The Tableau
Rules
(Note that it is not necessary to put in the vertical line
when drawing a tableau.)
The
Propositional Rules
The
Predicate Rules
where there
is a designator D which has already occurred in the branch to which y is added, and y is the result of replacing each free occurrence
of "x" in j by an occurrence of D. (i.e. the rule is
that we can add "y" to any open branch in which ""xj" occurs where this condition is satisfied.)
where there
is a proper name D which has not occurred anywhere in the branch to which
y is added, and y is the result of replacing each free occurrence
of "x" in j by an occurrence of D.
Identity
rules
where the
designator D occurs in j, and y is the result of replacing one or more occurrence
of D in j by E.
Rules
for closing a branch
Any branch
containing both j and ¬j closes.
Any branch
containing ¬D=D, where D is a designator, closes.
Which
rules can I use?
With the
exception of the identity rules, there is at most one rule for extending
a branch which can be applied to any formula.
If the formula
begins with "¬¬" the only rule that can be used is the
¬¬j rule.
If the formula
begins with "[", the only rule that can be used is the rule
for the truth functor to which this "[" belongs.
If the formula
begins with "¬[", the only rule that can be used is the
rule for the negated truth functor to which this "[" belongs.
If the formula
begins with """,
the only rule that can be used is the "xj rule.
If the formula
begins with "¬"",
the only rule that can be used is the ¬"xj rule.
If the formula
begins with "$", the only rule that can be used is
the $xj rule.
If the formula
begins with "¬$", the only rule that can be used is
the ¬$xj rule
Which
rules are worth applying more than once?
The only
rules which are worth applying more than once to the same formula in the
same branch are the "xj rule and the identity rules. Sometimes these
have to be applied more than once (to the same formula in the same branch).
Though, of course, even these rules are worth reapplying only if they
result in the addition of something new to the branch.
Notice, in
particular, that there is never any point in applying the $xj rule twice to the same formula in the same
branch  even though applying it a second time is, of course, bound to
add something new to the branch.
Ticking
Ticking is optional if you are doing a predicate tableau, but you may
find that, used with caution, it helps you to keep track. You should,
however, tick a formula only if you have applied a rule which is not worth
applying a second time to the same formula in the same branch, and have
applied it in every open branch in which the formula occurs. So:
·
do not tick a formula to which you have applied the "xj rule;
· do not tick a formula to which you have applied an identity rule;
· remember that you may want to use a ticked formula again in conjunction
with one of the identity rules. (Though this is never actually
necessary.)
