that the lemma holds for any
wff
A
b
in which
b
occurs. And if
b
does not occur in
A
b
,
we know that the lemma holds for
A
b,
which
then falls under the special case. This completes the proof of the
substitution lemma.
It
follows as a corollary of the substitution
lemma that substitution preserves validity (details are left to the
reader).
28.1.
Independence of Axioms and Rules
Other things being equal, elegance is preferable to inelegance. Ac
cordingly, in logic and mathematics some emphasis is put on the
creation of elegant primitive bases for axiomatic systems. One
feature that contributes to the elegance of a primitive basis is the
independence of its axioms and rules of inference. An axiom or rule
of inference of a system L is
dependent in
L if the systemthat results
by dropping the given axiom or rule from the primitive basis of L
has the same set of theorems (actual output) that L has. So, if an
axiom of a logistic system L is independent in L, then its omission
from the primitive basis would diminish the set of theorems of L.
The same holds for an independent rule of inference.
We will show that all the axioms and rules of system P are in
dependent (in system P). Hence none of the axioms or rules of
system P could be dropped from its primitive basis except on pain
of denying theoremhood to some of the theorems (the valid wffs)
of system P. First, the rules of inference.
If
the rule of substitution
were dropped from the primitive basis of system P, no theorem of
the resulting system would be longer than its longest axiom, axiom
2. The reason is that the conclusion of an application of
modus
ponens
is always shorter than the longer premiss. But there are
theorems of system P longer than axiom 2. So it follows from the
definition of dependence that substitution is an independent rule
in system P.
Just as substitution is needed in system P to obtain theorems
longer than the longest axiom, so
modus ponens
is needed to obtain
theorems shorter than the shortest axiom. (Full details are left to
the reader.) There are theorems of system P shorter than its short
est axiom, for example,
'[p
:)
p]',
so it follows that
modus ponens
is an independent rule in system P.
To prove that an axiom of a logistic system L is independent
1
Any three objects would have done just as well as the numbers 0, 1, and 2.
135
28.
Metatheory of System P (I)
A
",A
A
B
A:)B
0
1
0
0
0
1
1
0
1
2
2
1
0
2
2
1
0
2
1
1
2
1
2
0
2
0
0
2
1
0
2
2
0
We call 0 the
designated
truth value, and we refer to 1 and 2 as
undesignated
truth values. By a
valid wff
we now understand a wff
that has a designated truth value under every assignment of values
to its variables. For example, as the tables below show, axiom 2
and axiom 3 are valid in the sense just defined.
(in L), it suffices to find some property which the given axiom lacks
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 Spring '07
 FITELSON
 Logic, Axiom, primitive basis

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