In Episode 47, Baltag and Matt briefly discuss what they call the ‘KK principle,’ or the ‘principle of positive introspection.’ The basic formulation of this principle is:
(KK): If I know that p, then I know that I know that p. (Where ‘p’ is some proposition.)
For example, if I know that 2+2=4, then I know that I know that 2+2=4. A close cousin of the ‘KK principle’ is what we’ll call the ‘K-not-K principle,’ or the principle of negative introspection. The basic formulation of this principle is:
(K-not-K): If I don’t know that p, then I know that I don’t know that p.
For example, if I don’t know who the prime minister of England was in 1965, then I know that I don’t know who the prime minister of England was in 1965. (Harold Wilson, for those who are curious.)
Baltag and Matt quickly proceed to show the implausibility of these principles for our ordinary notion(s) of knowledge. For example, if K-not-K were a principle of the ordinary notion of knowledge concerning the participants in Socratic dialogues, the whole procedure would be unnecessary. That is, Socrates’ interlocutors would already know that they did not know what justice was (for example)—there would be no need to show them this by means of Socratic dialogue.
Likewise, we can note that there are occasions on which the KK principle itself does not seem to reflect our ordinary notion of knowledge. To take an example that some readers may have experienced, recall a particularly painful physics exam from high school or college. You may have left the room exclaiming either to yourself or others: “Oh man, I totally tanked that exam—I had no clue what I was doing!” Now, in some cases, we might imagine that this is simply a defense mechanism; you have some degree of confidence that you solved the problems correctly, but you’re not absolutely sure, and so are trying to diffuse your own (or others’ expectations). But we can imagine a case in which you genuinely have no idea whether you solved the problems correctly, and let’s focus on this case. Then the exam is returned, and you discover that you did very well—perhaps you in fact aced it, and solved every problem perfectly. You may remark to yourself (or others): “Well, I guess I knew the answers to these problems all along.” But, before got the graded exam back, you didn’t know that you knew how to solve the problems. So it seems that the KK principle may not reflect our ordinary notion(s) of knowledge either.
However, Hintikka, who introduced the KK principle, does not take either the Socratic dialogue or physics exam cases as counterexamples to the principle—simply because these are examples of the notion of knowledge that the principle is meant to apply to. The question arises, then, what sorts of knowledge does (or should) the KK principle apply to? This is addressed briefly in the podcast, but I’d like to draw things out a bit more. (I should note that I’m drawing heavily from Hintikka’s “’Knowing that One Knows’ Reviewed” (Synthese 21, 1970), which is a response to his introduction of the KK principle in his earlier book Knowledge and Belief.)
Baltag and Matt briefly gesture towards one sort of knowledge that the KK principle might apply to—namely, knowledge of mathematical truths. Hintikka offers a more general label for this notion of knowledge: ‘the philosophers’ strong sense of knowledge.’ What is this sense of knowledge? Well, to simply sketch an intuitive picture, we might consider a case of mathematical proof. In a mathematical proof, one proceeds from a set of axioms and inferences rules and arrives at a theorem, which is derived from the axioms by means of the inference rules. Crudely put, one might say that ‘knowing that a mathematical theorem is the case’ means: assuming a set of axioms and inference rules, the theorem necessarily follows, so that upon applying the inference rules in such a way to the axioms, I acquire knowledge of the theorem.
What makes such cases knowledge in the strong sense? Well, we might say that this sort of knowledge is indefeasible. That is, knowing that a mathematical theorem is the case (in the strong sense) entails that there is no evidence that one would admit against the proposition of the knowledge claim (i.e.: the mathematical theorem). The possibility of a counterexample is ruled out in virtue of knowing that the mathematical theorem is the case. This contrasts with our ordinary notion of knowledge. For example, I might say that I know (in the ordinary sense) that the bus is going to arrive at 4:45, but my knowing this does not rule out evidence that the bus won’t come at 4:45 (perhaps the best evidence being the failure of the bus to arrive at this time). Knowing in the ordinary sense doesn’t rule out the possibility of counterexamples, while the ‘strong sense’ does. (I won’t attempt to offer conditions for when one is justified in laying claim to the strong sense of knowledge—I just want to get the notion out there.)
How does this link up with the KK principle? This strong sense of knowledge amounts to the claim that all the possibilities of, say, the mathematical theorem being false are ruled out—the strong sense of knowledge is ‘conclusive.’ But if this is the case, then there is no possibility of my knowing that the mathematical theorem is the case being false—for what would make this false is for the mathematical theorem itself to be false, and we’ve already ruled this out by stipulating that we know it in the strong sense: we get the same ‘conclusiveness’ about my knowing as we do about the mathematical theorem itself. So knowing in the strong sense seems to imply knowing that one knows (in the strong sense).
Of course, one immediate worry is that there is in fact nothing that we really know in this strong sense (worries can be constructed for the case of mathematical truths that have been mentioned above and in the podcast—I’ll leave these worries aside for now.). If so, it becomes unclear what the value of the KK principle really is: if the only sense of knowledge that it applies to is so strong that we can never attain knowledge of this sort, it seems to render the principle idle. One neat (at least, I think so) way of responding to this worry is to show that the KK principle also obtains for the weakest possible notion of knowledge: true belief (this position was originally advocated by Barense, and is discussed in section XIV of the Hintikka paper).
True belief is the weakest sense of knowledge because it demands no evidence whatsoever; in order for me to know something in this sense, I simply need to have a belief that is true (which we might take to mean: ‘corresponding to the world in an appropriate way’). Hintikka notes that a literal interpretation of ‘knowledge as true belief’ means that no evidence whatsoever is needed to ground it (I simply need to believe something that turns out to be the case). This puts knowledge as true belief in parallel with the strong sense of knowledge discussed above, as both rule out the possibility of being overturned by evidence: the strong sense because my knowledge is ‘conclusive,’ the weak sense because there is no notion of evidence at play. A parallel argument for the KK principle in the case of knowledge as (merely) true belief can thus be provided (though I will leave this as an exercise for the reader).
We have seen that if I know that p, I know that I know that p, if we construe knowledge either in a very strong or very weak sense. Of course, as Matt and Baltag point out in the episode, neither of these correspond to our ordinary (moderate) sense of knowledge. Why take interest in the KK principle (or the K-not-K principle)? I’ll leave Hintikka to answer this question: “The ultimate court of appeal in deciding whether a logical principle governing some give concept is acceptable is not ordinary usage, however regimented, but rather whether the principle helps the concept in question to serve the purpose or purposes it in fact is calculated to serve in our conceptual repertoire, and whether these purposes are themselves worth our effort. By spelling out these purposes and by using them to evaluate various logical principles an analyst can perfectly well disagree with ordinary usage and even attempt to reform it” (“‘Knowing that One Knows’ Reviewed,” 141).