There are conferences, workshops, centers, and jobs in formal epistemology, and several institutions offer courses or seminars in the field. In doing so, they have more than earned the title of Assistant Editors of Readings in Formal Epistemology.
Bayesian Epistemology
Agency and interaction: what we are and what we do in formal epistemology,” Journal for the Indian Council of Philosophical Research, vol.
Belief Change
We would like to thank the authors, editors, publishers, copyright holders, permission holders, and other parties who allowed us to reprint the works found in this volume. What We Are and What We Do in Formal Epistemology," Journal for Indian Council of Philosophical Research, Vol. XXVII Special Issue on Logic and Philosophy Today, invited by Amitabha Gupta and Johan van Benthem.
Decision Theory
Logics of Knowledge and Belief
Interactive Epistemology
Horace Arlo-Costa (natay) Unibersidad ti Carnegie Mellon, Pittsburgh, PA, E.U.A. Unibersidad ni Frank Arntzenius ti Oxford, Oxford, UK. Dretske(natay) Unibersidad ti Wisconsin, Madison, WI, E.U.A. Unibersidad ti Haim Gaifman Columbia, Nueva York, NY, E.U.A.
Agency and Interaction What We Are and What We Do in Formal Epistemology
Introduction
The following is an attempt to locate the dominant paradigms in formal epistemology - e.g. probability theory, belief revision theory, decision theory, the logic of knowledge and belief, and finally interactive epistemology – in the context of action and interaction.
Probability
Where the agent is not committed to full belief in a given proposition, the negation of that proposition is a serious possibility for the agent. A consequence of such a requirement is that, for a given pair of serious possibilities, the agent either judges one of the possibilities to be more likely than the other, or the agent considers the possibilities to be equally likely.
Belief Change
How should the potential belief states be interpreted? The belief states might be thought to represent partial beliefs; for example, the agent has some degree of belief in proposition P. Alternatively, one may be interested in states of complete belief, expectation, or plain belief; e.g. the agent fully believesP, expectsP, etc.
Decision Theory
Logics of Knowledge and Belief
On the other hand, there is much more to what agents are today, but also much more to what they do dynamically (as opposed to statically in the form of e.g. (van Ditmarsch et al. 2008) epistemic axioms that describe rationality of single agents). Dynamic epistemic logic is a rich mix of studies spanning multi-agent axiomatizations of knowledge, belief, common knowledge and belief (Barwise 1988), certainty, uncertainty, doubt, ignorance and a host of other epistemic attitudes; models of the interaction between knowledge and games (Benthem knowledge and justification in general epistemology (Artemov and Nogina2005), social software (Parikh2002), knowledge and public announcement of information (Baltag et al. 2002), knowledge intertwined with preferences, actions and decisions ( Liu 2011);.
Interactive Epistemology
Formal epistemology is right in the middle; between mainstream epistemology's fairly abstract theories on the one hand and the more concrete cognitive sciences. This is an edited and reorganized version of the paper "Agency and Interaction: What We Are and What We Do in Formal Epistemology", Journal for the Indian Council of Philosophical Research, no.
The contraction of belief in the context of the general theory of rational choice. The Journal of Symbolic Logic. Fine Opinion, Probability, and the Logic of Absolute Belief. Journal of Philosophical Logic Probability, Statistics and Truth (Revised English ed.).
Bayesian Epistemology
Furthermore, the notion of belief is pragmatically analyzed in the direction suggested by Ramsey: “the type of measurement of belief with which probability is concerned is. And Ramsey's essay ends with programmatic ideas that distinguish what he called the "logic of consistency" from the "logic of truth."
Suggested Further Reading
Another important formal epistemological issue is investigated by Timothy Williamson in his paper, "Conditionalizing on Knowledge", British Journal for the Philosophy of Science, which aims to integrate probability theory and probability kinematics, with other epistemological notions such as the notion i know Patrick Maher revives this in his article, "Probability Captures the Logic of Scientific Confirmation," in Contemporary Debates in the Philosophy of Science, ed.
Truth and Probability
So we define the degree of belief in a way that presupposes the use of the mathematical expectation. We are also able to define a very useful new idea - the 'degree of belief in pgiven q'.
Probable Knowledge
A Pragmatic Analysis of Belief
Ramsey's basic idea was that the desirability of a venture G is a weighted average of the desirability of winning and losing in which the weights are the probabilities of winning and losing. Thus, if the desirability of losing and winning happen to be 0 and 1, we probably have ADdes G, as illustrated in Fig.4.1, for the case in which the probability of winning is assumed to be 34.
Probabilistic Acts and Observations
Conversely, if the observer's belief function after the observation is probE and probE is not identical to prob, then the direct effect of the observation will be to change the observer's degree of belief in Eto 1. My degree of belief in the proposition that the sun is shining has two distinct characteristics, (a ) It is 1, as close as gives no odds.
Discussion
If he then hears the warden tell him that he is one of the persons to be released, he will have good reason to feel happy. As for utilities, it is implicit in the argument that they are linear functions of the probabilities that will be released given the custodian's answer.
Fine-Grained Opinion, Probability, and the Logic of Full Belief
Full Belief and Personal Probability
I'll take it for granted that the probability of a proposition is a real number in the interval [0, 1], with the empty propositionƒ(self-contradiction) getting 0 and the universal propositionU(tautology) getting 1. So my probability is 100% that the number lies in the set [a,b] fxg, for each such numberx.
Supposition and Two-Place Probability
The proposition that conditional probability is irreducible means that two-place probability P(j) – the probability of one thing given (assuming) another – is autonomous and cannot be defined from the usual one-place (“absolute” ) probability. If P(j A) is a (1-place) probability measure, I will call A normal(forP), and otherwise abnormal. Absurd” might have been a better name; it is clearly a notion associated with contradiction.) The definition of 2-place probability allows for the totally anomalous attitude state (P(Aj B)D1 for all AandB).
Full Belief Revisited
However, the following elementary results show that all a-priorities belong to the propositions implied by cores of belief. Very little in this discussion of complete faith depends on the peculiarities of probability.
Identification of Full Beliefs
Therefore, K* is also not empty and thus normal, because it is a subset of at least one belief kernel. T5.3] If (*) holds and there is at least one belief kernel, then the intersection of all belief kernels is also a belief kernel.
To What Extent Does Belief Guide Opinion?
Assume also that the belief kernels are countable and form a chain (the latter from [T5.1]), and call the intersectionK*. Hence there is no group whose subsets are all normal, and hence no kernel of confidence.
The Multiplication Axiom Visualized
This makes it quite clear that (in this case) belief drives opinion, for the probabilities conditional on the residuals of belief are virtually all the conditional probabilities there are.
Equivalence Relations on Propositions
Consequently, Btoo is normal and P.BjA[B/is positive; the same argument leads mutatis mutandistoP(AjB)D1. Instead of elevating a proposal to the status of subjective certainty, it elevates it to subjective priority.
Implication Relations; Superiority Is Transitive
If we add this to the fact that P> is reflexive, we conclude that P> is a partial ordering of the field of propositions. The anomalous statements form an hPi equivalence class at the very bottom of this partial ordering.
A Large Class of Models
On the other hand, suppose that P is a two-digit probability measure in the sense of fulfilling I–II, defined on the Fin space SD hU; Fi. A special case whose existence depends on the axiom of choice is this: let SEQ contain all one-digit probability measures defined on a given domain F.
Belief in a Lexicographic Model
Definition model M D hS; in withS D hU; Fiis lexicographically, there is a sequence (well-ordered class) SEQ of 1-place probability measures defined on the whole of F such that B.A/D q.A\B/ =q.B/ for the first member q of the sequence SEQ such thatq( B) > 0;Bis undefined when there is no suchq. Note that one of the defining conditions for a belief core is that it must be very fine on K. T11.1] IfPis represented by lexicographic modelMD hS;pidedefined by w.o. sequence SEQ, andA, Just disjoint normal sets for P, then the following are equivalent:.
Appendix
The two-site probability theory presented here is essentially as explored in my book (1979), but with considerable improvement in the characterization of the described classes of models. See Birkhoff (1967), XI, 5 and Kappos (1969), II, 4 and III, 3.) The difficulty with this approach is that a probability algebra does not have the structure usually required of a propositional algebra.
A Theory of Higher Order Probabilities
In general, the representative will consider the expert to be more knowledgeable than himself, but not omniscient; for example, the expert may know the true bias of the coin but not the results of future tosses, or he may have statistical information to estimate the bias that the agent does not. To present knowledge at different levels, PR is made into a 3-digit operator: PR.A;t; /is an event in which the degree of probability Aat. Stages can be points in time at which the case extends through some ordered sequence.
Simple HOPs
The meaning of px is obvious: it is the probability corresponding to the maximum state of knowledge in worldx – the distribution chosen by the expert of that world. Say his additional information is the value of N – the number of "heads" in 50 independent tosses.
Probability Logic
It can be shown that for D series of rational numbers the set of p-valid wffs is recursive. Other validity notions can be considered (e.g. that jjH always contains all connected worlds in the HOP), as well as other interpretations of the necessity operant (e.g. asV.
General HOPs
The logic PRL,T (corresponding to HOPs with a set of stagesT) is defined in the same way as PRL, except that PR has an additional argument in the range overT. It is defined as p-valid if it obtains a probability of 1 in all HOPs in which the set of T ranks.
On Indeterminate Probabilities
The problem of revising credo states is thus reduced to the problem of revising corpora of knowledge. According to Bayes' answer to the problem of rational plausibility, no shift can be justified.
VIII
A hint as to the source of the difficulty can be obtained by noting that if 'E-admissible' is replaced by 'optimal' in the various formulations of the principle given by Luce and Raiffa, p. I have scratched the surface of some of the questions raised by the proposals in this essay.
It is a curious fact that the procedures Bayesians use for empirically determining subjective degrees of confidence are an example of the general strategy described in Glymour1981, ch. In the limit approaching infinity, the probability that the given probability x of any element of the partition it actually receives is greater than ˛ is 1.
Discussion: A Mistake in Dynamic Coherence Arguments?
Static Coherence of Degrees of Belief
Dynamic Coherence for Updating Rules
The Dynamic Dutch Book
The Converse
Sequential Analysis 1: A Mistake in the Dynamic Coherence Argument?
Sequential Analysis 2: A Mistake in the Mistake?
Thus, the value is at the node where the bookie rejected the bet today and where it is observed tomorrow. Then the value at the node where the bookmaker rejected the bet today is not 0, but before.
Sequential Analysis 3: What Makes the Cunning Bettor Tick?
Note that the bookkeeper knows that ifeis is observed, she will accept the offer the day after tomorrow for the same reason on the lower path as on the upper. It is just the same as the value at the node where the bookie agrees to bet today.
Strategic Rationality
This bettor's threat to hold and bet the day after tomorrow, even if the bookie refuses to bet today, is perfectly credible and consistent with sequential rationality. If he is asked to carry out the threat, he maximizes expected utility by doing so.
The Bottom Line
Postscript: Conditioning, Coherence and Rationality
Slightly more realistic personal probability. Philosophy of Science Subjective Probability: Critique, Reflections and Problems. The Journal of Philosophical Logic. Reexamining the perfectness concept of equilibrium in extended form games. International Journal of Game Theory, 4, 25-55.
Some Problems for Conditionalization and Reflection
Two Roads to Shangri La
Then, as you travel along the Mountains and you look at them, your degree of faith in heads will be 1. Nevertheless, you think you will act in a fully rational way when you acquire those future degrees of faith.
The Prisoner
After midnight, there are two possible ways in which your degree of belief distribution can develop. It is true, as I suspected, that your degree of belief in tails will have increased at 23.59.
John Collins’ Prisoner
Because, in the first place, at 18:00 the prisoner knows that by 23:30 his degrees of belief in heads will be less or equal to what they are now, and cannot be greater. His current expectation of what his degrees of belief in heads will be at 23:30 is therefore less than his current degree of belief in heads.
Sleeping Beauty
Similarly, if the real world is W3, at 11:30 PM the prisoner will have the degree of belief 0 in W2 and the degree of belief in 1/3 of each of the remaining options. If SB then pinches herself and finds herself awake, she must condition and then have degree of belief 1/3 in each of the remaining 3 possibilities (Monday&Tails&Awaken, Monday&Heads&Awaken, Tuesday&Tails&Awaken).
Duplication
Consequently, you should now have a degree of belief for 1/3 that the roll landed heads and 2/3 tails. This makes no relevant difference and you should now believe 1/3 that the coin has landed heads.
Diagnosis
The case of John Collins' prisoners involves just such a change of degrees of belief in one's self-determination. It is not too surprising that the spread of degrees of confidence can lead to a violation of Reflection.
Conclusions
However, the arguments of sections IV and V show that one can in any case interpret such distribution of self-localizing degrees of belief in violations of Reflection on such ordinary beliefs or a coin lands heads or tails. Such violations of conditioning can be interpreted as violations of reflection, and lead to a distrust of one's future degrees of belief.
Stopping to Reflect
Time is indexed for the agent by the number of recent coin flips. According to Pt(A)Dr, the added information obtained by A is relevant to the agent's current probability when it happens later.
