Symmetric relations and logic

Suppose Alice and Bob are friends, and that friendship is a fundamental relation. Consider the facts expressed by these two sentences:

1. Alice and Bob are friends.

2. Bob and Alice are friends.

It is implausible that these are different facts. For if they were different facts, they would both be fundamental facts (otherwise, which of them would be the fundamental one?), and we would be multiplying fundamental facts beyond necessity—only one of the two is needed in the totality of fundamental facts.

Furthermore, I think that the propositions expressed by (1) and (2) are the same. Here’s one reason to think this. Imagine a three-dimensional written language where plural symmetric predicates like “are friends” are written (say, laser-inscribed inside a piece of glass) with “and are friends” on one horizontal layer, with “Alice” on a layer below the “and” and “Bob” on a layer above it. If (1) and (2) express different propositions, we would have to ask which of them is a better translation of the three-dimensional language. But surely there is no fact about that.

If this is right, then First Order Logic (FOL) fails to accurately represent propositions about fundamental relations, by having two atomic sentences, F(a,b) and F(b,a), where there is only one fundamental fact. Moreover, FOL will end up having non-trivial proofs whose conclusion expresses the same proposition as the premise, since we will presumably have an axiom like ∀x∀y(F(x,y)→F(y,x)) that lets us prove F(b,a) from F(a,b). This is not the only example of this phenomenon. Take the proof that ∀xF(x) follows from ∀yF(y), even though surely the two express the same proposition, namely that everything is F.

In particular, the logic of sentences appears to differ from the logic of propositions, since the proposition that Bob and Alice are friends follows by reiteration from the proposition that Alice and Bob are friends if they are the same proposition, but sentence (2) does not follow from sentence (1) by reiteration (nor is this true for the FOL versions).

If we think there is a One True Logic, it presumably will be a logic of propositions rather than sentences, then. But what it will be like is a difficult question, to answer which we will have to a worked out theory of when we have the same proposition and when we have different ones.

Aquinas on God's knowledge of propositions

Does God know that the sky is blue?

That seems like a silly question. It’s not like we’re asking whether God knows future contingents, or counterfactuals of freedom. That the sky is blue is something that it is utterly unproblematic for God to know.

Except that it is tempting to say that God has no propositional knowledge, and knowing that the sky is blue is knowing a proposition.

It seems that Aquinas answers the question in Summa Theologiae I.14.14: “God knows all the propositions that can be formulated” (that’s in Freddoso’s translation; the older Dominican translation talks of “enunciable things”, but I think that doesn’t affect what I am going to say). It seems that God does have propositional knowledge, albeit not in the divided or successive way that we do.

But what he is up to in I.14.14 is not what it initially sounds like to the analytic philosopher’s ear.

For consider Thomas’s argument in I.14.14 that God knows all formulable propositions:

Since (a) to formulate propositions lies within the power of our intellect, and since (b), as was explained above (a. 9), God knows whatever lies within either His own power or the power of a creature, it must be the case that God knows all the propositions that can be formulated.

But now notice an ambiguity in “God knows the proposition that the sky is blue.” In one sense, which I will call “alethic”, this just means God knows that the sky is blue. In another sense, the “objectual”, it means that God knows a certain abstract object, the proposition that the sky is blue. In the objectual sense, God also knows the proposition that the sky is green—God fully knows that proposition, just as he knows other objects, like the person Socrates. But God does not, of course, have the alethic knowledge here—God does not know that the sky is green, because the sky is not green.

If it was the alethic sense that Thomas was after, his argument would be invalid. For in article 9, the discussion clearly concerns objectual knowledge. Exactly the same argument establishes that God knows the proposition that the sky is green as that he knows the proposition that the sky is blue. Furthermore, the Biblical quote Thomas gives in support of his view is “The Lord knows the thoughts of men” (Psalm 93:11). But the Lord doesn’t know all of them to be true, doesn’t know all of them alethically, because not all of the thoughts of humans are true.

Furthermore, if it was alethic knowledge that Aquinas was after, it would be inaccurate to say God knows all propositions. For only “half” of the propositions can be known alethically—the true ones!

All that said, I think we can still bootstrap from the objectual to the alethic knowledge. God’s knowledge of objects is perfect (Aquinas relies on this perfection multiple times in Question 14) and hence complete. If God knows something, God also knows all of its properties, intrinsic and relational. Thus, if God knows a proposition objectually, and that proposition has a truth value, God knows that truth value. In particular, if that proposition is true, God knows that it is true. And that seems to suffice for counting as knowing the proposition alethically.

So, it looks like Aquinas is committed to God objectually knowing both the propositions that the sky is green and that the sky is blue, and also knowing that the former is false and the latter is true—which seems to be enough for God to count as knowing that the sky is blue. (Though I could see this last point getting questioned.)

Dog whistles

From time to time I’ve had occasion to make use of examples where someone says different things to two different interlocutors in a single utterance. My favorite examples were pointing to a bottle and saying “Gift!”, which would mean a very different thing to a German speaker and to an English speaker, or using coded language while speaking to someone while knowing a spy is overhearing. Such examples illustrate the interesting fact that we cannot identify propositions with equivalence classes of utterance tokens, because a single utterance token can express different propositions.

But arguments based on such contrived cases have a tendency to be less than convincing. However, it has just occurred to me that dog whistles in politics are a real-life example of the same phenomenon, and one technically within a single language.

By the way, if we’re looking for equivalence classes that function like propositions, I guess instead of looking at equivalence classes of tokens utterances, we should look at equivalence classes of context-token pairs, where a context includes the language and dialect as well as the (actual? intended?) audience.

A near lie

Alice knows that her friend Bob has no pets and no experience with birds. While recommending Bob for a birdkeeping job at a zoo and having discovered or to be surprisingly ignorant about birds, she says:

1. Bob has a fine collection of Southern yellow-beaked triggles.

It seems that Alice is lying. Yet it seems that to lie one must assert, and to assert one must express a proposition. But Alice’s sentence does not express a proposition since “triggle” is meaningless.

Sentence (1) seems to entail the falsehood:

2. Bob owns some birds.

But entailment is a relation between propositions, and (1) neither is nor expresses a proposition. We might want to say that if it did express a proposition, it would express a proposition entailing (2). But even that isn’t so clear. After all, maybe a world where “triggle” denotes a science-fictional beaked reptile is closer than a world where it denotes a kind of bird (imagine that some science-fiction writer almost wrote Southern yellow-beaked triggles as reptiles into a story but stopped themselves at the last moment).

Here is what I think I want to say about what Alice did. According to Jorge Garcia, what makes lying bad one linguistically solicits trust that what one is saying is true, while at the same time betraying that trust. Alice did exactly that, but without asserting. So, while Alice did not lie, she did something that is wrong for the same reason that lying is.

Can one lie without asserting a proposition?

I am starting to think that one can lie without asserting a proposition.

Let’s say that a counterintelligence agent tells an enemy spy that a new weapons technology has just been deployed, in order to dissuade the enemy from invading. The description of the technology contains nonsensical technobabble. This seems to be a lie. If it is, my argument is complete, because nonsense does not express a proposition.

But suppose we say it’s mere BS. Let’s now complicate the case. The counterintelligence agent passes to the enemy spy a fake classified document saying “We have just built a weapon that shoots three simultaneous hyperquark beams.” The spy is taken in by the BS, but also wishes to deter war. And thus the spy reports to her government: “The enemy has just built a weapon that shoots ten simultaneous hyperquark beams.” It is clear that the spy is not merely engaging in BS. The spy sure seems to be lying. But the spy is no more asserting a proposition than the counterintelligence agent did.

If we say that the counterintelligence agent is lying, then we have to allow that one can lie without even taking oneself to assert a proposition.

If we think that the counterintelligence agent is only BSing, but that in my more complicated case the enemy spy is lying to her government, then we should say that to lie one needs to take oneself to be asserting a false proposition, but one need not be actually asserting a proposition, true or false.

In either case, one can lie without asserting a proposition.

Perhaps I am wrong. Perhaps what the spy does in the more complicated case is neither BS nor a lie, but engaging in a verbal deceit we don’t have a good name for.

Dr. Smith ate a banana

Suppose you receive this trustworthy report:

1. Dr. Smith ate a banana.

You are now in a position to learn this additional fact:

2. Someone whose last name is “Smith” ate a banana.

But (2) does not logically follow from (1). So how do we learn (2) from the report?

Knowledge of English tells us that “Dr. Smith ate a banana” has “Dr. Smith” as the subject and that this sentence attributes eating a banana to the subject of the sentence. Assuming defeasibly that the the use of English in the report is correct, we conclude that someone correctly styled “Dr. Smith” was reported to have eaten a banana. And assuming defeasibly that the report itself is factually correct, we conclude that:

3. Someone correctly styled “Dr. Smith” ate a banana.

Knowing English, we also know that anyone correctly styled “Dr. Smith” has the last name “Smith”, so we get (2). We also know that anyone correctly styled “Dr. Smith” has a doctorate, so:

4. Someone with a doctorate ate a banana.

These are instances of the familiar fact that what we learn from receiving a report goes beyond the propositional content of the report.

Non-propositional representations

I used to think that it’s quite possible that all our mental representations of the world are propositional in nature. To do that, I had to have a broad notion of proposition, much broader than what we normally consider to be linguistically expressible. Thus, I was quite happy with saying that Picasso’s Guernica expresses a proposition about war, a proposition that cannot be stated in words. Similarly, I was quite fine—my Pittsburgh philosophical pedigree comes out here—with the idea that an itch or some other quale might represent the world propositionally.

That broad view of propositions still sounds right. But I am now thinking there is a different problem for propositionalism about our representational states: the problem of estimates. A lot of my representations of the world are estimates. When I estimate my height at six feet, there is a proposition in the vicinity, namely the proposition that my height is exactly six feet. But that proposition is one that I am quite confident is false. There are even going to be times when I wouldn’t even want to say that my best estimate of something is approximately right—but it’s still my best estimate.

The best propositionally-based of what happens when I estimate my height at six feet seems to me to be that I believe a proposition about myself, namely that my evidence about my height supports a probability density whose mean is at six feet. But there are two problems with this. First, the representational state now becomes a representation of something about me—facts about what evidence I have—than about the world. Second, and worse, I don’t know that I would stick my neck out far enough to even make that claim about evidence unequivocally—my insight into the evidence I have is limtied. Moreover, even concerning evidence, what I really have is only estimates of the force of my evidence, and the problem comes back for them.

So I think that estimating is a way of representing that is not propositional in nature. Notice, though, that estimates are often well expressible through language. So on my view, linguistic expressibility (in the ordinary sense of “linguistic”—maybe there is such a thing as the “language of painting” that Picasso used) is neither necessary for a representation of the world to be propositional in nature.

I now wonder whether vagueness isn’t something similar. Perhaps vague sentences represent the world but not propositionally. But just as we can often—but not always—reason as if sentences expressing estimates expressed propositions, we can often reason as if vague sentences expressed propositions. The “logic” of the non-propositional representations is close enough to the logic of propositional ones—except when it’s not, but we can usually tell when it’s not (e.g., we know what sorts of gruesome inferences not draw from the estimate that a typical plumber has 2.2 children).

The A-theory and a countably infinite fair lottery

Let’s suppose that the universe has a beginning and the tensed theory of propositions (which is accepted by most A-theorists) is true. Then consider for each n the proposition d_n that n days have elapsed from the beginning of the universe. This proposition is contingent on a tensed theory of propositions. Exactly one of the propositions d_n is true. No one of the propositions d_n is more likely to be true than any other. So, it seems, we have a countably infinite fair lottery. But such are, arguably, impossible. See Chapter 4 of my infinity book. (E.g., it’s fun to note that on the tensed theory of time we should be incredibly surprised that it’s only 13 billion years since the beginning of the universe.)

Since the universe does have a beginning (and even if it does not, we can still run the argument relative to some other event than the beginning of the universe), it seems we should reject the tensed theory of propositions.

Do all positive truths have truthmakers?

Consider this thesis:

1. Every positive true proposition has a truthmaker.

This seems plausible. But I think it is only reasonable to accept (1) if one accepts:

2. Any plurality of objects has a mereological sum or fusion which essentially has the members of the plurality as parts.

To see this, consider some plurality, the xs of existing things. Then, surely:

3. The proposition, E!xx, that the xs exist is positive.

But what object is suited to be the truthmaker of E!xx? The truthmaker of E!xx will have to be some object o with the property that, necessarily, if o exists, so do all the xs. Our best candidate for that object is some object that has all the xs as essential parts. But we also don’t want to include irrelevancies in the truthmaker, so we shouldn’t include in o anything that overlaps none of the xs. In other words, o will very plausibly be the mereological sum of the xs.

Since I don’t believe in fusions, I have to deny (1). But at least I may be able to accept:

4. Every positive true proposition has a plural truthmaker,

where a plural truthmaker of p is a plurality of objects that collectively make p true. Note that pluralities need not in general be objects themselves, so we do not have the same problem as above.

The present king of Ruritania

Suppose I am a quack and I announce:

1. These green pills cured the king of Ruritania of lung cancer.

I am lying, of course. The green pills never cured anyone of lung cancer.

But wait. To lie, I have to assert. To assert, there has to be a proposition that is being expressed. But (1) doesn’t express any proposition, because “Ruritania” is a non-referring name.

Maybe, then, (1) is not a lie, but something that is wrong for the same reason that a lie is wrong. For instance, on Jorge Garcia’s account, lying is wrong as it’s a betrayal of the trust solicited by the very same act. If so, then my pretend assertion of (1) might be wrong for exactly the same reason as a lie.

The point can also be made without relying on non-referring proper names. Suppose Jones has lied, cheated, stolen, plagiarized and defenestrated his friends, but reporting doesn’t make his character black enough for my purposes. So I say:

2. Dr. Jones has lied, cheated, stolen, plagiarized, defenestrated his enemies, and garobulated his friends.

This doesn’t express a proposition. But it’s just as bad as a lie.

If classical theism rules out open theism, then classical theism rules out presentism

If presentism and most, if not all, other versions of the A-theory are true, then propositions change in truth value. For instance, on presentism, in the time of the dinosaurs it was not true that horses exist, but now it is true; on growing block, ten years ago the year 2019 wasn't at the leading edge of reality, but now it is. The following argument seems to show that such views are incompatible with classical theism.

1. God never comes to know anything.

2. If at t₁, x doesn’t know a proposition p but at t₂ > t₁, x knows p, then x comes to know p.

3. If propositions change in truth value, then there are times t₁ < t₂ and a proposition p such that p is not true at t₁ and p is true at t₂.

4. It is always the case that God knows every true proposition.

5. It is never the case that anyone knows any proposition that isn’t true.

6. So, if propositions change in truth value, then there are times t₁ < t₂ and a proposition p such that God doesn’t know p at t₁ but God does know p at t₂. (by 3-5)

7. So, if propositions change in truth value, God comes to know something. (by 2 and 6)

8. So, propositions do not change in truth value. (by 1 and 7)

I think the only controversial proposition is (1). Of course, some non-classical theists—say, open theists—will deny (1). But non-classical theists aren’t the target of the argument.

However, there is a way for classical theists to try to get out of (1) as well. They could say that the content of God’s knowledge changes, even though God and God’s act of knowing are unchanging. The move would be like this. We classical theists accept divine simplicity, and hence hold that God would not have been intrinsically any different had he created otherwise than he did. But had God created otherwise than he did, the content of his knowledge would have been different (since God knows what he creates). So the content of God’s knowledge needs to be partially constituted by created reality. (This could be a radical semantic externalism, say.) Thus, had God created otherwise than he did, God (and his act of knowledge which is identical to God) would have been merely extrinsically different.

But exactly the same move allows one to reconcile the denial of (1) with immutability. The content of God’s knowledge is partially constituted by created reality, and hence as created reality changes, the content of God’s knowledge changes, but the change in God is merely extrinsic, like a mother’s change from being taller than her daughter to being shorter than her daughter solely due to her daughter’s growth.

I agree that denying (1) is compatible with God’s being intrinsically unchanging. For a long time I thought that this observation destroyed the argument (1)-(8). But I now think not. For I am now thinking that even if (1) is compatible with immutability, (1) is a part of classical theism. For it is a part of classical theism that God doesn’t learn in any way, and coming to know is a kind of learning.

Here is one way to see that (1) is a part of classical theism. Classical theists want to reject any open theist views. But here is one open theist view, probably the best one. The future is open and propositions reporting what people will freely do tomorrow are now either false or neither-true-nor-false, but tomorrow they come to be true. An omniscient being knows all true propositions, but it is no shortcoming of omniscience to fail to know propositions that aren’t true. Then, our open theist says, God learns these propositions as soon as they become true. This is all that omniscience calls for.

Now, classical theists will want to reject this open theist view on the grounds of its violating immutability. But they cannot do so if they themselves reject (1). For the presentist (say) classical theist can reject (1) without violating immutability, so can our open theist. Indeed, our open theists can say exactly the same thing I suggested earlier: God changes extrinsically as time progresses, and the content of God’s knowledge changes, but God remains intrinsically the same.

So, what do I think the classical theist should say to our open theist? I think this: that God doesn’t come to know is not just a consequence of the doctrine of immutability, but is itself a part of the doctrine of immutability. A God who learns is mutable in an objectionable way even if this learning is not an intrinsic change in God. But if we say this, then of course we are committed to (1), and we cannot be presentists or accept any other of the theories of time on which propositions change in truth value.

I think the best response on the part of the classical theist who is an entrenched presentist would be to deny (1) and concede that classical theism does not rule out open theism. Instead, open theism is ruled out by divine revelation, and revelation here adds to classical theism. But it seems very strange to say that classical theism does not rule out open theism.

Presentism and haecceities

Suppose that times are maximal consistent present-tense propositions. Then if we are to make sense of eternal recurrence—reality being exactly alike at two different times—it seems we need haecceities for events or tropes. Thus, a certain kind of presentist needs haecceities.

Non-propositional conveyance

One sometimes hears claims like:

1. There are things that can be conveyed through X (poetry, novels, film, art, music, etc.) that cannot be conveyed propositionally.

But what kind of a thing are those things? Facts? Not quite. For while some of the “things that can be conveyed … that cannot be conveyed propositionally” are in fact real and true, some are not. Leni Riefenstahl’s Triumph of the Will and Fritz Lang’s M are both good candidates for conveying “things … that cannot be conveyed propositionally”. But Triumph in doing so conveys falsehoods about the Nazi Party while M conveys truths about the human condition. But facts just are. So, the “things” are not just facts.

What I said about Triumph and M is very natural. But if we take it literally, the “things” must then be the sorts of things that can be true or false. But the primary bearers of truth are propositions. So when we dig deeper, (1) is undermined. For surely we don’t want to say that Triumph and M convey propositions that cannot be conveyed propositionally.

Perhaps, though, this was too quick. While I did talk of truth and falsehood initially, perhaps I could have talked of obtaining and not obtaining. If I did that, then maybe the “things” would have turned out to be states of affairs (technically, of the abstract Plantinga sort, not of the Armstrong sort). But I think there is good reason to prefer propositions to states of affairs here. First, it is dubious whether there are impossible states of affairs. But not only can X convey things that aren’t so, it can also convey things that couldn’t be so. A novel or film might convey ethical stuff that not only is wrong, but couldn’t be right. Second, what is conveyed is very fine-grained, and it seems unlikely to me that states of affairs are fine-grained enough. The right candidate seems to be not only propositions, but Fregean propositions.

But (1) still seems to be getting at something true. I think (1) is confusing “propositionally” with “by means of literalistic fact-stating affirmative sentences”. Indeed:

2. There are things that can be conveyed through X (poetry, novels, film, art, music, etc.) that cannot be conveyed by means of literalistic fact-stating affirmative sentences.

(Note the importance of the word “conveyed”. If we had “expressed”, that might be false, because for any of the “things”, we could stipulate a zero-place predicate, say “xyzzies”, and then express it with “It xyzzies.” But while that sentence manages to express the proposition, it doesn’t convey it.)

Vague propositions

Suppose Jim says, in English, “2+2=4”. Then:

1. What Jim said is such that it is contigent that it is true, because it is contingent that “4” means four rather than five

but:

2. What Jim said is a necessary truth, because it cannot but be true that 2+2=4.

Here the apparent contradiction is resolved by disambiguating “what Jim said” between the uttered sounds and the expressed meaning.

But when talking about vagueness, this straightfoward point can be a bit less clear. Suppose that it’s vaguely true that “4” in Jim’s dialect means four, rather than five, and Jim says “2+2=4” (and suppose that all the other relevant stuff is definite). Then:

3. What Jim said is vaguely true, because it’s vaguely true that “4” is four.

4. What Jim said is not vaguely true, because what Jim said is definitely true or definitely false, depending on what “4” means.

Again, make the same move as in (1)-(2): in (3), “what Jim says” is the uttered sounds or words and in (4) it’s the proposition.

This line of thought suggests one of two possibilities. Either, propositions are never vague, or there are two interestingly different kinds of vagueness. If propositions are never vague, then in the proposition sense of “what was said” it is never correct to say that what was said is vague. That’s a bit counterintuitive, but some counterintuitive things are true.

But if some propositions are vague, then it seems that we have two interestingly different kinds of vagueness an utterance could suffer from. It could be vague which non-vague proposition an utterance expresses or it could be definite which vague proposition an utterance expresses—or one could have combinations, as when it’s vague which vague proposition is expressed. In the case above, I claimed that it was vaguely the case that Jim expressed the non-vague proposition that 2+2=4. But presumably if there are vague propositions, there will be one that has the kind of vagueness that makes the non-vague propositions that 2+2=4 and that 2+2=5 be its admissible precifications.

So now we would have this interesting question: What determines whether Jim’s case was a case of vaguely expressing a non-vague proposition or non-vaguely expressing a vague proposition or some combination? Maybe there is a good answer to this question, but I have some doubts. In light of these doubts, I think that the proponent of vague and non-vague propositions should say is something like this. There are at least three senses of “what was said”: the sounds or words (and that makes for two, but I won’t be interested in this distinction in this post), the non-vague proposition and the vague proposition. What Jim said is vaguely true in the first and third sense, but not in the second. This is sufficiently complicated that one might prefer to go back to the less intuitive option, that in the proposition sense “what was said” is never vague.

I am dreadfully confused.

What can we learn from the Contingent Liar?

Start with this:

• The last bulleted item in this post is not true.

Call this token bulleted linguistic item p. Then p is, in fact, the last--and the only--item prefixed with a bullet point in this text. If it's not true, then it seems it's true. But if it's true, then it seems it's not. Oops! That's a contradiction in classical logic. But, famously, sentences like this are only contingently paradoxical. If I were to add a bullet point followed by "2+2=4" at the end of the post, then p would be unparadoxically false, while if I were to end the post with a bullet point followed by a piece of nonsense or a falsehood, then p would be unparadoxically true (nonsense is not true).

Here is an assumption that I think is implicit in the above derivation:

1. The item p is true if and only if the last bulleted item in this post is not true.

The argument needs some bridge like this between the truth of the linguistic item p and the last bulleted item not being true. Once we have (1), then the argument is quick, using only uncontroversial premises. If item p is true, then the last bulleted item is not true by (1). But empirically the last bulleted item is p. So if p is true, then it's not true. But if it's not true, then by (1) it's not the case that the last bulleted item is not true. Since empirically the last bulleted item is p, it follows that it's not the case that p is not true, i.e., that p is true. So p is true if and only if it's not true, a contradiction in classical logic.

Since we should not deny classical logic or obvious empirical truths, it follows that (1) is false. Now, if p expresses a proposition, then it's got to express a proposition that makes (1) be true--that's both intuitively obvious and a consequence of the Tarski T-schema. (Doesn't (1) follow from the T-schema absent the expression assumption? It had better not. If "s" is meaningless, then the instance "'s' is true if and only if s" does not express a proposition, too, and hence is not true. So the T-schema had better apply only to meaningful items.) So p doesn't express a proposition. But that's a contingent fact, since in another possible world I screw up and end this post with a bulleted "2+2=5" thereby making p both meaningful and true.

So, whether a linguistic item expresses a proposition is in general a contingent matter. We already should have known this in the case of linguistic items using names, indexicals and demonstratives, and indeed p contains the demonstrative "this". But nothing hangs on p containing the demonstrative "this"--one could just replace it with some complex definite description--so I will ignore this demonstrative. If we think that whether a linguistic item expresses a proposition determines whether it's a meaningful sentence, then it follows that whether a linguistic item is a meaningful sentence is contingent, even in the absence of names, indexicals and demonstratives.

Further, not only is it a contingent matter whether a linguistic item expresses a proposition, but whether it does so can vary from token to token, again in the absence of names, indexicals and demonstratives. After all, I just gave a conclusive argument p does not express a proposition, and hence that the last bulleted item in this post does not express a proposition, and thus is not true:

2. The last bulleted item in this post is not true.

Item token (2) is true (note that numerals aren't bullets) and hence expresses a proposition. But s, which is a token of exactly the same type, does not. And that's not due to names, indexicals or demonstratives.

These conclusions are interesting independently of the paradox. But somehow it feels wrong to use the paradox to reach them. Is it?

An argument against heavy-weight Platonism

Heavy-weight Platonism explains (or grounds) something's being green by its instantiating greenness. Light-weight Platonism refrains form making such an explanatory claim, restricting itself to saying that something is green if and only if it instantiates greenness. Let's think about a suggestive argument against heavy-weight Platonism.

It would be ad hoc to hold the explanatory thesis for properties but not for relations. The unrestricted heavy-weight Platonist will thus hold that for all n>0:

1. For any any n-ary predicate F, if x₁,...,x_n are F, this is because x₁,...,x_n instantiate Fness.

(One might want to build in an ad hoc exception for the predicate "instantiates" to avoid regress.) But just as it was unlikely that the initial n=1 case would hold without the relation cases, i.e., the n>1 cases, so too:

2. If (1) holds for each n>0, then it also holds for n=0.

What is the n=0 case? Well, a 0-ary predicate is just a sentence, a 0-ary property is a proposition, the "-ness" operator when applied to a sentence yields the proposition expressed by the sentence, and instantiation in the 0-ary case is just truth. Thus:

3. If (1) holds for each n>0, then for any sentence s, if s, then this is because because of the truth of the proposition that s.

(The quantification is substitutional.) For any sentence s, let <s> be the proposition that s. The following is very plausible:

4. For any sentence s, if s, then <s> is true because s.

But (4) conflicts with (3) (assuming some sentence is true). In fact, to generate a problem for (3), we don't even need (4) for all s just for some, and surely the proposition <The sky is blue> is true because the sky is blue, rather than the other way around: the facts about the physical world explain the relevant truth facts about propositions. Thus:

5. It is false that (1) holds for each n>0.

The above argument is compatible, however, with a restricted heavy-weight Platonism on which sometimes instantiation facts explain the possession of attributes. Perhaps, for instance, if "is green" is a fundamental predicate, then Sam is green because Sam instantiates greenness, but this is not so for non-fundamental predicates. And maybe there are no fundamental sentences (a fundamental sentence would perhaps need to be grammatically unstructured in a language that cuts nature at the joints, and maybe a language that cuts nature at the joints will require all sentences to include predication or quantification or both, and hence not to be unstructured). If so, that would give a non-arbitrary distinction between the n>0 cases and the n=0 case. There is some independent reason, after all, to think that (1) fails for complex predicates. For instance, it doesn't seem right to say that Sam is green-and-round because he instantiates greenandroundness. Rather, Sam is green-and-round because Sam is green and Sam is round.

Fundamental logical relations

One might think the fundamental logical relations are between propositions. But I now think they are between relations (and propositions are a special case: 0-ary relations). Why? Well, in quantified logic (think: universal introduction and existential elimination) we need to talk about logical relationships between either (a) sentences with arbitrary names, (b) sentences with irrelevant names or (c) open formulas. Now ordinary sentences represent propositions, and the logical relations between sentences plausibly are grounded in the logical relations between the corresponding propositions. But sentences with arbitrary names don't represent anything, and so we have this unsatisfying grounding discontinuity: some logical relations between sentence-like entities in proofs are grounded in relations between proposition-like entities and others aren't. For somewhat similar reasons, if we want our logic to mirror the logical structure of reality, (b) isn't an option.

So we have philosophical reason to use a logic where there are logical relationships between open formulas. But an open formula represents a relation of arity equal to the number of free variables. It seems, thus, that some logical connections between propositions hold in virtue of logical connections between relations. Thus, the proposition that all humans are mortal follows from the propositions that all humans are animals and that all animals are mortal because the property (a property is a unary relation) of being mortal-if-human follows from the properties of being animal-if-human and being mortal-if-animal.

OK, time to stop procratinating grading the modal logic assignments!

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.

Illocutionary force and propositions

Suppose I say to Bill: "Make all of your papers be between two and four pages." Bill hands in an eight page paper for his first assignment. I rebuke him and he apologizes. He then hands in another eight page paper for his second. When I rebuke him, he says: "You told me to bring it about that all my papers be between two and four pages. With my first paper I ensured that the proposition that all my papers are between two and four pages is false. Sorry! By the time of my second paper, it was too late to undo this: no matter what length of paper I wrote, that proposition would still be false. So I might as well write the length that I like."

Bill's mistake was thinking that the content of my command was the proposition that all his papers be between two and four pages. I didn't command that proposition. Rather, I commanded distributively of each of his papers that it be between two and four pages.

This means that we should not analyze my speech act as having a propositional content plus an illocutionary force. The content of the speech act wasn't a proposition, but something else. Perhaps the content of the speech act was an ordered pair of properties, the property P of being one of Bill's papers, and the property L of being between two and four pages in length. And the illocutionary force was of something one might call distributive command. Successful distributive command in respect of a pair of properties P and L creates for each instance x of P a reason to make x have L.

There are, I think, assertion-like speech acts that also have such a non-propositional content. For instance, assertoric endorsement. A paradigm case: I endorse what you are about to assert. The content of assertoric endorsement is a property which is supposed to be had by one or more propositions—say, the property of being soon asserted by you—and when successful, the assertoric endorsement makes you stand behind each of these propositions as if you asserted it. This kind of assertoric endorsement is distributive.

I wish I knew what kinds of entities can be contents of speech acts. The above suggests that some speech acts have propositions as contents, some have pairs of properties, some have single properties. There must be many other options.

Reducing sets

I find sets to be very mysterious candidates for abstract entities. I think it's their extensionality that seems strange to me. And anyway, if one can reduce entities to entities that we anyway want to have in our ontology, ceteris paribus we should. I want to describe a three-step procedure—with some choices at each step—for generating sets. I will use plural quantification quite a lot in this. I am assuming that one can make sense of plural quantification apart from sets.

Step 1: The non-empty candidates. The non-empty candidates, some of which will end up counting as sets in the next step, will be entities that stand in "packaging" relation to a plurality of objects, such that for any plurality, or at least for enough pluralities, there is a candidate that packages that plurality. There are many options for the non-empty candidates and the packaging relation.

Option A: Plural existential propositions, of the form <The Xs exist>, where a plural existential proposition p packages a plurality, the Xs, provided that it attributes existence to the Xs and only to the Xs.

Option B: Plural existential states of affairs (either Armstrong or Plantinga style), i.e., states of affairs of the Xs existing, where a plural existential state of affairs e packages a plurality, the Xs, provided that it is a state of affairs of the Xs existing. I got this option from Rob Koons.

Option C: This family of options generates the candidates in two sub-steps. The first is to have candidates that stand in a packaging relation to individuals, such that each candidates packages precisely one individual. Call these "singleton candidates". For brevity if x is a singleton candidate that packages y, I will say x is a singleton of y. The second step is to take our non-empty candidates to be mereological sums of singleton candidates, and to say that a mereological sum m packages the Xs if and only if m is a mereological sum of Ys such that each of the Ys is a singleton of one of the Xs and each of the Xs is packaged by exactly one of the Ys. We need the singleton packaging relation to satisfy the condition (*) that a mereological sum of singletons of the Xs has no singletons as parts other than the singletons of the Xs. (In particular, no singleton of y can be a part of any singleton of x if x and y are distinct.)

We get different instances of Option C by considering different singleton candidates. For instance, we could have the singleton candidates be individual essences, and a singleton candidate then packages precisely the entity that it is an individual essence of. I got this from Josh Rasmussen. Or we might use variants of Options A and B here: maybe a proposition attributing existence to x or a state of affairs of x existing will be our candidate singleton. (Whether the state of affairs option here differs from Option B depends on whether the state of affairs of a plurality existing is something different from the mereological sum of the states of affairs of the individuals in the plurality existing.)

There are many other ways of packaging pluralities.

Step 2: The empty candidate. We also need an empty candidate, which will be some entity that differs from the non-empty candidates of Step 1. Ideally, this will be an entity of the same sort as the non-empty candidates. For instance, if our non-empty candidates are propositions, we will want our empty candidate to be a proposition, say some contradictory proposition.

Step 3: Pruning the candidates. The basic idea will be that x is a member of a candidate y if and only if y is one of the non-empty candidates and y packages a plurality that has x in it. But the above is apt to give us too many candidates for them all to be sets. There are at least two reasons for this. First, on some of the options, there won't be a unique candidate packaging any given plurality. For instance, there might be more than one proposition attributing existence to the same plurality. Thus, the propositions <The Stagirite and Tully exist> and <Aristotle and Cicero exist> will be different propositions if Millianism is false, but both attribute existence to the same plurality. Second, some of the candidates will be better suited as candidates for proper classes than for sets and some candidates may be unsuitable either as sets or as proper classes. For instance, there might be a proposition that says that the plural existential propositions exist. Such a proposition packages all the candidates, including itself, and will not be a good set or proper class on many axiomatizations.