Cartesian-style ontological arguments

Cartesian-style ontological arguments run like this:

1. God has all perfections.

2. Existence is a perfection.

3. So, God exists.

These arguments are singularly unconvincing. Here is a simple reason they are unconvincing. Suppose we are undecided on whether there are any leprechauns and, if so, whether they have a king, and someone tells us:

4. The leprechaun king is very magical.

This sure sounds plausible in a certain frame of mind, and we may accept it. When we accept (4), while remaining undecided on whether there are leprechauns and, if so, whether they have a king, what we are accepting seems to be the conditional:

5. If the leprechaun king exists, he is very magical.

By analogy, when the agnostic accepts (1), it seems they are accepting the conditional:

6. If God exists, God has all perfections.

Given premise (2), we can conclude:

7. If God exists, God exists.

But every atheist accepts (7).

It seems to make little difference if in (2) we replace “existence” with “necessary existence”. For then we just get:

8. If God exists, God necessarily exists.

That’s not quite as trivial as (7), but doesn’t seem to get us any closer to the existence of God.

The above seems to perfectly capture why it is that Cartesian-style ontological arguments are unconvincing.

Even if the above is adequate as a criticism of Cartesian-style ontological arguments, I think there is still an interesting question of what sort of a conditional we have in (5)–(8)?

It’s not a material conditional, for then (5) would be trivially true given that there are no leprechauns, while (5) is non-trivially true.

Should it be a subjunctive conditional, like “If the leprechaun king existed, he would be very magical”? I don’t think so. For suppose that in the closest possible leprechaun world to ours, for some completely accidental reason, the leprechaun king is very magical, but in typical possible worlds with leprechauns, leprechaun kings are are actually rather a dud with regard to magicality. Then it’s true that if the leprechaun king existed, he would be very magical, but that shouldn’t lead us to say that the leprechaun king is very magical.

Perhaps it should be a strict conditional: “Necessarily, if the leprechaun king exists, he is very magical.” That actually sounds fairly plausible, and in light of this we would actually want to deny (4). For it is not necessary that the leprechaun king be very magical. But if we take it to be a strict conditional, we still have a triviality problem. Imagine an atheist who thinks that God is impossible. Then the strict conditional

9. Necessarily, if God exists, God has all perfections

is true, but so is:

10. Necessarily, if God exists, God has exactly 65% of the perfections.

But while it seems that our atheist would be likely to want to say that God has all perfections (indeed, that might be a part of why the atheist thinks God necessarily does not exist, for instance because they think that the perfections are contradictory), it doesn’t sound right to say that God has exactly 65% of the perfections, even if you think that necessarily there is no God.

I think the best bet is to make the conditional be a strict relevant conditional:

11. Necessarily and relevantly, if God exists, God has all perfections.

It is interesting to ask whether (11) helps Cartesian-style ontological arguments. Given (11), if all goes well (it’ll depend on the modal relevance logic) we should get:

12. Necessarily and relevantly, if God exists, God exists.

That sounds right but is of no help. We also get:

13. Necessarily and relevantly, if God exists, God necessarily exists.

Again, that sounds right, and is less trivial, but still doesn’t seem to get us to the existence of God, barring some clever argument.

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.

Towards a counterexample to Weak Transitivity for subjunctives

Transitivity for a conditional → says that if A→B and B→C, then A→C. For subjunctive conditionals this rule is generally taken to be invalid. If I ate squash (B), I would be miserable eating squash (C). If I liked squash (A), I'd eat squash (B). But it doesn't follow that if I liked squash, I'd be miserable eating squash.

Weak Transitivity says that if A→B, B→A and A→C, then A→C. The squash counterexample fails, for it's false that if I were eating squash (B), I'd like squash (A).

I don't know whether Weak Transitivity is valid. But here's something that at least might be a counterexample. Suppose a heavy painting hangs on two strong nails. But if one nail were to fail, eventually--maybe several days later--the other would fail. The following seem to be all not unreasonable:

1. If the right nail failed (B), the left nail would fail because of the right's failure (C).
2. If the left nail failed (A), the right nail would fail because of the left's failure (D).

So, by Weakening (if P→Q and Q entails R, then P→R):

3. If the left nail failed (A), the right would fail (B).
4. If the right nail failed (B), the left would fail (A).

If Weak Transitivity holds, then:

5. If the left nail failed (A), the left nail would fail because of the right's failure (C).

But surely (2) and (5) aren't true together.

As I said, I am not sure if Weak Transitivity is valid. If it is, then there is something wrong with (1)-(4), probably with (1) and (2). Maybe there is. But the example should at least give one reason not to be very confident about Weak Transitivity. (There is another reason: Weak Transitivity is incompatible with the non-triviality of the Adams Thesis for subjunctives.)

"Even if" clauses in promises

If I promise to visit you for dinner, but then it turns out that I have a nasty case of the flu, I don't need to come, and indeed shouldn't come. But I could also promise to meet you for dinner even if I have a nasty case of the flu, then if the promise is valid, I need to come even if I have the flu. I suspect, however, that typically such a promise would be immoral: I should not spread disease. But one can imagine cases where it would be valid, maybe say if you really would like to get the flu for a serious medical experiment on yourself.

In my previous post, I gave a case where it would be beneficial to have a promise that binds even when fulfilling it costs multiple lives. Thus, there is some reason to think that one could have promises with pretty drastic "even if" clauses such as "even if a terrorist kills ten people as a result of this." But clearly not every "even if" clause is valid. For instance, if I say I promise to visit you for dinner even if I have to endanger many lives by driving unsafely fast, my "only if" clause is not valid under normal circumstances (if we know that my coming to dinner would save lives, though, then it might be).

One can try to handle the question of distinguishing valid from invalid "only if" clauses by saying that the invalid case is where it is impermissible to do the promised thing under the indicated conditions. The difficulty, however, is that whether doing the promised thing is or is not permissible can depend on whether one has promised it. Again, the example from my previous post could apply, but there are more humdrum cases where one would have an on balance moral reason to spend the evening with one's family had one not promised to visit a friend.

Maybe this is akin to laws. In order to be valid, a law has to have a minimal rationality considered as an ordinance for the common good. In order to be valid, maybe a promise has to have a minimal rationality considered as an ordinance for the common human good with a special focus on the promisee? To promise to come to an ordinary dinner even if it costs lives does not have satisfy that condition, while to promise to bring someone out of general anesthesia even if a terrorist kills people as a result could satisfy it under some circumstances. It would be nice to be able to say more, but maybe that can't be done.

Nonpropositional views of conditionals, and lying

1. Some conditional claims are lies. ("If you show this car to any mechanic in town, he'll tell you it's a great deal.")
2. Conditional claims do not express propositions (but, say, conditional probabilities).
3. Assertions express propositions.
4. So, not all lies are assertions.

Of course, (2) is a quite controversial theory of conditionals. And one can turn the argument around: All lies are assertions, so conditional claims express propositions (at least in those cases; but one can generalize from them). But if one thinks that the argument should be turned around in this way, then one must make the same move for every non-cognitivist theory, since it takes only one non-cognitivist theory in whose domain lies can be made to yield conclusion (4). For instance, one must reject non-cognitivism in metaethics and aesthetics. So far so good. One probably doesn't need to reject non-cognitivism about requests, on the other hand, since one doesn't lie with requests: "I'll have fries with that" isn't a lie when you don't want fries.

Are there domains where one can lie but where non-cognitivism is clearly right? I am not sure. Maybe something like talk of the spooky? One certainly can lie in something is spooky (e.g., to scare off a competing house buyer). But even there I am not convinced that the right conclusion is that not all lies are assertions. The right conclusion there may still be that we do in fact make assertions when we attribute spookiness.

Vagueness cases are another case to think about. I think propositions are always sharp, but we lie with claims that clearly do not express sharp propositions ("He was bald two days ago, but washed his hair with this shampoo, and now he's not"). But I don't think vagueness cases give one reason to accept (4). Rather, they lead to a refinement of (3): assertions don't express individual propositions, but something like a vague assignment of weights to propositions.

So overall, I don't know what to make of the argument.

More on the Adams Thesis

The Adams Thesis for a conditional → says that P(A→B)=P(B|A). There are lots of theorems, most notably due to Lewis, that say that this can't be right, but they all make additional assumptions. On the other hand, van Fraassen has a paper arguing that any countable probability space can be embedded in a probability space that has a conditional → which satisfies the Adams Thesis and a whole bunch of axioms of conditional logic. The proof in the paper appears incomplete to me (it is not shown that all necessary conditions for the choice of [A,B] are met). Anyway, over the last couple of days I've been working on this, and I think I have a proof (written, but needing proofreading) of a generalization of van Fraassen's thesis that drops the countability assumptions (but uses the Axiom of Choice).

The conditional logic one can have along with the Adams Thesis is surprisingly strong. In my construction, for each A, the function C_A(B)=(A→B) is a boolean algebra homomorphism. Thus, we have Weakening, Conjunction of Consequents, Would=Might, and the Conditional Law of Excluded Middle. The main plausible axioms that we don't get are Weak Transitivity and Disjunction of Antecedents (can't get in the former case; don't know about the latter).

The proof isn't that hard once one sees just how to do it, but it ends up using the Maharam Classification Theorem, the von Neumann-Maharam Lifting Theorem and oodles of Choice, so it's not elementary.

Conditional commitments

Compare:

1. Assuming you pass at least one of your classes this spring, we will hire you in May.
2. We will hire you in May.

To a literalist it sounds like 2 makes the stronger commitment than 1.

But suppose that you get the lowest passing grade in one of your classes, and Fs in all the others. Then if I said 2, I could say: "Well, of course, but I was assuming half-decent performance." But if I said 1, I can't say that!

What's going on? Normally when I say what I will do, there are some unstated conditions. But when I get into the business of stating conditions, I had better list all of them, or at least all the ones that are likely to be as relevant as the ones I list.

The Adams thesis reconsidered

The following is known as the the Adams Thesis for a conditional →:

1. P(A→B)=P(B|A).

This is very plausible. However, Brian Weatherson expresses a widely shared conviction when he says:

As with so many formal theories, accepting this thesis leads to paradox. Lewis (1976) showed that any probability function Pr satisfying [(1)] would be trivial in the sense that the domain of the function could not contain three possible but pairwise incompatible sentences.

And indeed in the literature, the Lewis result gets used to argue that a conditional cannot have a truth value, since if it had one, that value would have to satisfy the Adams Thesis.

But what Lewis actually showed was somewhat weaker. Lewis showed that triviality results from:

2. P(A→B|C)=P(B|AC).

Now, Lewis perhaps correctly concludes from this that the Adams Thesis can't hold for subjective probabilities. For given a probability distribution P satisfying (1) and any C with P(C)>0, we could imagine another rational agent, or the same one at a later point, who has conditionalized her subjective probabilities on C, and when we apply (1) to her newly conditionalized probabilities we get (2).

But suppose that the probabilities we are dealing with are objective chances. Then one might well accept (1) for the objective chance probability function, without insisting on (2) in general. For instance, a reasonable restriction on (2) would match the restriction on the background knowledge in Lewis's Principal Principle, namely that the background knowledge is admissible, i.e., does not contain any information about B or stuff later than B.

Perhaps, though, clever people will find triviality results for (1), much as Lewis did for (2)? I doubt it. My reason for doubt is that I think I can prove the following two results which show that any probability space, no matter how nontrivial, can be extended to an Adams Thesis verifying probability space for all A with P(A)>0.

Proposition 1: Let <P,F,Ω> be any probability space. Then there is a probability space <P',F',Ω'> that extends <P,F,Ω> in the sense that there is a function e:F→F' that preserves intersections, unions and complements and where P'(e(A))=P(A), and such that for every A in F with P(A)>0 and every B in F, there is an event A→B in F satisfying P'(A→B)=P(B|A).

This result only yields a probability space verifying the Adams thesis for conditionals where neither the antecedent nor the consequent contains a conditional. Since conditionals that have conditionals in the antecedent and consequent can be at least somewhat hairy, this restriction may not be so bad. And one can iterate the Theorem n times to get an extension that allows the antecedent and consequent to have n conditional arrows in them. But if we are willing to allow merely finite additivity, then we have:

Proposition 2: Let <P,F,Ω> be any probability space and assume the Axiom of Choice. Then there is a finitely additive probability space <P',F',Ω'> (in particular, F' is a field, perhaps not a sigma-field) that extends <P,F,Ω> and is such that for any events A and B with P'(A)>0 there is an event A→B such that P'(A→B)=P'(B|A).

To prove Theorem 2 from Theorem 1, let <P_n,F_n,Ω_n> be the probability space resulting from applying Theorem 1 n times. Let N be an infinite hypernatural number. Then there will be a hyperreal-valued *-probability <*P_N,*F_N,*Ω_N>, and when we restrict this appropriately to include only finitely many iterations of arrows in an event and take the standard part of *P_N we should be able to get <P',F',Ω'>.

From ease to counterfactuals?

Consider the concept of how easy it is for a proposition to be made true, given how things are. It is by far easiest for propositions that are already true: nothing more needs to happen. It is hardest for self-contradictory propositions, like that Socrates is not Socrates: there is no way at all for it to happen. Contingently false propositions that require changes that go far back in time are going to be harder to be made true than ones that don't. And we can talk of the ease of p being made false as just the ease of not-p being made true. So, we can offer this account of counterfactuals:

• p→q holds if and only if it is easier for p to be made true than for the material conditional p⊃q to be made false.

This yields the Lewis-Stalnaker account of counterfactuals provided that we stipulate that a is easier to be made true than b if and only if there is a world where a holds which is closer than every world where b holds.

But we need not make this stipulation. We might instead take the easier to be made true relation as more fundamental. (And while we might define a closeness relation in terms of it—say, by saying that w₁ is closer than w₂ iff <w₁ is actual> is easier to be made true than <w₂ is actual>—depending on which axioms easier to be made true satisfies, that might not yield an account equivalent to the Lewis-Stalnaker one.)

On some assumptions, this is a variant of the central idea in yesterday's post.

A sufficient condition for a subjunctive conditional

Start with the idea of grades of necessity. At the bottom, say[note 1], lie ordinary empirical claims like that I am typing now, which have no necessity. Higher up lie basic structural claims about the world, such as that, say, there are four dimensions and that there is matter. Perhaps higher, or at the same level, there are nomic claims, like that opposite charges attract. Higher than that lie metaphysical necessities, like that nothing is its own cause or that water is partly composed of hydrogen atoms. Perhaps even higher than that lie definitional necessities, and higher than that the theorems of first order logic. This gives us a relation: p<q if and only if p is less necessary than q.

Let → indicate subjunctive conditionals. Thus "p→q" says that were it that p, it would be that q. Let ⊃ be the material conditional. Thus "p⊃q basically says that p is false or q is true or both. Then, the following seems plausible:

1. If ~p<(p⊃q), then p→q.

I.e., if the material conditional has more necessity than the denial of its antecedent, the corresponding subjunctive conditional holds.

Suppose it's a law of nature that dropped objects fall. Then the material conditional that if this glass is dropped, then it falls is nomic and hence more necessary than the claim that this glass is not dropped, and the subjunctive holds: were the glass dropped, it would fall.

Moreover, the subjunctives that (1) can yield hold non-trivially, if there are grades of necessity beyond metaphysical necessity (on my view, those are somewhat gerrymandered necessities), and this yields non-trivial per impossibile conditionals. Let p be the proposition that water is H₃O, and let q be the proposition that a water molecule has four atoms. Then ~p<(p⊃q), because p⊃q is a definitional truth while ~p is a merely metaphysical necessity. Hence were p to hold, q would hold: were water to be H₃O, a water molecule would have four atoms.

I wonder if the left-hand-side of (1) is necessary for the non-trivial holding of its right-hand-side.

Gentler structuralisms about mathematics

According to some standard structuralist accounts, a mathematical claim like that there are infinitely many primes, is equivalent to a claim like:

1. Necessarily, for any physical structure that satisfies the axioms A₁,...,A_n, the structure satisfies the claim that there are infinitely many primes.

There are two main motivations for structuralism. The first motivation is anti-Platonic animus. The second is worries about uniqueness: if there are abstract objects, there are many candidates for, say, the natural numbers, and it would be arbitrary if our mathematical language were to succeed in picking out on particular family of candidates.

The difficulty with this sort of structuralism is that while it may be fine for a good deal of "ordinary mathematics", such as real analysis, finite-dimensional geometry, dealing with prime numbers, etc., it is not clear that there are enough possible physical structures to model the axioms of such systems as transfinite arithmetic. And if there aren't, then antecedents in claims like (1) will be false, and hence the necessary conditional will hold trivially. One could bring in counterpossibles but that would be explaining the obscure with the obscurer.

I want to drop the requirement that the structures we're talking about are physical structures. Thus, instead of (1), we should say:

2. Necessarily, for any structure that satisfies the axioms A₁,...,A_n, the structure satisfies the claim that there are infinitely many primes.

If we do this, we no longer have a physicalist reduction. But that's fine if our motive for structuralism is worries about arbitrariness rather than worries about abstracta.

Next, restrict the theory to being about what modern mathematics typically means by its mathematical claims. If we do this, the claim becomes logically compatible with Platonism about numbers. Let us suppose that there really are numbers, and our ordinary language gets at them. Nonetheless, I submit, when a modern number theorist is saying that there are infinitely many primes, she is likely not making a claim specifically about them. Rather, she is making a claim about every system that satisfies the said axioms. If the natural numbers satisfy the axioms, then her claims have a bearing on the natural numbers, too.

Here is one reason to think that she's saying that. Mathematical practice is centered on getting what generality you can. What mathematician would want to limit a claim to being about the natural numbers, when she could, at no additional cost, be making a claim about every system that satisfies the Peano axioms?

Now, if we go for this gentler structuralism, and allow abstract entities, we can easily generate structures that satisfy all sorts of axioms. For instance, consider plural existential propositions. These are propositions of the form of the proposition that the Fs exist, where "the Fs" directly plurally refers to a particular plurality. We can define a membership relation: x is a member of p if and only if x is said by p to exist. Add an "empty proposition", which can be any other proposition (say, that cats hate dogs) and say that nothing is its member. Then plural existential propositions, plus the empty proposition, with this membership relations should satisfy the axioms of a plausible set theory with ur-elements. If all one wants is Peano axioms, we can take them to be satisfied by the sequence of propositions that there are no cats, that there is a unique cat, that there are distinct cats x and y and every cat is x or is y, that there are distinct cats x and y and z and every cat is x or is y or is z, and so on.

I am not completely convinced that this sociological thesis about modern mathematics is correct. Maybe I can retreat to the claim that this is what modern mathematics ought to claim.

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.

Non-triviality of conditionals

Here's a rough start of a theory of non-triviality of conditionals.

A material conditional "if p then q" is trivially true provided that (a) the only reason that it is true is that p is false or (b) the only reason that it is true is that q is true or (c) the only reasons that it is true are that p and q are true.

A subjunctive conditional "p □→ q" is trivially true provided that (a) the only reason that it is true is that p and q are both true or (b) the only reason that it is true is that p is impossible or (c) the only reason that it is true is that q is necessary or (d) the only reasons that it is true are that p is impossible and q is necessary.

For instance, "If it is now snowing in Anchorage, then it is now snowing in the Sahara" understood as a material conditional is trivially true, because the falsity of the antecedent (I just checked!) is the only reason for the conditional to be true. The contrapositive "If it not now snowing in the Sahara, then it is not now snowing in Anchorage" is trivially true, since it is true only because of the truth of the consequent. On the other hand, "If I am going to meet the Queen for dinner tonight, I will wear a suit" is non-trivially true. It is true not just because its antecedent is false--there is another explanation.

Likewise, "Were horses reptiles, then Fermat's Last Theorem would be false" and "Were Fermat's Last Theorem false, horses would be mammals" are "Were I writing this, it would not be snowing in Anchorage" are trivially true, in virtue of impossibility of antecedent, necessity of consequent and truth of antecedent and consequent, respectively. But "Were horses reptiles, either donkeys would be reptiles or there would no mules" is non-trivally true--there is another explanation of its truth besides the impossibility of antecedent, namely that reptiles can't breed with mammals and mules are the offspring of horses and donkeys.

Two fun counterfactuals

1. If I were a better football player than everybody else, I would be very strong.
2. If everyone else were a worse football player than I, nobody would be very strong.

Both of these conditionals are true. But their antecedents are logically equivalent. This shows[note 1] that one cannot substitute logical equivalents for logical equivalents in the antecedents of counterfactuals while preserving truth value, even when one restricts one's consideration to counterfactuals with possible antecedents—i.e., counterfactuals are hyyperintensional. And this, in turn, shows that possible worlds and probabilistic accounts of counterfactuals fail.

I am not happy with this argument. I want to say that the antecedents of (1) and (2) describe families of possible worlds. So we need a interpretation of the antecedents of (1) and (2) on which, although seeming logically equivalent, these antecedents rigidify different features. Thus, the antecedent of (1) rigidifies the range of others' abilities, while the antecedent of (2) rigidifies my abilities.

It is tempting to do this with the overused distinction between semantics and pragmatics: the antecedents of (1) and (2) implicate non-equivalent things, though their propositional content is the same. But if we did that, then either we need to depart from the possible worlds or probabilistic analysis (since that analysis is in terms of truth, not implicature), or we would have to say that although (1) is true and (2) is false, or (1) is false and (2) is true, the real communication goes on at the level of implicature. But the view that (1) is true and (2) is false is implausible, as is the view that (1) is false and (2) is true. (Lewis's closeness account forces one to keep everyone else's abilities constant, so I guess he has to say that (2) is false, but surely (2) is true—not just something that implicates truly.)

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.

Eternal significance

I find myself pulled to the following two claims:

1. If nothing lasting can come from human activity (think of Russell's description of everything returning "again to the nebula"), then no human life has much meaning.
2. If nothing lasting can come from human activity, some human lives (e.g., lives lived in loving service to others) still have much meaning.

I don't think I am alone in finding myself pulled in these two directions. It would be nice if one could reconcile these two intuitions.

If the conditionals in (1) and (2) are material, then there is an easy way to reconcile these two intuitions. For if they are material conditionals, then (1) and (2) together entail:

3. Something lasting can come from human activity.

And given (3), there is no contradiction between (1) and (2)—both are trivially true because their antecedents are false.

This seems too facile. (Maybe only because I am not sufficiently convinced by my arguments here. But I also think that this interpretation ignores the anti-material marker "still" in (2).) But here is a more sophisticated hypothesis about these two intuitions. Suppose that God has designed our world so that only events that can have eternal significance are deeply morally significant. Then it is contingently true that:

4. Nothing that lacks eternal significance has deep moral significance.

Moreover, suppose that God implants in us a strongly engrained intuition that (4) is true. He does this in order to set our sights on eternity and to comfort us under the slings and arrows of outrageous fortune. (I think here of Boethius' Consolation of Philosophy.) At the same time, the moral significance of events does not entirely come from their eternal significance. Thus, such counterfactuals as

5. If lives of loving service to others lacked eternal significance, they would still have deep moral significance

are true, and moral reflection can discover these truths.

This hypothesis would explain why we are drawn to (1). We are drawn to (1) because we have a deep divinely implanted intuition that (4) is true, and (4) makes (1) very plausible. Moreover, the hypothesis can explain why we are drawn to (2), namely that with reflection we discover (5) to be true. (Contrary to what the name "subjunctive conditional" suggests, we do use the indicative mood for subjunctive conditionals sometimes.)

The hypothesis also explains why it is hard to find arguments for (1), why belief in (1) is more of a gut feeling than an argued position, but nonetheless a gut feeling that it is hard to get rid of.

Finally, the hypothesis is compatible with the possibility of there being non-theists like Russell who overcome their pull to (1). The intuition isn't irresistable. The only plausible story as to how (4) can be true is that, in fact, God makes all morally significant things have potential eternal effects. So a non-theist is likely to realize that (4) fits poorly with her overall view, and hence get rid of (4).

This hypothesis about (1) and (2) charitably does about as much justice as can be done to both intuitions simultaneously. This gives us not insignificant reason to think the hypothesis is true, and hence that there exists a God who makes morally significant events have potentially eternal effects.

Of course, one might come up with naturalistic explanations of the pull to (1) and (2). But I suspect that these naturalistic explanations will end up simply denying one of the two intuitions, and then explaining why we have this mistaken view. An explanation of our intuitions on which the intuitions are true is to be preferred for anti-sceptical reasons.

Indicative and material conditionals

I will use "p→q" for the indicative conditional "if p, then q". I will use "p⊃q" for the material conditional "(not p) or q". I will say that "indicatives are material" providing that p→q and p⊃q are logically equivalent for all p and q, where a and b are logically equivalent if and only if it is necessary that (a if and only if b). I will say that p entails q provided that it is necessary that p⊃q.

Almost no philosopher thinks indicatives are material. There are very plausible counterexamples. For instance, suppose it is lightly raining in Seattle and Seattle is not having a drought. Let p be "Seattle is having heavy rain" and let q be "Seattle is having a drought". Then p⊃q, since p is false. But it seems quite wrong to say that if Seattle is having heavy rain, then Seattle is having a drought, so p→q doesn't seem to be true.

I am going to offer some arguments that indicatives are material. Say that → is non-hyperintensional provided p→q and p*→q* are logically equivalent whenever p and p* are logically equivalent and q and q* are logically equivalent. Consider the following two theses:

1. For any possible world w: (p at w) → (q at w) if and only if (p→q at w).
2. For any predicates F and G, from "Every F is a G" (where "x is an F" is more euphonious way of saying that x satisfies F) together with the assumption that c exists, it logically follows that if c is an F, then c is a G.

I will argue in S5, and using some fairly uncontroversial further premises:

3. If (1) is true and → is non-hyperintensional, then indicatives are material.
4. If (2) is true and → is non-hyperintensional, then indicatives are material.

Moreover, I will try to make plausible:

5. If (2) is true, then one has to assign the same truth value as the material conditional does to a number of paradoxical-sounding examples of indicative conditional sentences that are relevantly just like the standard alleged counterexamples to the thesis that all indicatives are material.

I think (1) and (2) are quite plausible. I don't know, however, how plausible it is that → is non-hyperintensional. My argument for (3) is very similar to more general arguments in Williamson. However, given (5), even if we don't assume that → is hyperintensional, there seems to be little advantage to denying the elegant and simple view that indicatives are material.

Argument for (5): Take my heavy rain and drought in Seattle case. Suppose that as it happens, there is no place where there presently is heavy rain. Let Fx say that x is having heavy rain. Let Gx say that x is having drought. Then all Fs are Gs. (If you think, with Aristotle, that "All Fs are Gs" requires there to be an F, then add the premise that on Venus somewhere right now there is a drought but a very, very brief heavy rain is currently occurring. I will leave out such modifications in the future.) Then by (2), we have to say that if F(Seattle), then G(Seattle):

6. If Seattle is having heavy rain, then Seattle is having drought.

And that is pretty much a standard alleged counterexample to the view that indicatives are material, of the false-antecedent sort. The case divides into two: we might suppose that Seattle is having neither drought nor heavy rain, in which case (6) is false-antecedent, false-consequent, or we might suppose that Seattle is having drought and (unsurprisingly) no heavy rain, in which case we have false-antecedent, true-consequent.

We can also use (2) to manufacture a true-antecedent, true-consequent case. Suppose that it is raining in both Seattle and the Sahara. Then the following is a standard alleged counterexample of the true-antecedent, true-consequent sort:

7. If it's raining in Seattle, then it's raining in the Sahara.

Let Fx say that x is a planet on which it is raining in Seattle, and let Gx say that x is a planet on which it is raining in the Sahara. Then every F is a G, since the only F is earth. By (2):

8. If earth is a planet on which it is raining in Seattle, then earth is a planet on which it is raining in the Sahara.

And that sounds about as paradoxical as (7). That completes my argument for (5).

Argument for (3): First we need a special case:

9. If p and q are non-contingent and → is non-hyperintensional, then p→q is logically equivalent to p⊃q.

To argue for (9), consider the following four sentences:

10. 2+2=4→2+3=5. (necessary, necessary)
11. 2+2=5→2+3=6. (impossible, impossible)
12. 2+2=5→ (2+2=5 or 1+1=2 or both). (impossible, necessary)
13. 2+2=4→2+2=5. (necessary, impossible)

If p and q are non-contingent, then they are respectively logically equivalent to the antecedent and consequent of exactly one of (10)-(13). By non-hyperintensionality of →, it follows p→q must have the same truth value as the conditional in that line. But the truth values of (10)-(13) are just as the material conditional says they are: thus, clearly, (10)-(12) are (necessarily) true and (13) is (necessarily) false. So, p→q must have the same truth value as p⊃q, assuming p and q are non-contingent.

The argument for (3) is now easy. Observe that (p at w) and (q at w) are non-contingent, even if p and q are contingent. So,

14. (p at w) → (q at w) is logically equivalent to (p at w) ⊃ (q at w).

But, plainly:

15. (p⊃q at w) is logically equivalent to (p at w) ⊃ (q at w).

From (1), (14) and (15) we conclude that:

16. (p→q at w) is logically equivalent to (p⊃q at w)

and hence indicatives are material.

Argument for (4): The most intuitive form of the argument is to assume theism, and let Fx say that x is an omniscient being that knows that p, and let Gx say that x knows that q. Then as long as p⊃q, it will be the case that every F is a G (just think about the four possible truth-value combinations). Hence:

17. If God is an omniscient being that knows that p, then God knows that q.

Since God's existence and omniscience are necessary, the antecedent and consequent are logically equivalent to p and q, respectively, and so we get p→q by non-hyperintensionality.

If we don't want to suppose there is a God, let's suppose that numbers and sets exist necessarily. Let P be the singleton set whose only member is p. Let Q be the singleton set whose only members is q. Then, let Fx say that x is greater than zero and x equals the number of truths in P. Let Gx say that x is greater than zero and x equals the number of truths in Q. Then, if p⊃q, it is easy to see that all Fs are Gs, so:

18. If one is greater than zero and one equals the number of truths in P, then one is greater than zero and one equals the number of truths in Q.

But the antecedent and consequent are logically equivalent to p and q respectively, so by non-hyperintensionality we get p→q, once again.

Moral and conditional realism

1. (Premise) I know that if I am human, I am mortal.
2. (Premise) If I know something, it's true.
3. So, it is true that if I am human, I am mortal.
4. (Premise) If something is true, it has truth value.
5. So, that if I am human, I am mortal has truth value.
6. (Premise) That if I am human, I am mortal is a conditional.
7. So, some conditionals have truth value.

Continuing:

8. (Premise) I know that it is wrong to torture the innocent for fun.
9. So, it is true that it is wrong to torture the innocent for fun.
10. So, that it is wrong to torture the innocent for fun has truth value.
11. (Premise) That it is wrong to torture the innocent for fun is a moral claim.
12. So, some moral claims have truth value.

Another argument:

13. (Premise) No one is morally to blame for violating a moral rule that no one could know.
14. (Premise) One is only to blame for violating a moral rule.
15. (Premise) Someone is to blame for something.
16. So, someone is to blame for violating a moral rule.
17. So, some moral rule can be known.
18. (Premise) Necessarily, only truths are known.
19. So, some moral rule can be true.
20. (Premise) Necessarily, anything that is true has truth value.
21. So, some moral rule can have truth value.

This is, of course, a problem for non-realist accounts of conditionals and morals.

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).