Initial approaches relied on the results of Herbrand and Skolem to convert a first-order formula into successively larger sets of propositional formulae by instantiating variables with terms from the Herbrand universe.

The propositional satisfiability problem ( PSAT ), which decides whether a given propositional formula is satisfiable, is of central importance in various areas of computer science, including theoretical computer science, algorithmics, artificial intelligence, hardware design, electronic design automation, and verification.

Particularly in hardware design and verification applications, satisfiability and other logical properties of a given propositional formula are sometimes decided based on a representation of the formula as a binary decision diagram ( BDD ).

A validity is a formula that is true under any possible interpretation, e. g. in classical propositional logic validities are tautologies.

As in the disjunctive normal form ( DNF ), the only propositional connectives a formula in CNF can contain are and, or, and not.

Every propositional formula can be converted into an equivalent formula that is in CNF.

where is any formula of first-order-logic ( or whatever logic the deduction system applies to, e. g., propositional calculus or a higher-order logic or a modal logic ).

A simple model-checking problem is verifying whether a given formula in the propositional logic is satisfied by a given structure.

# Symbolic algorithms avoid ever building the graph for the FSM ; instead, they represent the graph implicitly using a formula in quantified propositional logic.

* Each propositional variable is, on its own, a formula.

# If is a propositional variable, then is a formula.

# If is a propositional constant, then is a formula.

A modal sentence is a modal formula that contains no propositional variables.

If is a modal formula with only one propositional variable, then a modal fixed point of is a sentence such that

For example, propositional logic is decidable, because the truth-table method can be used to determine whether an arbitrary propositional formula is logically valid.

* A boxed atom is a propositional atom preceded by a number ( possibly 0 ) of boxes, i. e. a formula of the form ( often abbreviated as for < math > 0

* Conjunctive Normal Form Satisfiability: SAT solvers returns an assignment to the variables of a propositional formula that satisfies it if such an assignment exists.

This is the problem of establishing whether a formula follows from the result of a revision, that is,, where,, and are propositional formulae.

A Horn formula is a propositional formula formed by conjunction of Horn clauses.

Depending on the underlying logic, the problem of deciding the validity of a formula varies from trivial to impossible.

Propositional satisfiability has various generalisations, including satisfiability for quantified Boolean formula problem, for first-and second-order logic, constraint satisfaction problems, 0-1 integer programming, and maximum satisfiability problem.

Since mathematics is related to logic, he cites an example from mathematics: If we have a formula like ( a + b )( a-b )= a²-b² it does not tell us how to think mathematically.

If some specific deductive system of first-order logic is sound and complete, then is it " perfect " ( a formula is provable iff it is a semantic consequence of the axioms ), thus equivalent to any other deductive system with the same quality ( any proof in one system can be converted into the other ).

This diagram shows the Syntax ( logic ) | syntactic entities which may be constructed from formal language s. The symbol ( formal ) | symbols and string ( computer science ) | strings of symbols may be broadly divided into nonsense and well-formed formula s. A formal language can be thought of as identical to the set of its well-formed formulas.

* Disjunctive normal form, a standardization of a logical formula in boolean logic

Then the formula Cold ( r )→ High ( t ) is true for any t and therefore any t gives a correct control given r. A rigorous logical justification of fuzzy control is given in Hájek's book ( see Chapter 7 ) where fuzzy control is represented as a theory of Hájek's basic logic.

* Sentence ( mathematical logic ), a formula with no free variables

* " Undecidable " is sometimes used as a synonym of " independent ", where a formula in mathematical logic is independent of a logical theory if neither that formula nor its negation can be proved within the theory

In boolean logic, a disjunctive normal form ( DNF ) is a standardization ( or normalization ) of a logical formula which is a disjunction of conjunctive clauses.

In Boolean logic, a formula is in conjunctive normal form ( CNF ) if it is a conjunction of clauses, where a clause is a disjunction of literals.

In first order logic, conjunctive normal form can be taken further to yield the clausal normal form of a logical formula, which can be then used to perform first-order resolution.

A set of formulas in first-order logic is consistent ( written Con ) if and only if there is no formula such that and.

The Wittgensteinian philosopher Peter Winch discussed the paradox in The Idea of a Social Science and its Relation to Philosophy ( 1958 ), where he argued that the paradox showed that " the actual process of drawing an inference, which is after all at the heart of logic, is something which cannot be represented as a logical formula ... Learning to infer is not just a matter of being taught about explicit logical relations between propositions ; it is learning to do something " ( p. 57 ).

So to move on to a more significant example, in logic the formula α can refer to any formula, so it could be β, γ, δ, ... or η → π, ς ∨ σ ,....

Another relationship is given by the Gödel – Gentzen negative translation, which provides an embedding of classical first-order logic into intuitionistic logic: a first-order formula is provable in classical logic if and only if its Gödel – Gentzen translation is provable intuitionistically.

The German mathematician Gottlob Frege ( 1848 – 1925 ) presented an independent development of logic with quantifiers in his Begriffsschrift ( formula language ) published in 1879, a work generally considered as marking a turning point in the history of logic.

Actually, only two men know what the formula is, Blake and '' -- He stopped and looked at Thor's body.

First, the present situation is too varied to be systematized by any single formula.

These factors practically insure that no single financing formula is feasible ; ;

As is the case with the allotment provisions for support of vocational rehabilitation services, the matching requirements are also based on a statutory formula.

A mathematical formula is nothing more than a pattern for solving a specific problem.

The first step in using a formula is to insert the numerical values of the factors involved in their correct positions in the formula.

A constant is a number that remains the same regardless of the other numbers used in the formula and the resultant equation.

When it is needed, one formula is as important as another.

Actually, the engine displacement formula is the standard formula for computing the volume of a cylinder of any type with an added factor that represents the number of cylinders in the engine.

The formula is: Af.

Cylinder volume is determined in exactly the same manner as for the displacement formula: Af.

The only precaution is that all volumes used in the formula be quoted in the same terms.

The formula for this step is: Af.

When tire size is measured in this manner a constant of 168 is used in the formula.

To determine car speed for a given combination of engine speed, gear ratio, and tire size, the formula is: Af.

To determine engine speed for a given combination of the other three factors the formula is: Af.

To determine the rear axle gear ratio for a combination of the other three factors, the formula is: Af.

This formula is given by Rumscheidt and Mason.

If it is assumed that the formula given by Lodge of cosec Af applies, the pressure difference along the major axes can be calculated from the angle of inclination of the major axis, and from this the interfacial tension can be calculated.

Since the complex of singular lines is of order K and since there is no complex of invariant lines, it follows from the formula Af that the order of the involution is Af.