In mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, the rules of passage govern how quantifiers distribute over the basic logical connective

Logical connective

In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...

s of first-order logic

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

. The rules of passage govern the "passage" (translation) from any formula of first-order logic to the equivalent

Logical equivalence

In logic, statements p and q are logically equivalent if they have the same logical content.Syntactically, p and q are equivalent if each can be proved from the other...

 formula in prenex normal form

Prenex normal form

A formula of the predicate calculus is in prenex normal form if it is written as a string of quantifiers followed by a quantifier-free part .Every formula in classical logic is equivalent to a 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:

• 

• 

• 

• 

• 

The following conditional sentences can also be taken as rules of passage:

• 
• 
• 

"Rules of passage" first appeared in French, in the writings of Jacques Herbrand

Jacques Herbrand

Jacques Herbrand was a French mathematician who was born in Paris, France and died in La Bérarde, Isère, France. Although he died at only 23 years of age, he was already considered one of "the greatest mathematicians of the younger generation" by his professors Helmut Hasse, and Richard Courant.He...

. Quine employed the English translation of the phrase in each edition of his Methods of Logic, starting in 1950.

External links

• Stanford Encyclopedia of Philosophy
  Stanford Encyclopedia of Philosophy
  The Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...
  
  : "Classical Logic -- by Stewart Shapiro.