Thirders factor in the number of wakenings into the probability Sleeping Beauty should invest in heads, which seems counterintuitive when you make the number of wakenings extremely high. If thirderism is wrong, than what should we say about the Bayes’ net Maybe we should revise the Baye’s net with new probabilities: 1 given H or […]

# Sleeping Beauty

Sleeping Beauty Riddle On sunday, someone will flip a coin. Tails, she’s given a draught to forget, and wakes up Tuesday feeling like Monday. What chance she should give a probability that heads was rolled on a given morning? Seems like 1/2 at first, obviously, because she doesn’t affect the roll chance. But it also […]

# Bayes Nets

Bayes nets topographically encode information about causal probabilities. But just because there’s no connection directly doesn’t mean two nodes are independent. We should really be thinking in terms of ‘paths.’ Just because there’s a path doesn’t mean there’s a dependence. So what paths indicate dependence and which don’t? You can go against the arrows to […]

# Euclidity and Different Systems of Logic

Goal of epistemic logic: make it so simple and accurate a machine could copy it. This begins with a syntax, aka, which strings are grammatical: axioms and rules of inference. Two rules: Substitution: we can substitute any valid formula with a another valid formula if it is syntactically the same: r->r, then (p->q)->(p->q) Modus Ponens: […]

# Proof of Consistency of KT5

Completeness will be an online course. Consistency will be proved in class. Consistency: a contradiction cannot be proven from one logical system. E.g. a and ~a cannot be proven from the system KT4. What is mathematical truth? What makes 2+2=4 true? Answer: it’s a stipulation of the ‘game’ of math, the syntactical relations in that […]

# Axioms, Theorems, and Logical Omniscience

We shall move on to Bayesian Logic after this lecture. They function numerically (probabilistically). How much confidence should you put into a proposition given your confidence in another proposition(s). There’s a question of whether it is proscriptive or descriptive. Knights and Kaves. Knights speak truth, Knaves speak falsely. Is a married couple both Knights or […]

# Proofs by Induction

Proofs by Induction Something is true of an infinite number of cases by proving it true in a base case that leads to infinite regress. Usually one proves the ‘lowest’ case of something, and then the ‘inductive’ case (if it’s true of n, then it’s true of n+1). Say 0 is happy, and if n […]

# G-Reachability

Modeling a Card Game is situationepistemically possible from? No, because it is combining the knowledge of two different agents. The superimposed graphs imply this as a knowledge, but it is not so because it is really two superimposed graphs of two different agents. What’s wrong with the question: nothing is epistemically possible relative to anything […]

# Kripke Structures Expanded

4/25 Knolwedge represented formally: For every w’ (M,w’) |= (phi) if (w, w’) E Ki Don’t forget agents are representations of relations between worlds which are represented with ordered pairs such as

# Kripke Structures

Arrows represent possible worlds. The world Jones is in has a reflexive relation to itself, in that for all Jones knows he could be in that world. Equivalence relation: the agent represented by a binary relation. The agent is just the arrows (until we do this with graphs, but that’ll be later) that represent epistemic […]

# Hintikka’s Rules with Defensibility/Defendedness

Molyneux wants to take back: on his best reading, the analogue of consistency is defensibility. But what is the analogue of truth? Not defensibility, it seems, (Molyneux’s interpretation), but defendedness. Rather than interpreting Modus Ponens as being true, rather the components are defended. A.PPK*: if you know something, then you know that you know it. […]

# Defensibility Saves Closure!

Epistemic Logic (Apr. 18 ’12) Attempts to keep Closure One (normal gloss): if you deduce a conclusion from premises you know, and if you remember these premises, thereby you come to know conclusion. Two (Hintikka’s move): we should replace our notions of truth and consistency with something called defensibility. Example: Going from set L to […]