Another definition defines a cause to be:
If P(A) 0 then P(B |A)= P(A ∧ B) / P(A)
As concerns differences between induction and deduction, one of theseis dramatically illustrated in the problems with Williams' thesisdiscussed in This is that inductive conditionalprobability is not monotonic with respect to conditions: Addingconditions may increase or decrease the value of a conditionalprobability. The same holds for non-probabilistic induction: Addingpremises to a good induction may weaken its strength: That the patientpresents flu-like symptoms supports the hypothesis that he has theflu. When to this evidence is added that he has been immunizedagainst flu, that support is undermined. A second difference concernsrelativity to context, to which induction but not deduction issusceptible. We return to this question in
Sentences A, B are said to be independentin P if
Our deep and extensive understanding of deductive logic, in particularof the first-order logic of quantifiers and truth functions, ispredicated on two metatheorems; semantical completeness ofthis logic and the decidability of proofs and deductions.The decidability result provides an algorithm which when applied to a(finite) sequence of sentences decides in finitely many steps whetherthe sequence is a valid proof of its last member or is a validdeduction of a given conclusion from given premises. Semanticalcompleteness enables the easy and enlightening movement betweensyntactical, proof theoretic, operations and reasoning in terms ofmodels. In combination these metatheorems resolve both themetaphysical and epistemological problems for proofs anddemonstrations in first-order logic: Namely, what distinguishes validfrom invalid logical demonstration? and what are reliable methods fordeductive inference? (It should however be kept in mind that neitherlogical validity nor logical implication is decidable.) Neither ofthese metatheorems is possible for induction. Indeed, if Hume'sarguments are conclusive then the metaphysical problem, to distinguishgood from bad inductions, is insoluble.
where p is some quantity between zero and one inclusive.
The probability P is said to be regular iff the condition ofP3 is also necessary, i.e., iff no contingent sentence has probabilityone.
P(σ(k)| H2) =(2/3)n(1/3)(k − n)
Conditional probability may also be taken as fundamental and simpleprobability defined in terms of it as, for example, probabilityconditioned on a tautology (see, for example, Hajek 2003).
And hence that for each hypothesis Hj,
P(B1 ∧ …∧ Bn) =P(B1) P(B2)… P(Bn)