1 Introduction
You might believe that Plato wrote the Republic; alternatively, you might wonder or be curious about who wrote it. In the former case, your attitude would be propositional; in the latter it would be inquiring. This paper is about inquiring attitudes; those attitudes take questions (instead of propositions) as their objects and (in some sense) aim at answering or settling those questions. In addition to curiosity and wonder they include contemplation, deliberation, and perhaps more besides. We'll henceforth use “wonder” as a general stand-in for them.
These attitudes are the subject of a rich debate.Footnote 1 In this debate, some argue that it is irrational to wonder a question Q (e.g., whether it's raining) while knowing a complete answer to Q. Others argue similarly but replace knowledge with other states, like belief. Remarkably, all these claims target people who do wonder something. What about people who don't? When are they being irrational? Consider a dialogue between Authades (“obstinate one”) and Zetegetes (“leader of an inquiry”):
Authades: I don't know who wrote the Republic, and I wonder who it was.
Zetegetes: It's the same person who wrote the Meno.
Authades: Who cares? I wonder who wrote the Republic, not who wrote the Meno.
Zetegetes: Um… Did you believe what I just said?
Authades: Yep. The writer of the Republic also wrote the Meno. I know that.
Zetegetes: But then given what you know, the answer to Who wrote the Meno?—whatever it is—entails the answer to Who wrote the Republic?—whatever it is.
Authades: I know. It's not that I don't understand logic. It's just that I don't care. I wonder who wrote the Republic, and I know that the writer of the Republic also wrote the Meno, but I still don't wonder who wrote the Meno.
Zetegetes: That's irrational!
Zetegetes is right: Authades is being irrational. But why? For an explanation, we might look to norms for wondering discussed in recent epistemology, e.g., the “Don't Believe and Inquire” norm DBI:
“One ought not…have an interrogative attitude towards Q at t while believing [a complete answer to Q] at t.” (Friedman, Reference Friedman2019: 303).
Friedman here treats “one ought not” and “it is irrational for one to” interchangeably. Thus DBI says (e.g.) that it is irrational to wonder whether P when you believe that P.
DBI is what we'll call a “norm of restriction”—a principle saying it's irrational to wonder a question Q given certain conditions. Principles saying it's irrational not to wonder Q given certain conditions, are “norms of expansion”. (Motivating this terminology: one kind of norm requires us to restrict our attitudes by not wondering certain things given certain conditions, while another requires us to expand our attitudes by wondering certain things given certain conditions.) For simplicity, we assume that all norms of both sorts are “wide scope” in the sense that they tell us it is irrational to: c 1, and …, c n for some (possibly singleton) set of conditions {c 1, …, c n}. With norms of restriction, one of those conditions involves wondering something; with norms of expansion one of them involves not wondering something.
DBI is a norm of pure restriction because it is a norm of restriction that isn't also one of expansion. In fact, all of the norms for wondering thus far discussed in print are of this sort (though Rosa (Reference Rosamanuscript), helpfully discusses some norms of expansion). Still, there can be norms that are both. Indeed, many of the norms we'll discuss are; they say it's irrational to wonder certain things while not wondering others.
Norms of pure restriction don't explain Authades’ irrationality. If they did, he wouldn't be able to ameliorate his condition by expanding his wondering. And, he is able to do so—by coming to wonder who wrote the Meno. To explain his irrationality, then, we need a norm of expansion.
As it happens, a certain body of logical theory inspires such norms: inferential erotetic logic (IEL). We'll explain IEL and use it for two tasks: reverse-engineering some already-discussed norms of restriction, and forward-engineering some new norms of expansion. These tasks don't close down the territory. Rather, they open it up by starting a research program of connecting IEL to interrogative epistemology.
2 Inferential Erotetic Logic (IEL) Primer
Inferential Erotetic Logic (IEL) studies erotetic arguments: arguments whose conclusions (and sometimes premises) are questions (Wiśniewski, Reference Wiśniewski1991, Reference Wiśniewski1995, Reference Wiśniewski2013; Leszczyńska-Jasion & Chlebowski, Reference Leszczyńska-Jasion and Chlebowski2015; Pelis̆, Reference Pelis2016; Cordes, Reference Cordes2020). Each question has a set of “direct answers” consisting in the propositions among which it calls for us to choose. For example, Who won the race: Smith or Jones? calls for us to choose among the propositions Smith won the race and Jones won the race; those are its direct answers. Partial answers are disjunctions of some but not all of a question's direct answers, and eliminative answers are negations of direct answers. Now to an erotetic argument:
1. Someone stole the tarts.
2. So, who was it?
Here, the conclusion-question in some sense follows from the premise. We might capture that sense by saying that Someone stole the tarts raises the question of who that person was. In IEL, this kind of raises-relation is called “evocation” and there is a standard attempt to define it. To state that definition, we'll need some additional vocabulary. A proposition P is a presupposition of Q just in case P is entailed by each of Q's direct answers. A question is sound just in case it has at least one true direct answer; and sound relative to a set of propositions Γ just in case Γ entails that it is sound. Notice that if a question is sound then all of its presuppositions are true.
For illustration, consider Who wrote the Meno?. Its direct answers are propositions like Plato wrote the Meno and Aristotle wrote the Meno. Its partial answers include such propositions as Either Plato wrote the Meno or Aristotle wrote the Meno. Its eliminative answers are propositions like Aristotle didn't write the Meno. Its presuppositions include Someone wrote the Meno. Since Who wrote the Meno? has a true direct answer, it's sound. In contrast, consider Is the present king of France bald?. Its direct answers are The present king of France is bald and The present king of France is not bald. Since there is no present king of France, neither of these is true; the question is therefore unsound. Still, it is sound relative to the proposition that there is a present king of France.
Many theorists—“partitioners”—think of questions as partitions of logical space, each element of which they call a “complete answer” and each proper subset of which they call a “partial answer” (Groenendijk & Stokhof, Reference Groenendijk and Stokhof1984). IEL's framework is broader; it associates every (nonsingleton) set of propositions with a question, whether or not its elements form a partition. Still, there is a workable translation scheme: roughly speaking, complete answers are direct answers to questions whose direct answers form a partition; and partial answers (in the partitioners’ sense) are partial answers (in the IEL sense) to these questions.
Partitioners can thus make use of IEL, as can many others, including those with differing views about (declarative) entailment. While the latter figures in many of its definitions, IEL can treat it in numerous ways. We'll treat entailment in the manner most familiar to philosophers—i.e. as the relation between premises and conclusions in classically valid arguments. Logicians have explored several alternatives (see, e.g., Leszczyńska-Jasion & Chlebowski (Reference Leszczyńska-Jasion and Chlebowski2015)). Perhaps entailment can even be usefully replaced by probabilistic relations. This flexibility is a feature, not a bug. IEL is an intentionally modular tool, designed to be embeddable into many theoretical frameworks.
Now we can state the standard definition of evocation in IEL. That definition—which comes from Wiśniewski (Reference Wiśniewski1991, Reference Wiśniewski1995) via Belnap (Reference Belnap and Lambert1969) and Bromberger (Reference Bromberger and Mandelbaum1971)—has it that Q is evoked by Γ just if the following two conditions hold:
Relative Soundness: Q is sound relative to Γ.
Affirmative Openness: Γ does not entail any direct answer to Q.
To illustrate this, notice that Fiona is smiling evokes only one of the questions below.
Q1. Is Fiona smiling?
Q2. Is the present king of France smiling?
Q3. Is Fiona happy?
From now on, we'll mean by “evocation” only what this definition picks out. We won't claim that this is a good explication of the notion of propositions “raising” questions. Rather, we'll take the relation as-defined and do some philosophical work with it.
Evocation connects questions to propositions. Another relation, erotetic implication, connects questions to questions (or to questions and propositions). Another argument illustrates it:
1. Who wrote the Republic?
2. The writer of the Republic also wrote the Meno.
3. So, who wrote the Meno?
Erotetic implication comes in several varieties. The above example features “strong regular” erotetic implication, which we'll call resolution (see Wiśniewski (Reference Wiśniewski2013: 76) and Millson (Reference Millson2019, Reference Millson2021)). This relation requires three conditions. First, the premises must secure the soundness of the conclusion-question in the sense that, if the premise-propositions are all true and the premise-question is sound, then the conclusion-question is also sound.
Security: Q 2 is secured by Q 1 given Γ iff : if Q 1 is sound given Γ then so is Q 2.
Second, the conclusion-question must effectively answer the premise-question given the premise-propositions, in the sense that each direct answer to the conclusion-question entails (given the premise-propositions) a direct answer to the premise-question.
Effectiveness: Q 2 is effective for Q 1 given Γ iff , given Γ, each direct answer to Q 2 entails a direct answer to Q 1.
Third, the premise-propositions must leave the premise-question and (thus) the conclusion-question affirmatively open in the sense that they don't entail any of their direct answers. The definition of resolution is thus as follows:
Resolution: Q 2 resolves Q 1 given Γ iff
1. Q 2 is secured by Q 1 given Γ,
2. Q 2 is effective for Q 1 given Γ, and
3. Γ leaves Q 1 (and hence Q 2) affirmatively open.
We've described IEL in terms of propositions. But, in full disclosure, it's usually formulated in terms of wffs—strings of symbols—some of which are declarative (the formulas of propositional logic, e.g., “p”) and others erotetic (declarative wffs inside brackets with a question mark, e.g., “?{p, ¬p}”). In speaking of propositions, then, we impose a philosophical interpretation on IEL. Though not uncontroversial, this interpretation is traditional and modest; we assume only that propositions can be believed, known, and asserted (and adequately modeled by the kind of propositional logic that's in the Appendix). These assumptions are compatible with many theories about propositions’ nature.
3 Proof of Concept for a Research Program
It's one thing to define logical relations like evocation and resolution; it's quite another to state norms of rationality and wondering. To move from the definitions to the norms we need “bridge principles” that connect the two. There's a tradition exploring principles that bridge declarative logic to rational belief. We'll explore some similar bridges from erotetic logic to rational wondering.
In this section, we examine one such principle with two important features. First, it appeals to a central IEL relation: evocation. Second, it entails some norms of restriction for wondering that have appeared in recent debates among interrogative epistemologists. Our interest is not to adjudicate the plausibility of this principle (or the other norms it entails). Instead, we offer it up as evidence that IEL is a fruitful resource for theorizing norms of wondering—i.e., as our research program's proof of concept.
The aforementioned bridge principle is the Evoked Question Norm:
EQN
It is irrational to: wonder Q when your knowledge doesn't evoke Q.
Bridge principles have alterable parameters. EQN gives a requirement (rather than a permission or a pro tanto reason); its term of criticism is “rational” (instead of “moral,” “epistemic,” etc.); that term takes wide, rather than narrow, scope. EQN is synchronic (applying only at a given time) instead of diachronic (applying across times); it applies given our knowledge (rather than our beliefs, our certainties, etc.); it enjoins us to restrict our wonderings (instead of expanding them); and it applies irrespective of whether a subject knows about the logical facts at issue.
EQN offers evaluations rather than blame. All it claims is that certain people are being irrational—those who wonder a question their knowledge doesn't evoke. While irrationality may provide evidence of blameworthiness, it does not entail it. Maybe a person is being irrational but has an excuse, and so is blameless. Like the norms we'll discuss later, EQN concerns how we ought to be and not how we ought to be held accountable.
Now, there are two ways for Q to fail to be evoked by the propositions you know: it might be (affirmatively) unopen relative them or it might be unsound relative to them. Considering the first possibility, notice that EQN entails an “Open Question Norm”:
OQN
It is irrational to: wonder Q when your knowledge entails a direct answer to Q.
One way to violate OQN is to know a direct answer to Q. Thus, OQN entails another norm—it is irrational to: wonder Q when you know a direct answer to Q. This other norm is a natural bridge from IEL to rationality and wondering. Interestingly, it's nearly identical to certain principles discussed in recent epistemology:
“It is illegitimate to be curious about a question when you know its answer.” (Whitcomb, Reference Whitcomb2010: 674).
“Necessarily, if one knows Q at t, then one ought not have [an interrogative attitude] towards Q at t”. (Friedman, Reference Friedman2017: 311).
“Interrogative attitudes… are never compatible with knowledge of the question's answer.” (Sapir and van Elswyk, Reference Sapir and van Elswyk2021: 1).
The similarity between bridge principles inspired by IEL, and principles discussed in recent epistemology, does not end there. Recall that the other way to violate EQN is to wonder Q when it's unsound relative to your knowledge—i.e., when that knowledge doesn't entail all of its presuppositions. Thus we have a “Sound Question Norm”:
SQN
It is irrational to: wonder Q while your knowledge does not entail all of Q's presuppositions.
A nearby norm that's more demanding replaces entailment with inclusion, yielding:
KNI
It is irrational to: wonder Q while your knowledge does not include all of Q's presuppositions.
Here again, the norm (the “knowledge norm of inquiry”) is defended by an epistemologist: Willard-Kyle (Reference Willard-Kyleforthcoming). So evocation, a relation broached decades ago by erotetic logicians, is closely related to several norms proposed recently by epistemologists.
None of those epistemologists mention IEL. How did that happen? We think the two literatures pursued similar issues and generated similar sets of ideas independently. As a result, each set can help “reverse engineer” the other. This point is not just historical. It is also proof of concept for a research program: the program of bringing new material into each of the literatures via the other. We'll now take that program's first steps.
4 A Research Program's First Steps
We seek an explanation of why Authades counts as being irrational. Consider a “Resolution Norm”:
RN
It is irrational to: wonder Q 1 but not Q 2 when, given what you know, Q 2 resolves Q 1.
This norm leaves fixed all the parameters from the Evoked Question Norm (EQN), save two: it is a norm of expansion and it focuses on cases where the logical relation does obtain. These changes yield an explanation of Authades’ irrationality—he's being irrational because he violates RN. He (a) fails to wonder who wrote the Meno while also (b) wondering a question that, given his knowledge, is resolved by Who wrote the Meno?. RN deems these states jointly irrational.
RN doesn't deem irrational Authades’ failure to wonder who wrote the Meno. That claim would “detach” one of the states RN targets jointly, invalidly inferring that it targets that state singly. Still, RN entails that if Authades wonders a question that's resolved by Who wrote the Meno given his knowledge, he's doing something irrational if he fails to wonder who wrote the Meno: he's in jointly irrational states. So RN requires him—on pain of being in jointly irrational states—to wonder who wrote the Meno whenever he wonders a question that, given his knowledge, is resolved by Who wrote the Meno?.Footnote 2
In sum, RN yields the explanation we seek. Similar explanations are available in declarative cases, as another dialogue (adapted from dialogues in Hawthorne (Reference Hawthorne2004) and Carroll (Reference Carroll1895)) illustrates:
Achilles: I'm agnostic about whether Plato wrote the Meno.
Tortoise: Do you know that Plato wrote the Republic?
Achilles: Of course.
Tortoise: And do you know that the writer of the Republic also wrote the Meno?
Achilles: Oh yeah. I've known that for a long time.
Tortoise: Well there you go then. You should believe that Plato wrote the Meno.
Achilles: Not so fast, turtle. I'm still unconvinced.
Achilles is being irrational. This can be explained by an “Entailed Belief Norm”:
EBN
It is irrational to: know that P and not believe that R when, given what you know, P entails R.
Achilles and Authades fail to expand their attitudes in ways deemed irrational by the RN and EBN, respectively. What Achilles does with his propositional attitudes, Authades does with his inquiring attitudes. RN is an interrogative analogue of EBN.
It is an open question which kinds of irrationality Achilles and Authades exhibit. Perhaps they exhibit epistemic irrationality, like people whose beliefs are unjustified. Or, perhaps it's instrumental irrationality, like people who don't take the means to their ends. Or, perhaps it's structural irrationality, like people whose mental states don't “fit together” (e.g., people who want to ϕ while wanting to not want to ϕ). Or, perhaps it's zetetic irrationality, a putative species of irrationality specific to inquiry.
We suspect the irrationality at issue is, at least, structural. That's because it features incoherence. The relevant mental states do not maximally fit together. This kind of fit, and its relationship to irrationality, are both subjects of extensive discussions (e.g., Worsnip (Reference Worsnip2021)). In the absence of a developed account of these features, the best we can do is register our suspicion that the irrationality at issue is structural. You might view that irrationality differently; that would be fine for our purposes. As would be replacing irrationality in toto with something else, like incuriosity or uninquisitiveness or obtuseness. What's essential is that there's some way in which our characters’ mental states are defective or suboptimal or inappropriate (here compare Friedman (Reference Friedman2019: 303)).
4.1 Problems for the Resolution Norm (RN)
RN faces two kinds of problems: old and new.
4.1.1 Old Problems
The old problems are versions of standard worries about declarative bridge principles — worries discussed by Harman (Reference Harman1986), MacFarlane (Reference MacFarlane2004), and Steinberger (Reference Steinberger2019) among others. We'll say two things about them. First, they aren't knock-down arguments against RN—if they were, they'd also knock down many popular declarative bridge principles, which they don't. Second, those who think the problems do knock down RN can avoid them by toggling its parameters.
For instance, here's an old problem for (“closure”) principles like EBN. That principle deems irrational people who fail to believe a proposition entailed by what they know—even if they don't know the entailment obtains. Similarly, RN deems irrational those who fail to wonder a question that (given their knowledge) resolves a question they wonder—even if they don't know the resolution obtains. Call this the problem of logical ignorance.Footnote 3 We are unmoved by it, for two reasons. First, RN issues evaluations; it deems certain people irrational. It doesn't issue blame. If it did then we might include a logical knowledge requirement (since ignorance often mitigates blameworthiness); but again, it doesn't. Second, many plausible norms of evaluation are comparably independent of one's logical (or other formal) knowledge. Plausibly, people with inconsistent beliefs are being irrational even if they don't know about the inconsistency. Plausibly, people with probabilistically incoherent credences are being irrational even if they don't know about the incoherence. Similarly with RN.
Here's a related worry. “Coarse” theories of content deem Batman is humorless identical to Bruce Wayne is humorless. If your knowledge leaves affirmatively open Is Batman humorless? and Is Bruce Wayne humorless?, these theories make RN deem you irrational if you wonder one of them but not the other. Irrational and also impossible: on these theories they're the same question (Teague, Reference Teague2022). That's worrisome, but here again RN is in good company. Coarse content brings similar worries to many popular norms, for instance probabilism (see, e.g., Christensen (Reference Christensen2004: 16)). We submit that whatever solution applies to the other norms—perhaps a finer individuation of content—applies to RN too.
This isn't to say that RN is as plausible as the other norms, or that arguments for those other norms transfer over into arguments for RN. It's just to say that a certain problem that doesn't knock down those other norms, doesn't knock down RN either.
While some theorists would restrict RN to cases where you know the relevant formal facts, others would restrict it to cases where your wonderings (and lacks-of-wonderings and knowledge) are “occurrent”, rising to the level of consciousness. Call the resulting norm RNC. Applied generally, its internalism is objectionably silent. Unclosed or inconsistent beliefs, incomplete or otherwise-probability-theory-violating credences, unconnected or intransitive preferences: whenever these are backstage at the Cartesian theatre, the internalism behind RNC declines to deem them irrational. We think this loss of informativeness outweighs the plausibility added by dropping RN for (the logically weaker) RNC.
Some might weigh things differently and opt for RNC—especially since, in our dialogue, Authades knows about the formal facts and his mental states are easily construed as occurrent. If that's how you weigh things, don't stop reading! You can still profit from the discussion. Move forward with us mutatis mutandis, applying our points in your restricted domain.
Actually, you can toggle all of RN's parameters. For instance, you can (as we've said) replace irrationality with incuriosity. The resulting principle is sometimes very plausible, including in certain cases where your mental states are non-occurrent. But compatibly with this, we think those cases also feature irrationality. Here again we follow probabilism, on which similar cases would feature irrationality due to (non-occurrent) credences in Plato wrote the Republic and The writer of the Republic wrote the Meno but no credences (not even zero-valued ones) in Plato wrote the Meno. The cases here seem similar enough to merit at least one shared evaluation. We favor irrationality. Still, that parametric setting is just a starting point. While we think it's defensible, we've chosen it partly because we've got to start somewhere. Feel free to start elsewhere: we've designed our discussion for easy retrofitting.
4.1.2 New Problems
At least two problems for RN are new. The first we'll call the problem of eliminated conjuncts. Suppose that a disease is afoot and that your student James wants to visit your office hours. Let A = James is allowed to visit, V = James is vaccinated, and E = James has an exemption from the vaccine requirement. Suppose that nothing you know settles whether A, V, or E obtain, but that you do know that [A ↔ (V∨E)]. Finally, suppose that you wonder Q A = Is James allowed to visit?.
Holding these suppositions fixed, RN requires you to wonder the conjunctive question of whether James is vaccinated and whether he's exempt—the question $Q_V\& Q_E$
with direct answers $V\& E$
, $V\& \neg E$
, $\neg V\& E, \;$
and $\neg V\& \neg E$
. That's a welcome result. But now suppose you learn that James is not exempt (¬E). RN then still requires you to wonder $Q_V\& Q_E$
. That's not a welcome result. You know that James isn't exempt. Why should you have to wonder Whether he's vaccinated and whether he's exempt?.
RN also faces what we'll call the problem of irrelevant conjuncts. First a preliminary point. Conjunctive questions “resolve their evoked conjuncts” in the sense that, if Γ evokes both Q 1 and Q 2, then $Q_1\& Q_2$
resolves Q 1 given Γ. We prove this in the Appendix (Theorem 1), but here an illustration should suffice. Suppose that your knowledge evokes Is Ted alive? and Is Ted asleep?. Then, given your knowledge, Is Ted alive and is he asleep? resolves Is Ted alive?.
To see the problem of irrelevant conjuncts, hold fixed the supposition that Authades’ knowledge evokes Who wrote the Republic?, a question he wonders. Let Q n be any other question his knowledge evokes. Since conjunctive questions resolve their evoked conjuncts, RN requires him to wonder Who wrote the Republic and Q n?. To see how objectionable this can seem, add it to our stock of fixed suppositions that his knowledge evokes What is the 104th digit in the phone book?. RN then requires him to wonder Who wrote the Republic and what is the 104th digit in the phone book?.
4.2 From RN to FRN
You might approach the problem of eliminated conjuncts by deeming it irrational to wonder questions while knowing any of their eliminative answers, and applying RN subject to that constraint. But why should we have to stop wondering a question just because we rule out one of its direct answers? Alternatively you might dig in, endorsing the requirement to continue wondering whether James is vaccinated and whether he's exempt after learning he's not exempt. But while rationality may allow that kind of continuation, we think requiring it is a bridge too far. A better approach narrows RN via the following notion:
Full Openness: Γ leaves Q fully open iff Γ does not entail any direct or eliminative answer to Q.
This notion yields limited versions of the resolution relation and RN:
Full Resolution: Q 2 fully resolves Q 1 given Γ iff
1. Q 2 resolves Q 1 given Γ, and
2. Γ leaves Q 2 fully open.
FRN
It is irrational to: wonder Q 1 but not Q 2 when, given what you know, Q 2 fully resolves Q 1.
FRN solves the problem of eliminated conjuncts via the full openness condition, which keeps it from requiring that we wonder questions with direct answers that are ruled out by our knowledge. But the problem of irrelevant conjuncts remains, at least when our knowledge leaves the questions at issue fully open.
4.3 From FRN to MRN
The problem of irrelevant conjuncts applies to both RN and FRN. How bad is it? Some theorists might view it as a reductio. Others might find it unfortunate but tolerable. Others still might deny that it is even a problem. We aren't sure which of these responses is best. Assuming the problem is a genuine one, where should we go next?
One option is to rebuild RN and FRN using nonclassical consequence relations such as relevance entailment. Another is to continue using classical consequence and build from it norms that restrict RN even more than FRN restricts it. Both of these paths are worth navigating; we'll navigate the latter.
Suppose that you wonder whether Jerry has any malevolent friends, that you know that Jerry's only friend is Tom, and that your knowledge leaves fully open the questions of whether Tom is malevolent and whether Susan (Jerry's enemy) is malevolent. Under these assumptions, FRN requires you to wonder whether Tom is malevolent. It also requires you to wonder whether Tom is malevolent and whether Susan is. Here we see an instance of the problem of irrelevant conjuncts: why should you have to wonder the Tom-and-Susan question, and not just the Tom-question? We also see a way forward.
The Tom-question “weakens” the Tom-and-Susan question. In general, one question weakens another if you can obtain the former by starting with the latter and replacing at least one of its direct answers with a proposition that is logically weaker given your knowledge. In the case at hand, not just some but all of the direct answers get replaced in this way (see the figure below).
So the Tom-question weakens the Tom-and-Susan question. It has two other very important features (given your knowledge) as well. First, like the Tom-and-Susan question, the Tom-question fully resolves your original question—the question of whether Jerry has any malevolent friends. Second, and unlike the Tom-and-Susan question, the Tom-question can't be further weakened compatibly with continuing to fully resolve your original question. We'll combine these last two points by saying that the Tom-question minimally resolves your original question (given your knowledge). Here's the definition:
Minimal Resolution: Q 2 minimally resolves Q 1 given Γ iff, given Γ,
1. Q 2 fully resolves Q 1, and
2. No weakening of Q 2 fully resolves Q 1.
This relation suggests a new norm:
MRN:
It is irrational to: wonder Q 1 but not Q 2 when, given what you know, Q 2 minimally resolves Q 1.
To see MRN's payoff, return to Authades and his original question Q R (Who wrote the Republic?) along with the irrelevant conjunct Q D (What is the 104th digit in the phone book?) and the question Q RD which conjoins them. MRN does not require Authades to wonder Q RD when he wonders Q R. For we can take any direct answer to Q RD (say, Aristotle wrote the Republic and the digit at issue is “7”), replace it with one that is logically weaker given his knowledge (here, Aristotle wrote the Republic), and get a question that still fully resolves Q R. Hence, Q RD doesn't minimally resolve Q R; and MRN doesn't require Authades to wonder the former when he wonders the latter.
Yet it does require him to wonder who wrote the Meno (Q M) when he wonders Q R. For Q M, unlike Q RD, minimally resolves Q R given his knowledge—it fully resolves Q R and no weakening of it does. MRN thus explains why Authades in our dialogue counts as being irrational, without objectionably requiring him to wonder questions like Q RD.
This is a success story—and it generalizes over at least one large and interesting class of cases. These are the cases where (a) your knowledge leaves Q 1 affirmatively open and (b) there's at least one direct answer to Q 2 that isn't entailed by at least one direct answer to Q 1 (given your knowledge). In these cases, certain ways of settling Q 2 aren't delivered by certain ways of settling Q 1. The required disconnect is slim: only one way of settling Q 2 needs to be left out, and only one way of settling Q 1 needs to leave it out. In these cases, Q 1Q 2 can be weakened while still fully resolving Q 1; just eliminate from $Q_1\& Q_2$
's direct answers those conjuncts from Q 2 that aren't entailed (given your knowledge) by their corresponding conjuncts from Q 1. MRN thus permits you to wonder Q 1 without wondering $Q_1\& Q_2$
. A large part of the problem of irrelevant conjuncts is duly boiled off.
There does remain a residue. Suppose we start with Q R and add to one of its direct answers a conjunct consisting in a proposition Authades knows. Then MRN requires him to wonder the resulting question if he wonders Q R. This kind of case is a residue of the problem of irrelevant conjuncts, and a task for future work.
MRN makes significant progress on the problem of irrelevant conjuncts. And it solves the problem of eliminated conjuncts—in the same way as FRN. And it does these things while offering an explanation of why Authades counts as being irrational, at least if his knowledge leaves the questions at issue fully open. Thus, while not perfect, MRN has bone fide credentials. It merits our theoretical attention.
5 Generalizing the Norms
The Minimal Resolution Norm (MRN) gives us the explanation we seek, solves the problem of eliminated conjuncts, and makes significant progress on the problem of irrelevant conjuncts. These are successes; but they come at a price. MRN is narrower than RN and FRN, targeting a smaller group of would-be violators than either of them. Narrow norms tend to avoid error. But they purchase their error-avoidance with reduced informativeness, rendering fewer (correct or incorrect) verdicts. Is there a way to lower the cost, to make MRN more informative?
5.1 From Resolution to Helps-resolve
Recall that each direct answer to a resolving question entails (given Γ) a direct answer to the resolved question. By replacing this effectiveness condition with the following helpfulness condition, we can generate a broadened version of the resolution relation:
Helpfulness: Q 2 is helpful for Q 1 given Γ iff , given Γ, each direct answer to Q 2 entails a partial answer to Q 1.
We can further broaden the resolution relation by removing its affirmative openness condition. The result is a new relation, the helps-resolve relation.
Helps-resolve: Q 2 helps resolve Q 1 given Γ iff
1. Q 2 is secured by Q 1 given Γ, and
2. Q 2 is helpful for Q 1 given Γ.
This relation is none other than general erotetic implication, the most extensively studied relation in IEL — a relation that (among other things) underwrites Socratic Proofs and Erotetic Search Scenarios, methods for reducing one set of questions to another (Wiśniewski, Reference Wiśniewski2003, Reference Wiśniewski2004; Leszczyńska-Jasion & Chlebowski, Reference Leszczyńska-Jasion and Chlebowski2015). It inspires several norms that are more informative than those we've thus far discussed. For example:
HRN:
It is irrational to: wonder Q 1 but not Q 2 when, given what you know, Q 2 helps resolve Q 1.
This norm attributes irrationality to everyone our previous resolution norms did, plus many about whom the latter are silent. It is thus a natural place to begin the search for more informative norms. It is also a natural norm to explore for practitioners of IEL, built as it is from IEL's most extensively studied relation.
Is HRN plausible? We'll argue that it is not, because it leads to a dilemma. Having done that, we'll ask whether the dilemma can be averted by reintroducing the affirmative or full openness conditions. We'll argue that it can't; then we'll draw out a general lesson.
5.2 The Difficult Dilemma
Two steps lead from HRN to an unhappy result. Let K be the propositions that you know, and let Q big be The Big Question—What are all the facts?.
The first step is to note that Q big helps resolve Q rain (= Is it raining?) given K. Since the direct answers to Q big form a partition, it has a true direct answer no matter what (and, thus, a true direct answer given K). So, Q big is secured by Q rain given K. Q big is also helpful for Q rain given K, since any direct answer to Q big will (given K) either entail it is raining or entail it is not raining (each of those propositions being direct answers and, thus, partial answers to Q rain). Since Q big is secured by and helpful for Q rain given K, it helps resolve Q rain given K. Conditional on your wondering whether it's raining, then, HRN requires you to also wonder what all the facts are.
The second step is to note that Q even (= Are the sand grains even?) helps resolve Q big given K. Like Q big itself, Q even is a partitioning question and is thus secured by Q big given K. Now notice that both of the direct answers to Q even entail partial answers to Q big. For instance, if there is an even number of sand grains, then the true direct answer to Q big must be a disjunct in the disjunction of those direct answers (to Q big) that contain the conjunct the sand grains are even. So Q even is both helpful for and secured by, and thus helps resolve, Q big given K. Conditional on your wondering what all the facts are, then, HRN requires you to also wonder whether the sand grains are even.
And now we've got the unhappy result: conditional on your wondering whether it's raining, HRN requires you to also wonder whether the sand grains are even. If you find yourself wondering whether it is raining, then, you've got two options: stop wondering that question, or start wondering whether the sand grains are even.
The dilemma generalizes. Conditional on your wondering any question at all, HRN requires you to also wonder every other question secured by Q big given your knowledge. All partitioning questions, many of which are wildly trivial, meet those conditions no matter what you know. How many beige things crossed Poland's border a prime number of times in 1981? What proportion of those things were chihuahuas? Either you become a dullard by not wondering anything at all, or you become an inquiry-pump by wondering the foregoing questions and countless similar others.
Poisonous options both. And yet, if one were to start with IEL and try to glean norms of expansion from it, HRN would be a wholly sensible candidate. The helps-resolve relation on which it is built has been studied extensively and for good reason. On its face, that relation seems apt to make for a plausible norm of expansion.
Can the dilemma be averted by adding affirmative or full openness conditions to the helps-resolve relation and rebuilding HRN accordingly? Sadly, no: the resulting norms would still yield a version of the dilemma. To see why, just replace Q big with the slightly smaller question Which propositions, among those neither affirmed nor denied by what I know, are true?. You'll then end up with (slightly restricted versions of) the same poisonous options we've described.
5.3 A Lesson
FRN and MRN don't lead—at least not via the path we've charted—to the choice between dullard and inquiry-pump. MRN blocks the path's first step, because Q big can be weakened in the relevant way. And FRN blocks the path's second step, because Q even does not resolve Q big (it merely helps resolve it). Are there other paths from FRN or MRN to the difficult dilemma? We don't think so. Given any background knowledge whatsoever, Q big helps resolve Q rain, and Q even helps resolve Q big. But given that same background knowledge, it need not be the case that Q even helps resolve Q rain. The helps-resolve relation is therefore intransitive. This is likely why HRN leads to the difficult dilemma.
In contrast, we prove in the Appendix that resolution, full resolution, and minimal resolution are transitive. These proofs show that if it's only norms based on intransitive relations that lead to the difficult dilemma, RN and FRN and MRN evade its grip. Perhaps the dilemma can be reached in other ways, but we can't see how. Thus we conjecture that no path leads from RN or FRN or MRN to the difficult dilemma. This conjecture, if correct, is an important general lesson.
6 Outro
We've broached the research program of connecting IEL to interrogative epistemology, first by using IEL to reverse engineer several extant norms of restriction and second by using it to forward engineer several new norms of expansion. The new norms can explain why certain cases feature irrationality. Winner winner chicken dinner, but the food's not free. The strongest new norm, RN, yields implausible results. Some attempts to fix it gave us FRN and MRN. These norms add plausibility but subtract informativeness. The obvious way to bring some informativeness back, HRN, leads to a difficult dilemma.
Now to our conclusion. On brand, it consists in some questions. Given the foregoing points, it seems sensible to search for a minimal version of HRN. So, what would such a norm say? Would it evade the difficult dilemma? What other features would it have?
Finally, a metapoint. We've been, for philosophers, unusually noncommittal. Instead of staking out our ground and fortifying its defenses we've openly explored some uncharted territory, traversing our preferred path while marking out other paths too. This kind of theorizing is not always called for, but sometimes it is.Footnote 4
A Appendix
A.1 Preliminaries
Our language, ${\cal L}$
, is composed of ${\cal L}_d$
and ${\cal L}_e$
. ${\cal L}_d$
consists of the declarative wffs of classical propositional logic (hereafter: “d-wffs” or “propositions”), built by applying the familiar connectives ($\neg , \;\wedge , \;\vee , \;\to , \;\leftrightarrow$
) to a countable set of atoms (p, q, r, …). We use A, B, C, D, sometimes with subscripts, for arbitrary propositions and Γ, Δ, Σ for (possibly empty) sets of propositions. Let ${\rm \models }$
be the classical declarative consequence relation defined over ${\cal L}_d$
, so that ${\rm \models }\;\subseteq {\cal P}( {{\cal L}_d} ) \times {\cal L}_d$
. ${\cal L}_e$
consists of erotetic wffs (hereafter: “e-wffs” or “questions”), built by adding the question mark, left and right brackets, and the comma to a sequence of at least two syntactically distinct d-wffs constituting the resulting question's direct answers—for instance, $?\{ {p, \;\neg p} \} \in {\cal L}_e$
. (Note that the question mark is not a set-theoretic operator—e.g. ?{p, q} and ?{q, p} are distinct e-wffs.) Arbitrary questions are represented by Q, often with subscripts. We use = for both set-theoretic identity and syntactic identity between wffs.
A.2 Basic Definitions
We begin with some basics.
Definition 1 (Syntax of ${\cal L}$
). ${\cal L}$
is the set such that
(i) If $A\in {\cal L}_d$
, then $A\in {\cal L}$.(ii) If $A_1, \;\ldots , \;A_n( {n > 1} ) \in {\cal L}_d$
and A i ≠ A j for all 1 ≤ i, j ≤ n, then $?\{ {A_1, \;\ldots , \;A_n} \} \in {\cal L}$.(iii) Nothing else is an element of ${\cal L}$
.
Fact 1 (Properties of Classical Declarative Consequence).
1. $\Gamma \mathop \cup \{ A \} \ {\rm \models }\ A$
(Reflexivity)2. If $\Gamma \mathop \cup \{ A \} \ {\rm \models }\ B$
and $\Gamma \mathop \cup \{ B \} \ {\rm \models }\ C$, then $\Gamma \mathop \cup \{ A \} {\rm \models }C$ (Transitivity)3. If $\Gamma \mathop \cup \{ A \} \ {\rm \models }\ B$
, then $\Gamma \mathop \cup \{ A \} \mathop \cup \{ C \} \ {\rm \models }\ B$ (Monotonicity)
Definition 2 (Direct Answers and the d( ⋅ )-function). Let dQ be the function that maps Q to the set of its direct answers. So, if Q = ?{A 1, …A n}, then dQ = {A 1, …A n}. We apply disjunction as follows: If Q = ?{A 1, …, A n} then ∨ dQ = A 1∨…∨A n.
Definition 3 (Partial Answers). Let dQ = {A 1, …, A n}. B is a partial answer to Q iff B = A i∨…∨A j where {A i, …, A j} ⊂ {A 1, …, A n}.
Definition 4 (Eliminative Answers). B is an eliminative answer to Q iff B = ¬A for some A ∈ dQ.
(Note that Definitions 3 and 4 differ from the typical ones; see Wiśniewski (Reference Wiśniewski2013: 43–44).)
Definition 5 (Presuppositions). B is a presupposition of Q iff $A{\rm \models }B$
for all A ∈ dQ.
Definition 6 (Sound Questions). Q is sound iff at least one of its direct answers is true, i.e. iff ∨ dQ is true. Q is sound relative to Γ iff $\Gamma \ {\rm \models }\, \vee dQ$
. It follows from Definition 5 that Q is sound only if all of its presuppositions are true.
Definition 7 (Openness). Γ leaves Q fully open iff it leaves Q affirmatively and negatively open—i.e. iff
(i) $\Gamma \nvDash A{\rm \;for\;all\;}A\in dQ$
(affirmative openness), and(ii) $\Gamma \nvDash \neg A{\rm \;for\;all\;}A\in dQ$
(negative openness).
Definition 8 (Conjunctive Questions). Let $Q_1\& Q_2$
be the question such that $d( {Q_1\& Q_2} ) = \{ {A\wedge B\,{\rm \mid }\,A\in dQ_1, \;B\in dQ_2} \}$
. Since disjunction distributes over conjunction, ${\vee} d( {Q_1\& Q_2} )$
is equivalent to ${\vee} dQ_1\wedge{\vee} dQ_2$
.
A.3 Evocation, Helpful Resolution, and Resolution
Now to some inferential relations.
Definition 9 (Evocation). Γ evokes Q iff Q is sound and (affirmatively) open relative to Γ—i.e. iff
(i) $\Gamma {\rm \models }\vee dQ$
(relative soundness), and(ii) $\Gamma \,\nvDash A{\rm \;for\;all\;}A\in dQ$
(affirmative openness).
Definition 10 (Helpful Resolution, a.k.a General Erotetic Implication). Q 2 helps resolve Q 1 given Γ iff Q 2 is secured by Q 1 given Γ and every direct answer to Q 2 entails a partial answer to Q 1 given Γ—i.e. iff
(i) $\Gamma \mathop \cup \{ {\vee dQ_1} \} {\rm \models }\vee dQ_2$
(security), and(ii) for all B ∈ dQ 2 there is some Δ ⊂ dQ 1 such that $\Gamma \mathop \cup \{ B \} \, {\rm \models }\vee \Delta$
(helpfulness).
Narrower than helpful resolution is the following relation.
Definition 11 (Resolution, a.k.a. Strong Regular Erotetic Implication). Q 2 resolves Q 1 given Γ iff Q 2 is secured by and effective for Q 1 given Γ, and Γ leaves Q 1 affirmatively open—i.e. iff
(i) $\Gamma \mathop \cup \{ {\vee dQ_1} \} \,{\rm \models }\vee dQ_2$
(security),(ii) for all B ∈ dQ 2 there is some A ∈ dQ 1 such that $\Gamma \mathop \cup \{ B \} \,{\rm \models }\,A$
(effectiveness),(iii) $\Gamma \,\nvDash A{\rm \;for\;all\;}A\in dQ_1$
(affirmative openness).
Definition 12 (Abbreviations). When a relation obtains between a set of propositions Γ, a question Q 2, and a question Q 1, we say that Q 2 Γ-relates to Q 1. So, if Q 2 resolves Q 1 given Γ, we say that Q 2 Γ-resolves Q 1. Similarly, when $\Gamma \mathop \cup \{ A \} {\rm \models }B$
but $\Gamma \mathop \cup \{ B \} \,\nvDash\ A$
, we say that B is Γ-weaker than A, which we express as $A > _\Gamma B.$
On to some lemmas and theorems involving resolution and evocation.
Lemma 1 (Resolving Questions are Affirmatively Open). If Q 2 resolves Q 1 given Γ, then Γ leaves Q 2 affirmatively open.
Proof. Suppose, for reductio, that Q 2 Γ-resolves Q 1 but that there is some B ∈ dQ 2 such that $\Gamma {\rm \models }B$
. From Definition 11, it follows that $\Gamma \,\nvDash A$
for all A ∈ dQ 1 and that for all B ∈ dQ 2 there is some A ∈ dQ 1 such that $\Gamma \mathop \cup B{\rm \models }A$
. So, if $\Gamma {\rm \models }B$
for some B ∈ dQ 2, then, by transitivity (Fact 1), $\Gamma {\rm \models }A$
for some A ∈ dQ 1—a contradiction. □
Theorem 1 (Conjunctive Questions Resolve their Evoked Conjuncts). If Γ evokes Q 1 and Q 2, then $Q_1\& Q_2$
resolves Q 1 given Γ.
Proof. Assume that Γ evokes Q 1 and Q 2. We now show that $Q_1\& Q_2$
satisfies each of the three conditions in Definition 11 for Q 1. Since Q 1 and Q 2 are sound relative to Γ, we know that $\Gamma {\rm \models }\vee Q_1$
and $\Gamma {\rm \models }\vee Q_2$
. So, $\Gamma {\rm \models }\vee dQ_1\wedge{\vee} dQ_2$
. Distributing the conjunction we obtain $\Gamma {\rm \models }\vee d( {Q_1\& Q_2} )$
, and from monotonicity it follows that $\Gamma \mathop \cup \{ {dQ_1} \} {\rm \models }\vee d( {Q_1\& Q_2} )$
. Thus, Q 1 Γ-secures $Q_1\& Q_2$
. By classical logic, we know that for any propositions A and B, $\Gamma \mathop \cup \{ {A\wedge B} \} {\rm \models }A$
. So, $Q_1\& Q_2$
is Γ-effective for Q 1. Since Γ evokes Q 1, it follows that it leaves Q 1 affirmatively open. Thus, $Q_1\& Q_2$
resolves Q 1. □
Theorem 2 (Resolution is Transitive). If Q 2 resolves Q 1 given Γ and Q 3 resolves Q 2 given Γ, then Q 3 resolves Q 1 given Γ.
Proof. Assume that Q 2 Γ-resolves Q 1 and Q 3 Γ-resolves Q 2. It follows that $\Gamma \mathop \cup \vee dQ_1{\rm \models }\vee dQ_2$
and $\Gamma \mathop \cup \vee dQ_2{\rm \models }\vee dQ_3$
. By transitivity, $\Gamma \mathop \cup \vee dQ_1{\rm \models }\vee dQ_3$
and so Q 3 is Γ-secured by Q 1, satisfying condition (i) in Definition 11. Likewise, since Q 3 Γ-resolves Q 2, it follows that Γ leaves Q 3 affirmatively open, satisfying condition (iii) in Definition 11. Lastly, by hypothesis, Q 2 is Γ-effective for Q 1—i.e. for all B ∈ dQ 2 there is some A ∈ dQ 1 such that $\Gamma \mathop \cup B{\rm \models }A$
—and Q 3 is Γ-effective for Q 2—i.e. for all C ∈ dQ 3 there is some B ∈ dQ 2 such that $\Gamma \mathop \cup C{\rm \models }B$
. By transitivity, we obtain $\Gamma \mathop \cup C{\rm \models }A$
for all C ∈ dQ 3 and some A ∈ dQ 1, and thus, Q 3 is Γ-effective for Q 1. So, Q 3 satisfies all the conditions in Definition 11 and Γ-resolves Q 1. □
A.4 Full Resolution
While resolution is narrower than helpful resolution, the following is narrower still.
Definition 13 (Full Resolution). Q 2 fully resolves Q 1 given Γ iff Q 2 resolves Q 1 given Γ and Γ leaves Q 2 negatively open—i.e. iff
(i) Q 2 resolves Q 1 given Γ, and
(ii) $\Gamma \nvDash \neg B{\rm \;for\;all\;}B\in dQ_2$
(negative openness)
Again we offer some proofs involving the relation at hand.
Lemma 2 (Fully Resolving Questions are Fully Open). If Q 2 fully resolves Q 1 given Γ, then Γ leaves Q 2 fully open.
Proof. Assume that Q 2 fully Γ-resolves Q 1. From Definition 13 it follows that $\Gamma \nvDash \neg B$
for all B ∈ dQ 2 and from Lemma 1, it follows that $\Gamma \nvDash B$
for all B ∈ dQ 2. □
Lemma 3 (Full Resolution is Reflexive given Full Openness). If Γ leaves Q fully open, then Q fully resolves itself given Γ.
Proof. Assume that Γ leaves Q fully open. From reflexivity and monotonicity, it follows that every question is Γ-secured by itself—i.e. $\Gamma \mathop \cup \{ {A_1\vee \ldots \vee A_n} \} {\rm \models }A_1\vee \ldots \vee A_n$
—and is Γ-effective for itself—i.e. $\Gamma \mathop \cup \{ {A_i} \} {\rm \models }A_i$
for all A i ∈ dQ. So, Q Γ-resolves itself. Thus, Q meets both conditions in Definition 13 and fully Γ-resolves itself. □
Theorem 3 (Full Resolution is Transitive). If Q 2 fully resolves Q 1 given Γ and Q 3 fully resolves Q 2 given Γ, then Q 3 fully resolves Q 1 given Γ.
Proof. Suppose that Q 2 fully Γ-resolves Q 1 and that Q 3 fully Γ-resolves Q 2. Since full resolution entails resolution, it follows from Theorem 2 that Q 3 resolves Q 1. So, all that remains is to establish that Γ leaves Q 3 negatively open, i.e. $\Gamma \nvDash \neg C$
for all C ∈ dQ 3, and this follows from the hypothesis that Q 3 fully Γ-resolves Q 2. □
A.5 Minimal Resolution
Our final relation, minimal resolution, is narrower than even full resolution.
Definition 14 (Weakening Questions). Q 2 weakens Q 1 given Γ iff there is some subset Δ of dQ 1 such that dQ 2 is the result of replacing each element of Δ with a proposition that's logically weaker given Γ — i.e. iff, for some Δ⊆ dQ 1,
(i) there is some surjection f:Δ → Σ such that $A > _\Gamma f( A )$
for every A ∈ Δ, and(ii) dQ 2 = $( {dQ_1\backslash \Delta } ) \mathop \cup \Sigma$
.
Definition 15 (Minimal Resolution). Q 2 minimally resolves Q 1 given Γ iff
(i) Q 2 fully Γ-resolves Q 1, and
(ii) No Γ-weakening of Q 2 fully Γ-resolves Q 1.
The following notion of counter-minimals will become helpful momentarily.
Definition 16 (Counter-minimals). Q 3 is a Γ-counter-minimal to Q 2 for Q 1 iff
(i) Q 2 and Q 3 both fully resolve Q 1 given Γ, and
(ii) for some B ∈ dQ 2 and some C ∈ dQ 3, $B > _\Gamma C$
.
Now we show that Definition 15 can be reformulated by substituting for (ii) a certain condition, (ii ′), which is equivalent to it given condition (i).
Lemma 4 (Equivalencies). If Q 2 fully Γ-resolves Q 1, then (ii) is equivalent to (ii ′):
(ii) No Γ-weakening of Q 2 fully Γ-resolves Q 1,
(ii′) There is no Γ-counter-minimal to Q 2 for Q 1.
Proof. Assume that Q 2 fully Γ-resolves Q 1.
(ii ↠ ii ′). For contraposition, let Q 3 be a Γ-counter-minimal to Q 2 for Q 1. Then there is some B ∈ dQ 2 and some C ∈ dQ 3 such that $B > _\Gamma C$
. Now let $dQ_4 = ( {dQ_2\backslash \{ B \} } ) \mathop \cup \{ C \}$
. By Definition 14, Q 4 is a Γ-weakening of Q 2. We'll show that it also fully Γ-resolves Q 1. First note that, given how Q 4 is defined, it follows from $\Gamma \mathop \cup \{ B \} {\rm \models }C$
that $\Gamma \mathop \cup \{ {\vee dQ_2} \} {\rm \models }\vee dQ_4$
. Thus, since Q 1 Γ-secures Q 2 (i.e. $\Gamma \mathop \cup \{ {\vee dQ_1} \} {\rm \models }\vee dQ_2$
), transitivity ensures that $\Gamma \mathop \cup \{ {\vee dQ_1} \} {\rm \models }\vee dQ_4$
. So, Q 1 Γ-secures Q 4. Next note that, since Q 2 and Q 3 are Γ-effective for Q 1, Q 4 is Γ-effective for Q 1. Finally, since Γ leaves Q 2 and Q 3 fully open, it leaves Q 4 fully open. Thus, Q 4 is a Γ-weakening of Q 2 that fully Γ-resolves Q 1.
(ii ′ ↠ ii). Assume, for contraposition, that some Q 3 fully Γ-resolves Q 1 and Γ-weakens Q 2. By Definition 14 then, $B > _\Gamma C$
for some B ∈ dQ 2 and some C ∈ dQ 3. □
Theorem 4 (Minimal Resolution is Transitive). If Q 2 minimally resolves Q 1 given Γ and Q 3 minimally resolves Q 2 given Γ, then Q 3 minimally resolves Q 1 given Γ.
Proof. Assume that Q 2 minimally Γ-resolves Q 1 and that Q 3 minimally Γ-resolves Q 2. By Definition 13, Q 2 fully Γ-resolves Q 1 and Q 3 fully Γ-resolves Q 2. By Theorem 3 then, Q 3 fully Γ-resolves Q 1. All that remains is to show that there is no Γ-weakening of Q 3 that fully Γ-resolves Q 1—which, by Lemma 4, is equivalent to showing that there is no Γ-counter-minimal to Q 3 for Q 1. To do so, observe that our hypothesis entails that Γ leaves Q 2 fully open. So, Lemma 3 ensures that Q 2 fully Γ-resolves itself. But since Q 2 cannot be a Γ-counter-minimal to Q 3 for Q 2, it follows from Definition 16 that, for all C ∈ dQ 3 and all B ∈ Q 2, if $\Gamma \mathop \cup \{ C \} {\rm \models }B$
, then $\Gamma \mathop \cup \{ B \} {\rm \models }C$
.
Now suppose, for reductio, that Q 4 is a Γ-counter-minimal to Q 3 for Q 1. Then there is some C i ∈ dQ 3 and some D k ∈ dQ 4 such that $\Gamma \mathop \cup \{ {C_i} \} {\rm \models }D_k$
but $\Gamma \mathop \cup \{ {D_k} \} {\rm \mathrel{\nvDash}} C_i$
. By hypothesis, Q 3 is Γ-effective for Q 2; so, there is some B j ∈ Q 2 such that $\Gamma \mathop \cup \{ {C_i} \} {\rm \models }B_j$
. It follows from our points above that $\Gamma \mathop \cup \{ {B_j} \} {\rm \models }C_i$
. Since $\Gamma \mathop \cup \{ {C_i} \} {\rm \models }D_k$
, we obtain $\Gamma \mathop \cup \{ {B_j} \} {\rm \models }D_k$
by transitivity. If $\Gamma \mathop \cup \{ {D_k} \} {\rm \models }B_j$
, then transitivity yields $\Gamma \mathop \cup \{ {D_k} \} {\rm \models }C_i$
—a contradiction. If $\Gamma \mathop \cup \{ {D_k} \} {\rm \mathrel{\nvDash}} B_j$
, then Q 4 is a Γ-counter-minimal to Q 2 for Q 1—a contradiction. □
