Timeline for Bayesian conditional probability and material implication
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Apr 19 at 6:50 | comment | added | Bumble | I don't think Boole says that. Truth and probability are distinct things. Probability is not a degree of truth. In its Bayesian interpretation, probability is a rational degree of credence. Two different propositions may be true, but I may have different degrees of credence in them, so they can have different probabilities. The important thing is that an uncertain conditional has a probability typically given by P(Y|X) not by P(¬X ∨ Y), so the two must not be confused. It seems to me that it is Jaynes who is confusing them, not Boole. | |
| Apr 18 at 17:33 | vote | accept | adoan | ||
| Apr 18 at 16:13 | comment | added | adoan | @ac15 this is definitely beyond my familiarity with logic. I graphed the equation for the implication, and I see that when y>x the implication is certain, otherwise we have some probability, 0<=p<1, for the implication that holds whenever y=x-(p-1). Intuitively, this does not make sense to me, but I see the graph of f(x,y)=min(1,1-x+y) is the Łukasiewicz t-norm. The paper posted by a commenter above addresses my exact confusion and relates it to t-norms, so hopefully I will learn more there. | |
| Apr 18 at 15:49 | comment | added | adoan | @Bumble Thank you for your response. I actually just came across your reply thread here. Would you say that Boole is misguided in thinking this means two statements with the same truth value have differing probabilities? Really, there is only one statement, the probability of which either refers to the conditional or probability of material implication. Am I just getting into semantics or is there genuine substance to this clarification? | |
| Apr 18 at 14:46 | comment | added | ac15 | possibly of interest: en.wikipedia.org/wiki/… | |
| Apr 18 at 12:51 | history | became hot network question | |||
| Apr 18 at 10:01 | comment | added | Bumble | I don't think Boole is making a mistake. He shows that P(¬X ∨ Y) coincides with P(Y|X) when P(Y|X) is either one or zero. I.e. the probability of the conditional proposition of "pure Logic" as he puts it, which we now call the material conditional, agrees with the conditional probability in the case when both are certainly true or certainly false. That is not a confusion; it is correct. In practice, uncertain conditionals are nearly always correctly represented by P(Y|X) not by P(¬X ∨ Y). It is one of the limitations of the material conditional that it does not handle uncertainty correctly. | |
| Apr 18 at 6:33 | answer | added | causative♦ | timeline score: 4 | |
| Apr 18 at 5:04 | comment | added | Conifold | Your interpretation is correct, see Nguyen et al., Probability of Implication:"There exist two different approaches to defining this probability, and these approaches lead to different probabilistic inference rules: We may interpret the probability of an implication as the conditional probability P(A|B), in which case we get Bayesian inference. We may also interpret this probability as the probability of the material implication "A or not B", in which case we get different inference rules." | |
| S Apr 18 at 4:50 | review | First questions | |||
| Apr 18 at 10:06 | |||||
| S Apr 18 at 4:50 | history | asked | adoan | CC BY-SA 4.0 |