In the case of inconsistent preferences, the inference rule to reject would be Pref(¬φ) ⊢ ¬Pref(φ). After all, a case of inconsistent preferences just is a case where that rule is false.
If you're willing to endorse trivialism, or at least trivialism about preferences, then rejecting (N) might make sense. But the whole point of this exercise was to allow for inconsistent preferences without collapsing into trivialism. Simply discarding (N) seems to remove any meaningful structure from preferences, making them completely unconstrained. That’s not a position I, or probably most people, would endorse. Unless you have a way to reject (N) without making preference logic trivial?
Presumably, all the rules about preferences should be internal to the Pref predicate, rather than treating Pref φ logically in the same way that you would treat φ. You don't have to reject classical logic to talk about inconsistent preferences. The preferences themselves don't follow classical logic in the sense that you can't infer some preferences from others in the way that you can with propositions, but statements about the preferences should still follow classical logic. It doesn't violate classical logic, for instance, to say, "He prefers that φ&¬φ," or, "He prefers φ but also prefers ¬φ," since nothing about classical logic says that you can't prefer a contradiction, just that a contradiction can't actually be true.
As for how we can reason internally about preferences, I'm not sure if there are any universal rules, rather than just generalizations that are usually true. But I would definitely suggest that disjunction introduction should be rejected: It doesn't follow from Pref(φ) that Pref(φ∨Ψ). In fact, I'm not even sure if Pref(φ∨Ψ) is a meaningful statement in most cases. Consider, for example, the case where φ is that you get $1000 and Ψ is that you lose $1000. Do you prefer for φ∨Ψ to be true? Obviously, that depends on which one of the disjuncts is true.
Are you suggesting that we should reject both (N) and (DI)? You wrote that
> It doesn't follow that from Pref(φ) that Pref(φ∨ψ)
but I didn't use that inference in my post. Instead, I inferred Pref(φ) ∨ Pref(ψ) from Pref(φ), which seems logically valid: If 'he prefers getting $1000' is true, then 'he either prefers getting $1000 or prefers losing $1000' must also be true, simply by disjunction introduction on the meta-level (statements about preferences).
Would you argue that even this inference should be rejected when dealing with preferences?
I'm not suggesting that we reject DI. DI is a tautology - it's a law of classical logic. What I'm saying is that once you reject N, the laws of classical logic are not going to get you an explosion. The reason that you got an explosion in classical logic is because you assumed two things:
1. There can be inconsistent preferences.
2. N is a valid inference rule.
But applying N to inconsistent preferences immediately gives a contradiction, which always leads to explosion in classical logic. That's why I think we must reject N if we're assuming (1). N just is the claim that there can be no inconsistent preferences. It says, "If you prefer not-A, then you don’t prefer A," which is just a restatement of, "You can't prefer both A and not-A".
What I was actually suggesting we reject is what you might call "internal DI", the rule that Pref(A) entails Pref(A or B) (as opposed to Pref(A) or Pref(B), which no longer causes any problems). I think the problem with the explosion argument is that you're treating logical operators as if they commute or distribute with the preference operator when they don't. N is a particular instance of that, assuming that negation and preference commute. Once that assumption is rejected, you can't derive anything from inconsistent preferences. You now have the question of, what rules does Pref follow? If I know that Pref(A), what other preferences can I derive from that? I think the answer is most likely, "None, there are going to be rare exceptions to any rule you can come up with," but in any case, it's certainly not going to be, "You can derive Pref(C) for any C entailed by A". One particular instance of this is that, while A does entail (A or B), Pref(A) does not entail Pref(A or B). That is enough to block explosion, since it would be a necessary step if we were to recast the proof of explosion internally to A.
You still do need a paraconsistent logic for this, it's just that, rather than deriving true propositions from other true propositions, as classical logic loes, this paraconsistent logic derives preferences from other preferences. Thus, it's paraconsistent not in the sense that it allows true contradictions (it doesn't, since the paraconsistent logic is internal to the preference operator and so says nothing about what is true), but that it allows preferred contradictions without a preference explosion.
> None, there are going to be rare exceptions to any rule you can come up with," but in any case, it's certainly not going to be, "You can derive Pref(C) for any C entailed by A". One particular instance of this is that, while A does entail (A or B), Pref(A) does not entail Pref(A or B). That is enough to block explosion, since it would be a necessary step if we were to recast the proof of explosion internally to A.
In the case of inconsistent preferences, the inference rule to reject would be Pref(¬φ) ⊢ ¬Pref(φ). After all, a case of inconsistent preferences just is a case where that rule is false.
If you're willing to endorse trivialism, or at least trivialism about preferences, then rejecting (N) might make sense. But the whole point of this exercise was to allow for inconsistent preferences without collapsing into trivialism. Simply discarding (N) seems to remove any meaningful structure from preferences, making them completely unconstrained. That’s not a position I, or probably most people, would endorse. Unless you have a way to reject (N) without making preference logic trivial?
Presumably, all the rules about preferences should be internal to the Pref predicate, rather than treating Pref φ logically in the same way that you would treat φ. You don't have to reject classical logic to talk about inconsistent preferences. The preferences themselves don't follow classical logic in the sense that you can't infer some preferences from others in the way that you can with propositions, but statements about the preferences should still follow classical logic. It doesn't violate classical logic, for instance, to say, "He prefers that φ&¬φ," or, "He prefers φ but also prefers ¬φ," since nothing about classical logic says that you can't prefer a contradiction, just that a contradiction can't actually be true.
As for how we can reason internally about preferences, I'm not sure if there are any universal rules, rather than just generalizations that are usually true. But I would definitely suggest that disjunction introduction should be rejected: It doesn't follow from Pref(φ) that Pref(φ∨Ψ). In fact, I'm not even sure if Pref(φ∨Ψ) is a meaningful statement in most cases. Consider, for example, the case where φ is that you get $1000 and Ψ is that you lose $1000. Do you prefer for φ∨Ψ to be true? Obviously, that depends on which one of the disjuncts is true.
Are you suggesting that we should reject both (N) and (DI)? You wrote that
> It doesn't follow that from Pref(φ) that Pref(φ∨ψ)
but I didn't use that inference in my post. Instead, I inferred Pref(φ) ∨ Pref(ψ) from Pref(φ), which seems logically valid: If 'he prefers getting $1000' is true, then 'he either prefers getting $1000 or prefers losing $1000' must also be true, simply by disjunction introduction on the meta-level (statements about preferences).
Would you argue that even this inference should be rejected when dealing with preferences?
I'm not suggesting that we reject DI. DI is a tautology - it's a law of classical logic. What I'm saying is that once you reject N, the laws of classical logic are not going to get you an explosion. The reason that you got an explosion in classical logic is because you assumed two things:
1. There can be inconsistent preferences.
2. N is a valid inference rule.
But applying N to inconsistent preferences immediately gives a contradiction, which always leads to explosion in classical logic. That's why I think we must reject N if we're assuming (1). N just is the claim that there can be no inconsistent preferences. It says, "If you prefer not-A, then you don’t prefer A," which is just a restatement of, "You can't prefer both A and not-A".
What I was actually suggesting we reject is what you might call "internal DI", the rule that Pref(A) entails Pref(A or B) (as opposed to Pref(A) or Pref(B), which no longer causes any problems). I think the problem with the explosion argument is that you're treating logical operators as if they commute or distribute with the preference operator when they don't. N is a particular instance of that, assuming that negation and preference commute. Once that assumption is rejected, you can't derive anything from inconsistent preferences. You now have the question of, what rules does Pref follow? If I know that Pref(A), what other preferences can I derive from that? I think the answer is most likely, "None, there are going to be rare exceptions to any rule you can come up with," but in any case, it's certainly not going to be, "You can derive Pref(C) for any C entailed by A". One particular instance of this is that, while A does entail (A or B), Pref(A) does not entail Pref(A or B). That is enough to block explosion, since it would be a necessary step if we were to recast the proof of explosion internally to A.
You still do need a paraconsistent logic for this, it's just that, rather than deriving true propositions from other true propositions, as classical logic loes, this paraconsistent logic derives preferences from other preferences. Thus, it's paraconsistent not in the sense that it allows true contradictions (it doesn't, since the paraconsistent logic is internal to the preference operator and so says nothing about what is true), but that it allows preferred contradictions without a preference explosion.
> None, there are going to be rare exceptions to any rule you can come up with," but in any case, it's certainly not going to be, "You can derive Pref(C) for any C entailed by A". One particular instance of this is that, while A does entail (A or B), Pref(A) does not entail Pref(A or B). That is enough to block explosion, since it would be a necessary step if we were to recast the proof of explosion internally to A.
Aaaah gotcha. Yeah that's a valid position.