As noted in section5.5, the ruleL0⊃from G4ipprecludes that calculus from the analysis in the previous sections. Here, we relax the condition that the active part of rules may only have one compound formula in their conclusions. Now, however,every metaformula which appears in the conclusion of the active part is principal for the rule. As an example, both P andP⊃φare principal forL0⊃.
We can then reconstruct similar lemmata to those of the previous sections, however we must be careful with respect to atoms and⊥. We have the following:
Lemma 19 (Multiprincipal Rules) Let R be a set of decomposable rules. Then, the rule:
Γ,Γ00⇒∆,∆00 Γ,Γ0⇒∆,∆0
is height preserving admissible if:
1. For every φ∈Γ00,∆00, we have Γ0 ⇒∆0 is the active part of a premiss of every rule with φprincipal.
2. Every atom variable inΓ00 (∆00) occurs inΓ0 (∆0). 3. If⊥ ∈Γ00, then⊥ ∈Γ0.
Proof. Let Γ + Γ00⇒∆ + ∆00 be an instantiation of the premiss. We proceed by induction on the height,n, of the derivation of the premiss. Ifn= 0, then the premiss was an axiom. Conditions 2 and 3 guarantee the conclusion is also an axiom.
Ifn >0, letr be the last inference used in the derivation. Then there are four cases:
1. r is an instance of a normal rule, and there is someA∈Γ00,∆00 which is principal for r.
2. r is an instance of an IW rule, and there is someA∈Γ00,∆00 which is principal forr. 3. r is an instance of a normal rule, and there is noA∈Γ00,∆00 which is principal forr. 4. r is an instance of an IW rule, and there is noA∈Γ00,∆00which is principal forr. Case 1. Γ0⇒∆0 is the active part of a premiss of r. If the active part ofr had conclusion Γ00⇒∆00, then the result is immediate. There may be other inferences with active parts of
the form:
· · · Γ0 ⇒∆0 · · ·
Γ00,Γ000⇒∆00,∆000
In these cases, we have Γ000⊆Γ and ∆000⊆∆. So, there exist ¯Γ and ¯∆ such that Γ = Γ000+ ¯Γ and ∆ = ∆000+ ¯∆. We can then rewriter as:
· · · Γ,¯ Γ0⇒∆,¯ ∆0 · · ·
¯
Γ,Γ00,Γ000 ⇒∆,¯ ∆00,∆000
Weakening with Γ000 and ∆000gives the required result; the premisses are derivable at height n−1.
Case 2. This case is similar to the equivalent case in the proofs of lemmata 8(p. 82),9(p.
85) and case 1.
Case 3. Since no formula from Γ00,∆00 is principal for r, all such formulae must be in the context ofr. Then, we reason in the same fashion as in the proofs of lemmata 8and9. Case 4. We know no formulae from Γ00,∆00 are principal for r. We have r will be of the
form:
Γ1,Γ001 ⇒∆1,∆001 · · · Γ1,Γ00n⇒∆1,∆00n Γ1,Γ2,Γ00? ⇒∆1,∆2,∆00?
Either Γ00 is in Γ1, or Γ2, or partially contained in both. We reason by cases. Either,
Γ00FΓ2, and ∆00F∆2, or at least some part of Γ
00 or ∆00is in Γ
1 or ∆1, respectively.
In the former case, the situation is similar to that in lemma 8 and lemma 9; we use a new inference which is the same asrexcept that the implicit weakening part of the inference contains Γ0 and ∆0, instead of Γ00 and ∆00.
In the other case, we perform two steps. Firstly, we weaken every premiss of r so that the context of each premiss contains Γ00and ∆00. This will be the same weakening for every premiss, because every premiss contains Γ1and ∆1, and hence the same elements of Γ00and
∆00. Then, we apply the induction hypothesis to each of these new premisses, so that we remove Γ00 and ∆00and replace them with Γ0 and ∆0 in the context of each premiss.
Then, we remove from Γ2and ∆2 any formulae which occur in Γ00and ∆00, respectively,
information will yield Γ + Γ0⇒∆ + ∆0as derivable at heightn, and this completes the case,
and the proof. a
In particular, the results from the previous sections are now specialisations of this result.
6.7.1
Examples
When restricted to single formulae on the right in an analogous way to section 6.3, G4ip is a calculus in which all rules have the form given above. The rule which did not allow G4ip to be classified as a uniprincipal calculus was L0⊃, shown above. Now we can see, however, that it is indeed invertible; the atom variablePis retained in the premiss, and this rule is the only rule whereP⊃φwill be principal on the left. In what follows, the notation
HA1, . . . , AnIis used to denote the multiset containing the formulaeA1, . . . , An.
Consider the calculus G3-LCfor G¨odel-Dummett logic from [Sonobe, 1975]. Axioms are given as normal, the only logical connective is⊃and the rules for it are:
Γ, φ⊃ψ⇒φ,∆ Γ, ψ⇒∆ Γ, φ⊃ψ⇒∆ L⊃ {Γ, φi⇒ψi,∆i} Γ⇒∆ R⊃ wherei= 1, . . . , mand: ∆i=Hφ1⊃ψ1, . . . , φi−1⊃ψi−1, φi+1⊃ψi+1, . . . , φm⊃ψmI
∆ contains this multiset and may contain other formulae as well. Whenm= 2, for instance, the rule will be:
Γ, φ1⇒ψ1, φ2⊃ψ2 Γ, φ2⇒ψ2, φ1⊃ψ1
Γ⇒φ1⊃ψ1, φ2⊃ψ2,∆0
For a given m, this rule is an IW multiprincipal rule. In the case where ∆0 is empty, then the rule is invertible. In the case where ∆0is non-empty, then we can use lemma19to show that each premiss, weakened on the right with ∆0, is derivable without an increase in height. As a more involved example, consider the calculus with the two binary connectives◦and ?, and the six rules:
Γ, φ, ψ⇒∆ Γ, φ◦ψ⇒∆ L◦ Γ⇒φ, ψ,∆ Γ⇒φ◦ψ,∆ R◦ Γ, φ⇒ψ,∆ Γ, φ ? ψ⇒∆ L? Γ, φ⇒∆ Γ⇒ψ,∆ Γ⇒φ ? ψ,∆ R? Γ, φ, ψ⇒∆ Γ⇒ψ,∆ Γ, φ◦ψ⇒φ ? ψ,∆ L◦R? Γ, ψ⇒φ,∆ Γ, φ ? ψ⇒φ◦ψ,∆ L?R◦ The following rules are height preserving admissible:
Γ, φ◦ψ⇒∆ Γ, φ, ψ⇒∆
Γ⇒φ ? ψ,∆ Γ⇒ψ,∆
Take the left-hand rule. φ◦ψwill be left principal in rulesL◦andL◦R?. For both of these rules φ⊕ψ ⇒ ∅is the active part of a premiss. If the last inference was based on L◦R?, then A ? B (the instantiation of φ ? ψ) would be part of ∆, and so would disappear from the premiss. We would thus weaken withA ? B(as in the latter part of case 1 of the proof, above).
The rule:
Γ⇒φ ? ψ,∆ Γ, φ⇒∆
cannot be shown admissible by use of lemma19, forφ⇒ ∅is not the active part of a premiss ofL◦R?. Neither of the multiprincipal rules are invertible.