite-valued (infinitely-many-valued), such as fuzzy logic and probability logic.