Modal Logics With Provability Interpretations

In this section we give tableau calculi for the modal logics that have im-portant readings as logics of \provability" where²Ais read as \it is prov-able in Peano Arithmetic thatAholds"; see Fitting [Fit83, page 241] and Boolos [Boo79]. These systems are obtained either by adding the axiom G:²(²A^!A)^!2A, named after Godel-Lob and sometimes calledGL, or adding the axiomGrz:²(²(A^!2A)^!A)^!A, named after Grzegor-czyk, or adding the axiom 4 and the axiomGo:²(²(A^!2A)^!A)^!2A, to

K

^.

It is known that both Gand Grzimply the transitivity axiom 4 when they are respectively added to

K

[vB78]. But the logic

K4G

^owhose frames share some of the properties of

G

-frames and

Grz

-frames, explicitly

con-(G) ²X;^:2P

X;²X;^:P;²P ⁽Grz) ²X;^:2P X;²X;^:P;²(P^!2P)

Fig. 9.

Tableau Rules for logics of provability tains 4 as an axiom. It is also known thatGrz implies reexivity.

Once again, all the tableau calculi contain the rules of^C

PC

^{and one or}

more logical rules from Figure 9 on page 52 as shown below:

C

L

Static Rules Transitional Rules X^CL

C

G

^C

PC

(G) X^e

C

K4G

^{o C}

PC

(Grz) Sf²(X^{e !2}X^e)

C

Grz

^C

PC

, (T) (Grz) Sf²(X^{e !2}X^e) The semantic and axiomatic intuitions behind these rules are more en-lightening than any technical proof (of soundness) so we present these as well.

Intuitions for

(G)

:

We know that axiomatically formulated logic

G

is characterised by

G

-frames. Therefore, axiom G must be valid on any

G

-frame; hence true in any world of any

G

-model. The axiomGis

2(²A^!A)^!2A:

Its contrapositive is

:2A^!:(²(²A^!A)) which is the same as

:2A^!3(²A^{^:}A):

Thus, if the numerator represents a worldwwhere^:2P is true, then there exists another worldw⁰ where²P is true andP is false, andw⁰is reachable fromw. The denominator represents this world.

Intuitions for

(Grz)

:

The axiomGrz is

2(²(A^!2A)^!A)^!A:

It is known that 4 and T are theorems of

Grz

[HC84, page 111], hence

S4

Grz

. Segerberg [Seg71, page 107], and more recently Gore et al [GHH95], show that

Grz

=

S4Grz

=

S4G

^o whereGois

2(²(A^!2A)^!A)^!2A

which gives the following (contraposed formulae) as theorems of

Grz

:

:2A ^{! :2}(²(A^!2A)^!A)

:2A ^{! 3}(²(A^!2A)^{^:}A):

Thus, if ^:2P is true at the numerator, then there exists some world where ²(P ^{! 2}P)^{^:}P eventually becomes true. The denominator of (Grz) represents this world.

Theorem 4.16.1 (Soundness).

The calculi ^C

G

^,C

Grz

^andC

K4G

^oare

sound with respect to

G

^-frames,

Grz

-frames and

K4G

^o-frames respec-tively.

Proof Outline :

For each rule in ^C

L

we have to show that if the numerator of the rule is

L

-satisable then so is at least one of the denomi-nators.

Proof of

^C

G :

Suppose^M =^hW;R;Vⁱ is a

G

-model, w^{0 2}W and w^{0 j}= ²X;^:2P: Thus there exists some w^{1 2} W withw⁰Rw¹ and w^{1 j}= X;²X;^:Pby the transitivity ofR:SinceRis irreexive,w⁰⁶=w¹:Suppose w^{1 6j}=²P: Then w^{1 j}= ^:2P and there exists some w^{2 2}W with w¹Rw² andw^2j=X;²X;^:P by transitivity ofR:SinceR is irreexive,w¹⁶=w²: Since R is transitive, w² = w⁰ would give w¹Rw⁰Rw¹ implying w¹Rw¹ and contradicting the irreexivity ofR;hencew⁰⁶=w²:Supposew^26j=²P then ::: Continuing in this way, it is possible to obtain an innite path of distinct worlds in^Mcontradicting the

G

-frame condition on^M. Thus there must exist somewi²W withw⁰Rwi andwi^j=X;²X;^:P;²P and we are done.

Proof of

(T)

for

^C

Grz :

The (T) rule is sound for

Grz

-frames since every

Grz

-frame is reexive.

Proof of

(Grz)

for

^C

K4G

^o

:

Suppose ^M = ^hW;R;Vⁱ is a

K4G

^o -model, then R is transitive, there are no proper clusters, and there are no proper ¹-R-chains. Suppose w^{0 2} W is such that w^{0 j}= ²X;^:2P. We have to show that there exists somew_n ²W withw⁰Rw_n andw_n ^j= X;²X;^:P;²(P ^!2P). SinceRis transitive,w^0j=²Xmeans that⁸w² W;w⁰Rw impliesw ^j=X;²X. Thus our task is reduced to showing that there exists somewn ²W such thatw⁰Rwn andwn ^j=^:P;²(P ^!2P): Suppose for a contradiction that no such world exists in W. That is,

(a) ⁸w²W;w⁰Rwimpliesw^6j=^:P;²(P ^!2P).

Since w^{0 j}=^:2P, there exists some w^{1 2} W with w⁰Rw¹ and w^{1 j}=

:P. By (a), w^{1 6j}= ²(P ^{! 2}P) and hence w^{1 j}= ^:2(P ^{! 2}P): Thus there exists some w^{2 2} W with w¹Rw² and w^{2 j}= ^:(P ^{! 2}P); that is, w^2j=P^{^:2}P. Sincew^1j=^:P,w¹⁶=w²and since

K4G

^o-models cannot contain proper clusters, w^{0 6}= w². Since w^{2 j}= ^:2P there exists some w^{3 2} W with w^{3 j}= ^:P. Since w^{2 j}= P, w^{3 6}= w². And w^{3 6}= w⁰ and w³⁶=w¹ as either would give a proper cluster. By (a),w^36j=²(P ^!2P)

and hencew^3j=^:2(P ^!2P). Continuing in this way, we either obtain an innite path of distinct points, giving a proper¹-R-chain, or we obtain a cycle, giving a proper cluster. Both are forbidden in

K4G

^o-frames. Hence (a) cannot hold and ⁹w ²W;w⁰Rw and w ^j=^:P;²(P ^!2P): That is, the desiredwn exists.

Proof of

^(Grz)

for

^C

Grz :

^Every

Grz

-frame is a

K4G

^o-frame, hence the proof above suces.

As we saw in Subsection 4.11, proving completeness boils down to prov-ing the followprov-ing: ifX is a nite set of formulae and no^C

L

-tableau for X is closed then there is an

L

^{-model forX on an}

L

^-framehW;Ri.

Lemma 4.16.2.

If there is a closed^C

L

-tableau forX then there is a closed

C

L

-tableau for X with all nodes in the nite setX^CL .

Proof:

Obvious from the fact that all rules for^C

L

operate with subsets ofX^CL only.

Lemma 4.16.3.

For each ^C

L

-consistentX there is an eective procedure to construct some nite^C

L

-saturatedX^swith XX^sX^CL .

Theorem 4.16.4 (Completeness).

If X is a nite set of formulae and X is^C

L

-consistent then there is an

L

-model forX on a nite

L

-frame.

As usual we construct some^C

L

^{-saturatedw0 fromX with X w0}

X^CL .

Proof for

^C

G:

^{If no :2P occurs in w0 thenhfw0g;;i} is the desired model graph as (i)-(iii) are satised. Otherwise, letQ¹;Q²;;Qm be all the formulae such that ^:2Qi ² w⁰. Create a ^C

G

^-saturatedQi-successor for eachQi using () and (G) giving the nodes vi of level one. Repeating this construction on the nodes of level one gives the nodes of level two, and so on for other levels. Consider any sequencewiwi⁺¹wi⁺². Since wihas a successor, there is some^:2Q²wiand²Q²wi⁺j for allj1 by (G). Thuswi⁶=wi⁺jfor anyj1 and each such sequence must terminate sinceX^CG is nite. LetRbe the transitive closure of; that is putwRw⁰ ifww⁰ and putwRvifww⁰ v. The resulting tree is a model graph

hW⁰;RⁱforX which is also a

G

-frame.

Proof for

^C

Grz:

If no^:2P occurs inw⁰then^hfw^0g;^f(w⁰;w⁰)^giis the desired model graph as (i)-(iii) are satised. Otherwise, letQ¹;Q²;;Qm

be all the formulae such that^:2Qi ²w⁰ and ^:Qi⁶²w⁰. Create a^C

Grz

-saturatedQi-successor for eachQiusing () and (Grz) giving the nodesvi

of level one, and so on for other levels. Consider any sequencewiwi⁺¹

wi⁺². Sincewi has a successor, there is someQ such that^:2Q²wi,

:Q ⁶² wi, and by (Grz), ²(Q ^{! 2}Q) ² wi⁺j for all j 1. Suppose wi⁺j =wi, then²(Q^!2Q)²wi and henceQ^!2Q²wi by (T). Since Q^!2Qis just abbreviation for ^:(Q^{^:2}Q), we know that ^:Q²wi or

::2Q²wi. We created a successorwi⁺¹forwiprecisely because^:Q⁶²wi

and so the rst case is impossible. And if^::2Q²withen²Q²wiby (^:),

contradicting the

Grz

-consistency of w_i since^:2Q ²w_i by supposition.

Thus each such sequence must terminate (without cycles). Let R be the reexive and transitive closure of to obtain a model graph ^hW⁰;Rⁱfor X which is also a

Grz

^-frame.

Proof for

^C

K4G

^o

:

^{If no :2P occurs in w0 then hfw0g;f;gi is the}

desired model graph as (i)-(iii) are satised. Otherwise, letQ¹;Q²;;Qm

be all the formulae such that ^:2Qi ² w⁰. A ^C

K4G

^o-saturated set v is reexive i ²A ² v implies A ²v. If v is non-reexive then there exists some²B²v butB⁶²v.

Ifw⁰ is reexive then create a^C

K4G

^o-saturatedQi-successor for each

:2Qiwith^:Qi⁶²w⁰, otherwise ifw⁰is non-reexive then create a^C

K4G

^o -saturated Qi-successor for each ^:2Qi, 1 i m. This gives the nodes of level one. Continue creating successors in this fashion for these nodes using () and (Grz).

Consider any sequence wi wi⁺¹wi⁺². Sincewi has a successor, there is some^:2Q²w_i that gives rise tow_i⁺¹. Also,²(Q^!2Q)²w_i⁺_j for allj1.

If w_i is reexive then ^:Q ⁶² w_i, and yet ^:Q² w_i⁺¹ by (Grz); hence wi ⁶= wi⁺¹. Suppose wi⁺j = wi, j 2. That j 2 is crucial! Then

2(Q ^{! 2}Q)² wi and Q^{! 2}Q ² wi by (Grz). Since Q ^{! 2}Q is just abbreviation for^:(Q^{^:2}Q), we know that^:Q²wior^::2Q²wi. Since wiis reexive, we created a successorwi⁺¹forwiprecisely because^:Q⁶²wi

and so the rst case is impossible. And if^::2Q²withen²Q²wiby (^:), contradicting the

K4G

^o-consistency ofwisince^:2Q²wi by supposition.

If wi is non-reexive then there is some ²B ² wi, withB ⁶² wi, and yet both ²B and B are inwi⁺j by (Grz), for all j 1; hencewi ⁶=wi⁺j, j1.

Thus each such sequence must terminate (without cycles). Let R be the transitive closure of and also putwRw ifwis reexive to obtain a model graph^hW⁰;RⁱforX which is also a

K4G

^o-frame.

As Amerbauer [Ame93] points out, this means that

K4G

^o is charac-terised by nite transitive trees of non-proper clusters refuting the conjec-ture of Gore [Gor92] that

K4G

^o is characterised by nite transitive trees of degenerate non-nal clusters and simple nal clusters.

4.16.1 Bibliographic Remarks and Related Systems

The tableau system^C

G

is from Fitting [Fit83] who attributes it to [Boo79], while^C

Grz

is from Rautenberg [Rau83]. Rautenberg gives a hint on how to extend these to handle ^C

K4G

^o but Gore [Gor92] is unable to give an adequate system for^C

K4G

^o, leaving it as further work. The given^C

K4G

^o

is due to Martin Amerbauer [Ame93] who following suggestions of Rauten-berg and Gore also gives systems for

KG

^:

2

^and

KGL

(which Amerbauer calls

K4

:

3G

).

Provability logics have also been studied using Gentzen systems, and

ap-propriate cut-elimination proofs have been given by Avron [Avr84], Bellin [Bel85], Borga [Bor83], Borga and Gentilini [BG86], Sambin and Valentini [SV80, VS83, SV82], and Valentini [Val83, Val86].

No documento Rajeev Gore (páginas 52-57)