    Modal Logic via Global Consequence

In modal logic, semantic consequence is usually defined locally by truth preservation at all worlds in all models (with respect to a class of frames). It can also be defined globally by truth preservation in all models (with respect to a class of frames). The latter is called global consequence, which is much less studied than the standard local one. In this paper we first study the relationship between local and global consequence. Then we give some correspondence results for global consequence. Finally, we illustrate two applications of global consequence, connecting it with informational consequence and update consequence proposed in formal semantics. Some results in the paper are already known, which are collected in the paper for the sake of completeness. The others appear to be new. We suggest that global consequence is not only interesting theoretically, but also useful for application.

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1 Introduction

Given a class of frames , the inference from to is valid with respect to , if for every world in every model such that is a frame in , if all formulas in are true at in then is also true at in . This is called the local consequence (or local validity) in modal logic, which is the standard one. Another notion called global consequence (or global validity) in modal logic is also defined in the literature (e.g. in ). The inference from to is globally valid with respect to , if for every model such that is a frame in , if all formulas in are true in then is also true in , where a formula is true in a model if it is true at all worlds in the model. Compare to local consequence, global consequence is much less studied. Notable exceptions include, ,  and . Kracht  studied global consequence from an algebraic point of view systematically. Fitting  integrated local and global consequence into a ternary relation, and proved completeness for various kinds of proof systems. Ma and Chen  presented Gentzen-style sequent calculi for global consequence. This paper studies global consequence within the standard relational semantics of modal logic, emphasizing its connection with local consequence and some other consequence notions, which were proposed for natural language arguments.

In the sequel, we consider only normal modal logics. Let be the classical propositional language, the basic modal language. We use (with or without subscripts) for local consequence and (with or without subscripts) for global consequence, respectively. We use for satisfaction relation. We write if for all . We write if for all in , and if for all based on . We denote by the (local) syntactic consequence for the axiomatic system . We denote by be the class of all frames, and the class of all models. We assume the readers are familiar with notations for typical classes of frames and axiomatic systems. For example, refers to the class of transitive frames, and the class of frames with equivalent relations; and denote their corresponding axiomatic systems, respectively. Some other notations: , , , , , , .

The remaining part of the paper is organized as follows. Section 2 shows the relationship between local consequence and global consequence. Section 3 gives a general correspondence result for global consequence and its typical instances. Section 4 illustrates two applications of global consequence, connecting it with informational consequence and update consequence proposed in formal semantics. Section 5 concludes the paper. Some results in the paper are already known, which are collected in the paper for the sake of completeness. The others are supposed to be new.

2 Relationship Between Local and Global Consequence

For a start, the following are well known results that connect local and global consequence.

Fact 1.

For any class of frames , for any ,

1. iff ;

2. implies .

Since local and global valid formulas coincide, we are more interested in global consequence rather than globally valid formulas. The following fact can be easily verified, which says that local consequence and global consequence coincide for modal-free formulas. This may be the reason why for modal-free reasoning, we do not distinguish local and global consequence.

Fact 2.

Let . Then for any class of frames, iff .

The following two known results show that if we add some global operators in the language, then global consequence can always be defined by local consequence. Before that we need two definitions for the global operators.

Definition 3.

Given a model , define the operator as follows,

where is the reflexive and transitive closure of .

Definition 4.

Given a model , define the universal operator as follows,

 M,w⊩Aφ iff for all u∈W,M,u⊩φ.
Proposition 5 (, p. 159).

For any class of frames , iff .

Proposition 6 (, Proposition 2.1).

For any class of frames , iff iff .

If we consider only the class of frames , then global consequence can be defined by local consequence within the basic modal language, as the following proposition shows.

Proposition 7 (, p. 32).

For any , iff .

The proposition appears as an exercise in . Instead of proving it directly, we generalize it as follows.

Theorem 8.

Let be any class of frames that is closed under point generated subframes. Then for any , iff .

Proof.

Suppose . Then there exist a frame in , a valuation on , and a world in such that but . Let be the model generated by from . Then and . From the former, it follows that , since all worlds in are accessible from in finite (including zero) steps. From the latter, it follows that . Since is closed under subframes, is also in . Thus, .

Suppose . Then there exists a frame in and a valuation on such that but . From the latter, it follows that there exists a world in such that . From the former, it follows that every is true at all worlds in . Thereby, it can be easily verified by induction that is true at all worlds in for all and . In particular, . Hence, . ∎

Note that the direction from right to left does not require to be closed under point generated subframes. The other direction, however, does not hold for all , as the following fact shows.

Fact 9.

There exist a class of frames and formulas such that but .

Proof.

Let with . Then for any valuation on , , since . Hence, . On the other hand, given any valuation on , for all , but . Hence, . ∎

In [4, p. 425], the authors claim that the equivalence between and holds for all , which is incorrect by the above fact. But the closure under point generated subframes is not a necessary condition for the equivalence in Theorem 8, as the following fact shows.

Fact 10.

There exists a class of frames that is not closed under point generated subframes such that for any , iff .

Proof.

Consider with . Obviously, is not closed under point generated subframes. The direction from right to left is easy. For the other direction, suppose . Then there exists a valuation on such that either and , or and . W.l.o.g., suppose the former holds. Then . Let be a valuation such that . It is easily verified that iff iff for all . Hence, and . Thereby, . ∎

If we consider transitive frames, then the biconditional between local consequence and global consequence can be further simplified, as the following corollary shows.

Corollary 11.

Let be any class of transitive frames that is closed under point generated subframes. Then for any , iff .

Proof.

It follows from Theorem 8, noting that for any transitive frame , for . (Recall that denotes .) ∎

To define global consequence by local consequence using only rather than , we could add another constraint for the class of frames.

Definition 12.

A class of frames is closed under irreflexive point extension, if for any frame in , for any with , any point extension of for by is also in , where is defined as follows:

 W′ =W∪{u} R′ =R∪{(u,w)}∪{(u,w′)∣(w,w′)∈R}
Theorem 13.

Let be any class of transitive frames that is closed under point generated subframes and irreflexive point extension. Then for any , iff .

Proof.

Suppose . Then there exit a frame in , a valuation on , and a world in such that and . From the latter, it follows that there exists such that . Then from the former, it follows that , noting that is transitive. Let be the submodel of generated by . Then and hence . Since is transitive, every world in is either or accessible from . Thus for all in . Then we have . Since is closed under point generated subframes, the frame underlying is also in . Therefore, .

Suppose . Then there exist a frame in and a valuation on such that and . From the latter, it follows that there exists in such that . If , then . Since , we also have . Hence . If , let be a point extension of for by . Then it can be verified that and . Since is closed under irreflexive point extension, is also in . Hence, . ∎

Corollary 14.

Let be any class of reflexive and transitive frames that is closed under point generated subframes. Then for any , iff iff .

Proof.

Note that any class of reflexive frames is also closed under irreflexive point extension. Thus we have the first biconditional from Theorem 13. The direction from left to right of the second biconditional follows from the fact that in any reflexive frame , . The other direction follows from the fact that for any class of frames , implies , and hence implies . Then using the fact that for any transitive frame , , we obtain the final result. ∎

Remark 15.

The above corollary can also be derived from Theorem 8, noting that in any reflexive frame , .

Corollary 16.

For any , for any in , iff .

Proof.

Straightforward from Theorem 13, noting that all in are closed under point generated subframes and irreflexive point extension. ∎

The following proposition shows that to define global consequence by local consequence, sometimes various classes of frames are attainable.

Proposition 17.

For any , iff iff iff iff

Proof.

The first two ‘iff’s follow from Corollary 14. The direction from right to left of the third ‘iff’ is easy. For the other direction, suppose . Then there exist a transitive and Euclidean model and a world such that and . From the latter, it follows that there exists such that . Since is transitive, we also have . Let be the point generated submodel of by . Then it can be verified that is reflexive, transitive and Euclidean. Moreover, and . Therefore . The last ‘iff’ can be proved analogously. ∎

If we restrict premises to be modal-free formulas, then global consequence can always be defined by local consequence (within the basic modal language), as the following proposition shows.

Proposition 18.

Let and . Then for any class of frames , iff .

Proof.

Suppose . Then there exist a frame in , a valuation on , and a world in such that but . Let be the model generated by from . Let . Then and . From the former, it follows that , since all worlds in are accessible from in finite (including zero) steps. From the latter, it follows that . Noting that is satisfiable and contains no modal formulas, we can define a valuation on such that for all worlds in , coincides with , and for all worlds in , for every atom , iff . Then , but . Thus, .

The same as that in the proof of Theorem 8. ∎

Some of the above results can also be given syntactically. Before that, we need some definitions. We define local syntactic consequence in an eliminational way, as in most textbooks in modal logic (e.g.  and ) , i.e. iff there is a finite subset such that . The gist of this definition is to prevent the application of the rule of necessitation to the premises in , since the inference from to is generally not valid under local semantic consequence. On the contrary, since we have , given a standard axiomatic system, the global syntactic consequence can be defined in the same way as in classical propositional logic, i.e. iff there is finite sequence of formulas such that and for each either , or is an instance of an axiom scheme, or is obtained from previous formulas in the sequence by applying the rule(s) of the system. As a result, under global syntactic consequence, the rule of necessitation is applicable to the premises. Now we have the following result.

Proposition 19.

Let be any axiomatic extension of . Then for any , iff .

Corollary 20.

Let be any axiomatic extension of . Then for any , iff .

Corollary 21.

Let be any axiomatic extension of . Then for any , iff iff .

These results can be obtained by the completeness of the axiomatic systems as well as the above semantic results. They can also be proved directly by induction on the length of proofs. We omit it here.

Conversely, local consequence can also be defined by global consequence, but much harder. We need a local operator.

Definition 22.

Given a model , define the ‘only’ operator as follows:

 M,w⊩Oφ iff M,w⊩φ and% for all w′≠w,M,w′⊮φ.

Venema gave the following result in  (without proof).

Proposition 23 (, p. 159).

For any class of frames , for any , for any

 Γ⊨Fφ iff {EOp}∪{p→γ∣γ∈Γ}⊨gFp→φ,

where is the dual of the universal operator in Definition 4.

We summarize the results in this section as follows. Those with bold fonts are supposed to be new.

Local by Global Global by Local
restricting , for all Fact. 2 Fact. 2, Prop18
beyond , for all Prop. 23 Prop. 5, Prop. 6
within , for some Thm8, Thm13, Prop. 7, Cor. 11, Cor. 14, Cor. 16, Prop17

Though within global consequence can not be reduced to local consequence generally, many properties for local consequence are preserved for global consequence. See [8, 9, 10].

3 Global Correspondence

If we consider the correspondence between modal formulas and first-order frame properties, then there is nothing new for global consequence, since globally valid formulas coincide with locally valid formulas. But if consider the correspondence between modally valid inferences and first-order frame properties, then it turns out to be much different for global consequence.

First, we have the following obvious fact.

Fact 24.

for any class of frames , in particular, we have

In contrast, if and only if is transitive, and if and only if is Euclidean.

Fact 25.

iff is globally isolated, i.e. for every in , .

Proof.

Given any globally isolated frame in , given any valuation on , suppose . Given any , since is globally isolated, there exists s.t. for all , if then . Since , we have . Then there exists s.t. and . By the property of , we have . Hence, . Since is arbitrary, we have , as required.

Suppose in is not globally isolated. Then there exists s.t. for all there exists s.t. and . Let . Then and hence . Given any , by the property of , there exists s.t. . Hence, and thus . Since is arbitrary, we have . Therefore, . ∎

Fact 26.

iff is globally transitive, i.e. for every in , .

Proof.

Given any globally transitive frame in , given any valuation on , suppose . Given any , since is globally transitive, there exists s.t. for any if and then . By , we have . Then there exists s.t. , , and . By the property of , . Hence, . Since is arbitrary, , as required.

Suppose in is not globally transitive. Then there exists s.t. for all , there exist s.t. , , and . Let . Then and hence . Given any , since there exist s.t. , , and , we have and hence . Since is arbitrary, we have . Therefore, . ∎

Fact 27.

iff is globally Euclidean, i.e. for every in , .

Proof.

Given any globally Euclidean frame in , given any valuation on , suppose . Given any , suppose . Since is globally Euclidean, there exists s.t. for all if and then . By , we have . Then there exists s.t. . By the property of , we have . It follows that . Thus, . Since is arbitrary, we have , as required.

Suppose in is not globally Euclidean. Then there exists s.t. and for all there exists s.t. , and . Let . Then and hence . Given any , by the property of , there exists s.t. , and . Hence, and . Since is arbitrary, we have . Therefore, . ∎

Fact 28.

iff is globally reflexive, i.e. for every in , .

Proof.

Given any globally reflexive frame in , given any valuation on , suppose . Given any , since is backward serial, there exists s.t. . Since , we have . Hence, . Since is arbitrary, we have , as required.

Suppose in is not globally reflexive. Then exists s.t. for all , . Let . Then and hence . Given any , since , we have . Since is arbitrary, we have . Therefore, . ∎

Fact 29.

iff is globally inverse reflexive, i.e. for every in , .

Proof.

Given any globally inverse reflexive frame in , given any valuation on , suppose . Then for all . Since is globally inverse reflexive, for all , i.e. .

Suppose in is not globally inverse reflexive. Then there exists such that . Let . Then but . Thus . Therefore, . ∎

Fact 30.

iff is globally serial, i.e. for every in , .

Proof.

Given any globally serial frame in , given any valuation on , suppose . Given any , since is globally serial, there exist s.t. and . By , we have . Hence, . By , we have . Since is arbitrary, we have , as required.

Suppose in is not globally serial. Then there exists s.t. for all if then . Let . Then and hence . Given any , suppose , by the property of , we have . Hence, . Thus . Since is arbitrary, we have . Therefore, . ∎

Fact 31.

iff is globally symmetric, i.e. for every in , .

Proof.

Given any globally symmetric frame in , given any valuation on , suppose . Given any , suppose . Since is globally symmetric, there exists s.t. . Since , we have . Thus . Hence, . Since is arbitrary, we have , as required.

Suppose in is not globally symmetric. Then there exists s.t. and . Let . Then and . Thus and hence . Therefore, . ∎

Fact 32.

iff is globally inverse symmetric, i.e. for every in , .

Proof.

Given any globally inverse symmetric frame in , given any valuation on , suppose . Given any , by the property of , there exists s.t. for all if then . Since , we have . Then there exists s.t. . By the property of , we have . Hence, . Since is arbitrary, we have , as required.

Suppose in is not globally inverse symmetric. Then there exists s.t. for all there exists s.t. and . Let . Then and hence . Given any , by the property of , there exists s.t. and . Hence, and . Since is arbitrary, we have . Therefore, . ∎

Note that for local consequence, a valid inference often has an equivalent dual version. For example, iff . This equivalence, however, does not hold for global consequence. For example, though holds for any class of frames, its dual holds only for globally transitive frames. This is a notable contrast between local and global consequence.

Parallel to a famous general correspondence result for local consequence, we give a general correspondence result for global consequence, of which the above facts are all instances.

Theorem 33.

iff every frame in satisfies the following condition

 ∀w∀x∃y∀z∃u(Rkwx∧Riyz→Rlxu∧Rjzu).
Proof.

Given any frame in that satisfies the above property, given any valuation on , suppose . Given any , suppose . Then by the property of , there exists s.t. for all if then there exists s.t. and . By , we have . Then it follows that there exists s.t. and . By the property of , there exists s.t. and . Thus and . Hence, . Since is arbitrary, we have , as required.

Suppose in does not satisfy the above property. Then there exists s.t. and for all there exits s.t. and . Let . Then and . Hence, and . Given any , by the property of , there exists s.t. and . Thus and . Since is arbitrary, we have . Therefore, . ∎

4 Applications

4.1 Informational Consequence

In  Yalcin advocated a non-classical consequence relation, called informational consequence. Yalcin noticed that if denotes epistemic ‘might’ or ‘may’, then saying both and seems inconsistent, which is not reflected in standard modal logic. So he proposed domain semantics and informational consequence (details below) to formalize this phenomenon. We will soon find that informational consequence is intimately related to global consequence.

Definition 34.

A domain model is a pair , where and is a valuation on . Given a domain model , that is true at in , denoted , is inductively defined as follows, where means for all , :

• iff

• iff

• iff and

• iff

Definition 35 (Informational consequence).

The inference from to is informationally valid, denoted , if for all domain models and , implies .

It can be easily shown that under domain semantics, . But this can also be achieved by global consequence for free.

Fact 36.

for any class of frames .

Proof.

Suppose . Then and . The former implies that , which contradicts the latter. ∎

In , Bledin convincingly argued that the rule of reduction to absurdity and constructive dilemma are not generally valid for natural language arguments. Rather, their correct forms should add some modal operators. More precisely, Bledin suggests that

• , instead we have ;

• , instead we have .

Bledin argued that informational consequence can perfectly predict the above desiderata. But global consequence can do the same job as well.

Fact 37.

, instead for any reflexive and transitive , we have .

Proof.

By Fact 36, we have for any class of frames . But by Fact 25, holds only for that is globally isolated. For the remaining part, suppose . Then there exists a model with its underlying frame in such that and . By the latter there exists in such that , i.e. . Let be the subframe of generated by . Then . Since is reflexive and transitive, we have . Thus . ∎

Fact 38.

, instead for any reflexive and transitive , we have .

Proof.

By Fact 24, we have and for any class of frames . But it is easily verified that , where . For the remaining part, suppose and . Let be any model with its underlying frame in . Suppose and . Given any in , we have . Then either or . Since is reflexive and transitive, if the former holds, then . By , we have . Thus . If the latter holds, then . By , we have . Thus . Hence, . Since is arbitrary, we have , as required. ∎

Indeed, Schulz proved the following general result.

For any