Some Remarks and ExtraCredit Exercises Concerning the Deductive System of Hi˙z
Branden Fitelson
03/04/05
Hi˙z’s system (H) for propositional logic consists of the following axiom and inference rule
schemata
for P:
Axiom schemata for H
:
(HA1)
A
B
A
(HA2)
A
B
B
Inference Rule schemata for H
:
(HR1) From
H
A
B
and
H
B
C
, infer
H
A
C
.
(HR2) From
H
A
B
C
and
H
A
B
, infer
H
A
C
.
(HR3) From
H
A
B
and
H
A
B
, infer
H
A
.
Note
: the premises of these rules must all be
theorems
of H! This is different than MP in PS!
Some interesting facts about H, and some extracredit exercises concerning H
:
1.
H
is weakly
semantically sound
. For all formulae
A
of P, if
H
A
, then
P
A
. Prove this!
2.
Question: Is H
strongly
semantically sound
? That is, are there sets of formulae Γ and formulae
A
of P such that Γ
H
A
, but Γ
P
A
? Settle this question with a proof!
3.
H
is weakly
semantically complete
.
For all formulae
A
of P, if
P
A
, then
H
A
.
Note: One
cannot use Henkin’s method to prove this! This proof is worth a
serious
amount of extra credit!
1
You
may
assume
(3) for the other exercises below. [Note: (1) and (3) imply:
H
A
P
A
(
PS
A
).]
4.
H is
not strongly
semantically complete
. That is, there are sets of formulae Γ and formulae
A
of
P such that Γ
P
A
, but Γ
H
A
. Why? One way to see why is to note that:
5.
Modus Ponens is
not
(in at least one sense) an inference rule schemata of H
. That is, there
exist formulae
A
and
B
of P such that
A
B, A
H
B
. Exercise: prove this! [Hint: choose a
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 Spring '07
 FITELSON
 Logic, Axiom, H H H H H H H H

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