Here is a puzzle. Well, it's probably not really a puzzle but merely a display of my lack of understanding of a standard Kripke model. However, here goes:
In a standard possible world Kripke model, in normal modal logics, S denotes the entire set of worlds and R denotes the binary relations that obtain among these worlds depending on the system in question. Now, following standard notation, a binary relation is written R(w, w1) - which is read something along the lines of "world w has access to world w1". In epistemic logic (as in standard athletic logic), the possible worlds are alternatives that could obtain, and thus be the actual world. In other words, if an agent A is considering a proposition p, and is unable to decide whether or not p obtains, then both the p world and the not-p world are possible with respect to A’s knowledge. That is, the agent A has access to both the p world and the not-p world.
Now a paradigm of epistemic logic is that factivity only obtains insofar as a reflexivity assumption is imposed on the accessibility relation. Although logicians and epistemologist quarrel over which axioms and intuitions an epistemic logic should satisfy, factivity is probably the most uncontroversial of these (for instance, compare to the rule of necessity, closure / the axiom for knowledge, KK, Negative Introspection and so on.) Now, the question (and my puzzle) is the following: Why is reflexivity needed to obtain factivity.
Consider an example: Mary is wondering whether or not it is raining in Paris and this is the only thing Mary is considering. As such, two worlds are possible with respect to Mary's knowledge, namely world w1 (the world where it is raining in Paris) and world w2 (the world where it is not raining in Paris). These are epistemic alternatives open to Mary, and as such, she is unable to decide which of these worlds, she is presently in.
Now, what would the semantics for this example look like. What does S contain in this particular example? Let's suppose that S contains only the two worlds that are possible with respect to Mary's knowledge, such that S={w1, w2}. What will the accessibility relation (lets for simplicity's sake reason in T) look like then? Let me start by suggesting the following: R={(w, w1)(w, w2)(w,w)(w1,w1)(w2,w2)}. However, there is obviously a problem here? Which world is the world w? w is not a member of the set S. In other words, there is no base world from which Mary is reasoning? And if we stipulate that there is such a world, what is the truth-value of "it is raining in Paris" in that world w? Nonetheless, having a base world (or perhaps “an actual world”), would help explain why reflexivity is needed for factivity. For instance, Mary knows that her name is Mary, and by the definition of knowledge, it obtains in all worlds accessible to her. If the only worlds in the set are w1 and w2, then Mary has direct access to both worlds, and thus does not need reflexivity. Conversely, the reflexivity assumption seems necessary to guarantee that in the world from where she is reasoning (the stipulated world w), it also obtains that her name is Mary. This is the world she needs reflexivity to access. But, there is obviously something wrong here. If we accept that there is a base world, from which Mary is reasoning, then it remains entirely unclear what the truth-value of "It is raining in Paris" is? Since all propositions are assigned a truth value in all worlds (given the interpretation function i), this is obviously wrong. However, if we do not accept a base world, from which Mary is reasoning, then there is no clear answer as to why reflexivity is needed for factivity?
A friend of mine proposed the following answer: There are two worlds, w1 and w2. Depending on which world she is presently in (which is undeterminable to her), reflexivity is needed to access that particular world. For instance, if it is in fact raining in Paris, then she is in w1, and then she needs reflexivity to ensure, that the proposition "My name is Mary" obtains in that world. However, I find this suggestion inadequate. That would entail that in order to know that her name is Mary, she must also know whether or not it is raining in Paris. If she cannot determine whether or not it is raining in Paris, then she can potentially be said to know something which is false – which is of course absurd? An even further consequence of this suggestion seems to be that the world w1 (where it is raining in Paris) has access to the world w2 (where it is not raining in Paris) and this also seems quite strange. How can these worlds be accessible to one another?
But perhaps, the friend of mine was on the right track, I am not sure. There must be something that I've misunderstood. Perhaps, my understanding of the accessibility relation is misguided, but I cannot find any clear answer to this problem.
Maybe a more complex example could illustrate why reflexivity is needed to ensure factivity, but I cannot come up with any example that demonstrates this.
Any suggestions are more than welcome.
I've started writing my paper without really knowing where it will end. I just needed to get something on paper to make sure that I've finally understood the basic framework of epistemic logic. However, one problem is still nagging me, and I hope that someone out there can suggest an answer.
It seems entirely unclear to me, why one would want to include the axiom of positive introspection in an epistemic logic modeling human knowledge (let alone the axiom of negative introspection, but leave that be - that is obviously an axiom adopted by computerscientist for pragmatic reasons). Given that the KK-thesis faces a variety of difficulties, why not just leave it out, why not give up transitivity? The only remarks I have found on why some logicians are inclined to add transitivity to their system is that "this will add the positive introspection axiom, which seems intuitively plausible". Well does it? I'm not so sure. Certainly in some circumstances, when I know, I also know that I know. But I think there are plenty of counter examples, depending on what (tripartite) analysis one employs to define knowledge. I can't see the intuitive plausibility in an entirely unconstrained KK-axiom in epistemic logic. So I'm thinking that there must be a better reason for wanting to incorporate transitivity into a formal system of knowledge, but I really have no clue what it is?
Could the reason be that since we sometimes do know that we know, then logicians feel that the system becomes to weak if transitivity is left out? Suggestions are more than welcome.
So, tomorrow I’m leaving for Stockholm for the weekend. I haven’t been to Stockholm since I was 15, so it should be good. I’m still searching desperately for relevant literature on the KK-thesis and still without any luck. Although there is a lot of literature on epistemic logic, a great deal of the articles are excruciating technical with a primary focus on knowledge distribution and computer related attributions. In general those papers concern epistemology as a philosophical discipline very little – at least in my impression. More specifically I’m thinking of Vardi, Halpern, Moses, Fagin, van der Hoek, J.-J. Ch. Meyer, and so on. On the other hand, there is the more old school stuff from Hintikka, Lenzen, Lemmon, Hilpinen, Castañeda, Ginet and probably some more. This stuff certainly has a stronger relation to epistemology as well as philosophy of language and philosophical logic. However, it is quite old, which sometimes becomes somewhat evident (for instance, I think it was Rilpinen who writes “In a very recent paper by Edmund Gettier” – recent paper. Well things have certainly changed drastically since then). However, all these philosophers and logicians probably have much more to offer than I am able to appreciate, so my somewhat harsh comments should be read as a consequence of frustration.
All I really want, is some up to date literature on the KK-thesis along the lines of Tim Williamson’s argument in chapter 5 (I think) of “Knowledge and Its Limits”. But, as I said, I haven’t been able to find anything – yet. If you have any suggestions, please let me know, I would be very grateful.
Today, I gave my presentation of Dretske. It went pretty well, all things being equal, although I found myself somewhat out of place presenting Dretske to a bunch of linguistic students. I have a strong sense that majority of the faculty (and as such the grad students as well) at the linguistics department at the University of Copenhagen just find what philosophers are doing in language, logic and epistemology to be absolutely ridiculous. Here I was hoping for other formal approaches to language, but guess not. I’ll have to wait until next semester when I’ll be at UCLA (hopefully).
This LaTeX-thing is finally working out pretty well for me. Although I still have great difficulties understanding how to install new packages, except when it can be done just by using the TeX live - 'Add Tex Package' option in my start menu. However, I've mastered some of the basic stuff - or perhaps not mastered, but understood - and I'm really starting to appreciate it. The bibtex feature is just great, and features such as \tableofcontents are really convenient.
The last couple of days, I've been trying to figure out how the type-setting program Latex works. I really like the lay-out on papers written in Latex, but my god - what a hazzle. It has taken me like five hours to write the few notes that I have prepared for my talk on Dretske this Thursday. I don't know if any of you out there actually use this program, but is there really no easier way than to write the entire code yourself?
Even though MS Word et al. cannot typeset as fancy as Latex, WYSIWYG programs do have very obvious qualities. Still, I can't help feeling somewhat impressed with myself.
If there are any experts out there, I would like to know one thing.
How do I divide the text into three separate columns, but only for perhaps a single paragraph. To spell it out: When writing some logical proof or argument, I want line numbers 1,2,3.. in column A) and in column B) I want my premises and inferences, in the last column C) I want the premise and inference notation. The problem as it is now is that the space between A, B, and C varies for each line. Three perfectly left-aligned columns would solve this problem, such that each premise is written in the exact same position as the previous only on the next available line. Anyone, who understood this? Anyone who even has a possible solution?
By the way. Those notes on Dretske are for a lingustics course I'm attending this semester. None in the class has ever been taught any logic, which explains why I included the basic stuff. Half the arguments are not written out in full form, but they should (hopefully) be valid. I just didn't want to add to the complexity by making the arguments even longer or by being forced to explain inference rules for semantic tableau or the likes. Intuitions will do.
Last week, Blackwell finally got round to delivering Tim Williamson's 'Knowledge and Its Limits', which I ordered approximately two weeks earlier. I have heard a variety of people (DeRose, Patrick Greenough and a couple of my own professors) speak very highly of it, and at the Namicona Conference, Tim Williamson's talk, was without a doubt the highpoint.
I've been reading fairly intensely in it for the last few days. Now, this book is by no means an introduction to epistemology. To keep up pace with Williamson one needs a fairly comprehensive understanding of a variety of topics within epistemology, logic etc. And thus, at some stages during my reading, I must admit that I lack the expertise to understand everything being said.
However, leaving aside my incapabilities in various philosophical areas, this is probably this most exciting book I've ever read in epistemology. Williamson writes elegantly and in general with great clarity. I haven't read the whole thing yet, but especially the chapters on Anti-Luminosity, Margins and Iterations, Sensitivity, and Scepticism are excellent. Undoubtedly, anyone with an interest in contemporary epistemology ought to read this book.
I offered to give a presentation of his arguments against luminosity and his rejection of the KK-principle to my study group this Tuesday. I probably shouldn't have, or at least given myself a better time frame. Despite it being a great and especially interesting read, it takes a long, long time for me to get things right. Hopefully, my presentation won't suck too much.
In January at the Namicona Conference on Mentalism and Modalism in Contemporary Epistemology, Jonathan Kvanvig gave a talk on ’Counterfactuals, Closure and the Value of Knowledge’. During the talk, Kvanvig made a short reference to some lectures that Kripke supposedly gave at some point during his time at Princeton. Kvanvig said that these were now referred to as 'The Nozick-Bashing Lectures' amongst the people who had actually read or heard the lectures.
Since I'm probably writing a paper on Nozick this semester, this obviously stirred my curiosity. After the conference, I checked with a couple of my professors, and the few people I know at various universities abroad, but only few of them had ever heard of these lectures, and none had ever seen them.
Finally, I decided to e-mail Kvanvig about it. He was very kind to reply that the lectures had, as expected, never been published, and that Kripke sometimes made copies for people, but made them promise not to distribute the material. My best chance, he wrote, is just to e-mail Kripke himself.
Now, it might seem a bit compulsive to hunt this desperately for these lectures, seeing as I have no idea if the content of them is actually worth the trouble. I know that there is plenty of material and discussions of Nozick's suggestions in epistemology, but the suggestive name of the lectures just makes me damn curious. Perhaps these lectures will be just as much an eye-opener as 'Naming and Necessity' was for me.
The whole thing also has an Indiana Jones feel to it. The hunt for the lost scrolls. But perhaps the hunt just ends here. I have not been able to find Kripke's e-mail anywhere. Yet another great mystery - unsolved...
I've just found a great paper on the history of Possible World Semantics (via Kai von Fintel). It's from the Journal of Philosophical Logic 2002, 31:2 - so you need access to Electra or something similar in order to read it. However, it's certainly worth the effort.
Jack Copeland: The Genesis of Possible World Semantics
Some of the paragraphs are fairly demanding and there is a lot of technical stuff - but the frequent anecdotes of various kinds make for a really entertaining and illuminating read.
I have been considering for some time whether I should start blogging in English instead of Danish. Since most of the papers I write are in English, and I’m going to California next semester, it seems natural to do this – if only to practice my English skills. However, as you’ve probably noticed, I’ve now decided to do so. Blogging in English on a .dk site might seem strange to some - and it probably is. But student loans don't add up to .com sites, so that just how it's going to be. The risk is of course that this will severely delimit the number of posts that I will be inclined to write. It does take an extra effort of me to write in English. Furthermore, my writing skills in English are of a questionable quality, so I hope my readers (non-existing or perhaps, subsisting) will bear with me.
Honestly one of the reasons, although a quite idiosyncratic one, for writing in English, is that I just got immensely tired of the Danish word “filosofi”. It has an irritatingly mundane and yet flaky sound to it. When Danes speak of philosophy, it is usually associated with a wide range of things that has absolutely nothing to do with the things that I am in fact interested in, nor the things that I actually spend time studying. The English word “philosophy” has a more scientific ring to it, at least in my ears. However, admittedly, this is almost certainly only due to my inexperience of the common sense association of the word among English speakers. But, ignorance is bliss.
Yet, this should not be understood as though I want the word to have a more romantic or intellectual ring to it. All I want is for the word to be intuitively understood as general term for a science. I want to avoid questions like ‘oh, so you’re philosophy student, well then what are your thoughts on Nietzsche?’ I’m sorry, but I don’t have any thoughts on Nietzsche, at least none that could possibly be of any interest to you, let alone anyone. Imagine the same happening to a student in physics or biology? ‘Oh a biology student, so what’s your opinion on photosynthesis?’ Unlikely indeed.
As for the vulgar headline, I'm sorry, it was just too obvious.
Thoughts, Arguments and Rants.
- Brian Weatherson (Philosophy, Brown - snart Cornell):
Små anekdoter og en række posts om nuværende forskning og interesser. En masse spændende artikler om semantik, logik, mind, game theory mm.
- Kai von Vintel (Linguistics, MIT):
Posts om semantik, pragmatik, modalitet, logik mm. Masser af gode noter fra hans undervisning og links til en masse spændende sekundære artikler. Mest logik og semantik herinde, men spændende.
- Brian Leiter (Law & Philosophy, Texas)
Brian Leiter skriver en masse om rankings af universiteter i USA, ajourfører hvilke professorer (i filosofi og jura) der får tilbud fra andre universiteter, hvem der accepterer tilbud, og hvem der afslår. Det kan jo være meget sjovt at følge med i.. eller?
- Online Papers in Philosophy - (Weatherson)
Denne side opdaterer dagligt, hvilke forskere der udgiver nye artikler papers mm. Man kan også følge med i de fleste store journalers (Mind, Ethics, Philosophical Quarterly, Australasian mm) seneste udgivelser osv. Altid godt lige at checke, om der er kommet noget nyt spændende.
- The Epistemology Page (Keith DeRose)
Nyheder om konferencer, nye papers og artikler, noter fra kurser i epistemologi, og rankings af engelsksprogede epistemologi departments. Der er også links til en række praktiserende epistemologer.
Gennem de seneste par uger har jeg virkelig fået læst en del. Jeg føler hele tiden, at jeg har totalt travlt, men jeg er allerede en måned eller noget lign. foran i mit pensum, så det har jeg egentlig ikke. Men problemet er, at jeg skal skrive to store opgaver det her semester, og jeg kan ikke rigtig slappe af, før jeg i det mindste er kommet i gang med at skrive. Desværre kan jeg ikke bare begynde at skrive, da jeg ikke rigtig ved noget om de emner, som jeg skal skrive om – ergo stresser jeg lidt allerede, hvilket egentlig er absurd.
Men jeg er dog så småt ved at spore mig ind på et par emner.
Det ser ud til, at jeg skal skrive et eller andet forhåbentlig intelligent om Nozick’s tracking teori. Jeg havde egentlig fået det indtryk i mine unge dage (red. unge studie dage) – at ingen rigtig tog teorien alvorligt, men det er tilsyneladende ikke tilfældet. Det gik op for mig på Namiconas konference (Modalism and Mentalism in Contemporary Epistemology), at den faktisk stadig er ret omdiskuteret. Åbenbart, fordi mange har haft det indtryk, at Nozicks teori var en effektiv strategi mod skeptikeren, dog med den lidt dyre omkostning at skulle afvise closure. Og det har vist været den primære fokus på teorien – altså hvordan det kunne undgås.
I sidste uge læste jeg Fred Dretske’s ’Epistemic Operators’ fra 1970 (’The Journal of Philosophy’), og den åbnede virkelig mine øjne for emnet. Dretskes strategi er nemlig den samme som Nozick’s: at undgå skeptikeren ved at benægte closure. Eller, det vil mere præcist sige, at Dretske benægter anden præmis i det klassiske skeptiske scenarie (nr. 1 i det lille illustrative billede). At, hvis jeg ved, at jeg sidder foran computeren, så følger det, at jeg også ved, at jeg ikke er en hjerne i et kar. Men jeg kan, indlysende nok, ikke vide, at jeg ikke er en hjerne i et kar, altså – ved modus tollens – kan jeg ikke vide at jeg sidder foran min computer. 
Uden overhovedet at have sat mig udførligt ind i diskussionerne, synes jeg dog det lugter noget uldent at afvise closure. Umiddelbart forekommer det nemlig, at der er masser af sammenhænge, hvor der følger ting af ens viden, som man ganske nemt kan indse og som man endvidere ofte handler på baggrund af. Men det bliver jeg forhåbentlig klogere på, i løbet af det næste stykke tid.
I det fag jeg følger på engelsk/lingvistik (modalitet, evidentialitet og epistemicitet) tror jeg til gengæld, at jeg vil skrive noget om konstekstualistiske forsøg på at forklare sætninger med epistemiske operatorer. Bl.a. læste jeg for et par dage siden et paper af Keith DeRose, som ser ud til at være oplagt at tage udgangspunkt i. Spørgsmålet er nok mest, om jeg kan få en tilstrækkelig sprogvidenskabelig vinkel på emnet, og ikke fortabe mig for meget i epistemologiske og filosofiske problemstillinger. Det er jeg desværre ikke så sikker på. Men vi får se. Her på falderebet vil jeg varmt anbefale alle med bare en smule interesse i epistemologi at læse David Lewis’ paper ’Elusive Knowledge’, som jeg også omsider fik læst for et par dage siden. Den var virkelig velskrevet, også selvom man måske får en umiddelbar fornemmelse af, at kontekstualister gør tingene meget svære for sig selv.
