nLab propositional logic

Idea

Propositional logic, also called $0$th-order logic and sentential logic, is that part of logic that deals only with propositions with no bound variables.

Compare predicate logic, or $1$st-order logic, and higher-order logic. Note that while one can have free variables in $0$th-order logic, one cannot really do anything with them; each $P(x)$ in a $0$th-order proposition might as well be thought of as atomic.

This can be understood more cleanly in the language of many-sorted logic, where each variable has to have a specified sort. Then ordinary predicate logic has exactly one sort, usually unnamed. Propositional logic is for a signature with no sorts, hence no variables at all.

A propositional calculus, also called sentential calculus, is simply a system for describing and working with propositional logic. The precise form of such a calculus (and hence of the logic itself) depends on whether one is using classical logic, intuitionistic logic, linear logic, etc; see those articles for details.

Notation

In classical / intuitionistic logic

$\phantom{-}$notation$\phantom{-}$	$\phantom{-}$in propositional classical logic/intuitionistic logic$\phantom{-}$
$\phantom{A}$$\top$$\phantom{A}$	$\phantom{A}$true$\phantom{A}$
$\phantom{A}$$\bot$$\phantom{A}$	$\phantom{A}$false$\phantom{A}$
$\phantom{A}$$\wedge$	$\phantom{A}$logical conjunction, AND operator
$\phantom{A}$$\vee$	$\phantom{A}$logical disjunction, OR operator
$\phantom{A}$$\veebar$	$\phantom{A}$exclusive disjunction, XOR operator
$\phantom{A}$$\Rightarrow$	$\phantom{A}$implication, conditional, implies operator
$\phantom{A}$$\Leftrightarrow$	$\phantom{A}$logical equivalence, biconditional, equals operator
$\phantom{A}$$\not$	$\phantom{A}$negation, NOT operator
$\phantom{A}$$\not \not$	$\phantom{A}$negation of negation$\phantom{A}$

In linear logic

$\phantom{-}$symbol$\phantom{-}$	$\phantom{-}$in linear logic$\phantom{-}$
$\phantom{A}$$\top$	$\phantom{A}$additive truth
$\phantom{A}$$\bot$	$\phantom{A}$additive falsehood
$\phantom{A}$$0$	$\phantom{A}$multiplicative falsehood
$\phantom{A}$$1$	$\phantom{A}$multiplicative truth
$\phantom{A}$$\multimap$$\phantom{A}$	$\phantom{A}$linear implication$\phantom{A}$
$\phantom{A}$$\otimes$$\phantom{A}$	$\phantom{A}$multiplicative conjunction$\phantom{A}$
$\phantom{A}$$\oplus$$\phantom{A}$	$\phantom{A}$additive disjunction$\phantom{A}$
$\phantom{A}$$\&$$\phantom{A}$	$\phantom{A}$additive conjunction$\phantom{A}$
$\phantom{A}$$\invamp$$\phantom{A}$	$\phantom{A}$multiplicative disjunction$\phantom{A}$
$\phantom{A}$$\;!$$\phantom{A}$	$\phantom{A}$exponential conjunction$\phantom{A}$
$\phantom{A}$$\;?$$\phantom{A}$	$\phantom{A}$exponential disjunction$\phantom{A}$
$\phantom{A}$$^\bot$$\phantom{A}$	$\phantom{A}$negation$\phantom{A}$

Related concepts

• logic

  • propositional logic (0th order)

  • predicate logic (1st order)

  • higher order logic

  • modal logic

  • propositional logic as a dependent type theory