Next: Proof syntax
Up: Natural Deduction Language
Previous: Terms
  Contents
Formulas
There are three types of formulas in NDL: atomic formulas
(or simply atoms); sentential formulas; and quantified
formulas. We explain each in turn:
- An atom is either the constant true; or the constant false;
or else it is of the form
(
,
,
), where
is a relation symbol
of arity
and
are terms. An example of an atom is
between(x,y,z). Infix notation is allowed in the case of binary
relation symbols, for instance
foo = 2 |
x * 3 < 15 |
father(joe) loves mary |
- A sentential formula is either:
- a negation for any formula
;
- a conjunction for any formulas
;
- a disjunction for any formulas
;
- a conditional
==>
for any formulas
; or
- a biconditional
<==>
for any formulas
.
The negation sign ~ binds tighter than all other four connectives.
Conjunctions and
disjunctions have equal precedence, bind tigher than conditionals
and biconditionals, and associate to the right. Conditionals and
biconditionals have equal precedence and associate to the right.
Accordingly,
is parsed as
To override these conventions, use parentheses or square brackets.
- A quantified formula is either of the form
(forall
) (a universal quantification);
or (exists
) (a existential quantification)
for any variable
and formula
. Brackets can
also be used for grouping, so we can write
[exists
] instead of (exists
).
We will use the letters
and
as typical formulas.
We assume the reader is familiar with notions such as free and bound
variable occurrences. These are defined in the standard way.
Two formulas are said to be alphabetically equivalent iff
each can be obtained from the other by consistently renaming
bound variables. Alphabetically equivalent formulas will be
viewed as identical.
We write
to denote the formula obtained from
by replacing every occurrence of
in it by the term
,
renaming bound variables if necessary in order to avoid
variable capture.
Next: Proof syntax
Up: Natural Deduction Language
Previous: Terms
  Contents
2004-08-06