User:LBehounek/Sandbox

This is a personal sandbox for User:LBehounek.

A draft of Names of logical formulae

Redirect from: Names of logical formulas, Names of logical laws, Names of logical rules

Include in "See also" in: Formula (mathematical logic)

---

Many logical formulae (or logical laws, rules of inference, etc.) have traditional names under which they are known and referred to in the literature. This list gives the most common names of logical formulae.

Naming logical formulae

The following points have to be taken into account in the lists of named logical formulae:

• Some formulae have more than one alternative names, sometimes depending on the branch of logic in which they are used.
• The same name can refer to variant formulae, which are equivalent in some logical systems, but differ in others.
• The form of the named formulae naturally depends on the language used in the particular system. This list gives the most common variants and comments on their usage in particular logical systems or branches of logic.
• The names usually refer to any formula of a given form. Thus, they are rather the names of schemes of formulae rather than their particular instances. Nevertheless, any instance of the scheme can usually be called by the name of the scheme.
• The graphical form of the formula depends on the style of notation employed, which differs not only across branches of logic, but also across communities of logicians. In this list, a common (but by no means the only possible) style of logical notation is employed. See also Table of logic symbols for the most basic logical symbols (however, the table does not exhaust the symbols used in this list). Common notational conventions are applied in this article, including the usual rules of precedence for logical connectives (namely that implication and equivalence connectives have the lowest and unary connectives the highest priority).
• The names are established by tradition in various communities of logicians, therefore their application can vary. Although a reference to the literature where the name is used can be given, the prevalence of the name in common usage usually cannot be evidenced.

Propositional formulae

In substructural logics (and logics weaker than classical in general), the formulae listed below can have several forms, depending e.g. on:

• Which split connective is used (i.e., whether a lattice or residuated one, whether postnegation or retronegation, which implication, whether false or bottom, resp. true or top, etc.)
• The order of arguments of propositional connectives (e.g., in non-commutative conjunctions, disjunctions, etc.).

In some cases (indicated in the Comments column), the name may apply only to one of such variants.

Formula	Name	Comments
$A\vee \neg A$	The law of excluded middle (LEM), or simply excluded middle	In t-norm logics, the name LEM usually denotes the formula with lattice disjunction, as this form ensures bivalence.
$\neg (A{\mathbin {\And }}\neg A)$	The law of contradiction, as well as the law of non-contradiction	In t-norm logics, the version with strong (monoidal) conjunction is tautological, while the version with the lattice conjunction in general need not be.
$\bot \rightarrow A,$ or depending on the connectives available in the language, $(A{\mathbin {\And }}\neg A)\rightarrow B,$ $A\rightarrow (\neg A\rightarrow B),$ etc.	Ex falso quodlibet (EFQ, Latin for from false anything), ex contradictione quodlibet (ECQ, Latin for from contradition anything), ex impossibile quodlibet (EIQ, Latin for from impossible anything)	The Latin names come from mediaeval times and do not distinguish between falsity, contradiction, and impossibility.
$A\rightarrow A{\mathbin {\And }}A$	Contraction	Used mainly in the context of substructural logics, where this formula internalizes the rule of contraction.
$A{\mathbin {\And }}A\rightarrow A$	Expansion

Names according to the structure of the formula

Formulae with a principal propositional connective can be called by the name of the connective: e.g., a formula of the form $A\rightarrow B$ is an implication.

Predicate logics

In the table, $Q$ stands for one of the quantifiers (usually $\exists ,\forall$).

Form of the formula	Name	Comments
$(\forall x)\varphi$	Universal formula; also Π-formula.
$(\exists x)\varphi$	Existential formula; also Σ-formula.
$(\exists x_{1})(\forall x_{2})\dots (Qx_{n})\varphi$ (i.e., with alternating quantifiers), where $\varphi$ does not contain quantifiers	Σn-formula.
$(\forall x_{1})(\exists x_{2})\dots (Qx_{n})\varphi ,$ where $\varphi$ does not contain quantifiers	Πn-formula.
$(Q_{1}x_{1})(Q_{2}x_{2})\dots (Q_{n}x_{n})\varphi ,$ where $\varphi$ does not contain quantifiers	Prenexed formula, or formula in a prenex form.

See also

• List of rules of inference