This web page provides answers, or hints and clues, to exercises
from Chapter 3 of
The Mathematics of Logic and also some additional
information on some of the examples.
By induction on the length of a derivation, the iduction hypothesis being that is is derived from in steps or fewer then contains a .
For the Given rule, if by Given then so contains a zero, as all elements of do.
For the Shortening rule, if where the last step is Shortening then and by shorter derivations, so by the induction hypothesis both and contain zeros, in particular and hence does.
For the Lengthening rule, if where the last step is Lengthening then where or and in fewer steps. So by induction, contains hence does too.
If then by 3.14 would contain , but is the empty string so contains no characters at all. So .
Let be the statement
Suppose is a string of 1's.
is true as a string of 1's is and by the Given rule.
Suppose and holds. Write for a string of 1's. Then since we may write down: (Given); (Given, in ); then (Shortening); and follow this with the proof from of given by . Thus holds and so holds for all by induction.
Suppose contains . Look at the first in it. It follows that for some (possibly zero) and some .
where each is or we
have the derivation:
(Given, in );
Let . Let be the set . We claim that and is maximal.
For the first, each contains so and also for we have by 3.17, hence any proof of the form can be rewritten to make it a proof of . By we know by 3.15 so , hence .
If then it doesn't contain any hence for some (possibly zero). Hence since by 3.16 and . Thus , as required to show maximality.
It remains to show that is the only maximal element of . Suppose is maximal, so and . Then if we must have for so would imply that by replacing each instance of by Given in the proof by the proof of from . But hence so by maximality .
Now as each string containing is provable in (3.17) we have . By maximailty of it follows that as required.
Let . This is a path, as proved in the proof of the Completeness theorem, using the fact that is maximal. But is the set of containing a zero, so and this is indeed a path as you can check.
Let and define a path . Then so and is an infinite path containing so hence by the Soundness Theorem .
is consistent iff there is no proof of from obeying the rules. (This is the official definition.)
is consistent iff there is no proof of from a finite subset of obeying the rules. (This is from the official definition, observing all proofs are finie.)
is consistent iff there is a a partial order on making true. (Left to right is the completeness theorem; right to left is Soundness.)
Since by RAA iff
and are negations (or
of each other) it suffices (and is slightly easier) to
give an algorithm for when .
Suggestion: Separate the set of positive sentences in and the set of negative sentences in . (We may assume for if it is the answer is easy by Given.) Now define the transitive closure of to be the set of all sentences provable from by the Transitivity rule and the Given rule. Since is finite, the transitive closure of is also finite. (Why?) Explain why the transitive closure of can be computed from . The algorithm goes by computing the transitive closure of and checking none is of the form nor for some . If these checks work then and if not then . (Why?)
The easiest semantics is holds iff . The rest involves proving Soundness and Completeness Theorems.
Hint: Show by induction that the number of s is any derivable string cannot be divisible by 3. In particular cannot be derived. Also that in any derivable string there always exactly one at the beginning. Thus cannot be derived. The rest are either derivable or are not derivable by similar arguments.
No, I don't accept that. I think you're quibbling here
about semantics quite frankly.
I think you're the one who's quibbling with semantics
Alistair (Darling) was talking about having
reduced unemployment over a million over the last three and a half
years, that is actually a slower rate than over the last three and a
half years ours of our government. What we're talking about here, is
not the semantics, it is about what actual people realise is happening
in their own lives.