Rules of passage (logic)


Rules of passage (logic)

In mathematical logic, the rules of passage govern how quantifiers distribute over the basic logical connectives of first-order logic. The rules of passage govern the "passage" (translation) from any formula of first-order logic to the equivalent formula in prenex normal form, and vice versa.

The rules

See Quine (1982: 119, chpt. 23). Let "Q" and "Q" 'denote ∀ and ∃ or vice versa. β denotes a closed formula in which "x" does not appear. The rules of passage then include the following sentences, whose main connective is the biconditional:

* Qx [lnotalpha (x)] leftrightarrow lnot Q'x [alpha (x)] .

* Qx [eta or alpha (x)] leftrightarrow (eta or Qx alpha (x)).

*exist x [alpha (x) or gamma (x)] leftrightarrow (exist x alpha (x) or exist x gamma (x)).
* Qx [eta and alpha (x)] leftrightarrow (eta and Qx alpha (x)).

* forall x , [alpha(x) land gamma(x)] leftrightarrow (forall x , alpha(x) land forall x , gamma(x) ).

The following conditional sentences can also be taken as rules of passage:
*exist x [alpha (x) and gamma (x)] ightarrow (exist x alpha (x) and exist x gamma (x)).
*(forall x , alpha(x) or forall x , gamma(x)) ightarrow forall x , [alpha(x) or gamma(x)] .
*(exists x , alpha(x) and forall x , gamma(x)) ightarrow exists x , [alpha(x) and gamma(x)] .

"Rules of passage" first appeared in French, in the writings of Jacques Herbrand. Quine employed the English translation of the phrase in each edition of his "Methods of Logic", starting in 1950.

ee also

*First-order logic
*Prenex normal form
*Quantifier

References

*Willard Quine, 1982. "Methods of Logic", 4th ed. Harvard Univ. Press.
*Jean Van Heijenoort, 1967. "From Frege to Godel: A Source Book on Mathematical Logic". Harvard Univ. Press.

External links

*Stanford Encyclopedia of Philosophy: " [http://plato.stanford.edu/entries/logic-classical/ Classical Logic] -- by Stewart Shapiro.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • logic, history of — Introduction       the history of the discipline from its origins among the ancient Greeks to the present time. Origins of logic in the West Precursors of ancient logic       There was a medieval tradition according to which the Greek philosopher …   Universalium

  • Boolean algebra (logic) — For other uses, see Boolean algebra (disambiguation). Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of… …   Wikipedia

  • Aristotle’s logic and metaphysics — Alan Code PART 1: LOGICAL WORKS OVERVIEW OF ARISTOTLE’S LOGIC The Aristotelian logical works are referred to collectively using the Greek term ‘Organon’. This is a reflection of the idea that logic is a tool or instrument of, though not… …   History of philosophy

  • Dynamic logic (modal logic) — For the subject in digital electronics also known as clocked logic, see dynamic logic (digital electronics). Dynamic logic is an extension of modal logic originally intended for reasoning about computer programs and later applied to more general… …   Wikipedia

  • Modal logic — is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals words that express modalities qualify a statement. For example, the statement John is happy might be qualified by… …   Wikipedia

  • Predicate functor logic — In mathematical logic, predicate functor logic (PFL) is one of several ways to express first order logic (formerly known as predicate logic) by purely algebraic means, i.e., without quantified variables. PFL employs a small number of algebraic… …   Wikipedia

  • List of mathematics articles (R) — NOTOC R R. A. Fisher Lectureship Rabdology Rabin automaton Rabin signature algorithm Rabinovich Fabrikant equations Rabinowitsch trick Racah polynomials Racah W coefficient Racetrack (game) Racks and quandles Radar chart Rademacher complexity… …   Wikipedia

  • Ferdinand Canning Scott Schiller — Infobox Philosopher region = Western Philosophy era = 19th/20th century philosophy color = #B0C4DE image caption = Ferdinand Canning Scott Schiller name = F.C.S. Schiller birth = August 16 1864 death = August 9 1937 school tradition = Pragmatism… …   Wikipedia

  • List of important publications in mathematics — One of the oldest surviving fragments of Euclid s Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.[1] This is a list of important publications in mathematics, organized by field. Some… …   Wikipedia

  • William Delbert Gann — (June 6, 1878 ndash; June 14, 1955), also W. D. Gann, was a trader recognized not only for his trading abilities, but also for his financial market forecasts. The accuracy of his forecasts is still the subject of substantial debate. Gann s son… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.