More on grounding of universals

The standard First Order Logic translation of “All As are Bs” is:

1. ∀x(A(x)→B(x)).

Suppose we accept this translation and we further accept the principle:

2. Universal facts are always partially grounded in their instances.

Then we have the oddity that the fact that all ravens are black seems to be partially grounded in my garbage can being black. Let R(x) and B(x) say that x is a raven and black, respectively, and let g be my garbage can. Then an instance of ∀x(R(x)→B(x)) is R(g)→B(g), and the latter material conditional is definable as ¬R(g)∨B(g). But a disjunction is grounded in its true disjuncts, and hence this one will be grounded in B(g) (as well as in ¬R(g)).

There are three things to dispute here: the translation (1), the grounding principle (2), and the claim that a material conditional is grounded in its consequent whenever that consequent is true. Of these, I am most suspicious of the translation of the two-place universal quantifier and the grounding principle (2).

Intending material conditionals and dispositions, with an excursus on lethally-armed robots

Alice has tools in a shed and sees a clearly unarmed thief approaching the shed. She knows she is in no danger of her life or limb—she can easily move away from the thief—but points a gun at the thief and shouts: “Stop or I’ll shoot to kill.” The thief doesn’t stop. Alice fulfills the threat and kills the thief.

Bob has a farm of man-eating crocodiles and some tools he wants to store safely. He places the tools in a shed in the middle of the crocodile farm, in order to dissuade thieves. The farm is correctly marked all-around “Man-eating crocodiles”, and the crocodiles are quite visible to all and sundry. An unarmed thief breaks into Bob’s property attempting to get to his tool shed, but a crocodile eats him on the way.

Regardless of what local laws may say, Alice is a murderer. In fulfilling the threat, by definition she intended to kill the thief who posed no danger to life or limb. (The case might be different if the tools were needed for Alice to survive, but even then I think she shouldn’t intend death.) What about Bob? Well, there we don’t know what the intentions are. Here are two possible intentions:

1. Prospective thieves are dissuaded by the presence of the man-eating crocodiles, but as a backup any that not dissuaded are eaten.

2. Prospective thieves are dissuaded by the presence of the man-eating crocodiles.

If Bob’s intention is (1), then I think he’s no different from Alice. But Bob’s intention could simply be (2), whereas Alice’s intention couldn’t simply be to dissuade the thief, since if that were simply her intention, she wouldn’t have fired. (Note: the promise to shoot to kill is not morally binding.) Rather, when offering the threat, Alice intended to dissuade and shoot to kill as a backup, and then when she shot in fulfillment of the threat, she intended to kill. If Bob’s intention is simply (2), then Bob may be guilty of some variety of endangerment, but he’s not a murderer. I am inclined to think this can be true even if Bob trained the crocodiles to be man-eaters (in which case it becomes much clearer that he’s guilty of a variety of endangerment).

But let’s think a bit more about (2). The means to dissuading thieves is to put the shed in a place where there are crocodiles with a disposition to eat intruders. So Bob is also intending something like this:

3. There be a dispositional state of affairs where any thieves (and maybe other intruders) tend to die.

However, in intending this dispositional state of affairs, Bob need not be intending the disposition’s actuation. He can simply intend the dispositional state of affairs to function not by actuation but by dissuasion. Moreover, if the thief dies, that’s not an accomplishment of Bob’s. On the other hand, if Bob intended the universal conditional

4. All thieves die

or even:

5. Most thieves die

then he would be accomplishing the deaths of thieves if any were eaten. Thus there is a difference between the logically complex intention that (4) or (5) be true, and the intention that there be a dispositional state of affairs to the effect of (4) or (5). This would seem to be the case even if the dispositional state of affairs entailed (4) or (5). Here’s why there is such a difference. If many thieves come and none die, then that constitutes or grounds the falsity of (4) and (5). But it does not constitute or ground the falsity of (3), and that would be true even if it entailed the falsity of (3).

This line of thought, though, has a curious consequence. Automated lethally-armed guard robots are in principle preferable to human lethally-armed guards. For the human guard either has a policy of killing if the threat doesn’t stop the intruder or has a policy of deceiving the intruder that she has such a policy. Deception is morally problematic and a policy of intending to kill is morally problematic. On the other hand, with the robotic lethally-armed guards, nobody needs to deceive and nobody needs to have a policy of killing under any circumstances. All that’s needed is the intending of a dispositional state of affairs. This seems preferable even in circumstances—say, wartime—where intentional killing is permissible, since it is surely better to avoid intentional killing.

But isn’t it paradoxical to think there is a moral difference between setting up a human guard and a robotic guard? Yet a lethally-armed robotic guard doesn’t seem significantly different from locating the guarded location on a deadly crocodile farm. So if we think there is no moral difference here, then we have to say that there is no difference between Alice’s policy of shooting intruders dead and Bob’s setup.

I think the moral difference between the human guard and the robotic guard can be defended. Think about it this way. In the case of the robotic guard, we can say that the death of the intruder is simply up to the intruder, whereas the human guard would still have to make a decision to go with the lethal policy in response to the intruder’s decision not to comply with the threat. The human guard could say “It’s on the intruder’s head” or “I had no choice—I had a policy”, but these are simply false: both she and the intruder had a choice.

None of this should be construed as a defence in practice of autonomous lethal robots. There are obvious practical worries about false positives, malfunctions, misuse and lowering the bar to a country’s initiating lethal hostilities.

Material conditionals and quantifiers

From:

1. Every G is H

it seems we should be able to infer for any x:

2. If x is G, then x is H.

This pretty much forces one to read “If p, then q” as a material conditional, i.e., as q or not p. For the objection to reading the indicative conditional as a material conditional is that this leads to the paradoxes of material implication, such as that if it’s not snowing in Fairbanks, Alaska today, then it’s correct to say:

3. If it’s snowing in Fairbanks today, then it’s snowing in Mexico City today

even if it’s not snowing in Mexico City, which just sounds wrong.

But if we grant the inference from (1) to (2), we can pretty much recover the paradoxes of material implication. For instance, suppose it’s snowing neither in Fairbanks nor in Mexico City today. Then:

4. Every truth value of the proposition that it’s snowing in Fairbanks today is a truth value of the proposition that it’s snowing in Mexico City today.

So, by the (1)→(2) inference:

5. If a truth value of the proposition that it’s snowing today in Fairbanks is true, then a truth value of the proposition that it’s snowing today in Mexico City is true.

Or, a little more smoothly:

6. If it’s true that it’s snowing in Fairbanks today, then it’s true that it’s snowing in Mexico City today.

It would be very hard to accept (6) without accepting (3). With a bit of work, we can tell similar stories about the other standard paradoxes. The above truth-value-quantification technique works equally well for both the true⊃true and the false⊃false paradoxes. The remaining family of paradoxes are the false⊃true ones. For instance, it’s paradoxical to say:

7. If it’s warm in the Antarctic today, it’s a cool day in Waco today

even though the antecedent is false and the consequent is true, so the corresponding material conditional is true. But now:

8. Every day that’s other than today or on which it’s warm in the Antarctic is a day that’s other than today or on which it’s cool in Waco.

So by (1)→(2):

9. If today is other than today or it’s warm in the Antarctic today, then today is other than today or today it’s cool in Waco.

And it would be hard to accept (9) without accepting (7). (I made the example a bit more complicated than it might technically need to be in order not to have a case of (1) where there are no Fs. One might think for Aristotelian logic reasons that that case stands apart.)

This suggests that if we object to the “material conditional” reading of “If… then…”, we should object to the “material quantification” reading of “Every F is G”. But many object to the first who do not object to the second.

A Gricean theory of indicative conditionals

The theory consists of two theses and two definitions. I will use → for indicative conditionals. And all my disjunctions will be inclusive.

1. MatCond: "p→q" expresses the same proposition as "~p or q".
2. NonTriv: A use of "p→q" normally implicates that "~p or q" is an evidentially non-trivial disjunction for the speaker.
3. Definition: "a or b" is an evidentially non-trivial disjunction for an agent x if and only if x has non-negligible evidence for the disjunction that goes over and beyond evidence for ~p and evidence for q.

I don't here commit to any particular view of evidence, and if there are non-evidential justifications, one can probably easily modify the theory.

Here is an interesting consequence of the theory which I think is just right. When my evidence that at least one of ~p and q is true is simply the evidence for ~p (or for q), I don't get to say "If p, then q." But if I tell you that at least one of ~p and q is true, then normally you get to say "If p, then q". For when I tell you that at least one of ~p and q is true, then "~p or q" comes to be an evidentially non-trivial disjunction for you: my testimony is evidence for the disjunction and this evidence does not derive for you from evidence for the one or the other disjunct.

Notice that "has non-negligible evidence for the disjunction" has some vagueness to it. Moreover, negligibility is contextual, and that is how it should be. If I tell you that at least one of the following is true: snow is not purple and 2+2=4, then "If snow is purple, then 2+2=4" does not generally become assertible for you. For while you do gain additional testimonial evidence for the disjunction that snow is not purple or 2+2=4 from my speaking to you, the gain is normally negligible over and beyond your earlier evidence that 2+2=4. But if you respond to my assertion with "So, if snow is purple, then 2+2=4", you are speaking quite correctly, since the use of "So" and the conversational context makes the evidence I just gave you salient and hence non-negligible. (Perhaps "salient" or "relevant" could be used in place of "non-negligible" in (3).)

The theory explains why it is that paradoxes of material implication can almost always be made to cease to be paradoxes of material implication as soon as one fills out the evidential backstory in a creative enough way. Take, for instance, the paradox of material implication:

4. If the president will invite me for dinner tonight, I will have dinner with the president in my pajamas.

The antecedent is false, so the material conditional is true, but (4) sure sounds bad (it sounds bad to assert and seems to be saying something bad about my manners). Yes, but now suppose that an epistemic authority has just handed me two numbered and folded pieces of paper, with a sentence written on each and folded in half, and told me that either at least the first paper contains a falsehood or they both contain truths. I puzzle out what she says, and I conclude, very reasonably:

5. If the sentence on the first piece of paper is true, the sentence on the second piece of paper is true.

I then unfold the pieces of paper, and notice that the first piece contains the sentence "The president will invite me for dinner tonight" and the second contains "I will have dinner with the president in my pajamas." And so I reasonably infer from (5):

6. So, if the president will invite me for dinner tonight, I will have dinner with the president in my pajamas.

(And, moreover, I now gain a new piece of evidence that the president won't invite me for dinner tonight—for it would be absurd to suppose I'd have dinner with him in my pajamas.) With this epistemic backstory, the paradoxical conditional is quite unparadoxical. That's because with this epistemic backstory, the corresponding disjunction

7. The president won't invite me for dinner tonight or I will have dinner with the president in my pajamas (or both)

is epistemically non-trivial. But in normal circumstances, (7) is epistemically trivial, since my only evidence for (7) is evidence for the first disjunct.

A similar kind of epistemic backstory can be given for any of the standard paradoxes of material implication, thereby turning paradoxical sentences into non-paradoxical ones (cf. this post). Our Gricean theory (1)-(3) explains this phenomenon neatly. So do theories on which indicatives are non-cognitive and ones on which they are subjective. But the Gricean theory is, I think, simpler.

Notice that in this Gricean theory we haven't brought in non-material conditionals through any back door, because we have explained the implicated content entirely in terms of disjunctions. Furthermore, (2) is basically a consequence of (1) plus the very plausible claim that disjunctive sentences normally implicate the epistemic non-triviality of the disjunction.

An argument for the material conditional account of indicatives

The material conditional account of indicatives is that "If s, then u" is true if and only if s is false or u is true or both.

1. (Premise) If the indicative conditional has the same truth values as the material conditional in the standard cases which are alleged to be counterexamples to the material conditional account, then the material conditional account is correct.
2. (Premise) The indicative conditional has mind-independent truth value.
3. (Premise) If the indicative conditional has mind-independent truth value, then it has the same truth values as the material conditional in the standard cases which are alleged to be counterexamples to the material conditional account.
4. Therefore, the material conditional account is correct.

In this argument, I am convinced of premises (1) and (3), but not sure of premise (2). Consequently, what the argument convinces me of is that either (2) is false or (4) is true. Premise (1) is not that controversial, I think. The material conditional account is simple and elegant, verifies modus ponens and contraposition, is well-defined and mind-independent. The only problem is that it appears to give the wrong answers for certain standard cases. If this appearance were undercut, the material conditional account would be the winner.

The hard work is going to be justify (3). Let us start by giving three representative alleged counterexamples, classified by the truth values of the antecedent and consequent:

5. "If I will have dinner with the queen tonight, I will eat dinner tonight in my pajamas." (Antecedent and consequent are both false.)
6. "If I will have dinner with the queen tonight, everyone that I will have dinner with tonight will be a family member." (Antecedent is false and consequent is true.)
7. "If it is snowing in the United States, it is snowing in Central Texas." (Suppose this was uttered a couple of days ago when it was snowing in Central Texas. Antecedent and consequent were both true.)

The material conditional account says that all three conditionals are true. But all three conditionals sound wrong (assuming I am not a member of the royal family and that I wouldn't wish to insult the queen).

I will argue that:

8. If the indicative conditional has mind-independent truth value, then (5)-(7) are all true.

The method of argument generalizes to all the standard counterexamples, and thus yields (3).

Here's the way I will argue for (8). Let "a" be the antecedent in the alleged counterexample. Let "c" be the consequent. Suppose I have the belief, justified or not, that at least one of "not-a" and "c" is true, and I have no further, more specific beliefs about the matters in a and in c. Since I believe that at least one of "not-a" and "c" is true, I should be able to sincerely say to someone:

9. I may not know much about the queen, dinners, pajamas, snow, etc., but I do believe that at least one of "not-a" and "c" is true. Hence, if a, then c.

This seems very reasonable.

Suppose now that I learn all the relevant facts about the queen, dinners, pajamas, snow, etc. In particular, I learn such facts as that people tend not to wear pajamas for dinner with the queen, that central Texas is one of the somewhat less likely places in the US to have snow, etc. I also learn the truth values of "a" and "c". None of the things I learn gives me reason to retract the claim that at least one of "not-a" and "c" is true. And neither have I any reason to retract the conclusion I drew, that if a, then c.

Therefore, when I said (9), I said something true. If it wasn't true, I would have reason to withdraw it. But the difference between the circumstances in my story in which I said the conditional in (9) and standard circumstances was in my beliefs—when I said (9), I lacked various beliefs that normal people in our culture have. Thus, if the indicative conditional has mind-independent true value, I have to conclude that actually the conditional "if a, then c" is also true. And so we have an argument for (8).

Material conditionals

Say that a proposition p is weakly earthly provided that for every pair of worlds w₁ and w₂ which exactly match one another in respect of all states of affairs localized within a thousand lightyears of earth and all entities and events capable of causally affecting states of affairs localized within a thousand lightyears of earth, p has the same truth value at w₁ and at w₂. It is very plausible that just about all propositions used in everyday speech are weakly earthly. A sentence is weakly earthly provided it expresses a weakly earthly proposition.

Now, suppose that "If p, then q" is weakly earthly. I shall argue that "If p, then q" has the same truth value as the material conditional. First, observe that if the material conditional is false, so is "If p, then q", since otherwise modus ponens wouldn't work.

For the converse, suppose the material conditional p→q is true. Now imagine a possible world w* which is just like our world, but which also contains a one-way causally isolated island universe u, such that events in our universe can affect events in u but not conversely, and where u contains Frizzy, a being that knows the truth values of all weakly earthly propositions (maybe God has told them all to him), and that believes no contradictions. Moreover, Frizzy has the odd property that he always speaks sentences in pairs. First, he utters a claim with no regard for its truth. After that, if the first claim he had uttered turns out to be something he believes to be true, he utters a second claim that he believes to be true; otherwise, he utters another claim with no regard for its truth.

Now suppose Frizzy utters the pair of propositions a and b (in this order), and suppose a and b are propositions Frizzy knows the truth values of. Then, if a is true, so is b. And, hence, it is true that "if a, then b". But now as long as the material conditional p→q is true, i.e., as long as we do not have both p and not-q, it is coherent with the above description of w* that Frizzy utters the pair p and q. So let us suppose that. Then, as noted above, it follows that it is true in w* that "if p, then q". But "if p, then q" is weakly earthly. Hence, if it is true in w*, it is true in the actual world.

What I have shown is that for any weakly earthly indicative conditional, the truth value of the conditional is equal to the truth value of the corresponding material conditional. Moreover, this argument can be run in any possible world (in some possible worlds, of course, the claim is close to trivial because there are no contingent weakly earthly claims). Therefore, necessarily, a weakly earthly indicative conditional holds iff the corresponding material conditional does. Now, assuming indicative conditionals have truth value it would, I think, be very unlikely that there would be something special in this way about weakly earthly indicatives. (The assumption is needed, because if indicatives don't have truth value, there are no weakly earthly indicatives.)

So, we have very good reason to think that either indicatives lack truth value or else indicatives are logically equivalent to material conditionals. I don't know which disjunct to choose.

"If... then..." and material conditionals

I will argue that if the indicative "If p, then q" in English has mind-independent truth value (a somewhat vague phrase, admittedly), then this truth value is the same as that of (not-p or q) (i.e., the material conditional). The way I shall argue this is as follows. Assume that "If p, then q" has mind independent truth value. Now, I will show that (i) if (not-p or q) is false, then "If p, then q" is false, and (ii) if (not-p or q) is true, then "If p, then q" is true. Claim (i) is easy. For if (not-p or q) is false, then p is true and q is false, and it clearly cannot be the case that "If p, then q" (modus ponens would be violated).

I now argue for (ii). The easiest way to do this is to specialize to the case where p and q and their denials do not tell us anything about what beliefs and credences people have (the proposition that there are dogs satisfies this constraint; the proposition that nobody believes anything does not satisfy this constraint). If (ii) holds for propositions satisfying this constraint, it will hold in general, surely (assuming "If... then..." has mind-independent truth value). Suppose that you rationally assign a probability very close to 1/2 to p as well as to q, and neither believe nor disbelieve either of these, and rationally assign a probability very close to 1 to the claim that (not-p or q), and, moreover, you know this disjunction to be true. Given the constraint on p and q, it should be quite possible to have a set of evidence that makes one have these probability assignments, and having this set of evidence should not affect the truth values of p, q or the indicative "If p, then q".

You then reflect on the following valid argument:

1. p (premise)
2. not-p or q (premise)
3. Therefore, q.

You want to summarize what you've learned from this argument. You know (2) to be true, and you assign very high probability to it. You don't know (1) or (3) to be true, and you in fact do not have a belief either way about either of these. It seems quite right to summarize your current position as: "If (1) holds, then (3) holds." After all, you've got a valid argument from (1) to (3) given an auxiliary premise, namely (2), which you know to be true. But once we agree that "If (1) holds, then (3) holds", surely we likewise have to agree that "If p, then q." Hence, if the disjunction (2) is known with probability close to 1, and neither p nor q is known or has high or low probability, then "If p, then q" is true. But if "If... then..." has mind-independent truth value, then the assumptions about knowledge and probability are irrelevant to its truth value, and hence we can simply conclude that if (2) holds, then "If p, then q."

The conclusion might be taken as a reductio of the claim that "If... then..." has mind independent truth value.

Adequacy of language

What does it mean to say that language M is at least as "adequate" as language L? One option would be to say that any proposition that L can express is a proposition that M can express. This, I think, is too strong a requirement. For sometimes one language cannot express exactly the same proposition as another language does, but in some sense loses nothing thereby in adequacy, or at least the language is at least as good for theology, ethics, science and ordinary life (that's what I mean by "practically"!) For instance, suppose that L is English and M is a restriction of English to those sentences that end with the conjunct "and each thing is identical to itself". Then M cannot express the proposition that there are horses. But the speakers of M do just as well with respect of theology, ethics, science and ordinary life any way: M is just as good as L for theology, ethics, science and ordinary life. Where a speaker of L would say that there are horses, the speaker of M will say that there are horses and each thing is identical to itself.

I do not have a clear notion of "adequacy" here—suggestions are welcome. Here, for what it is worth, are two interesting examples of how the notion might be useful.

1. Detensing: It is well known that tensed sentences like "I am now in pain" cannot be translated into token-reflexive sentences like "My pain is simultaneous with this utterance" (for instance, the latter sentence entails the occurrence of an utterance). But perhaps one can say, more weakly, that a tenseless language that makes use of token-reflexive forms like this is just as adequate as the tensed language. Certainly, it is just as adequate for ethics, science, ordinary life and probably theology. Instead of saying a sentence of ethics, science, ordinary life and theology like "It is now time for me to partially fulfill my duty of thanking God for the nomic orderliness of the universe", we just say: "This utterance is simultaneous with the time for the partial fulfillment of my duty of thanking God for the nomic orderliness of the universe." In saying this, we are saying something different. Different, yes, but in practice just as useful for ethhics, science, ordinary life and theology.

2. Indicatives: Maybe

1. "If the Queen visits me today, I will be prepared"

does not just mean

2. "I will be prepared for the Queen's visit or the Queen won't visit me today or both."

Nor does it just mean

3. "P(I will be prepared | the Queen visits me today) is high"

(claim (3) does not give modus ponens). In fact, plausibly, there is no paraphrase of the indicative conditional except in terms of indicative conditionals (including ones involving "unless" and other variants). Fine. But one can still say that all indicative conditionals could be dropped from English, and the resulting language would be just as adequate. I would not be saying the same thing as (1) if I affirmed the conjunction of (2) and (3), but I would lose nothing by doing so.

Indicative conditionals

On the material conditional interpretation, the propositional content of the indicative conditional "If p, then q" is p→q, i.e., (not-p or q).

I claim that this is basically the right interpretation if "If p, then q" expresses a proposition whose truth-value is mind-independent (except for any mind-dependence in p and q themselves). You can take this as evidence that the material conditional interpretation is right—that is how I take it—or that English indicative conditionals do not express a mind-independent proposition.

The argument is simple. Suppose that p and q concern non-mental matters, and suppose that w is a world p→q holds, i.e., p is false or q is true or both. Then there is a world w* which is very much like w, except that it contains two persons, A and B, conversing about p and q, neither of whom has any false or misleading or unjustified beliefs, and neither of whom has any beliefs giving significant evidence for any of the propositions p, q, not-p and not-q. We could then imagine A learning that either p is false or q is true or both, and that then the conversation turns to the subject of p and q. I claim that it would then be appropriate for A to say: "Well, I don't have any idea which if any of p and q is true, but I now know that if p holds, so does q." This seems quite right. Moreover, in saying this, A would not be saying anything false. Therefore, if "If p, then q" expresses a proposition, it expresses a true proposition in w*. But if the proposition it expresses is mind independent, it is also true in w, since the two worlds differ only in respect of mind-dependent stuff.

Hence, p→q entails that if p, then q. The converse is easy. If p→q is false, then p is true and q is false, and it is clear that then if p, then q isn't true. Therefore, necessarily, p→q holds iff if p, then q does. Hence, the material conditional gets the truth conditions for the indicative "if... then..." right.

Could it be that there is still a difference in meaning? The only way I could see that would be if "If p, then q" said something additional, something entailed by p→q, but nonetheless added on to it. But I just cannot see what that could be, unless it be something mind dependent.

But perhaps there is a difference here like that between "p or q" and "q or p"? Maybe there really is a difference in the proposition expressed by these claims, even though neither adds anything to the other. If there really is a difference in the propositions expressed by "p or q" and "q or p", then I guess there might be a difference between those expressed by "p→q" and if p, then q. But if so, that difference is not very significant, it seems. Basically, the two say the same thing. Of course, even if there is no difference in proposition, there may be pragmatic differences.

What about standard counterexamples to the material conditional interpretation? For instance, could I say about a batch of cookies that I know to be poisoned
(*) "If George eats these cookies, he won't feel sick"
simply because I know that George won't eat them? Well, I think such counterexamples at most challenge the claim that the indicative conditional expresses a proposition, not the claim that if it expresses a proposition, the proposition it expresses either is or is basically the same as a material condition. Suppose that I don't know that the cookies were poisoned, but Patricia tells me: "An omniscient being either told me that George won't eat these cookies, or that he won't feel sick, but I can't remember which." It seems perfectly appropriate for me to utter (*), then. Suppose I later learn that the cookies are poisoned and that George won't eat them. Do I have any reason to say that I was mistaken when I uttered (*)? Surely not. I can say that what I said was misleading, but not that it was false. Whether (*) is appropriate to say depends on mind-dependent stuff. But if (*) expresses a proposition, then that proposition is mind-independent. Consequently, the intuitions about the appropriateness of saying (*) should not be taken as evidence about what propositional content (*) has if it has any.