Peano–Russell notation
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In
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
, Peano–Russell notation was
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...
's application of
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...
's logical notation to the logical notions of
Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...
and was used in the writing of ''
Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
'' in collaboration with
Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applica ...
:
"The notation adopted in the present work is based upon that of Peano, and the following explanations are to some extent modelled on those which he prefixes to his ''Formulario Mathematico''." (Chapter I: Preliminary Explanations of Ideas and Notations, page 4)


Variables

In the notation, variables are ambiguous in denotation, preserve a recognizable identity appearing in various places in logical statements within a given context, and have a range of possible determination between any two variables which is the same or different. When the possible determination is the same for both variables, then one implies the other; otherwise, the possible determination of one given to the other produces a meaningless phrase. The alphabetic symbol set for variables includes the lower and upper case Roman letters as well as many from the Greek alphabet.


Fundamental functions of propositions

The four fundamental functions are the ''contradictory function'', the ''logical sum'', the ''logical product'', and the ''implicative function''.


Contradictory function

The contradictory function applied to a proposition returns its negation. :\sim p


Logical sum

The logical sum applied to two propositions returns their disjunction. :p \lor q


Logical product

The logical product applied to two propositions returns the
truth-value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false''). Computing In some progra ...
of both propositions being simultaneously true. :p \cdot q


Implicative function

The implicative function applied to two ordered propositions returns the truth value of the first implying the second proposition. :p \supset q


More complex functions of propositions

''Equivalence'' is written as p \equiv q, standing for p \supset q \cdot q \supset p. ''Assertion'' is same as the making of a statement between two full stops. :\vdash p An asserted proposition is either true or an error on the part of the writer. ''Inference'' is equivalent to the rule ''modus ponens'', where p \cdot p \supset q . \supset q In addition to the logical product, ''dots'' are also used to show groupings of functions of propositions. In the above example, the dot before the final implication function symbol groups all of the previous functions on that line together as the antecedent to the final consequent. The notation includes ''definitions'' as complex functions of propositions, using the equals sign "=" to separate the defined term from its symbolic definition, ending with the letters "Df".Russell, p. 11


Notes


References

* Russell, Bertrand and Alfred North Whitehead (1910). ''Principia Mathematica'' Cambridge, England: The University Press.


External links

* {{DEFAULTSORT:Peano-Russell notation Proof theory