Maarten Maartensz:    Philosophical Dictionary | Filosofisch Woordenboek                      

 L - Logic notation


Logic notation: The way of writing of logical terms.

Formal logic is an artificial language like mathematics and it is so because, taken all together, the formalities make the subject easier to understand and reason about. This is not to say that all formalities in logic or mathematics are justified or clarify their subject, for many do not, nor is it to to say that all formalisms or notations are well chosen, for many are not. (Frege's Begriffschrift or (reverse) Polish notation leap to mind, for those who are somewhat familiar with these.)

An additional difficulty with computers is that many of the fonts used as standard, even in common formats like html, do not come with adequate notations for mathematics or logic. There are generally three ways of resolving this: To try to make do with the possibilities there are with the font one standardly uses; to supplement this with the possibilities in some other standard fonts; or to use special pictures in gif, jpg or png formats (these are formats for storing images).

The solution I have chosen in html on my site is a compromise of these methods:

  • The standard font I use is Verdana, and this has more or less adequate support for most standard logical symbols.
  • On Windows there is a standard font called StarMath that has most standard symbols for mathematics and mathematical logic.
  • There are various free collections of gif or png renderings of mathematical and logical symbols available on the internet.

Here is a listing of the most common symbols I use at various places in the Philosophical Dictionary. If your computer's operating system does not have both the Verdana and the StarMath fonts available what follows will probably not be rendered correctly.

Notation for unordered and ordered collections

 ' ( [a1 .. an] ) ' for 'the ordered collection made up of a1 .. an' in that order'
 ' ( {a1 .. an} ) ' for 'the unorderd collection made up of a1 .. an'

These are fundamental, and indeed the first seems more so than the second, in as much as words are ordered sequences of letters, and statements ordered series of words, while unordered sequences arise from abstracting from order i.e. supposing that any ordering of all the elements of an unordered collection amounts to the same as any other.

In any case, I use square brackets to indicate that what is between them is an ordered collection or sequence, and curly brackets to indicate that what is between them may be arbitrarily re-ordered and still count for the same.

Also, the dots in ' a1 .. an' are used to indicate that this abbreviates a sequence of similar terms, as ' a1 .. a5 ' abbreviates ' a1 a2 a3 a4 a5 '. 

Notation for theorems and consequences

 ' (|- P ) '     for ' P is a theorem '
 ' ( P |- Q ) ' for ' Q is a consequence of P '

These are basic notations which may be extended in various ways like so:

 ' (|-S P ) ' for ' P is a theorem in system S '

while it should also be mentioned that often the left of the consequence relation is taken as a set of statements, that may be indicated thus:

 ' ( {P1 .. Pn} |- Q ) ' for 'Q is a consequence of the set P1 .. Pn'

where the curly brackets indicate that this is so for any re-ordering of P1 .. Pn

A useful distinction for theorems and consequences is to use |- for what is provably a theorem or consequence, in terms of its rules for proofs, and |=   for semantically a theorem or consequence, in terms of its interpretation or model.

The latter requires more, and often also shows more, than the former, which is essentially a mattter of systematic substitutions in formulas.

Notation for propositional logic

 ' (P) '       for ' not P '
 ' (P V Q) '   for ' P or Q'
 ' (P & Q) '   for ' P and Q'
 ' (P => Q) ' for ' P implies Q '
 ' (P iff Q) '  for ' P if and only if Q '

There are many alternatives, notably for 'not' which may also be indicated by

 ' (~P) '       for ' not P '
 ' (-P) '        for ' not P '

and for implication that may also be indicated by

 ' (P --> Q) ' for ' P implies Q '

Incidentally, the reading of this connective, that in standard propositional logic is equivalent with both ' => ' and ' --> ' is also a bit problematic, in that a common rendering is 'if P then Q'. I prefer 'implies' because it is one word, and also the logical implies has some properties the natural language 'if ... then ---' contexts do not always have. (See: Paradoxes of Implication).

There also are alternatives for iff that I will not remark upon since I hold that 'iff' is usual and adequate for ' (P => Q) & (Q => P) '  both as notation and also in common written English.

Notation for quantifiers and predicate logic

 ' (T) '   for  ' for all T '
 ' (ET) ' for  ' there is a T '

These are the basic quantifiers, which in standard logic are definable in each other's terms like so:

 ' ~(T)~ '   for ' some T' i.e. ' not for all T not '
 ' ~(ET)~ ' for ' all T'     i.e. ' not for some T not '  

There are many alternative readings for the standard quantiers like

' for all T ' or ' for each T ' or ' for every T ' or ' for any T '
' there is a T ' or ' for some T ' or ' there exists a T '

some of which may have a somewhat different import that I shall not discuss here.

A notation used with 'for all' that is useful should be mentioned, since it abbreviates a common usage of universally quantified statements:

' (x>y)(A[x,y]) ' for ' (x)(y)(x>y => A[x,y]) '

Thus, since often universal quantifiers are used in front of conditional statements, the condition may be included in the universal quantifier.The above gives an example, and the reader may imagine others.

 ' ( x=y ) '   for ' x is the same as y' or 'x is equal to y '
 ' ( x ≈ y ) ' for ' x is approximately the same as y '
 ' ( x ≤ y ) ' for ' x is smaller than or equal to y '
 ' ( x ≥ y ) ' for ' x is greater than or equal to y '

These come from or also occur in basic mathematics and may be extended in various ways such as notably

 ' ( x ≈d y ) ' for 'x differs no more than d from y '

which is useful in many contexts and for many purposes, like differential and integral calculus.

 ' ( A[x1 .. xn] ) '             for ' A is true of  x1 .. xn '
 ' (|x1 .. xn : (A[x1 .. xn])' for ' the x1 .. xn such that A '
 ' ( f(x1 .. xn)=y ) '           for ' y equals the f of x1 .. xn '

The first of these is basic notation for predicate logic, where a predicate is taken as that part of a statement that is not a variable. The second is notation for abstraction, and it should be noted that abstraction turns a formula into a term. The third is standard notation for functions.

Notation for set theory

In set theory everything is a element, a set or a class, with the elements for individual things and the other two as collections of things or collections. Here is some basic notation for it:

 ' (b i A) '   for ' b is an element of A '
 ' (A = ) '  for ' A equals the void set '
 ' (A = V) '  for ' A equals the universal set '
 ' (A = ∞) ' for ' A equals an infinite set or class '
 ' (A = B ) ' for ' A is the same set as B '
 ' (A ≠ B ) ' for ' A is a set different from B '
 ' (A C B ) ' for ' A is included in B '

These are mostly defined terms in set theory, in which the above are statements and the basic axiom for '=' is

A = B IFF (x)(x i A IFF x i B)

Sets are the same if and only if they have the same elements.

A common alternative for infinity is ' l ' (aleph). The lowest infinite number, the number of elements of the set of natural numbers is often styled ' lo ' (aleph zero).

The void set has no elements; the universal set (or class) contains everything considered in the domain, whether individual or collection; examples of infinite sets are the set of natural numbers and the set of real numbers; two sets are the same in set-theory if the contain precisely the same elements and else different; and a set is included in another set if all of the elements of the first are also element of the second.

It may be noteworthy that sets may be elements as well, as shown by '{, {}}' which is a set with as elements the void set and the non-void set which just contains the void set. In set theory this term is often identified with ' 2 ' on the ground that ' 0 ' is identified with ' ' and ' 1 ' with  ' {} ' and one then can start iterating without end on the pattern just indicated, with ' 3 ' for ' {, {, {}}} ' a.s.o.

Also there are these:

 ' (A U B ) ' for ' the union of A and B '
 ' (A . B ) ' for ' the intersection of A and B '
 ' (V - A) '   for ' the complement of A'

These expressions are not statements in set theory but sets, namely respectively for the elements in either A or B and the elements in both A and B and the elements in the universal set that are not in A. A normal abbreviation is ' -A ' read as 'non-A'. Thus we have V = (A N -A ) and ' = (A O -A ) = (A - A)'.

And also we have the definitiory statements for the above three sets:

' (x) (x i (A U B ) IFF x i A V x i B '
' (x) (x i (A . B ) IFF x i A & x i B )'
' (x) (x i ( V - A) ) IFF x i V & ~(x i A) )'



Also see: Basic Logic, Logical Terms


 Original: Mar 29, 2005                                                Last edited: 12 December 2011.   Top