Communication and Meaning: An Essay in Applied Modal Logic (Synthese Library)
Recapitulating, dynamic epistemic logics deal with modal operators for model operations, thus allowing the explicit representation of actions and the way they affect the concept under study. This has important repercussions, particularly complexity-wise , as will be discussed in section 3. The previous section focused on some of the ways one can take a system with a single modality and create a system with multiple modalities from it.
Another alternative to build a multi-modal system is to take existing uni-modal systems, and then put them together by using a particular strategy. This section contains a brief description of some of the possible techniques; for a deeper discussion, the reader is referred to the SEP entry on combining logics by Carnielli and Coniglio The method of fusion of modal logics introduced in Thomason was developed with the idea of combining relation-based hence normal modal logics in both a syntactic way by putting together their respective Hilbert-style axiom systems and a semantic way by taking the relations corresponding to the modality of each system, and putting them together in a single model.
When this strategy is followed, and leaving technical details aside, the most important decision is the possible introduction of bridge principles that link the main modalities of the systems to be combined. In order to provide a better explanation of this technique, here we will discuss the construction of a simple temporal epistemic logic. For the semantic interpretation, formulas of the new language are interpreted in the standard way, with each modality using its correspondent relation. With respect to axiom systems, it is enough to put together those of the individual logics.
But we are not done yet. As mentioned before, the most interesting part is the possible inclusion of bridge principles. So, which is the proper interaction between time and knowledge? This corresponds to the following bridge principle:. This is clearly an idealization, and as such it makes sense only under certain interpretations; still, it might imply more than meets the eye.
Is this within the expected consequences? This property corresponds to. A slight elaboration on these and related properties can be found on the discussion on time and knowledge section 4. The fusion method provides a simple yet powerful strategy for adding a further aspect to an existing modal system by using another already existing system dealing with this further aspect independently. Indeed, the discussed example can be seen as adding a temporal aspect to the standard study of the properties of knowledge. Besides the traditional epistemic questions e.
This idea of adding a temporal aspect makes sense not only for knowledge, but also for other modal concepts. Similarly, adding a temporal aspect to modal systems of deontic logic raises interesting concepts and questions, an example being the notion of deontic deadlines , discussed in section 4. The study of some concepts such as preferences or obligations gives raise to an epistemic concern: how much do the involved agents know about such preferences and obligations?
In most examples where such concepts play a role, whether or not the agent knows about the involved preferences or obligations makes an important difference. In case of the second, is an agent compelled to obey a duty even when she does not know what the duty is?
Join Kobo & start eReading today
More on these concerns under different formal systems can be found within the discussion on knowledge and obligations section 4. For a deeper study on the fusion of modal logics, the reader is referred to Wolter , Gabbay, Kurucz, Wolter, and Zakharyaschev Chapter 4 , and Kurucz Section 2. Further examples of fusion can be found in the SEP entry discussing methods for combining logics.
Still, before closing this subsection, we add a word of caution. One needs to be careful when building the fusion of modal systems.
This is because, in the system that results from the fusion, there are already formulas combining the modalities of its different fragments. Then, even if no particular bridge valid principles are enforced, the language might gain in expressive power , which might increase its complexity profile.
Section 3. The fusion of modal systems produces a rich language that allows us to express the different ways in which the involved modalities interact the bridge principles. Still, from a semantic perspective, there is still just one point of reference, as all formulas of this richer language are still evaluated on a single possible world.
But, on the semantic side, the approach is quite different: instead of working on a one-dimension domain, it works on a multi-dimensional domain that has one dimension for each one of the involved aspects i. For the relations, each one of them is given as in their original models, restricted now to their respective dimensions:. The product of modal logics is a many-dimensional modal logic Gabbay et al. This specific way of interpreting each one of the original modalities yields another crucial difference between the fusion and the product of modal systems: the latter enforces, by its own nature, certain bridge principles on top of those that might be added.
The product of modal systems, with its n -dimensional nature, is a very useful tool and, in particular, it has been of help for dealing with the philosophical semantics technique of two-dimensionalism see also Chalmers ; Stalnaker This technique has been applied in different fields. In philosophy of mind, two-dimensional semantics has been used by David Chalmers combining both epistemic and modal domains to provide arguments against materialism in philosophy of mind details can be found in Chalmers The setting can be seen as the combination of a standard single-agent epistemic logic and a modal logic for social networks cf.
These formulas are evaluated in pairs world , agent , with the key clauses being the following:. Note that the modalities are indexical in both the world and the agent. For more on the products of modal logics, the reader is referred to the already-mentioned combining logics , and also to Gabbay et al. The strategies of fusion and product for combining modal logics rely in merging both the languages and the semantic models of the modal logics to be combined.
Communication and Meaning: An Essay in Applied Modal Logic
In the fibring strategy called fibring by functions in Carnielli, Coniglio, et al. Formulas can be evaluated in pointed models of any of the original systems, in the following way. Thus, modal fibring requires a correspondence between the class of models of each one of the systems, and uses it to move between them when the modality under evaluation requires it.
For more details on the fibring of modal logics, the reader is referred to Gabbay Chapter 3. For other forms of fibring, see Section 4. So far, this text has described different ways in which a multi-modal system can emerge. We have briefly discussed how a single modal system can give rise to a multi-modal one by providing either syntactic or semantic definitions of new concepts section 2 , and also how two or more modal systems can be combined in order to produce a multi-modal one section 3.
But there is also another side to this coin. By adding modalities to a system, one increases its expressivity, but this may also have the undesirable consequence of raising its computational complexity. Indeed, a modal language allows us to describe a certain class of models. If the language is fairly simple, then deciding whether a given formula is true in a given world of a given model the model-checking problem and deciding whether a given formula is true in all worlds of all models in a given class the validity problem are simple tasks.
Now suppose a more expressive language is used.
However, intuitively, we might then need to make a stronger effort to see those differences: we might need more time to make the calculations, and we might need more space to save intermediate results. In a single sentence, expressivity and complexity go hand in hand, and an increase in the first typically produces an increase in the second. The simplest example of this phenomenon is given by the relationship between the two best-known logical languages, the propositional and the first-order predicate one, when used to describe first-order models.
The validity problem for the propositional language, which can be understood as one that only allows us to talk about the properties i.
Communication and Meaning: An Essay in Applied Modal Logic (Synthese Library) | KSA | Souq
The first-order predicate language can see much more all objects of the domain, together with their properties and their n -ary relations, among others , but this comes at a price: its validity problem is undecidable , as there is no effective procedure that can answer the validity question for all its formulas. In the modal realm there are also such cases, some of them in which seemingly harmless combinations produce dramatic results.
An example of this can be found in Blackburn, Rijke, and Venema Section 6. Even if the new multi-agent system turns out not to be undecidable, its complexity might be such that solving its validity problem for relatively small instances is, for all practical purposes, impossible. An example of such case is the basic multi-agent epistemic logic, with no requirements on the accessibility relations. A major methodological issue is then to strike a proper balance between expressive power and computational complexity, with the best multi-modal systems being those that manage to achieve a good compromise in this sense.
We end this section noticing briefly that, in the case of the combination of modal logics, complexity depends deeply on the assumed bridge axioms. The previous sections have described several ways of obtaining multi-modal systems.
The current one presents some of the most interesting examples, together with the discussions that arise from the interplay of the modalities involved. The interplay between knowledge and belief is an important topic in epistemology. However, can knowledge be truly defined as justified true belief? The examples provided among others in Gettier seem to go against this idea. This has lead to proposals that focus on the requirement of a correct justification the no false lemma : Clark ; Armstrong ; Shope Some others have used a stronger indefeasibility requirement, stating that knowledge is justified true belief that cannot be defeated by true information, i.
The aforementioned topological modal approach of Baltag, Bezhanishvili, et al. There are also other alternatives.