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I’m trying to respond to an argument. But I need this principle:

Consider a set of propositions P1, P2,...Pn that entails Q, where n>1. If

(1) any one of the Pk alone does not entail Q, and

(2) at least one of the P’s is dubitable,

then Q is not indubitable.

Is this principle correct? Intuitively, I think if p is dubitable and q is indubitable, I would be very surprised if p&q is indubitable.

  • Your (1) should be strengthened to: Q is not entailed if any one of Pk is dropped. You also need to say something about the type of inferences involved in the derivation of Q. Most substantive inferences are not "indubitable", they are only plausible, so the conclusion is open to doubt even if the premises are not. If you restrict to formal inferences this principle can be defended, but there isn't much that can be formally inferred outside of mathematics. – Conifold Feb 12 at 8:46
  • It is hard to believe that there is "something that is indubitable": philosophers and scientist in the past have doubted about everything. – Mauro ALLEGRANZA Feb 12 at 10:40
  • In principle: YES. See Ex falso: if P is false, we can validly infer anything from it, and thus also Q – Mauro ALLEGRANZA Feb 12 at 11:02
  • Couldn't Q be entailed by the 'indubitable' P_{n+1}, not in the original set? Would that be a counterexample to your principle? – Schiphol Feb 12 at 16:13
  • One cannot infer anything at all from a flawed set of Ps. Or, to put it another way, one can infer anything at all, including direct contradictions. This is a classic technique for proving that the set of Ps is flawed. One cannot say anything at all about Q. – Guy Inchbald Feb 12 at 19:33
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Here are some problems for your principle.

First, Q can be entailed by a subset of indubitable propositions. For example, suppose P1 and P1->P2 are indubitable. Then P2 is indubitable and entailed by them. But P2 is also entailed by the set {P1, P1->P2, P3}, for any P3, which may be dubitable. (Because entailment is monotonic, at least in classical logic.) So we have a set of propositions, none of which alone entails P2, and one of which is dubitable, that entails an indubitable proposition.

To fix this you might want to change your condition (1) to require that no proper subset of P1...Pn entails Q. (That is, there are no redundant premises.)

A second problem is that Q may be a tautology, in which case it is entailed by any set of propositions. Then in some cases both of your conditions would be satisfied, while the conclusion is indubitable. The above fix should counter this problem as well.

A third problem is that the premises could be inconsistent, in which case they (vacuously) entail anything, including indubitable propositions. So you may want to require non-vacuous entailment.

Finally, if you're thinking of indubitability in terms of probability, you might want to look at probability logic, which can shed some light on the relative probabilities of conclusions and premises.

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