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An axiom that is often included in standard modal logic is

Kp => KKp

If we use epistemic modal logic, so that K translates as 'he knows', then recalling p stands for a proposition, we have that Kp translates as

he knows proposition p

so Kp => KKp translates as

He knows proposition p implies that he knows that he knows proposition p

On the face of it, without further reflection, it seems a reasonable axiom to posit: If I know how to kick a football, then surely I know that I know that I can kick a football.

However - suprisingly, according to wikipedia its 'unclear' whether this axiom should be accepted in epistemic modal logic.

This suggests that there is a natural (or artificial) example of such a counter-example; one suggests itself (at least to me), Monsieur Jourdain in Molieres play The Bourgeois Gentleman, says delightedly on discovering what the word 'prose' stands for:

MONSIEUR JOURDAIN: Well, what do you know about that! These forty years now I’ve been speaking in prose without knowing it!

Is this a valid counter-example (perhaps with some semantic or syntactic re-jigging)? And does it exemplify what can go wrong in asserting the above axiom?

  • You may enjoy including a layer of doxastic logic in here too, as @virmaior mentions belief in Type 2. – bright-star Nov 27 '16 at 7:18
  • @bright-star: I think Monsieur Jourdain was contemplating temporal logic ('these forty years'); but doxastic as being modal seems apposite. – Mozibur Ullah Nov 27 '16 at 7:20
  • I guess it depends on whether or not you consider certainty a quality of belief removed one step from knowledge, or as a quality of knowledge directly. I believe Bayesian statisticians, for example, operate with the former view, since degrees of belief (including unshakable certainty) can be asserted in the absence of any evidence. Anyway, I don't mean to digress. – bright-star Nov 27 '16 at 7:43
  • @bright-star: belief is certainly important here, and degrees thereof; but one thing at a time! – Mozibur Ullah Nov 27 '16 at 10:12
  • @mobileink: I don't really see the relevance of the link. – Mozibur Ullah Nov 28 '16 at 20:15
5

Everything hinges on how you understand "knowing."

First off, you're going to need to decide whether there's a distinction between episteme, techne, and phronesis (or if you prefer Gilbert Ryle's "know-how" and "know-that"). It seems like if we accept this, then there's most definitely a distinction between "knowing how to kick a ball" (techne /know-how) and "knowing that you know how to kick a ball" (episteme concerning facts in the world / know that).

Presumably, the intent in axiom relates only to episteme / know-that. But even then, I don't think the axiom follows easily. In part, because I take these claims to be reflective judgments about the world. Using the "I know that ..." or "I believe that ..." structure, I would say there's a few different types of cases we can suggest as counter examples:

Type 1 - Failure to realize knowledge of all members set means knowledge of the whole set.

If I know separately that 2,3,5,7,11,13,17,19,23,29,31,37,41,43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, and 101 are each prime numbers AND that there are no intervening prime numbers. Then I can draw the implication that I know all prime numbers less than 100. If I do draw this implication, then I know that "I know all the prime numbers less than 100" but until I do, it's not clear that I know this.

Similarly, if I know that Elizabeth II is the current Queen of England and George VI , Edward VIII, George V, Edward VII, and Victoria precede her, then I do in fact know the names of all British monarchs in the 20th century. But do I know that I know that? Here, I would need to know additional information about them to infer that.

-- if you don't take knowing to be an action then it's not clear if this will convince.

Type 2 - failure to make a comparison

If I know that Hydrogen is lighter than Helium and that Helium is lighter than Gold, I still need to connect these bits of knowledge to know that Hydrogen is lighter than Gold.

--again, this hinges on believing to know something is an act, rather than that such information is somewhere in your mind/brain.

Type 3 - pure reflective failure

I know my passwords to login to various online systems. But I don't necessarily know that I know my linkedin password. I may in fact know it but I would need to reflect on this to know whether I do in fact know it.

I'm sure there's several other types but it all hinges on how we define knowledge. If knowledge = the brain contains this fact, then knowing the fact is identical to knowing that you know the fact. If knowledge is an action, then until you do the act of knowing, you don't know that you know that you know. This may be difficult to capture because it's often trivial to do so when prompted.

  • To go with the example I suggested, then knowing how to speak prose does not mean that I know that I speak prose; this follows your techne and epistemic distinction; but isn't there an issue of private/public knowledge too? If I know how, then I know that I know how; but I may not know the name of that 'know how' - in this case prose; but this is a public name, but simply because it is public does not mean that I, in my private person, know that public name and what it refers to. – Mozibur Ullah Nov 27 '16 at 5:02
  • I agree public / private names would also be true -- but then the question is whether we identifying knowing something based on content , terms, of the conjunction of both. – virmaior Nov 27 '16 at 5:33
  • I don't think it at all follows that if I know how that I know that I know how and not merely on the basis of terms. Perhaps the way I would cut short ribs off a steer is exactly the right way, but I have no knowledge that this is how I would cut it. Or alternately, perhaps I would conjugate a Swedish verb correctly but I don't in fact know any Swedish (though I've been learning Danish and know some German). – virmaior Nov 27 '16 at 5:35
  • Ignoring terms, then those examples seem to be examples of your 'reflective failure': I know how (to conjugate swedish verbs) does not neccessarily mean that I know that I know how (to conjugate swedish verbs); if here I can 'reflect', then I can and do know that I know how (to conjugate swedish verbs). Interestingly, I think its possible to know Swedish without knowing how to conjugate Swedish verbs - a native Swedish speaker who doesn't know any grammar may not even know what a conjugation table is. – Mozibur Ullah Nov 27 '16 at 6:22
  • When I ask him to conjugate, he merely looks at me blankly; but if I give him examples to fill in, we see that he does know this implicitly; but explicitly doesn't - it has to be drawn out of him. – Mozibur Ullah Nov 27 '16 at 6:25

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