In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word (i.e. a finite sequence of symbols from a given alphabet) which is part of a formal language.^[1] A formal language can be considered to be identical to the set containing all and only its formulas.

A formula is a syntactic formal object that can be informally given a semantic meaning.

Introduction[link]

A key use of formulas is in propositional logic and predicate logics such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have been instantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, and the final formula in the sequence is what is proven.

Although the term "formula" may be used for written marks (for instance, on a piece of paper or chalkboard), it is more precisely understood as the sequence being expressed, with the marks being a token instance of formula. It is not necessary for the existence of a formula that there be any actual tokens of it. A formal language may thus have an infinite number of formulas regardless whether each formula has a token instance. Moreover, a single formula may have more than one token instance, if it is written more than once.

Formulas are quite often interpreted as propositions (as, for instance, in propositional logic). However formulas are syntactic entities, and as such must be specified in a formal language without regard to any interpretation of them. An interpreted formula may be the name of something, an adjective, an adverb, a preposition, a phrase, a clause, an imperative sentence, a string of sentences, a string of names, etc.^{[broken citation]}. A formula may even turn out to be nonsense, if the symbols of the language are specified so that it does. Furthermore, a formula need not be given any interpretation.

Propositional calculus[link]

The formulas of propositional calculus, also called propositional formulas,^[2] are expressions such as Failed to parse (Missing texvc executable; please see math/README to configure.): (A \land (B \lor C)) . Their definition begins with the arbitrary choice of a set V of propositional variables. The alphabet consists of the letters in V along with the symbols for the propositional connectives and parentheses "(" and ")", all of which are assumed to not be in V. The formulas will be certain expressions (that is, strings of symbols) over this alphabet.

The formulas are inductively defined as follows:

• Each propositional variable is, on its own, a formula.
• If φ is a formula, then Failed to parse (Missing texvc executable; please see math/README to configure.): \lnot

φ is a formula.

• If φ and ψ are formulas, and • is any binary connective, then ( φ • ψ) is a formula. Here • could be (but is not limited to) the usual operators ∨, ∧, →, or ↔.

This definition can also be written as a formal grammar in Backus–Naur form, provided the set of variables is finite:

<alpha set> ::= p | q | r | s | t | u | ... (the arbitrary finite set of propositional variables)

<form> ::= <alpha set> | Failed to parse (Missing texvc executable; please see math/README to configure.): \neg

<form> | (<form>Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge <form>) | (<form>Failed to parse (Missing texvc executable; please see math/README to configure.): \vee <form>) | (<form>Failed to parse (Missing texvc executable; please see math/README to configure.): \rightarrow <form>) | (<form>Failed to parse (Missing texvc executable; please see math/README to configure.): \leftrightarrow <form>)

Using this grammar, the sequence of symbols

(((p Failed to parse (Missing texvc executable; please see math/README to configure.): \rightarrow

q) Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge
(r Failed to parse (Missing texvc executable; please see math/README to configure.): \rightarrow
s)) Failed to parse (Missing texvc executable; please see math/README to configure.): \vee
(Failed to parse (Missing texvc executable; please see math/README to configure.): \neg

q Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge

Failed to parse (Missing texvc executable; please see math/README to configure.): \neg

s)) is a formula, because it is grammatically correct. The sequence of symbols

((p Failed to parse (Missing texvc executable; please see math/README to configure.): \rightarrow

q)Failed to parse (Missing texvc executable; please see math/README to configure.): \rightarrow

(qq))p)) is not a formula, because it does not conform to the grammar.

A complex formula may be difficult to read, owing to, for example, the proliferation of parentheses. To alleviate this last phenomenon, precedence rules are assumed among the operators, making some operators more binding than others. For example, assuming the precedence (from most binding to least binding) 1. Failed to parse (Missing texvc executable; please see math/README to configure.): \neg

  2. Failed to parse (Missing texvc executable; please see math/README to configure.): \rightarrow

  3. Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge   4. Failed to parse (Missing texvc executable; please see math/README to configure.): \vee . Then the formula

(((p Failed to parse (Missing texvc executable; please see math/README to configure.): \rightarrow

q) Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge
(r Failed to parse (Missing texvc executable; please see math/README to configure.): \rightarrow
s)) Failed to parse (Missing texvc executable; please see math/README to configure.): \vee
(Failed to parse (Missing texvc executable; please see math/README to configure.): \neg

q Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge

Failed to parse (Missing texvc executable; please see math/README to configure.): \neg

s)) may be abbreviated as

p Failed to parse (Missing texvc executable; please see math/README to configure.): \rightarrow

q Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge
r Failed to parse (Missing texvc executable; please see math/README to configure.): \rightarrow
s Failed to parse (Missing texvc executable; please see math/README to configure.): \vee
Failed to parse (Missing texvc executable; please see math/README to configure.): \neg

q Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge

Failed to parse (Missing texvc executable; please see math/README to configure.): \neg

s This is, however, only a convention used to simplify the written representation of a formula. If the precedence was assumed, for example, to be left-right associative, in following order: 1. Failed to parse (Missing texvc executable; please see math/README to configure.): \neg

  2. Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge

  3. Failed to parse (Missing texvc executable; please see math/README to configure.): \vee   4. Failed to parse (Missing texvc executable; please see math/README to configure.): \rightarrow , then the same formula above (without parentheses) would be rewritten as

(p Failed to parse (Missing texvc executable; please see math/README to configure.): \rightarrow

(q Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge
r)) Failed to parse (Missing texvc executable; please see math/README to configure.): \rightarrow
(s Failed to parse (Missing texvc executable; please see math/README to configure.): \vee
((Failed to parse (Missing texvc executable; please see math/README to configure.): \neg

q) Failed to parse (Missing texvc executable; please see math/README to configure.): \wedge

(Failed to parse (Missing texvc executable; please see math/README to configure.): \neg

s)))

Predicate logic[link]

The definition of a formula in first-order logic Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{QS}

is relative to the signature of the theory at hand. This signature specifies the constant symbols, relation symbols, and function symbols of the theory at hand, along with the arities of the function and relation symbols.

The definition of a formula comes in several parts. First, the set of terms is defined recursively. Terms, informally, are expressions that represent objects from the domain of discourse.

1. Any variable is a term.
2. Any constant symbol from the signature is a term
3. an expression of the form f(t₁,...,t_n), where f is an n-ary function symbol, and t₁,...,t_n are terms, is again a term.

The next step is to define the atomic formulas.

1. If t₁ and t₂ are terms then t₁=t₂ is an atomic formula
2. If R is an n-ary relation symbol, and t₁,...,t_n are terms, then R(t₁,...,t_n) is an atomic formula

Finally, the set of formulas is defined to be the smallest set containing the set of atomic formulas such that the following holds:

1. Failed to parse (Missing texvc executable; please see math/README to configure.): \neg\phi

is a formula when Failed to parse (Missing texvc executable; please see math/README to configure.): \ \phi
is a formula

1. Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \land \psi)

and Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \lor \psi)
are formulas when Failed to parse (Missing texvc executable; please see math/README to configure.): \ \phi
and Failed to parse (Missing texvc executable; please see math/README to configure.): \ \psi
are formulas;

1. Failed to parse (Missing texvc executable; please see math/README to configure.): \exists x\, \phi

is a formula when Failed to parse (Missing texvc executable; please see math/README to configure.): \ x
is a variable and Failed to parse (Missing texvc executable; please see math/README to configure.): \ \phi
is a formula;

1. Failed to parse (Missing texvc executable; please see math/README to configure.): \forall x\, \phi

is a formula when Failed to parse (Missing texvc executable; please see math/README to configure.): \ x
is a variable and Failed to parse (Missing texvc executable; please see math/README to configure.): \ \phi
is a formula (alternatively, Failed to parse (Missing texvc executable; please see math/README to configure.): \forall x\, \phi
could be defined as an abbreviation for Failed to parse (Missing texvc executable; please see math/README to configure.): \neg\exists x\, \neg\phi

).

If a formula has no occurrences of Failed to parse (Missing texvc executable; please see math/README to configure.): \exists x

or Failed to parse (Missing texvc executable; please see math/README to configure.): \forall x

, for any variable Failed to parse (Missing texvc executable; please see math/README to configure.): \ x , then it is called quantifier-free. An existential formula is a formula starting with a sequence of existential quantification followed by a quantifier-free formula.

Atomic and open formulas[link]

An atomic formula is a formula that contains no logical connectives nor quantifiers, or equivalently a formula that has no strict subformulas. The precise form of atomic formulas depends on the formal system under consideration; for propositional logic, for example, the atomic formulas are the propositional variables. For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a term.

According to some terminology, an open formula is formed by combining atomic formulas using only logical connectives, to the exclusion of quantifiers.^[3] This has not to be confused with a formula which is not closed.

Closed formulas[link]

A closed formula, also ground formula or sentence, is a formula in which there are no free occurrences of any variable. If A is a formula of a first-order language in which the variables v₁, ..., v_n have free occurrences, then A preceded by Failed to parse (Missing texvc executable; please see math/README to configure.): \forall v₁ ... Failed to parse (Missing texvc executable; please see math/README to configure.): \forall v_n is a closure of A.

Properties applicable to formulas[link]

• A formula A in a language Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{Q}

is valid if it is true for every interpretation of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{Q}

.

• A formula A in a language Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{Q}

is satisfiable if it is true for some interpretation of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{Q}

.

• A formula A of the language of arithmetic is decidable if it represents a decidable set, i.e. if there is an effective method which, given a substitution of the free variables of A, says that either the resulting instance of A is provable or its negation is.

Usage of the terminology[link]

In earlier works on mathematical logic (e.g. by Church^[4]), formulas referred to any strings of symbols and among these strings, well-formed formulas were the strings that followed the formation rules of (correct) formulas.

Several authors simply say formula.^[5][6][7][8] Modern usages (especially in the context of computer science with mathematical software such as model checkers, automated theorem provers, interactive theorem provers) tend to retain of the notion of formula only the algebraic concept and to leave the question of well-formedness, i.e. of the concrete string representation of formulas (using this or that symbol for connectives and quantifiers, using this or that parenthesizing convention, using Polish or infix notation, etc.) as a mere notational problem.

However, the expression well-formed formulas can still be found in various works,^[9][10][11] these authors using the name well-formed formula without necessarily opposing it to the old sense of formula as arbitrary string of symbols so that it is no longer common in mathematical logic to refer to arbitrary strings of symbols in the old sense of formulas.

The expression "well-formed formulas" (WFF) also pervaded in popular culture. Indeed, WFF is part of an esoteric pun used in the name of "WFF 'N PROOF: The Game of Modern Logic," by Layman Allen,^[12] developed while he was at Yale Law School (he was later a professor at the University of Michigan). The suite of games is designed to teach the principles of symbolic logic to children (in Polish notation).^[13] Its name is an echo of whiffenpoof, a nonsense word used as a cheer at Yale University made popular in The Whiffenpoof Song and The Whiffenpoofs.^[14]

See also[link]

Logic portal

• Ground expression

Notes[link]

References[link]

• Allen, Layman E. (1965), "Toward Autotelic Learning of Mathematical Logic by the WFF 'N PROOF Games", Mathematical Learning: Report of a Conference Sponsored by the Committee on Intellective Processes Research of the Social Science Research Council, Monographs of the Society for Research in Child Development 30 (1): 29–41 
• Boolos, George; Burgess, John; Jeffrey, Richard (2002), Computability and Logic (4th ed.), Cambridge University Press, ISBN 978-0-521-00758-0 (pb.) 
• Ehrenberg, Rachel (Spring 2002). "He's Positively Logical". Michigan Today (University of Michigan). http://www.umich.edu/~newsinfo/MT/02/Spr02/mt9s02.html. Retrieved 2007-08-19. 
• Enderton, Herbert (2001), A mathematical introduction to logic (2nd ed.), Boston, MA: Academic Press, ISBN 978-0-12-238452-3 
• Gamut, L.T.F. (1990), Logic, Language, and Meaning, Volume 1: Introduction to Logic, University Of Chicago Press, ISBN 0-226-28085-3 
• Goble, Lou, ed. (2001), The Blackwell Guide to Philosophical Logic, Blackwell, ISBN 978-0-631-20692-7 
• Hofstadter, Douglas (1980), Gödel, Escher, Bach: An Eternal Golden Braid, Penguin Books, ISBN 978-0-14-005579-5 
• Kleene, Stephen Cole (2002) [1967], Mathematical logic, New York: Dover Publications, ISBN 978-0-486-42533-7, MR 1950307 
• Rautenberg, Wolfgang (2010), A Concise Introduction to Mathematical Logic (3rd ed.), New York: Springer Science+Business Media, DOI:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6, http://www.springerlink.com/content/978-1-4419-1220-6/ 

External links[link]

• Well-Formed Formula for First Order Predicate Logic - includes a short Java quiz.
• Well-Formed Formula at ProvenMath
• WFF N PROOF game site

Lucas di Grassi

Di Grassi at the 2010 Malaysian Grand Prix
Born	(1984-08-11) 11 August 1984 (age 27) São Paulo, Brazil
Formula One World Championship career
Nationality	Brazilian
Active years	2010
Teams	Virgin
Races	19 (18 starts)
Championships	0
Wins	0
Podiums	0
Career points	0
Pole positions	0
Fastest laps	0
First race	2010 Bahrain Grand Prix
Last race	2010 Abu Dhabi Grand Prix
2010 position	24th (0 points)

Lucas Tucci di Grassi (born 11 August 1984 in São Paulo) is a Brazilian racing driver who drove for Virgin Racing in the 2010 Formula One season. He is currently Pirelli's official test driver for the 2011 Formula One season and the 2012 Formula One season.^[1]

Career[link]

Early career[link]

Di Grassi's career in karting peaked in 2000, with a 5th place in the Formula A World Championship. Di Grassi began his formula racing career in Brazilian Formula Renault in 2002, finishing as runner-up to champion Sérgio Jimenez.

Formula Three (2003–06)[link]

In 2003 Di Grassi stepped up to Formula Three, driving a Dallara F301 Mugen-Honda for Avallone Motorsport in Formula Three Sudamericana. He finished as the championship runner-up behind Danilo Dirani, with one win and eleven other podium finishes. This came despite missing the last six races after suffering a major crash at the Curitiba round.^[2]

He also travelled to Europe to make four race starts in the F3 Euroseries with Prema Powerteam, achieving a best finish of fourth place.^[3] The following year, he made a permanent move to Europe and took a drive with Hitech Racing in the British F3 Championship. His season peaked with two wins, and he was classified in eighth place overall.^[4] Another of that year's highlights was a podium finish on his début at the Macau Grand Prix with Hitech.

In 2005, Di Grassi took another step up in a full-time return to the F3 Euroseries with Manor Motorsport. Against the dominant ASM pairing of Lewis Hamilton and Adrian Sutil he could only manage one win but ended the year third in the championship.^[2]

At the end of the season, Di Grassi returned to Macau, this time with Manor, and won the event from third on the grid,^[5] although runaway F3 Euroseries champion Hamilton missed the race.^[2]

GP2 Series (2006–09)[link]

Di Grassi driving for Campos Grand Prix at the Silverstone round of the 2008 GP2 Series season

2006 brought Di Grassi to the next stage in his career: promotion to the GP2 Grand Prix support series with Team Durango. It was a modest debut season, however, with only 8 points and 17th place in the championship standings.^[6]

For 2007 he joined reigning champions ART Grand Prix. He scored points consistently throughout the season, failing to score only once in the first 13 races. Despite not winning a race in that time, it put him in contention for the championship along with iSport's Timo Glock.^[2]

He scored his first win of the year in the 14th round of the championship at Istanbul, and took the lead of the championship,^[7] but Glock moved back ahead of him when he won the sprint race at the same event,^[8] and went on to win the title.

Di Grassi worked on testing the new-model GP2 car (to be used between 2008 and 2010), while testing for Renault F1. However, he resumed his GP2 career in 2008 by securing a drive at Campos Racing from round 4 onwards, replacing Ben Hanley.^[9] With three second places and one fourth place finish, he was the highest-scoring driver over the first two race meetings in which he took part. Two wins followed and he briefly looked set for a surprising championship challenge, before a final lap collision with Giorgio Pantano (who was disqualified for the incident) at Spa effectively ended his hopes. He finished an eventual third, ten points behind Pantano despite six fewer races.

He returned to the series for 2009 with Racing Engineering,^[10] and again finished third, finishing with the same number of points as he did in 2008.

Formula One[link]

Having previously been a test driver for Renault in 2005, di Grassi tested for the Honda Racing F1 Team at the end of 2008, alongside fellow Brazilian and GP2 rival Bruno Senna. Honda race driver Jenson Button tested a Honda at Circuit de Catalunya.

Renault (2008–2009)[link]

Di Grassi was then sent a contract by Renault to join them as a test driver, becoming the third GP2 runner-up to join Renault as a test driver after fellow Brazilian Nelson Piquet, Jr. and Heikki Kovalainen. However, di Grassi was not retained by the team for 2010.

Virgin (2010)[link]

Di Grassi took his and the Virgin team's first race finish at the 2010 Malaysian Grand Prix.

Di Grassi joined Timo Glock at new team Virgin Racing for 2010. The team, originally called Manor, was rebranded when Virgin Group owner Richard Branson purchased a 20% stake.^[11] Di Grassi finished in 24th position in the championship, with a best race finish of 14th at the 2010 Malaysian Grand Prix.

In Japan, he crashed on the way to the grid before the race had started.^[12] In December, di Grassi won the Desafio Internacional das Estrelas, a karting event organised by Felipe Massa.

On 21 December 2010, di Grassi was left without a drive, after Virgin announced Jérôme d'Ambrosio to partner Glock for the 2011 season.^[13]

Tyre Testing[link]

Pirelli (2011-)[link]

On 6 July 2011, di Grassi was appointed Pirelli's official tester for their development of Formula One tyres for the 2011 season,^[1] and will drive the Italian company's Toyota TF109 test car in five test sessions in order to develop the next generation of tyres, as well as attending several race weekends where he will collect information about tyre performance and attend technical briefings.^[1] He will remain with Pirelli for the 2012 Formula One season alongside Jaime Alguersuari to help develop tyres for 2013 and beyond using a Renault R30 chassis and it will be upgraded to the 2012 requirements for Alguersuari and Di Grassi to run the car across four development tests during the course of the season at Jerez, Spa, Monza and Barcelona to help Pirelli improve its selection of tyres.

Racing record[link]

Career summary[link]

Complete Formula Three Euroseries results[link]

(key) (Races in bold indicate pole position) (Races in italics indicate fastest lap)

Complete GP2 Series results[link]

(key) (Races in bold indicate pole position) (Races in italics indicate fastest lap)

Complete Formula One results[link]

(key) (Races in bold indicate pole position) (Races in italics indicate fastest lap)

References[link]

External links[link]

Wikimedia Commons has media related to: Lucas di Grassi

• Lucas di Grassi official website
• @lucasdigrassi on Twitter
• Lucas Di Grassi career statistics at Driver Database