VARIABLES.
a is a variable. If we're not being austere, so are b, e, d and e.
A variable followed by a prime is also a variable.
Examples: a b' e" d"' e'"'TERMS.
All numerals and variables are terms.
A term preceded by S is also a term.
If sand t are terms, then so are (s+ t) and (s' t).
Examples: 0 b SSa' (SO'(SSO+e)) S(Sa·(Sb·Se))
TERMS may be divided into two categories:
(1) DEFINITE terms. These contain no variables.
Examples: 0 (SO+SO) SS((SSO'SSO)+(SO'SO))
(2) INDEFINITE terms. These contain variables.
Examples: b Sa (b+SO) (((SO+SO)+SO)+e)
The above rules tell how to make parts of well-formed formulas; the
remaining rules tell how to make complete well-formed formulas.ATOMS.
If sand t are terms, then s = t is an atom.
Examples: SO=O (SSO+SSO)=SSSSO S(b+e)=((e·d)·e)
If an atom contains a variable u, then u is free in it. Thus there are
four free variables in the last example.
NEGATIONS.
A well-formed formula preceded by a tilde is well-formed.
Examples: -SO=O -3b:(b+b)=SO -<O=O~SO=O> -b=SO
The quantification status of a variable (which says whether the variable is
free or quantified) does not change under negation.
COMPOUNDS.
If x and yare well-formed formulas, and provided that no variable
which is free in one is quantified in the other, then the following
are all well-formed formulas:
<xl\y>, <xvy>, <x~y>.
Examples: <0=01\-0=0> <b=bv-3e:e=b>
<SO=0~Ve:-3b:(b+b)=e>
The quantification status of a variable doesn't change here.
QUANTIFICATIONS.
If u is a variable, and x is a well-formed formula in which u is free,
then the following strings are well-formed formulas:
3u: x and Vu: x.
Examples: Vb:<b=bv-3e:e=b> Ve:-3b:(b+b)=e -3e:Se=d
OPEN FORMULAS contain at least one free variable.
Examples: -e=e b=b <Vb:b=bl\-e=e>
CLOSED FORMULAS (SENTENCES) contain no free variables.
Examples: SO=O -Vd:d=O 3e:<Vb:b=bA-e=e>(^214) Typographical Number Theory