Notations, logical
Notations, logical
There follows a list of some of the logical symbols and notations found in contemporary usage. In each case the notation employed in articles in this dictionary is given first, afterwards alternative notations, if any.
PROPOSITIONAL CALCULUS (see Logic, formal, 1, and strict implication)
pq, the conjunction of p and q, “p and q.” Instead of simple juxtaposition of the propositional symbols, a dot is sometimes written between, as pq. Or the common abbreviation for and may be employed as a logical symbol, p & q. Or an inverted letter ?, usually from a gothic font, may be used. In the Lukasiewicz notation for the propositional calculus, which avoids necessity for parentheses, the conjunction of p and q id Kpq.
p ? q, the inclusive disjunction of p and q, “p or q.” Frequently the letter ? is from a gothic font. In the Lukasiewicz notation, Apq is employed.
~p, the negation of p, “not p.” Instead of ~, a dash – may be used, written either before the propositional symbol or above it. Heyting adds a short downward stroke at the right end of the dash (a notation which has come to be associated particularly with the intuitionistic propositional calculus and the intuitionistic concept of negation). Also employed is an accent after the propositional symbol (but this more usual as a notation for the complement of a class). In the Lukasiewicz notation, the negation of p is Np.
p ? q, the material implication of q by p, “if p then q.” Also employed is a horizontal arrow, p ? q. The Lukasiewicz notation is Cpq.
p = q, the material equivalence of p and q, “p if and only if q.” Another notation which has sometimes been employed is p ?? q. Other notations are a double horizontal arrow, with point at both ends, and two horizontal arrows, one above the other, one pointing forward and the other back. The Lukasiewicz notation is Epq.
p + q, the exclusive disjunction of p and q, “p or q but not both.” Also sometimes used is the sign of material equivalence = with a vertical or slanting line across it (non-equivalence). In connection with the Lukasiewicz notation, Rpq has been employed.
p|q, the alternative denial of p and q, “not both p and q.” — For the dual connective, joint denial (“neither p nor q”), a downward arrow has been used.
p 3 q, the strict implication of q by p, “p strictly implies q.”
p = q, the strict equivalence of p and q, “p strictly implies q and q strictly implies p.” Some recent writers employ, for strict equivalence, instead of Lewis’s =, a sign similar to the sign of material equivalence, =, but with four lines instead of three.
Mp, “p is possible.” This is Lukasiewicz’s notation and has been used especially in connection with his three-valued propositional calculus. For the different notion of possibility which is appropriate to the calculus of strict implication, Lewis employs a diamond.
CLASSES (see class, and logic, formal, 7, 9)
x?a, “x is a member of the class a,” or, “x is an a.” For the negation of this, sometimes a vertical line across the letter epsilon is employed, or a ~ above it.
a ? b, the inclusion of the class a in the class b, “a is a subclass of b.” This notation is usually employed in such a way that a ? b does not exclude the possibility that a = b. Sometimes, however, the usage is that a ? b (“a is a proper subclass of b”) does exclude that a = b; and in that case another notation is used when it is not meant that a = b is excluded, the sign = being either surcharged upon the sign ? or written below it (or a single horizontal line below the ? may take the place of =).
?!a, “the class a is not empty [has at least one member],” or, “a’s exist.”
?x, or ?x, or [x] — the unit class of x, i.e., the class whose single member is x.
V, the universal class. Where the algebra of classes is treated in isolation, the digit 1 is often used for the universal class.
?, the null or empty class. Where the algebra of classes is treated in isolation, the digit 0 is often used.
-a, the complement of a, or class of non-members of the class a. An alternative notation is a a’.
a ? b, the logical sum, or union, of the classes a and b. Alternative notation, a + b.
a n b, the logical product, or intersection, or common part, of the classes a and b. Alternative notation, ab.
RELATIONS (see Relation, and Logic, formal, 8; (where a notation used in connection with relations is here given as identical with a corresponding notation for classes, the relational notation will also often be found with a dot added to distinguish it from the one for classes)
xRy, “x has [or stands in, or bears] the relation R to y.”
R ? S, “the relation R is contained in [implies] the relation S.”
&esixt;!R, “the relation R is not null [holds in at least one instance].”
A downward arrow placed between (e.g.) x and y denotes the relation which holds between x and y (in that order) and in no other case.
V, the universal relation. Schrder uses 1.
?, the null relation. Schrder uses 0.
-R, the contrary, or negation, of the relation R. The dash may also be placed over the letter R (or other symbol denoting a relation) instead of before it.
R ? S, the logical sum of the relations R and S, “R or S.” Schrder uses R+S.
R n S, the logical product of the relations R and S, “R and S.” Schrder uses RS.
I, the relation of identity — so that xIy is the same as x = y. Schrder uses 1′.
J, the relation of diversity — so that xJy is the same as x ? y. Schrder uses 0′.
A breve ? is placed over the symbol for a relation to denote the converse relation. An alternative notation for the converse of R is CnvR.
R + S, the relative sum of R and S. Schrder adds a leftward hook at the bottom of the vertical line in the sign +.
R|S, the relative product of R and S. Schrder uses a semicolon to symbolize the relative product, but the vertical bar, or sometimes a slanted bar, is now the usual notation.
R2, the square of the relation R, i.e., R|R.
Similarly for higher powers of a relation, as R3, etc.
Ry, the (unique) x such that xRy, “the R of y.” Frequently the inverted comma is of a bold square (bold gothic) style.
Rb, the class of x’s which bear the relation R to at least one member of the class b, “the R’s of the b’s.” Then R?y, or R?y, is the class of x’s such that xRy, “the R’s of y.”
A forward pointing arrow is placed over (e. g.) R to denote the relation of R&slquo;?y to y. Similarly a backward pointing arrow placed over R denotes the relation of the class of y’s such that xRy to x.
An upward arrow placed between (e.g.) a and b denotes the relation which holds between x and y if and only if x?a and y?b.
The left half of an upward arrow placed between (e.g.) a and R denotes the relation which holds between x and y if and only if x?a and xRy, in other words, the relation R with its domain limited to the class a.
The right half of an upward arrow placed between (e.g.) R and b denotes the relation which holds between x and y if and only if xRy and y?b; in other words the relation R with its converse domain limited to b.
The right half of a double — upward and downward — arrow placed between (e.g.) R and a denotes the relation which holds between x and y if and only if xRy and both x and y are members of the class a; in other words, the relation R with its field limited to a.
DR, the domain of R.
(|R, the converse domain of R.
CR, the field of R.
Rpo, the proper ancestral of R — i.e., the relation which holds between x and y if and only if x bears the first or some higher power of the relation R to y (where the first power of R is R).
R* the ancestral of R — i.e., the relation which holds between x and y if and only if x bears the zero or some higher power of the relation R to y (where the zero power of R is taken to be, either I, or I with its field limited to the field of R).
QUANTIFIERS (see Quantifiers, and Logic, formal, 3, 6)
(x), universal quantification with respect to x — so that (x)M may be read “for every x, M.” An alternative notation occasionally met with, instead of (x), is (?x), usually with the inverted A from a gothic or other special font. Another notation is composed of a Greek capital pi with the x placed either after it, or before it, or as a subscript. — Negation of the universal quantifier is sometimes expressed by means of a dash, or horizontal line, over it.
(Ex), existential quantification with respect to x — so that (Ex)M may be read “there exists an x such that M.” The E which forms part of the notation may also be inverted; and, whether inverted or not, the E is frequently taken from a gothic or other special font. An alternative notation employs a Greek capital sigma with x either after it or as a subscript. — Negation of the existential quantifier is sometimes expressed by means of a dash over it.
?x, formal implication with respect to x. See definition in the article logic, formal, 3.
=x, formal equivalence with respect to x. See definition in logic, formal, 3.
?x. See definition in logic, formal, 3.
?xy or ?x,y — formal implication with respect to x and y. Similarly for formal implication with respect to three or more variables.
=xy, or =x,y — formal equivalence with respect to x and y. Similarly for formal equivalence with respect to three or more variables.
ABSTRACTION, DESCRIPTIONS (see articles of those titles)
?x, functional abstraction with respect to x — so that ?xM may be read “the (monadic) function whose value for the argument x is M.”
x, class abstraction with respect to x — so that x M may be read “the class of x’s such that M.” An alternative notation, instead of x , is x3.
xy, relation abstraction with respect to x and y — so that xyM may be read “the relation which holds between x and y if and only if M.”
(i x), description with respect to x — so that (i x)M may be read “the x such that M.”
E! is employed in connection with descriptions to denote existence, so that E!(ix)M may be read “there exists a unique x such that M.”
OTHER NOTATIONS
F(x), the result of application of the (monadic, propositional or other) function F to the argument x — the value of the function F for the argument x — ftF of x.” Sometimes the parentheses are omitted, so that the notation is Fx. — See the articles function, and propostttonal function.
F(x, y), the result of application of the (dyadic) function F to the arguments x and y. Similarly for larger numbers of arguments.
x = y, the identity or equality of x and y, “x equals y.” See logic, formal, 3, 6, 9.
x ? y, negation of x = y.
|- is the assertion sign. See assertion, logical.
Dots (frequently printed as bold, or bold square, dots) are used in the punctuation of logical formulas, to avoid or replace parentheses. There are varying conventions for this purpose.
? is used to express definitions, the definiendum being placed to the left and the definiens to the right. An alternative notation is the sign = (or, in connection with the propositional calculus, =) with the letters Df, or df, written above it, or as a subscript, or separately after the definiens.
Quotation marks, usually single quotes, are employed as a means of distinguishing the name of a symbol or formula from the symbol or formula itself (see syntax, logical). A symbol or formula between quotation marks is employed as a name of that particular symbol or formula. E.g., ‘p’ is a name of the sixteenth letter of the English alphabet in small italic type.
The reader will observe that this use of quotation marks has not been followed in the present article, and in fact that there are frequent inaccuracies from the point of view of strict preservation of the distinction between a symbol and its name. These inaccuracies are of too involved a character to be removed by merely supplying quotation marks at appropriate places. But it is thought that there is no point at which real doubt will arise as to the meaning intended. — Alonzo Church