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e a to b to a instance of a1 a a displaystyle a to a from 4 and 3 by modus ponens solvers edit one notable difference between propositional calculus and predicate calculus is that satisfiability of a propositional formula is decidable 120 81 deciding satisfiability of propositional logic formulas is an np complete problem however practical methods exist e g dpll algorithm 1962 chaff algorithm 2001 that are very fast for many useful cases recent work has extended the sat solver algorithms to work with propositions containing arithmetic expressions these are the smt solvers see also edit philosophy portal higher logical levels edit first order logic second order propositional logic second order logic higher order logic related topics edit boolean algebra logic boolean algebra structure boolean algebra topics boolean domain boolean function boolean valued function categorical logic combinational logic combinatory logic conceptual graph disjunctive syllogism entitative graph equational logic existential graph implicational propositional calculus intuitionistic propositional calculus jean buridan laws of form list of logic symbols logical graph logical nor logical value mathematical logic operation mathematics paul of venice peirce s law peter of spain author propositional formula symmetric difference tautology rule of inference truth function truth table walter burley william of sherwood notes edit many sources write this with a definite article as the propositional calculus while others just call it propositional calculus with no article zeroth order logic is sometimes used to denote a quantifier free predicate logic that is propositional logic extended with functions relations and constants 6 for propositional logic the formal language used is a propositional language not to be confused with the formal language s alphabet see all possible connectives on truth functional propositional logic with some of their properties the or both makes it clear 35 that it s a logical disjunction not an exclusive or which is more common in english the set of premises may be the empty set 38 39 an argument from an empty set of premises is valid if and only if the conclusion is a tautology 38 39 the turnstile for syntactic consequence is of lower precedence than the comma which represents premise combination which in turn is of lower precedence than the arrow used for material implication so no parentheses are needed to interpret this formula 45 a very general and abstract syntax is given here following the notation in the sep 2 but including the third definition which is very commonly given explicitly by other sources such as gillon 15 bostock 38 allen hand 39 and many others as noted elsewhere in the article languages variously compose their set of atomic propositional variables from uppercase or lowercase letters often focusing on p p q q and r r with or without subscript numerals and in their set of connectives they may include either the full set of five typical connectives displaystyle neg land lor to leftrightarrow or any of the truth functionally complete subsets of it and of course they may also use any of the notational variants of these connectives note that the phrase principle of composition has referred to other things in other contexts and even in the context of logic since bertrand russell used it to refer to the principle that a proposition which implies each of two propositions implies them both 53 the name interpretation is used by some authors and the name case by other authors this article will be indifferent and use either since it is collaboratively edited and there is no consensus about which terminology to adopt a truth functionally complete set of connectives 2 is also called simply functionally complete or adequate for truth functional logic 40 or expressively adequate 78 or simply adequate 40 78 see a table of all 16 bivalent truth functions some of these definitions use the word interpretation and speak of sentences formulas being true or false under it and some will use the word case and speak of sentences formulas being true or false in it published reliable sources wp rs have used both kinds of terminological convention although usually a given author will use only one of them since this article is collaboratively edited and there is no consensus about which convention to use these variations in terminology have been left standing conventionally φ displaystyle models varphi with nothing to the left of the turnstile is used to symbolize a tautology it may be interpreted as saying that φ displaystyle varphi is a semantic consequence of the empty set of formulae i e φ displaystyle models varphi but with the empty brackets omitted for simplicity 38 which is just the same as to say that it is a tautology i e that there is no interpretation under which it is false 38 to simplify the statement of the rule the word denial here is used in this way the denial of a formula φ displaystyle varphi 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