MODAL LOGIC

The term modal  logic denotes a whole range of logics designed to study various modes. For this investigation the interesting modes include:

  1. Alethic or metaphysical modes including metaphysical, empirical and temporal necessity operators prefixed formulas of a propositional logic.

  2. Temporal modes including tense operators F, G, P and H prefixed formulas of a propositional logic.

  3. Epistemic modes including a knowledge operator Kµ prefixed formulas of a propositional logic where denotes an arbitrary inquiry method of either assessment or discovery.

TABLE OF CONTENTS

The Formal Paradigm
Epistemic Axioms and Systems
Modal Logics
Combining Operators - An Example
Epistemic Axioms and Method
Further Reading

THE FORMAL PARADIGM

The formal paradigm of modal operator theory include:

  1. A definition of evidence streams.

  2. A space of theoretically possible worlds.

  3. A formal concept of background knowledge.

  4. A space of entertainable hypotheses.

  5. A definition of  correctness of a hypothesis in a possible world.}

  6. Methods for scientific discovery.

  7. Methods for scientific assessment.

  8. Definitions of methodological recommendations for scientific inquiry methods of discovery and assessment.

  9. Definitions of convergence.

  10. A definition of successful convergence to a correct hypothesis for discovery methods.

  11. A definition of successful convergence to a correct hypothesis for assessment methods.

  12. Definitions of (scientific) knowledge based on either assessment or discovery methods.

From the item list it follows that both agents/methods, tense aspects, alethics together with inductive complexity is inherent in the paradigm. Modal learning theory applies a version of possible  world semantics in which a possible world is a pair consisting of evidence stream and a state coordinate. This induces a branching time structure which is Ockhamistic. In turn, the modal logic in SYSTEM I-II must be able to incorporate both tenses and alethics together with method-sensitive epistemic operators in order to model the growth and strength of scientific knowledge.

EPISTEMIC AXIOMS AND SYSTEMS

The strength of an epistemic operator is typically measured by determining the epistemic axioms and systems corresponding to the operator:

Epistemic Axioms. Axioms designed to determine epistemic strength of a knowledge acquiring inquiry method. The axioms considered here most notably include:

  • (D) also called the axiom of consistency. If a method knows a hypothesis, then it does not also know its negation. 
    K    µh -> ¬Kµh.

  • (T)) also called the axiom of truth. If a method knows a hypothesis, then the hypothesis is true: 
    K    µh -> h.

  • (K) also called the axiom of deductive cogency. The knowledge of a method is closed under implication: 
    K    µ(h -> l) -> (Kµh -> Kµl).

  • (4) also called the KK-thesis or the axiom of self-awareness. If a method knows a hypothesis, then the method´knows that it knows the hypothesis: 
    K    µh -> KµKµh.

  • (5) also called the axiom of wisdom or negative introspection. If a method does not know a hypothesis, then the method knows that it does not know the hypothesis: 
    ¬K    µh -> Kµ¬Kµh.

  • (AFK) also called the axiom of futuristic knowledge. If a method knows a hypothesis, then the method knows the hypothesis in all future: 
    K    µh -> GKµh.

Epistemic Strength. Epistemic strength of a knowledge operator is determined by the epistemic system corresponding to the operator. The systems considered here most notably include the following in order of increasing strength:

  • KD4 is self-explanatory.

  • S4=(T)+(K)+{4}.

  • S5=(T)+(K)+(5).

There are also axioms and axiom systems for alethic and tense logic. These will be introduced later.

MODAL LOGICS

Modal logic including in particular, alethic, tense and epistemic logic has grown into a mature field of research with a wide range of applications in both philosophy, linguistics and computer science. Nevertheless 40 years ago, Dana Scott in his famous article "Advice on Modal Logic'' pointed out a problem for the entire modal logic endavour which still to this day largely holds true:

Here is what I consider one of the biggest mistakes of all in modal logic: concentration on a system with just one modal operator. The only way to have any philosophically significant results in deontic logic or epistemic logic is to combine these operators with: Tense operators (otherwise how can you formulate principles of change?); the logical operators (otherwise how can you compare the relative with the absolute?); the operators like historical or physical necessity (otherwise how can you relate the agent to his environment?); and so on and so on. (Scott 70)

The criticism is obviously quite severe both theoretically and for applications and thus one is in need of a system able to handle more than one operator. In recent years however modal logicians in general have begun to take Scott's perennial criticism into account. In branching tense logic one has begun to mix alethic and tense logical operators (Braüner 98a), Braüner 98b), (Zanardo 96) and the valauble and influential work by notably Fagin, Halpern, Moses and\ Vardi (Fagin et al. 95} combines epistemic operators with tense logic.

COMBINING OPERATORS - AN EXAMPLE

Let us take a closer look at the approach pursued by (Fagin et al. 95) as it relates directly to the second aim of this paper. In multi-agent systems agents may range from a collection of poker-players in a poker-game to robots on an assembly line. Now every agent in a given system is in a certain local state at a particular point in time where the local state includes all the information available to the agent modulus "now''. Thus the whole system is in some global state specified as the tuple of the agents' local states together with whatever additional information is relevant to the system not included in the tuple. This additional information is called the environment. Clearly, such a system is a dynamic entity given the local and global states at particular times. To model the dynamics of a system, a run is defined over the system. In particular, a run is defined as a function from time to global states where time ranges over the natural numbers. Then a run may be conceived as the description of the behavior of the system for some possible trial or execution. Points play a significant role as pairs consisting of a run and time. Then for every time, the system is in some global state which is a function of the particular time. But since the global state is defined as the tuple of local states, the local state of some particular agent can be extracted at that point. Formally the system can then be defined as a collection of runs rather than agents, where what is modelled are the possible behaviors of the system on possible executions or trials. It is often assumed that the system is synchronous in the sense that the agents involved know what time it is according to the global clock.

It turns out that such a system can be viewed as a Kripke-structure with an equivalence relation on points. Hence the accessibility relation is specified with respect to possible points: A point is possible given some current point, if the agent has the same local state at both points. So knowledge is determined by the agents' local state. Truth of a formula is now defined relative to a point. This feature also leaves room for the introduction of tense operators besides the epistemic operator KiA where i is an agent such that i knows proposition A. More specifically pertaining to the temporal dimension, in the framework specified they introduce the following tense operators: A universal future-tense operator G can be defined as truth of a formula modulus the current point and continuous truth of the formula for all later points; dually for the singular future-tense operator F defined as truth of a formula at some future point. Two other tense operators include the next-time operator which states that some formula is true at the next step and the two-place operator (_U_) until which for two arguments A and B says that A U B is true if A is true until B is true. No past-tense operators are introduced.

Again, the combination of knowledge and temporal operators then allows for claims about the development of knowledge in the system for a linear conception of time. A particularly interesting axiom that (Fagin et al. 95) consider in a synchronous system is the following one: 

KiA -> GKiA

which one for brevity may call the FHMV-axiom in reference to Fagin, Halpern, Moses and Vardi. The axiom says that if an agent i knows A at some particular point, then he will know A at all points in the future. We have independently entertained the same axiom called the AFK-axiom separating scientific realism (See  EPISTEMIC AXIOMS AND SYSTEMS above). There are theorems which go to show that the FHMV-axiom is possible to validate under exactly the same conditions under which the AFK-axiom is possible validate  - namely when attention is restricted to what (Fagin et al. 95) call stable hypotheses and what we call absolute time invariant empirical hypotheses (See SYSTEM I, AFK and Hendricks 00, The Convergence of Scientific Knowledge - a view from the limit, Hendricks & Pedersen 00, "Modal Logic, Operators and Methodology, Hendricks & Pedersen 01, Operators in Philosophy of Science).

Yet another feature of this epistemic logic should be noted. In the multi-agent systems it is also sometimes assumed that the agents possess certain epistemic properties - in particular what the authors call perfect recall. The idea of perfect recall is that the interacting agents' knowledge in the dynamic systems may grow while the agents still keep track of old information. In other words, an agent's current local state encodes all that has happend so far in the run or altenatively that the current local state of the respective agents encodes the entire local-state sequence or history. So perfect recall is sort of a methodological recommendation asking the method to remember it's earlier epistemic states. Thus, contemporary epistemic logic has started to consider the behavior of the agents involved in the knowledge acquistion process (See Methodology)

EPISTEMIC OPERATORS AND METHOD

Since it is a scientific method or agent which acquires knowledge it is indeed a natural consequence that whether the method obtains knowledge or not should be acutely sensitive to the methodological recommendations or programs commands that the method may choose to obey}. In classical epistemic logic (Hintikka62), (Lenzen78), the method does nothing besides indexing the accessibility relation but has otherwise no role to play in the actual validation process. In turn, the method is hardly epistemic at all since there is nothing particularly epistemic about indices.

Modal operator theory enjoys two desirable features pertinent to the issues raised immediately above:

Modal operator theory provides a new unifying modal logical framework in which both alethic, tense and epistemic operators can be defined painlessly and thus realizes what Dana Scott has long wished for and hence the reason for the label "modal operator theory".

Modal operator theory has at its base rather than as a derivative the idea that whatever epistemic axioms and epistemic systems are possible to validate for some epistemic operator is acutely sensitive to the methodological behavior of the agent involved.

FURTHER READING

There is jungle of literature on epistemic logic, especially from conference proceedings and transactions. Valuable classical books on the subject include (Hintikka 62) and (Lenzen78). In recent years epistemic logic has been studied quite intensively by computer scientists rather than philosophical logicians. For an overview, see (Gabbay et al. 95) and notably (Fagin et al. 95) the latter providing a combined textbook and research monograph with both introductory material and fairly advanced computational applications. 

Tense logic, just like epistemic logic has numerous sources also due to the many proceedings and transactions. Nevertheless, important books with overviews and not too technical include (Gamut 91), (Øhrstrøm & Hasle95). Then for a thoroughly technical treatment of tense logic refer to (Burgess  84). 

By far, alethic logic has been the field of modal logic which has received the greatest attention. This, once again, implies an endless list of valuable contributions. But perennial texts include (Hughes & Cresswell 68), (Chellas80), (Bull & Segerberg84) and a new technically very elegant presentation by (Chagrov & Zakharyaschev97) with many interesting novel applications. 

Below or above computational epistemology and modal learning theory lie some mathematical machinery. The mathematical machinery of the current book has, as opposed to other texts which draw inspiration from formal learning theory and computational epistemology, not been emphasized here because this volume is intended for a broader audience with more philosophical rather than technical interests. Nevertheless, the reader interested in the technical parts, but perhaps unfamiliar with the basic mathematical apparatus, should consult the following three disciplines for a thourough understanding of the whole subject matter and of how the philosophical results come about.

  • Mathematical Logic: There are many good texts in mathematical logic but a fine and complete presentation of the most important themes in mathematical logic is to be found in (Bell & Machover77) which include everything from soundness and (in-)completeness theorems for sentential logic and full first order logic over Boolean algebras, filters, ideals etc. to recursion theory.

  • Recursion Theory: As noted immediately above, recursion theory may be viewed as one of the main extensions of mathematical logic. An elementary introduction to recursion theory, the theory of computability and their relations to logic is to be found in (Epstein & Carnielli 89). Then, less wordy but with largely the same content refer to (Cutland 80). The two highly estimated and advanced texts in recursion theory include (Rogers 87}) and (Odifreddie89).

  • Topology: Part of the formal framework of this book rests on a particular topological space called the Baire space. It is not really used for anything here as it is in for instance (Kelly 96}) and (Hendricks & Pedersen 01). All the same and for a reader who appreciates a complete picture, an elementary introduction to topology is found in (Mendelson 87). More advanced texts, some of which are advanced way beyond what is needed here, include (Kelley 55}), (Cech 69), and (Moschovakis 94). Besides (Kelly 96), (Schulte & Cory96) likewise give the interpretation of point-set topology as the theory of inquiry for logically omniscient agents with no limitations on memory capacity. This view turns topology into a powerful tool for epistemology. The latter paper is a friendly introduction to the connection between topology and epistemology. The paper features topological interpretations of Popper's falsifiability criterion.