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In logic, WFF (pronounced "wiff") is an abbreviation for well-formed formula. Given a formal grammar, a WFF is any string that is generated by that grammar.

However, some logicians (Frederic B. Fitch, Alan R Anderson, and Nuel D. Belnap, Jr.) and lawyer-logician Layman E. Allen pronounce it "woof" (as in "wood").

For example, in propositional logic the sequence of symbols ${\displaystyle ((\alpha \rightarrow \beta )\rightarrow (\neg \beta \rightarrow \neg \alpha ))}$ is a WFF because it is grammatically correct. The sequence of symbols ${\displaystyle ((\alpha \rightarrow \beta )\rightarrow (\beta \beta ))\alpha ))}$ is not a WFF, because it does not conform to the grammar of propositional calculus.

In formal logic, proofs are sequences of WFFs with certain properties, and the final WFF in the sequence is what is proven.

In some formalizations of formal logic proofs are sequences of WFFs with certain properties, and the final WFF in the sequence is what is proven. However, in the powerful Frederic B. Fitch sub-proof formalizations of formal logic, proofs are sequences of items, where items are sequences of sub-proofs or WFFs, and the last item in the sequence is a WFF of what is proven. (See Frederic B. Fitch, SYMBOLIC LOGIC - AN INTRODUCTION, Ronald Press, New York, 1952, and Layman E. Allen, WFF 'N PROOF: The Game of Modern Logic, Ann Arbor, Michigan, 1968). Laymanal 00:42, 10 April 2007 (UTC)laymanal

Trivia

WFF is the basis for an esoteric pun used in the name of a product: "WFF 'n Proof: The Game of Modern Logic," by Layman Allen, a professor at the University of Michigan. The board game is designed to teach the principles of symbolic logic to children (in Polish notation), and its name is a pun on whiffenpoof, a nonsense word used as a cheer at Yale University made popular in The Whiffenpoof Song.

The above Trivia is fairly accurate, but only fairly so. But even trivia deserves a respectable correspondence to reality, even if involves only trivia about trivia. First, the trade-marked name of the game is "WFF 'N PROOF", all upper-case. Second, "WFF" alone was never the sole basis of the pun; its basis was the pair of names of two series of games about logic. Besides being the standard abbreviation in logic for "well-formed formula", "WFF" was the name adopted for a series of games dealing with the concept of "WFFhood" (or should that be "WFFship", or even "WFFness"?) that were created during the period when WFF 'N PROOF itself was evolving. There was an accompanying series of games created about the same time called the "PROOF" games that dealt with the concept of proof. I remember well what I believe to be the first utterance ever on planet earth of the term "WFF 'N PROOF". It occurred on a warm, sun-shiny afternoon in the Spring of 1960 on the back-yard lawn of a graduate law student named, "Charles Padden" when the two of us were at Yale Law School. Charles had just popped a can of cold beer for my benefit, and handed it to me with the question, "So, how are the games coming along?" My response was, "Well, the WFF and PROOF games are ... Oh, shit! we've given them the wrong names. They should be combined and called "WFF 'N PROOF".

That was the birth of the name of the first series of 24 games that dealt with 13 ideas: the concepts of WFF, PROOF, and 11 rule-of-inference schemas that formulate the basis of standard two-valued propositional logic in Jan Lukasciewicz notation using the powerful subordinate-proof technique of Frederic B. Fitch. So, clearly, it was the full name "WFF 'N PROOF" that was the intended pun with respect to the famous Yale singing group, the Whiffenpoofs. However, the names did lead to some confusion during the period when we both had mail-boxes at the Yale Station postal facility. The Whiffenpoofs and WFF 'N PROOF somewhat frequently received each other's mail. I, who am tone deaf, remember well one delightful invitation to come and sing in Denver. - Layman Allen

Laymanal 00:42, 10 April 2007 (UTC)laymanal

18:32, 9 April 2007 (UTC)18:32, 9 April 2007 (UTC)18:32, 9 April 2007 (UTC)18:32, 9 April 2007 (UTC)18:32, 9 April 2007 (UTC)18:32, 9 April 2007 (UTC)~~

See also

• Formula (mathematical logic)

External links

• [http://www.cs.odu.edu/~toida/nerzic/content/logic/pred_logic/construction/wff_intro.html

Well-Formed Formula for First Order Predicate Logic] - includes a short Java quiz.

• Well-Formed Formula at ProvenMath

• Logic