SYSTEM I - S4

The arrows from realism/anti-realism, AFK, epistemology and synchronic/ diachronic principles boxes point to the box of S4 ...

We are interested in epistemic strength. There exist 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 -> ¬K_µh.

• (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) -> (K_µh -> K_µl).

• (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 -> K_µK_µh.

• (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 -> K_µ¬K_µh.

• (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 -> GK_µh. ... (->)

Now the 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).

In (Hendricks 00, The Convergence of Scientific Knowledge - a view from the limit, Hendricks & Pedersen 01, Operators in Philosophy of Science) it is shown that the definitions of knowledge for both realism and anti-realism validate S4 depending on discovery and assessment methods respectively:

Theorem: The definitions of knowledge based on discovery validate S4 if the discovery method has consistent expectation.

Theorem: The definitions of knowledge based on assessment validate S4 if the assessment method is epistemically sound.

Thus again

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.

PERMISSION, RESTRICTION AND NEUTRALITY

Recall the distinction between

To recapitulate: 

• Permission: A methodological recommendation is} permissive if it is conducive towards validating epistemic axioms and systems for a one shot learning experience in the limit. 

• Restriction: A methodological recommendation is restrictive if it is an impediment to validating epistemic axioms and systems for a one shot learning experience in the limit. 

• Insufficiency: A methodological recommendation is insufficient if it is neither a persmission nor restriction for validating epistemic axioms and systems for a one shot learning experience in the limit.

A methodological recommendation is trivially permissive "(+)" if adding it to the the method makes no difference to the validity of the axiom. A methodological recommendation is genuinely permissive "+" if the recommendation is required for the validity of the axiom.

Now one may classify the methodological recommendations with respect to how they fair concerning the validation of epistemic axioms. 

The symbol (/) indicates that the principle is impossible to validate given the fundamental convergence thesis - see the box of Academic Skepticism ... (->)

1ST VS. 3RD AND SYNCHRONIC/DIACHRONIC:
AN EXAMPLE

Eecall the distinction between synchronic and diachronic implicational principles. From the convergence point of view, the KK-thesis is a complicated affair. Not having selfawareness initially supports William James' distinction between absolutist's philosophy and pragmatism. One may not infallibly know when one has converged to the fact that one has converged to the correct answer. Formal learning theorists like Martin and Osherson are of the same opinion:

This does not entail that knows he knows the answer, since (as observed above) µ may lack any reason to believe that his hypotheses have begun to converge. (Martin & Osherson98), p. 13.

Conversely, currently a concept of knowledge is on the one hand introduced based on the idea of limiting convergence, and yet on the other hand limiting convergence is often cited as one of the primary reasons for not validating the KK-thesis! But if one wants to validate the KK-thesis and simultanously entertain a limiting concept of knowledge how is it then possible to have it both ways? This paper is about how to have the cake and eat it too.With these two principles in hand return to the question of how to have the cake and eat it too: A notion of knowledge is introduced based on limiting convergence. Then again To recapitulate, the limiting convergence of knowledge is often cited as one of the primary reasons for not validating the KK-thesis and the reason is this: A method may of course know something in the limit of inquiry but since it may be unable to detect the modulus of convergence the method will accordingly know but not necessarily know that it knows in the limit. How to have the cake and eat it too?

To validate the KK-principle one may try to argue that the sentiment not to validate it is based on an equivocation between knowing and knowing when one begins to know. The idea would be that knowledge that one knows does not require the method to be able to say, reliably, when it has found the correct answer. Intuitively, knowledge that one knows at time n requires only that one has already stabilized to believing that one knows at time n and that one would have stabilized to the correct answer about whether one knows at n had things been otherwise. Thus, to validate the KK-thesis, it suffices that the method continues to believe that it knew A for as long as it believes that A. Then whatever the method knows, the method knows that it knows. This requirement is clearly stronger than Moore's autoepistemic principle (Moore 59), which requires that one believe that one knows what one currently believes. Thus, the KK-thesis is here a diachronic principle as opposed to Moore's synchronic principle. To that extent it indeed seems questionable. The agent learns that its earlier evidence was spurious but now receives convincing evidence so that his belief is uninterrupted. This line of defence for the validity of KK is up to method rather than the world, and hinges on what seems to an unwarranted confidence in the status of one's own earlier beliefs.

Our modal operator line of defense for KK is similar to the above and yet different. To have knowledge of a hypothesis is to have reached a modulus of convergence after which the agent continues to project the conjecture over all later times and possible worlds. If you know A, then A is true. So knowledge of A entails A. By iteration, knowledge of knowledge of A entails knowledge of A. Translating from a logical paradigm, set-theoretical relations among propositions (or sets of possible worlds) are represented by logical connectives and relations: Entailment is set inclusion, conjunction is intersection, etc. Hence 

[K_µK_µA] subset [K_µA] subset [A]        (**)

To have knowledge of knowledge of a hypothesis A must be to reach a modulus of convergence only after} convergence to knowledge of A has arisen. Given (**) knowledge of knowledge of a hypothesis is a subset of knowledge of a hypothesis so knowledge of knowledge can only happen once knowledge of the hypothesis has obtained. In other words, one has to force strategically to validate KK and hence consult methodology for a strategy.

Suppose the agent obeys a methodological recommendation asking it to have perfect memory in the sense that the agent remembers the past evidence. Then KK becomes impossible to learn. Perfect memory demands that the method starts to force for KK at a time l in worlds prior to the modulus of convergence has been reached for mere knowledge of A which only happens at a later stage n > l. Forcing for KK prior to knowledge of A is impossible because there may exist a world or circumstance w₂ required for KK veering off the actual world w₁ before knowledge of A has arisen. Since the agent has perfect memory he will attempt to crawl below n to get this world w₂ in the KK-conjecture. If he crawls below n and captures w₂, then w₂ will be in [K_µK_µA] but not necessarily in [K_µA] and thus violating (**) above. Suppose on the other hand, that the agent obeys the different methodological prescription of consistent expectation. Recall that perfect memory and consistent expectation are inconsistent, thus the two constraints cannot govern a method's behavior at the same time. Consistent expectation implies that any additional convergence and forcing in terms of knowing that one knows A takes place at a point in time n' > n > l where the method has converged to A and hence forces already. Adding knowledge to knowledge requires forcing and then later some more forcing. Thus, again the validity is up to the method rather than to the world, and hinges on the diachronic feature of the KK-principle but given consistent expectation not unwarranted confidence in the status of one's own earlier beliefs - the method already knows.

One would have to stand outside to see this - thus the KK-principle is validated given consistent expectation from the 3rd person perspective and in recognition of the fact that it is a diachronic as opposed to a synchronic implicational epistemic principle.