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24.244 Modal Logic: Epistemic Logic

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24.244 Modal Logic: Epistemic Logic 24.244 Modal Logic, Fall 2009 Prof. Robert Stalnaker Lecture notes 17: Epistemic Logic Epistemic logic began in 1962 with Jaakko Hintikka's classic book "Knowledge and Belief". The basic idea has been taken up in non-philosophical applications such as theoretical computer science, where it turns out to deliver a way to understand how a distributed system works at a certain level of abstraction. It is useful to think of such a system as a network of knowers sending signals – or package of information of some kind – to each other. Each member of the distributed system is in a certain information state which represents what it knows about the system and about what other members of the system know. One of the focuses of classical epistemology is the problem of error, or the relationship between knowledge and belief. Belief is a concept which is like knowledge, except that it does not have the success element built into it: belief can be false. One of the question of classical epistemology is: how do we protect ourselves against being wrong. In the computer context, there is a great interest in building systems that do not make errors. One of the main tasks in designing a system is error detection and error isolation: when something goes wrong, it should be discovered, and it should be isolated to avoid crashes so that other things can proceed. Thus, one wants to understand how it is that even when you're not perfect – even when you are wrong with respect to certain things – you still know something. The idea that one error leads to crash parallels the idea that if you are in error with respect to something then you don't know anything. When I was in grad school, the project of patching the justified true belief analysis of knowledge was fashionable, because it was so well defined as to what you have to do. People came up with byzantine counterexamples which motivate additional assumptions, to which people reacted with other counterexamples, until the theory became so complicated and the examples did not work anymore because no one had any intuition about them, and you were done. The project was interesting. Nice examples came out. But it came to have a bad name after a while, partly because it lost touch – as these counterexamples-patch games often go – with the original motivation: Why do we look for a definition of knowledge? What are the constraints on this definition? What problems do we seek to solve with our theory? Edmund Gettier's famous three-page paper – in which he challenges the justified true belief analysis of knowledge with counterexamples – was very close in time to Hintikka's work in formal semantics. Kripke developed modal logic and Hintikka was the one who applied framework to knowledge. Both formal epistemology and modal logic in its relevant application are concerned with the relationship between knowledge and belief. Both sought define knowledge in terms of belief. The idea is that knowledge is a concept that makes a truth claim, but at the same time also makes a claim about the subject's internal state. So it was a relationship between a subject with a certain perspective on the world on the one hand, and this world on the other. The attempt was to factor the epistemic state into two parts: the part that concerns the knower, i.e. the attitude holder, on the one hand and the part that concerns the way the world is, on the other. So we say that in order to know something, the world w has to be such that it answers to your internal state, and your internal state has to be a certain way. Thus, the intuitive idea behind the 'justtified true belief' analysis of knowledge is that it is an internal state: you could be wrong, but you can still justfify what you believe. So 'justified true belief' is the internal aspect and truth is the external aspect of knowledge. The deeper significance of the simple Gettier counterexamples was that you cannot think of knowledge as simply two factorable components separately met. Because you can design examples where the truth component is met for one reason and the justified belief component is met for another totally different reason, and intuitively it is just not knowledge. So there has to be some connection between truth and belief in order for it to count as knowledge. So that was that project. Now as for the formal project, the essential question is: what is the structure of the situation? What’s the structure of the concept of knowledge and the concept of belief and how are they structurally related. Let me first say that there is a extreme idealization when you use the modal logic framework to characterize knowledge. In a normal modal logic, if something is true in all possible worlds, then it is also necessary. Applying this to knowledge, it means that if something is logically true, then it is known, i.e. true in all epistemically accessible worlds. Also, the class of necessities is closed under deductive consequence. This means that the set of propositions that you know is closed under consequence. But that seems absurd. So the first move in applying modal logic to knowledge is to justify the application, i.e. to answer the question of why we should not give it up and go to something else as soon as we realize the unreality of this idealization. We should note that this idealization is a feature of any of the tools that are used in formal epistemology: in any application of probability theory to epistemology, all logical truths get probability 1, and any consequence of things that get probability 1 also get probability 1. So there is a kind of logical omniscience built into these kinds of formal tools. What is the object of knowledge? In our abstract model, the objects of knowledge are propositions, where a proposition is a set of possible worlds. Choosing propositions as objects of knowledge does not by itself entail logical omniscience. It doesn't imply the distribution principle or the necessitation principle. But it does imply that if two propositions are logically equivalent, they are the same object, no matter how opaque and deep the logical equivalence is. So if you represent the object of belief by a proposition then you go a long way to logical omniscience already: if you know p and p is logically equivalent to q then you know q. This is almost as much of an idealization as full logical omniscience. 24.244 Modal Logic, Fall 2009 Lecture Notes 17 Prof. Robert Stalnaker Page 2 of 3 So we need some other, more realistic, idea for the objects of knowledge. The problem is that it is completely unclear what it should be. One way to go is to go linguistic: knowledge/belief is a relation between individuals and sentences. To be disposed to assert or accept sentence S is to know that S. But if you do that you're leaving out the relationship between the sentence and what it is about. Knowledge is not a matter of accepting a sentence, it is accepting what a sentence says. So what we need is an account of what a sentence says which is more fine-grained than the representation in modal logic (set of possible worlds). But it's not clear how to do that. One of the strategies is to make use of the structure of the sentence, i.e. to say that a proposition is a complex, structured thing which reflects the structure of the sentence. That might be adequate for a very limited domain of places where we want to talk about knowledge. Imagine a game where the knower is simply someone who asserts things or denies them, i.e. a machine that says 'yes' or 'no' to our questions, for instance. In that case, the relationship between sentences and the world does not matter. What matters is the relationship between sentences themselves, i.e. between the questions and their answers. But we want a notion of knowledge that tells us something about the state of the knower in relation to the way the world is. If we don't, we're not really doing epistemology, we're not really addressing the problem, so to speak. Also, we're not only interested in knowledge in terms of what you say. We're interested in knowledge in terms of how you act in the world. There is knowledge which cannot be manifested in statements. For example knowledge of chess. The knowledge of chess might not contain the statement "move the queen to …", but the chess player does it anyway. The move might turn out to be a really clever move, but when asked why he did it, the chess player might not know how to put his reason into words. 24.244 Modal Logic, Fall 2009 Lecture Notes 17 Prof. Robert Stalnaker Page 3 of 3 MIT OpenCourseWare http://ocw.mit.edu 24.244 Modal Logic Fall 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. .
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