11 research outputs found
Cut-off points for the rational believer
I show that the Lottery Paradox is just a version of the Sorites, and argue that this should modify our way of looking at the Paradox itself. In particular, I focus on what I call âthe Cut-off Point Problemâ and contend that this problem, well known by Sorites scholars, ought to play a key role in the debate on Kyburgâs puzzle.
Very briefly, I show that, in the Lottery Paradox, the premises âticket n°1 will loseâ, âticket n°2 will loseâ⊠âticket n°1000 will loseâ are equivalent to soritical premises of the form â~(the winning ticket is in {âŠ, (tn)}) â ~(the winning ticket is in {âŠ, tn, (tn + 1)})â (where âââ is the material conditional, â~â is the negation symbol, âtnâ and âtn + 1â are âticket n°nâ and âticket n°n + 1â respectively, and â{}â identify the elements of the lottery ticketsâ set. The brackets in â(tn)â and â(tn + 1)â are meant to point out that in the antecedent of the conditional we do not always have a âtnâ (and, as a result, a âtn + 1â in the consequent): consider the conditional â~(the winning ticket is in {}) â ~(the winning ticket is in {t1})â).
As a result, failing to believe, for some ticket, that it will lose comes down to introducing a cut-off point in a chain of soritical premises.
In this paper I explore the consequences of the different ways of blocking the Lottery Paradox with respect to the Cut-off Point Problem. A heap variant of the Lottery Paradox is especially relevant for evaluating the different solutions.
One important result is that the most popular way out of the puzzle, i.e., denying the Lockean Thesis, becomes less attractive. Moreover, I show that, along with the debate on whether rational belief is closed under classical logic, the debate on the validity of modus ponens should play an important role in discussions on the Lottery Paradox
Cut-off points for the rational believer
I show that the Lottery Paradox is just a (probabilistic) Sorites, and argue that this should modify our way of looking at the Paradox itself. In particular, I focus on what I call âthe cut-off point problemâ and contend that this problem, well known by students of the Sorites, ought to play a key role in the debate on Kyburgâs puzzle
Qualitative probabilistic inference under varied entropy levels
In previous work, we studied four well known systems of qualitative probabilistic inference, and presented data from computer simulations in an attempt to illustrate the performance of the systems. These simulations evaluated the four systems in terms of their tendency to license inference to accurate and informative conclusions, given incomplete information about a randomly selected probability distribution. In our earlier work, the procedure used in generating the unknown probability distribution (representing the true stochastic state of the world) tended to yield probability distributions with moderately high entropy levels. In the present article, we present data charting the performance of the four systems when reasoning in environments of various entropy levels. The results illustrate variations in the performance of the respective reasoning systems that derive from the entropy of the environment, and allow for a more inclusive assessment of the reliability and robustness of the four systems
Relative Entailment Among Probabilistic Implications
We study a natural variant of the implicational fragment of propositional
logic. Its formulas are pairs of conjunctions of positive literals, related
together by an implicational-like connective; the semantics of this sort of
implication is defined in terms of a threshold on a conditional probability of
the consequent, given the antecedent: we are dealing with what the data
analysis community calls confidence of partial implications or association
rules. Existing studies of redundancy among these partial implications have
characterized so far only entailment from one premise and entailment from two
premises, both in the stand-alone case and in the case of presence of
additional classical implications (this is what we call "relative entailment").
By exploiting a previously noted alternative view of the entailment in terms of
linear programming duality, we characterize exactly the cases of entailment
from arbitrary numbers of premises, again both in the stand-alone case and in
the case of presence of additional classical implications. As a result, we
obtain decision algorithms of better complexity; additionally, for each
potential case of entailment, we identify a critical confidence threshold and
show that it is, actually, intrinsic to each set of premises and antecedent of
the conclusion
Against Belief Closure
I argue that we should solve the Lottery Paradox by denying that rational belief is closed under classical logic. To reach this conclusion, I build on my previous result that (a slight variant of) McGeeâs election scenario is a lottery scenario (see Lissia 2019). Indeed, this result implies that the sensible ways to deal with McGeeâs scenario are the same as the sensible ways to deal with the lottery scenario: we should either reject the Lockean Thesis or Belief Closure. After recalling my argument to this conclusion, I demonstrate that a McGee-like example (which is just, in fact, Carrollâs barbershop paradox) can be provided in which the Lockean Thesis plays no role: this proves that denying Belief Closure is the right way to deal with both McGeeâs scenario and the Lottery Paradox. A straightforward consequence of my approach is that Carrollâs puzzle is solved, too
Conditionals and modularity in general logics
In this work in progress, we discuss independence and interpolation and
related topics for classical, modal, and non-monotonic logics