Disjunctive predicates

I have found myself thinking these two thoughts, on different occasions, without ever noticing that they appear contradictory:

1. Other things being equal, a disjunctive predicate is less natural than a conjunctive one.

2. A predicate is natural to the extent that its expression in terms of perfectly natural predicates is shorter. (David Lewis)

For by (2), the predicates “has spin or mass” and “has spin and mass” are equally natural, but by (1) the disjunctive one is less natural.

There is a way out of this. In (2), we can specify that the expression is supposed to be done in terms of perfectly natural predicates and perfectly natural logical symbols. And then we can hypothesize that disjunction is defined in terms of conjunction (p ∨ q iff ∼(∼p ∧ ∼q)). Then “has spin or mass” will have the naturalness of “doesn’t have both non-spin and non-mass”, which will indeed be less natural than “has spin and mass” by (2) with the suggested modification.

Interestingly, this doesn’t quite solve the problem. For any two predicates whose expression in terms of perfectly natural predicates and perfectly natural logical symbols is countably infinite will be equally natural by the modified version of (2). And thus a countably infinite disjunction of perfectly natural predicates will be equally natural as a countably infinite conjunction of perfectly natural predicates, thereby contradicting (1) (the De Morgan expansion of the disjunctions will not change the kind of infinity we have).

Perhaps, though, we shouldn’t worry about infinite predicates too much. Maybe the real problem with the above is the question of how we are to figure out which logical symbols are perfectly natural. In truth-functional logic, is it conjunction and negation, is it negation and material conditional, is it nand, is it nor, or is it some weird 7-ary connective? My intuition goes with conjunction and negation, but I think my grounds for that are weak.

Probabilistic reasoning and disjunctive Gettier cases

A disjunctive Gettier case looks like this. You have a justified belief in p, you have no reason to believe q, and you justifiedly believe the disjunction p or q. But it turns out that p is false and q is true. Then you have a justified true belief in p or q, but that belief doesn’t seem to be knowledge.

Some philosophers, like myself, accept Lottery Knowledge: we think that in a sufficiently large lottery with sufficiently few winning tickets, for any ticket n that in fact won’t win, one knows that n won’t win on the probabilistic grounds that it is very unlikely to win.

Interestingly, assuming Lottery Knowledge, in at least some disjunctive Gettier cases one has knowledge of the disjunction. For suppose that 99.8% is a sufficient probability for knowledge in lottery cases. Consider a lottery with 1000 tickets, numbered 1–1000, and one winner. I will then have a justified belief that the winning ticket is among tickets 1 through 998 (inclusive). Let this be p. Suppose that unbeknownst to me, p is false and the winning ticket is number 999. Let q be the proposition that the winning ticket is number 999.

Then I have the structure of a disjunctive Gettier case: I have a justified belief in p, I have no reason to believe q, and I justifiedly believe p or q.

Now given Lottery Knowledge, I know that ticket 1000 doesn’t win. But p or q is equivalent to the claim that ticket 1000 doesn’t win, so I know p or q.

Thus, given Lottery Knowledge, I can have a case with the structure of a disjunctive Gettier case and yet know.

Note that usually one thinks in disjunctive Gettier cases that one’s belief in the true disjunction is inferred from one’s belief in the false disjunct p. But that’s not actually how I would think about such a lottery. My credence in the false disjunct p is 0.998. But my credence in the disjunction is higher: it’s 0.999. So I didn’t actually derive the disjunction from the disjunct.

So, someone who thinks probabilistically can have knowledge in at least some disjunctive Gettier cases.

Even more interestingly, the point seems to carry over to more typical Gettier cases that are not probabilistic in nature. Consider, for instance, the standard disjunctive Gettier case. I have good evidence that Jones owns a Ford. You have no idea where Brown is. But since I accept that Jones owns a Ford, I accept that Jones owns a Ford or Brown is in Barcelona. It turns out that Jones doesn’t own a Ford, but Brown is in Barcelona. So I have a justified true belief that Jones owns a Ford or Brown is in Barcelona, but it’s not knowledge.

However, if I think about things probabilistically, my belief in the disjunction is not simply derived from my belief that Jones owns a Ford. For my credence in the disjunction is higher than my credence that Jones own a Ford: after all, no matter how unlikely it is that Brown is in Barcelona, it is still more likely that Jones owns a Ford or Brown is in Barcelona than that Jones owns a Ford.

So it seems that I have a good inference that Jones owns a Ford or Brown is in Barcelona from the high probability of the disjunction. Of course, a good deal of the probability of the disjunction comes from the probability of the false disjunct. However, that doesn’t rule out knowledge if there is Lottery Knowledge: after all, a good deal of the probability of the disjunction in our lottery case could have been seen as coming from the false disjunct that the the winning number is between 1 and 998.

Perhaps the difference is this. In the lottery case, there were alternate paths to the high probability of the true disjunction. As I told the story, it seemed like most of the probability that the winning ticket was either from 1 to 998 (p) or equal to 999 (q) came from the first disjunct. But the disjunction is equivalent to many other similar disjunctions, such as that the ticket is in the set {2, 3, ..., 999} or is equal to 1, and in the case of the latter disjunction, the high probability disjunct is true. But in the Ford/Barcelona case, there doesn’t seem to be an alternate path to the high probability of the disjunction that doesn’t depend on the high probability of the false disjunct.

But it’s not clear to me that this difference makes for a difference between knowledge and lack of knowledge.

And it’s not clear that one can’t rework the Ford/Barcelona case to make it more like the lottery case. Let’s consider one way to fill out the story about how my mistake in thinking Jones owns a Ford came about. I’ve seen Jones driving a Ford F-150 at a few minutes past midnight yesterday, and I knew that he owned that Ford because I drove him to the car dealership when he bought it five years ago. Unbeknownst to me, Fred sold the Ford yesterday and bought a Mazda. Now, it is standard practice that when people buy cars, they eventually sell them: few people keep owning the same car for life.

So, my belief that Jones owned a Ford came from my knowledge that Jones owned a Ford early in the morning yesteray and my false belief that he didn’t sell it later yesterday or today. But now we are in the realm of a lottery case. For from my point of view, the day on which Fred sells the car is something random. It’s unlikely that that day was yesterday, because there are so many other days on which he could sell the car: tomorrow, the day after tomorrow, and so on, as well as the low probability option of his never selling it.

Now consider this giant exclusive disjunction, which I know to be true in light of my knowledge that Jones hadn’t yet sold the Ford as of early morning yesterday.

1. Jones sold the Ford yesterday and Brown is not in Barcelona, or Jones sold the Ford today and Brown is not in Barcelona, or Jones is now selling the Ford and Brown is not in Barcelona, or Jones will sell the Ford later today and Brown is not in Barcelona, or Jones will sell the Ford tomorrow and Brown is not in Barcelona, or … (ad infinitum), or Jones will never sell the Ford and Brown is not in Barcelona, or Brown is in Barcelona.

Each disjunct in (1) is of low probability, but I know some disjunct is true. This is now very much like a lottery case. Its being a lottery case means that I should—assuming the probabilities are good enough—be able to know that one of the disjuncts other than the first two is true. But if I can know that that one of the disjuncts other than the first two is true, then I should be able to know—again, assuming the probabilities are good enough—that Jones hasn’t sold the Ford yet or Brown is in Barcelona. And if I can know that, then there should be no problem about my knowing that Jones owns a Ford or Brown is in Barcelona.

So, it’s looking like I can have knowledge in typical disjunctive Gettier cases if I reason probabilistically.

Observation, collapse and circularity

The following four premises seem to be contradictory:

1. An observation of an event E is caused by the event E.

2. Observation causes collapse.

3. What is observed is the collapsed state.

4. There is no circular causation.

Here is my first attempt to get out of this, on behalf of those attracted to the observation-causes-collapse view. For concreteness, let’s suppose that we’re observing an electron in a mixed up-down spin state, and suppose that we observe that it’s in the up state. Distinguish these two events:

• O₁: Observing whether the electron is in an up or a down state.

• O₂: Observing the electron to be in an up state.

Then I think what the defender of observation-causes-collapse can say this: O₁ causes the collapsed state which in turn causes O₂. But this is rather strange. For O₁ and O₂ actually seem to be the same coarse-grained event, which makes that coarse-grained event be its own cause! Another way to see the problem is to note that O₁ is the disjunctive event of observing the electron to be in the up state or observing the electron to be in the down state. But then O₂ grounds O₁: disjunctions are grounded in their true disjunct(s). But then O₁ is causally prior to its grounds, which seems absurd.

A second attempt: deny (3). Compare Elizabeth Anscombe’s theory that an intention to ϕ in the successful case constitutes one’s knowledge that one will ϕ or Thomas Aquinas’s theory that God’s knowledge of the world is the cause of the world’s being as it is. On these cases, the direction of fit in the knowledge is reversed. Observation of quantum phenomena could be like that.

Third attempt: cut up an act of observation into two parts. Metaphorically speaking, we could imagine the mind querying the world: “Is the electron in an up or a down state?” In response, reality collapses, and the mind observes that reality is in an up state. Thus, we have a query event Q and an observation-proper O₂. It is Q that causes collapse, and P is then the observation of the collapse. This solves the circularity problem, but strictly speaking it’s incorrect to say that observation causes collapse. Rather, it is the pre-observation query event Q that causes collapse. And if simultaneous causation is possible, then Q and O₂ may be simultaneous.

I think the second and third attempts are the way to go, assuming we're keeping the basic idea behind observation-causes-collapse.

The search for new truths

I know that I have two hands. With a bit of thought, I now know a number of truths that it seems no ordinary person has ever known before:

• I have two hands or there is a palomino painted green and pink with someone in a Darth Vader costume on its back.

• I have two hands or the number of pigs born in 1745 is odd.

• I have two hands or Sir Patrick Stewart will consume a prime number of calories tomorrow.

• I have two hands or Donald Trump issued a series of anti-Klingon tweets yesterday.

And so on, ad infinitum. The search for new truths is thus really easy. I just need to search for silly propositions that no one has thought about, and disjoin them with something I know.

The logician's daughter

Yesterday, my eldest daughter burned the dessert she was baking. You see, the recipe at one point (after a fair amount of baking at a lower temperature) said to bake "for fifteen minutes or until golden brown." She knew the second disjunct was already true at the beginning of that period, but decided to opt for following the first disjunct, as apparently allowed by the recipe. The dessert was black when she was done (though some of the inside was edible).

Grounding graphs

Consider three propositions:

1. (2) or (3) is true.
2. (1) or (3) is true.
3. The sky is blue.

Then, clearly, (3) grounds (1) and (2). But there is also another path to grounding (1). We could say that (3) grounds (2), and then (2) grounds (1). But if (2) grounds (1), then by an exact parallel (1) grounds (2). And that violates the noncircularity of grounding.

What should we say about (1)-(3)? It was plausible to say that (3) grounds (1) and (2). But the line of thought that (3) grounds (2) and (2) grounds (1) was also plausible. We might say that there are three pathways to grounding among (1)-(3):

• (3) to both (1) and (2)
• (3) to (2) to (1)
• (3) to (1) to (2)

All pathways seem acceptable. But we had better not confuse the pathways, since if we mix up grounding claims that belong to the last two pathways, we get (2) grounding (1) and (1) grounding (2).

There are multiple grounding pathways. Here is one way to formalize this. Take as the primitive notion that of a grounding graph. A grounding graph encodes a particular mutually compatible grounding pathway. Each grounding graph is a directed graph whose vertices are propositions. It will often be a contingent matter whether a given graph is or is not a grounding graph: the same graph can be a grounding graph in one world but not in another. The notion is not a formal one. Moreover, grounding graphs will be backwards-complete: they will go as far back as possible. But their futures may be incomplete.

Say that a parent of a vertex b in a directed graph G is any vertex a such that a→b is an arrow of G, and then b is called a child of a. An ancestor is then a parent, or a parent of a parent, or .... An initial vertex is one that has no vertices.

We can say that a partly grounds b in G if and only if a is an ancestor of b in G and that a is fundamental in G if and only if a is initial in G. We say that a proposition a partly grounds b provided that there is a grounding graph G such that a partly grounds b in G, and that a proposition p is fundamental if and only if there is a grounding graph G such that p is fundamental in G. We say that the a partly grounds b compatibly with c partly grounding a provided that there is a single grounding graph in which both partial grounding relations hold.

We say that a finite or infinite sequence of vertices is a chain in G provided that there is an arrow from each element of the sequence to the next. We say that b is the terminus of a chain C provided that b is the last element of C.

We stipulate that a set S of vertices grounds b in G provided that (a) every vertex in S is an ancestor of b and (b) every chain whose terminus is b can be extended to a chain still with terminus b and that contains at least one member of S. In particular, the set of all the parents of b grounds b if it is non-empty.

We now have some bridge axioms that interface between the notion of a grounding graph and other notions:

• Truth: Every vertex of a grounding graph G is true.
• Explanation: Every non-initial vertex is explained by its parents.
• Partial Explanation: Every parent partly explains each of its children.

We add this very metaphysical axiom, which is a kind of Principle of Sufficient Reason:

• Universality: Every true proposition is a vertex of some grounding graph.

Now we add some structural axioms:

• Noncircularity: There is no grounding graph G in which a is a parent of b and b is a parent of a.
• Lower Bound: If C is a chain in a grounding graph G, then there is a vertex p of G which is the ancestor of all the vertices in C, other than p itself if p is in C.
• Wellfoundedness: No vertex of a grounding graph is the terminus of an infinite chain.
• Absoluteness of Fundamentality: No vertex is initial in one grounding graph and non-initial in another.
• Truncation: If G₁ is a grounding graph and G₂ is a subgraph of G₁ relatively closed under the parent relation (if b is in G₂ and a is a parent of b in G₁ then a is in G₂ and a is a parent of b in G₂), then G₂ is a grounding graph.

Absoluteness of Fundamentality says that if a proposition is fundamental, it is fundamental in every grounding graph where it is found. Of course Wellfoundedness entails Noncircularity and Lower Bound. And Noncircularity plus Absoluteness of Fundamentality entails that if a partly grounds b and b partly grounds a, then (a) these two grounding relations do not hold in the same grounding graph and (b) in every grounding graph where one of these relations holds, at least one of a and b is grounded in something other than a and b, so that there are no fundamental circles.

We can now add some "logical axioms". These are just a sampling.

• Disjunction Introduction: If a grounding graph G contains a vertex <p> but not the vertex <p or q>, then the graph formed by appending <p or q> to G together with an arrow from <p> to it is also a grounding graph.
• Conjunction Introduction: If a grounding graph G contains vertices <p> and <q> but not the vertex <p&q>, then the graph formed by appending <p&q> to G toegther with arrows from <p> and <q> to it is also a grounding graph.
• Existential Introduction: If a grounding graph G contains a vertex <Fa> but no vertex <(∃x)Fx>, then the graph formed by appending <(∃x)Fx> together with an arrow from <Fa> to <(∃x)Fx> is a grounding graph.
• Conjunctive Concentration: If a grounding graph G contains a vertex b with distinct parents <p> and <q> but no vertex <p&q>, then the graph formed by removing the arrows from <p> and <q> to b, adding the vertex <p&q> and inserting arrows from <p> and <q> to <p&q>, and from <p&q> to b is a grounding graph.
• No Disjunctive Overdetermination: If a grounding graph contains <p or q>, then it contains at most one of the arrows <p>→<p or q> and <q>→<p or q>.

Go back to our original example. There will be at least three distinct grounding graphs corresponding to the different grounding pathways. There will be a grounding graph where we have (3)→(2)→(1), and another where we have (2)→(3)→(1), and a third which contains (3)→(1) and (2)→(1). But there won't be a graph that contains both (2)→(1) and (1)→(2).

I don't really insist on this list of axioms. Probably the "logical axioms" are incomplete. Nor am I completely sure of all the axioms. But the point here is to indicate a way to structure further discussion.

[Definition of universality edited to fix problem pointed out in discussion.]

Remarks on the logic of commanding and permitting

Disjunctions

If I command you to do something, I thereby permit you to do it. But suppose I command you to do A or B or both. Then it seems that not only do I permit the disjunction, but I also permit each disjunct.

It is, I think, necessary that if I command you to do something, I also permit you to do it. Working out why exactly would be interesting.

But I do not think it is necessary that if I command you to do A or B or both, then I permit you to do A and I permit you to do B. Imagine a case where you are under all sorts of orders that I have no authority to override and which I do not know all of, but I know that you're not both prohibited from doing A and from doing B. I might then say: "Do A or B or both. Of course, stay within the scope of your other orders." If one of your other orders is never to do B, you can't say that my disjunctive command permitted you to do B. If this is right, then it's not part of the fundamental logic of commanding and permitting that by commanding a disjunction one permits the disjuncts.

Interesting question. Is it ever morally licit to issue the command to do A or B or both, when B is morally illicit? It is, I take it, always wrong to command or permit something wrong (I distinguish permission proper from waiving punishment). If commanding a disjunction always involves permitting the disjuncts, it follows that one may not licitly command a disjunction when one of the disjuncts in it is wrong. But if it is possible to command the disjunction without permitting each disjunct, then it may be licit to command a disjunction one disjunct of which is wrong, though not a disjunction both disjuncts of which are wrong. We can imagine a situation where very bad things will happen (to you and to your subordinates) if you refuse to issue an order you were commanded to issue, and the order is to do A or B or both, and B is morally wrong. In that case, it may be licit to say: "I command you to do A or B or both. And I forbid you from doing B." You've fulfilled your order to the letter and haven't commanded or permitted anything wrong. Still, in ordinary contexts, commanding A or B or both carries the implicated (and still real) permission of doing A and of doing B and of doing both.

Conjunctions

Suppose now I command you to do both A and B. Interestingly, while it does follow that I permit you to do both A and B, it does not follow that I permit you to do A. I may only be permitting you to do A if you're going to do B as well. So commands are not closed under logical entailment. For if they were, then in commanding A and B, I would be commanding A, and hence also permitting A.

Disjunction introduction and conditionals with disjunctive antecedents

[Note: In the original version of this post, I made the embarrassing false claim that relevance logic denies disjunction introduction. This claim will explain Brandon's and my exchange in the comments. I have since edited the post.]

Consider this argument, a version of which I've already discussed:

1. I won't write a blog post today mainly on French cooking.
2. Therefore: I won't write a blog post today mainly on French cooking or tomorrow the world will come to an end (or both).
3. Therefore: If I write a blog post today mainly on French cooking, tomorrow the world will come to an end.

Premise (1) is true. Conclusion (3) sounds false. There are a couple of things that one can do about this odd argument. One can embrace the conclusion but insist that the conditional is only used materially, and is trivially true because the antecedent is false. One can—and I think this is going to be the most common reaction among philosophers—reject the inference of (3) from (2). But a lot of ordinary people will balk at (2)—the disjunction introduction step, where from p, we conclude p or q for any q.

Denying disjunction introduction neatly undercuts the above argument, as well as removing the oddity that everything can be proved from a contradiction.

But blocking disjunction introduction is a mistake, because we need disjunction introduction. Suppose that we say:

4. One has committed a violation of a school safety zone if one is (a) driving a motor vehicle in a school safety zone and (b) talking on a cellphone or driving at more than 20 miles per hour or both.

Now suppose:

5. Sam is driving a motor vehicle in a school safety zone.
6. Sam is talking on a cellphone.

We obviously want to conclude that Sam has committed a violation of a school safety zone. But to do that with modus ponens, we need to establish that the antecedent of the conditional in (4) is true for Sam:

7. Sam is (a) driving a motor vehicle in a school safety zone and (b) talking on a cellphone or driving at more than 20 miles per hour or both.

We get (7a) from (5). But the only information relevant to (7b) is (6), and to get to (7b) from (6), one needs disjunction-introduction. One can imagine the sleazy lawyer who contends:

We grant that my client was driving a motor vehicle in a school safety zone. The evidence adduced by the state, we concede, shows that he was talking on a cellphone, but no evidence was adduced by the state that he was talking on a cellphone or driving at more than 20 miles per hour or both.

This is obviously bad. We use conditionals with disjunctions in their antecedents quite regularly and so denying disjunction introduction is not very tenable.

One might try, instead, having additional inference rules for conditionals with special antecedents. For instance, one might allow this

8. From (i) if p or q, then r, and (ii) p, infer r.
9. From (i) if s and (p or q), then r, and (ii) s, and (iii) p, infer r.

Rule (9) would take care of the school safety zone case. But, first of all, lots of such rules would be needed to handle all cases. And, second, once we allowed such a rule we would be liable to let disjunction introduction in through the backdoor. For instance, if we allow (8), we can prove disjunction introduction from the plausible axiom: if A, then A.

10. p. (Premise)
11. if p or q, then p or q. (Axiom)
12. p or q. (Rule (8)).

Jon Kvanvig suggests to me that one might take care of this problem by replacing conditionals with disjunctive antecedents by conjunctions of conditionals. On this proposal, we would replace (4) with:

13. One has committed a violation of a school safety zone if one is driving a motor vehicle in a school safety zone and talking on a cellphone, and one has committed a violation of a school safety zone if one is driving a motor vehicle in a school safety zone and one is driving at more than 20 miles per hour.

But while we could, indeed, stop using locution (4) and use (13) instead, that is a pretty revisionary proposal. We do think Sam has violated a school safety zone given (4)-(6)—we don't need (13) to get that conclusion.

So, the upshot is this: in this case we have a pretty good argument that we would be mistaken to deny disjunction introduction.

Quiz on "... or ..."

I am holding out to you my two closed fists. Let us suppose that I know that you know that I know which, if any, of my fists are empty and which are full (for simplicity, I take "full" to be the denial of "empty"). You don't know which, if any, of my fists are empty and which are full. In which of the following cases would I be telling you a lie if I said: "My left hand is full or my right hand is full" while competently using English? (Choose "depends" if you think the answer depends on factors that I didn't include in the description of the case.)

1. In fact my left hand is full and my right hand is full: lie not a lie depends don't know
2. In fact my left hand is empty and my right hand is full: lie not a lie depends don't know
3. In fact my left hand is full and my right hand is empty: lie not a lie depends don't know
4. In fact my left hand is empty and my right hand is empty: lie not a lie depends don't know