A puzzle

Suppose P and E are empirical claims and E does not entail P. Nonetheless suppose:

1. I know P precisely on the grounds of E.

Surely it is possible—maybe it even follows if some appropriate closure principle holds—that I am also in a position to know the disjunction P or not E. But on what grounds am I in a position to know the disjunction? Surely not on the grounds of E, since E is if anything evidence against the conjunction (after all the negation of E would be conclusive evidence against the disjunction). So, it seems, I must know it a priori. Cf. the Explainer in Hawthorne.

Here's an instance of the above (with some negations transposed). I know phlogiston theory, Q, to be false on the basis of some empirical evidence E. I surely also know that the conjunction of Q and E is false. On what grounds do I know that conjunction to be false? Surely not on the basis of E. But then what? The answer seems to be that I know it a priori, despite the conjunction being a contingent empirical claim.

Penal substitution theories of the atonement

According to the penal substitution theory of the atonement, Christ's sufferings satisfy justice in place of our being punished. That is, basically, the theory as found in Anselm's Cur Deus Homo.

Some contemporary Christians, mainly Protestant, add the claim that Christ was punished by the Father, and his punishment substitutes for our punishment. We can call the resulting theory punishment by punishment substitution (PBPS). PBPS isn't Anselm's theory, and as Mark Murphy has pointed out it may even be incoherent, since a part of punishing is the showing of disapproval at the person being punished, while God cannot show disapproval at an innocent person.

The Heidelberg Catechism explicitly only says that Christ satisfies for us. But it says in the answer to Question 14 that no mere creature can satisfy for us because "God will not punish any other creature for the sin which man has committed", which may implicate that satisfaction involves being punished. Still, it does not say that it does so in the case of Christ.

In any case, it seems to me that the biblical theory is not that the punishment of Christ substitutes for our punishment, but that the sacrifice of Christ substitutes for our punishment. Old Testament sacrifices for our sins were not punishments of the animals, except in the extended sense of the word as when we speak of "the punishing heat of Texas summer." It is central to the idea of sacrifice in the Old Testament that it is the best that is sacrificed. To sacrifice something is to treat it as the best that is available. But when someone is being punished, then he is far from being treated as the best—he is being treated as one of the worst. Thus, the biblical picture of Christ as sacrificed is in serious tension with PBPS.

That the sacrifice of Christ substitutes for our punishment isn't yet a theory of the atonement. To make it a theory of the atonement one would have to say how it does so.

A (failed?) Platonist deflationary theory of truth

Propositions are nullary relations, and properties are unary relations. Suppose the Fs are a plurality.

1. For any object x, there is the property D_x of being distinct from x.
2. For any n-ary relation P, there is an (n+1)-ary property P⁺ such that x₁,...,x_n,x_n+1 stand in P⁺ if and only if x₁,...,x_n stand in P.
3. For any pair of properties P and Q, there is the property P∨Q of having at least one of P and Q.
4. If for every x among the Fs there is a property P_x, then there is a property that is the conjunction of the properties P_x as x ranges over the Fs, a property that is had by y provided that y has P_x for every F x.

Now, for any proposition p, define the property P_p=D_p∨p⁺ (by 1-3; note that because a proposition p is a nullary relation, p⁺ is a property). Let T be the conjunction of the P_p over all propositions p. Then a proposition p has T if and only if it is true.

But we cheated in 2. For suppose P is nullary (i.e., a proposition). Then the second part of the biconditional says that an empty list of objects stands in P, which is just a fancy way of saying P is true.

Still, it is interesting that we can define a property of truth as long as we're given the arity-raising operator (·)⁺, plus some plausible (abundant) property formation rules.

Nullary predicates and truth

This post develops an ultimately unsatisfactory deflationary theory of truth. Feel free to skip.

A binary predicate needs two names, or two quantifiers, to make a sentence. A unary predicate needs one name, or one quantifier, to make a sentence. A nullary predicates needs no names—it by itself, with no arguments, makes a sentence. For instance, the English "It rains" is a nullary predicate. It pretends to be a subject-predicate sentence, with the subject "it", but the "it" has no reference.

English allows the stipulative introduction of new names and predicates. Thus, I can say things like:

1. Let "Cloak" denote Socrates' nose. It is notorious that Cloak is snub.
2. Let "tigging" denote that which is in fact Sam's favorite activity. There is then a possible world where Sam would rather eat spinach than tigg.

Extend English to allow the stipulative introduction of nullary predicates. Then a number of the tasks for which we use the word "truth" could be replaced by such stipulative introduction. For instance, we can replace:

3. Kathleen's theory about the origins of the universe is not true.

with:

4. Stipulate that "xyzz" is a nullary predicate expressing Kathleen's theory about the origins of the universe. It's not the case that xyzz.

By "we can replace", I mean that the communicative tasks accomplished with (3) could be accomplished with (4).

So far this strategy will only handle some uses of "true", namely those where the predicate "is true" is joined to a name or definite description. What about a more complex case?

5. At least one of Kathleen's astrophysical theories is true if string theory is true.

This we can handle as well with nullary predicate stipulation:

6. Stipulate that "xyzz" is a nullary predicate expressing the disjunction of Kathleen's astrophysical theories. Stipulate that "strig" is a nullary predicate expressing string theory. Xyzz if strig.

But what I cannot handle using this method are uses of "is true" embedded in modal operators, such as:

7. Kathleen could have come up with a true astrophysical theory.

Maybe we can handle this if we allow the "true in w" predicate and quantification over worlds:

8. There is a world w and a proposition p such that Kathleen comes up with p in w, and p is an astrophysical theory in w, and p is true in w.

Now, this seems to get us no further ahead—after all, we've only replaced the "true" in (7) with "true in w" in (8). Actually, we can make the theory stay alive a bit longer: "true in w" need not be a notion that depends on the notion of truth. Suppose, for instance, that worlds are maximal compossible sets of propositions. Then "p is true in w" can be replaced with "p is a member of w".

However, if that's how we understand worlds, then we need substitutional quantification to explain what it means to say that "s in w" (e.g., "Snow is white in w"), where "s" is a sentence. Our bet may be to say: for all s, s in w if and only if the proposition that s is a member of w. But the "for all" is substitutional quantification. But substitutional quantification and truth are probably interdefinable, so if we have to rely on substitutional quantification, the above account fails.

At this point my toy deflationary account of truth in terms of stipulation of nullary predicates comes to a halt. It is modal embedding that brings it to this halt.

It is interesting that modality does not seem to bring to a halt a similar view of A-predicates.

A failed grounding for the wrongness of killing

Some people may think that what makes killing the innocent is in part that the prospective victim has interests, and what in part grounds the prospective victim's having interests is her having desires. This makes for a neat argument against the claim that the prohibition extends to embryonic and early fetal humans, since these appear not to have desires.

But I think the position leads to absurdity. Suppose Martha is in a severe but temporary depressive episode such that she has only one desire: to die. In such a case, killing Martha is murder. (Some defenders of euthanasia may disagree in the case where the depression is permanent, but what I said is uncontroversial. If need be, add that Martha does not consent.) According to the above story:

1. Martha's one and only desire is to die.
2. That killing Martha is wrong[note 1] is partly grounded in the fact that she has interests.
3. That Martha has interests is partly grounded in the fact that she has desires.

Add the following plausible grounding principles:

4. If p is partly grounded in q and q is partly grounded in r, then p is partly grounded in r. (Transitivity)
5. That ∃x(Fx) is partly grounded in the fact that Fa, where a is some particular such that Fa. (Weak form of Existential Grounding)

Then:

6. That Martha has desires is grounded in the fact that she has a desire to die. (1 and 5)
7. That killing Matha is wrong is partly grounded in the fact that she has a desire to die. (2, 3, 6 and 4)

But this is absurd. It's wrong to kill Martha, but surely not even partly because she desires to die. Of course, one can imagine odd cases where the wrongness of killing someone is partly grounded in their desire to die. For instance, suppose Patricia is someone whom it is permissible to execute for a crime, but that the executioner has sworn an oath never to execute someone who desires to die. In such a case, it could be impermissible for the executioner to kill Patricia in part because she desires to die. But the Martha case isn't like that. In the Martha case her desire to die does not contribute to the wrongness of killing her—if anything, it slightly lessens the wrongness (though I am not sure I want to say that). Moreover, while it would be wrong for the executioner to kill Patricia, it wouldn't be murder (assuming this was a case where the death penalty is morally justified), and what I am interested in is the grounding of that wrongfulness of killing that makes the killing be a murder.

Now, the case of someone whose only desire is to die may seem odd, and maybe someone could say that in odd cases one gets odd conclusions like (7). I don't know that that would be a satisfactory answer. But let's give a less odd case. Marcus desires to die, but he also has a full suite of other desires. I shall assume the case is to be elaborated in such a way that to kill him would be murder. To run the argument in the case of Marcus we need this principle like (5):

8. That ∃x(Fx) is partly grounded in the fact that Fa, for any particular a such that Fa and such that the fact that Fa is not itself partly grounded in ∃x(Fx).

The latter proviso is needed to rule out some paradoxes. Add the fact that:

9. That Marcus desires to die is not itself partly grounded in the fact that he has desires.

We can now modify the argument about Martha to conclude:

10. That it is wrong to kill Marcus is partly grounded in the fact that Marcus desires to die.

And that's absurd in this case. And while the case of Martha was very odd, alas the case of someone who among with other desires has a desire to die is not so very odd, though deeply unfortunate.

[Edited. The first version attributed the view that right to life is grounded in having desires to Boonin. But I think that attribution may be incorrect--he may only be claiming that it is grounded in (ideally, dispositionally) having a desire for life. See here.]

Are infinite and infinitesimal sizes relational?

Suppose that Leibniz is right, and in addition us (call us "macros") there are infinitesimal embodied beings ("micros") and infinite embodied beings ("megas"). Suppose we are teaching English to one of the micros. Should we tell her to say:

1. The micros are infinitesimal, the macros are finite and non-infinitesimal, and the megas are infinite,

as we do, or should she say:

2. The micros are finite and non-infinitesimal, the macros are infinite, and the megas are infinitely many times bigger than the infinite macros?

I am inclined to think she should say (2), and so infinite and infinitesimal sizes are merely relational or indexical.

Meaning and use

Consider the thesis:

1. The meaning of a word is defined by how it is used.

This thesis seems false. I hereby stipulate that "Xozhik" is a name for me. As soon as I have made the stipulation, "Xozhik" has acquired a meaning. However the word did not get used, but was only mentioned, and it could be that nobody ever uses it.

This nit-picking point does not affect more sophisticated theses about meaning and use.

Modal realism and God

According to David Lewis's modal realism, every possible world exists as a concrete universe, and a proposition is possible provided it holds at some universe. But this seems incompatible with theism. For necessarily God believes every truth, and we can now run the following argument.

Necessarily, if p is true, God believes p. So, if p is possible, possibly God believes p. Thus, possibly, God believes that there are no horses, since the proposition that there are no horses is possibly true. So there is a universe, say u₁, at which God believes that there are no horses. Now God either actually has this belief or not. If he actually has this belief, then he actually has conflicting beliefs, since he actually believes that there are horses. But God does not have conflicting beliefs. So we have to say that while at u₁ God believes there are no horses, actually God instead believes there are horses. Thus, what propositions God believes differs between universes. But how could that make any sense? Granted, perhaps our beliefs can be localized to brain hemispheres and then at a location in my left hemisphere I believe p and at another I don't. If that can be made sense of, then one could give a sense to the locution "believes p at x". But God's beliefs surely do not have any such localization. Wherever God is present, he is wholly present. He is not a material being to have partial presence of the sort that might allow for a spatial distribution of our beliefs.

Paradoxes of comparison

There are three sets, A, B and C, each consisting of the same number of people, whose lives are endangered by the same sort of danger, and whose future prospects as far as you know are on par. There are also three hungry kids, x, y and z, who will survive if you don't give them breakfast, but who would benefit from your giving them breakfast once (you have no opportunity to do anything more for them). Suppose you have a choice between two actions:

1. Save all the people in A and feed nobody.
2. Save all the people in B and feed x.

Now, it sure seems like 2 is the better action than 1. One might even formulate a general principle:

• (*) If an action saves the same number of lives and feeds more hungry children, and all else is on par, then it's better.

But actually this is false. For suppose that B is a proper subset of A, and there are 100 people in A who are not in B. Since we said that A and B have the same number of people, this can only be the case when A and B are infinite sets. In this scenario, if one goes for 2, there will be 100 people whom one won't be saving. So we should modify (*), perhaps to:

• (**) If an action saves the same number of lives and feeds more hungry children, and the sets of lives saved and children fed are disjoint between the two actions, and all else is on par, then the action is better.

But paradox ensues when we specify that A and B have no people in common, but C is a subset of A missing 100 people, and then add the option:

3. Save all the people in C and feed y and z.

For by one application of (**), 2 is better than 1, and by another application of (**), 3 is better than 2. But 3 is not better than 1, because the 100 people in A who aren't in C die if you go for 3, and that's not balanced out by the two children fed.

(There is a literature on infinite utilities, and I am not claiming any originality for this case.)

One could take this as yet another argument against the transitivity of "better than". But that doesn't get us out of paradox, since denying that transitivity is itself paradoxical. Moreover, there is already something paradoxical in having to deny (*)—that principle sure seemed plausible.

We could conclude that one can't have infinite sets of people, and make this be one of the family of arguments against actual infinites. Maybe.

But I want to do something else here. I think this, like a number of other paradoxes (which need not all involve infinity; I have a hunch that White's puzzle, as per Joyce's reply discussed in the link, is in this family), is due to us having two ways of comparing. We have an uncontroversial and unproblematic inclusion or domination comparison. It is uncontroversial that all other things being equal, if you can save all the people in A or all the people in B, and the people in A are a proper subset of those in B, then you should save the people in B. It is uncontroversial that if p entails q, then q is at least as likely as p. And so on.

But we also insist on comparing apples to oranges, comparing where there is no inclusion or domination relation. Typically, five oranges are more valuable than one apple, and five apples are more valuable than one orange. To make such comparisons we often assign numbers—say, cardinalities, utilities, prices or probabilities—to the things we are comparing, but we can also just make ordinal comparisons without assigning numbers (I didn't assign any utilities when I gave the ethical story).

I think a lot of paradoxes have the consequence that comparisons without domination are fishy. They need not satisfy transitivity. They might suffer from some arbitrariness. In the ethical sphere, this can be manifested in incommensurability of options. In probability theory, this surfaces in difficulties surrounding infinite sample spaces or nonmeasurable sets (as in White's puzzle, since nonmeasurable sets and non-exact probabilities are of a piece, I think).

Yet we need comparisons-without-domination.

So what should we do? In the ethical sphere, perhaps what we need is basically what Aquinas says about the order of charity. Aquinas thinks that when choosing between an equal benefit to one's parent or to a stranger, one should bestow the benefit on one's parent. But what if the benefit to the stranger is greater? If only slightly greater, we should still benefit our parent. But if much greater, we should benefit the stranger. But where is the line drawn? Aquinas refuses to answer. There is no rule, it seems. Rather, this is just somehting for the wise and virtuous agent to know. And maybe there is an analogue to this answer in the case of the non-ethical paradoxes.

Grounding inhomogeneity and analogy

I ought to respect innocent human life. So I ought not feed cyanide to the innocent. I ought to respect the legitimate intellectual autonomy of others. So I ought not force my students to believe all my metaphysical views.

So, some ought claims are grounded, in part or whole, in other ought claims, and sometimes in further non-normative claims (such as that cyanide kills). This is familiar in many other cases. Thus, it's a standard libertarian view about freedom that some exercises of freedom are only derivately free: they are free insofar as they flow from a character that was formed by other free actions.

It would generate a vicious regress to suppose that all free actions are derivatively free. (In this case, the impossibility of the regress is obvious from the fact that we've only performed finitely many actions in our history.) Likewise, it would be a vicious regress to suppose all ought claims are grounded in further ought claims.

So there are some thing that are derivatively obligatory and some that are non-derivatively obligatory. (The two categories might overlap. For if I promise to fulfill a non-derivative obligation, then that obligation is both non-derivatively obligatory and obligatory by derivation from the duty to keep promises.) Likewise for freedom and many other properties. The non-derivative cases may be brute and ungrounded, or they may be grounded in a different kind of fact (e.g., maybe non-derivative freedom is grounded in alternate possibilities or non-derivative ought is grounded in divine commands—I am not advocating either option as it stands).

Here is a maxim I find plausible: Properties that exhibit this kind of grounding inhomogeneity—sometimes being grounded in one kind of fact and sometimes either ungrounded or grounded in a different kind of fact—are in fact non-fundamental.

This may lead one to say that properties that exhibit this kind of inhomogeneity are really disjunctive. That (or the related suggestion that they are existentially quantified) may be true, but I think it isn't the whole truth. Maybe freedom just is the disjunction of non-derivative and derivative freedom. But it's not a mere disjunction, in the way that being red or cubical is. It's a disjunction between related properties. In the cases of freedom and obligation, the relationship here seems to me to be precisely that of Aristotelian analogy: there is a focal sense of freedom and obligation—the non-derivative case—and there is a non-focal sense as well.

Conjecture: When we are dealing with a somewhat natural property that exhibits grounding inhomogeneity, we are precisely dealing with a disjunction (or quantificational combination) between analogous more fundamental properties.

Aquinas' thinking on divine names fits well here. St Thomas thinks that when we predicate wisdom of God and Socrates, we do so analogically, because God's wisdom is God and Socrates' wisdom is accidental to him. But this difference is precisely a grounding inhomogeneity in the property wisdom, with God being the focal case.

Van Fraassen's Reflection Principle

According to Van Fraassen's Reflection Principle, a rational agent's

subjective probability for proposition A, on the supposition that his subjective probability for this proposition will equal r at some later time, must equal this same number r.

So if you are rational and see that you will assign r to A later, you now already assign r to A.

It seems that Reflection can only be plausible if there is a uniquely rational subjective probability for every set of evidence where there is something one can rationally think (are there any where there isn't? interesting question). For suppose that given my evidence, I can be epistemically rational in assigning different probabilities r₁ and r₂ to some proposition A, but I have non-epistemic reasons for assigning r₁ to A today and r₂ tomorrow (they might actually be in a way epistemic--for instance, it could be that changing between these assignments will help me see things from different points of view). Then I am epistemically rational in assigning r₁ today and planning to assign r₂ tomorrow (I know I will get no new relevant evidence, let's say), and I could well be rational simpliciter. But my present subjective probability for A, on my prediction that I will assign r₂ tomorrow, is still r₁ and not r₂.

So Reflection needs to be restricted to cases where there is a uniquely rational answer. I am wondering if this doesn't vitiate one of White's arguments against imprecise probabilities (see here for a careful discussion of that argument).

Knowing what and knowing that

Consider this real-life sentence:

1. Once your dog knows what "sit" means, he will be happy to please you.

So it appears that the following sometimes happens:

2. The dog knows what "sit" means.

But surely dogs don't know semantic propositions. So, (2) does not entail that anything of the following form is true:

3. The dog knows that "sit" means ....

Dogs don't need to know semantic propositions to understand commands. Likewise, a small child don't need to know a proposition of the form <"Table" means ...> in order to know what "table" means.

Now it could be that this is something special about meaning, that we simply say that someone or something knows what something means provided simply that he, she or it grasps it, without him, her or its having to know any semantic proposition. But I've also toyed with the idea that when we say "x knows what/where/when n V" (where "n" is a noun and "V" is a verb), this should not be analyzed as attributing to x knowledge of the relevant proposition of the form <n V m>. (In the above case, n is a word and "V" is "means".)

Sam was Gettiered in his coming to believe that 9x8=72. His innumerate teacher was saying "7x10=70, and 9x8=70, too", and Sam heard it as "7x10=70, and 9x8=72." And Sam never acquired any other relevant evidence. Then Sam does not know that 9x8=72. But maybe we should say that Sam knows what 9x8 is. For it doesn't seem right to say Sam doesn't know what 9x8 is.

Or suppose Spike has just heard a genuinely powerful argument for external world scepticism. It hasn't made him lose his beliefs, but the argument provided a defeater for his knowledge. So Spike doesn't know he has ten fingers, though he correctly believes it. It doesn't seem right to say Spike doesn't know how many fingers he has (i.e., what the number of his fingers is, to put it in the form I used above).

There is another possibility. It could be that to know what/where/when n V does require knowing that n V m for an appropriate m, but that we use "doesn't know what/where/when" to indicate something stronger than the denial of this knowledge.

I am not very secure in my intuitions about Sam and Spike, actually.

Killing, letting die and ensuring death

Suppose my wife tells me to ensure that my son brushes his teeth. I go to his bathroom and see him brushing his teeth. I did not bring it about that he brushed his teeth. Did I ensure it?

I may or may not have. I might have ignored my wife's request and just happened to go to my son's bathroom to fill a bottle of water. Or her request might have simply triggered a curiosity about my son's brushing habits. In those cases, I did not ensure it.

What needs to be the case for me to count as having ensured that he brushed his teeth? Maybe it's some kind of a disposition to make him brush his teeth if he does not do so on his own. But even such a disposition is not quite enough. Suppose, for instance, I am a domestic tyrant and I enjoy making people do things. I go to my son's bathroom quickly with a hope that I will get there before he brushes his teeth, so I will have an opportunity to make him brush his teeth. But alas he has foiled me: he already started and by the time I open my mouth in command, he has finished. In this case, too, it seems incorrect to say that I ensured that he brushed his teeth. For if I ensured that he brushed his teeth, then I succeeded at ensuring that he brushed his teeth. But in this case there is no plan of action that I succeeded at—in fact, I failed. (Note: I am talking of here of intentional ensuring. We also sometimes speak of some action unintentionally ensuring a result. In that case, "ensuring" just means something like "causally necessitating".)

For me to count as having ensured that he brushed his teeth, his brushing has to be according to my plan. Thus, I need to form a plan that he brush his teeth, and a part of that plan is the forming of a disposition to make him brush his teeth if he doesn't do so on his own, but the plan's goal needs to be that he brush his teeth rather than that I make him brush his teeth. Embarking on this plan is a genuine action on my part, an action whose end is that he brush his teeth. When I embark on the plan, I form a disposition to make him brush his teeth if he doesn't do so on his own, but that is not all that happens.

Why does this matter?

Well, consider this famous case of Rachels:

Jones also stands to gain if anything should happen to his six-year-old cousin. Like Smith [who drowns his cousin], Jones sneaks in planning to drown the child in Ills bath. However, just as he enters the bathroom Jones sees the child slip and hit his head, and fall face down in the water. Jones is delighted; he stands by, ready to push the child's head back under if it is necessary, but it is not necessary. With only a little thrashing about, the child drowns all by himself, "accidentally," as Jones watches and does nothing.

Rachels thinks that this case shows that the distinction between killing and letting die is bogus. Jones is morally on par with Smith.

Rachels is probably right. But the reason for this isn't that there is no morally salient distinction between killing and letting die. It is, rather, that there is no morally salient distinction between killing and ensuring death. What Jones does is ensure death. This is a genuine action on his part. He forms a series of dispositions in himself aimed at ensuring death. This is just as much an action of his as it would be an action to program a robot to watch the child and drown him if the child didn't drown on his own. And Jones succeeds at ensuring death: he doesn't just attempt to ensure death, but he succeeds.

The death of Jones' cousin is according to his plan, albeit not his original plan, but the revised one he forms when he enters the bathroom. Compare this case. Jones comes into the bathroom. He sees his cousin drowning. He has a failure of nerve and gives up on his plan. (It doesn't matter if the failure of nerve comes after or before the observation of the drowning.) But he still doesn't go to the trouble of rescuing his cousin, which he easily could do, nor does he turn on the music to drown out the noise of the drowing lest someone else come to the rescue, though he does hope the cousin will drown. He is a wicked man, but he hasn't ensured his cousin's death.

The moral difference between watching the cousin die and ensuring death is slight in the above case, but it could be greater if Jones' reasons were different. Suppose, for instance, that upon entering the room Jones has a change of mind due to fear of getting caught. But he also notices that his cousin is a carrier of a disease that will kill Jones if Jones touches the cousin, and it is this that now is the primary reason why Jones does not pull out his cousin. Jones had a change of mind but no great change of heart. He still hopes his cousin drowns and is glad he does. But at this point, Jones' actions and inactions in the bathroom are morally defensible (though his action of going to that bathroom in order to ensure his cousin's death is not defensible). (Cf. Ian Smith's paper.)

If I am right, then when thinking about killing and letting die, we need to distinguish letting die proper from ensuring death.

An Aristotelian argument from a necessary being to a necessary concrete being

Suppose that none of the participants in World War II had ever existed. Then it would have been impossible for World War II to occur. Why? Because World War II's existence is solely grounded in the existence, activities, properties and relations of the participants, and

1. If an entity x's existence is solely grounded in the existence, activities, properties and/or relations of the Fs, then it is impossible for x to exist without at least one of the Fs existing.

Now add this Aristotelian axiom:

2. If x is abstract, then x's existence is solely grounded in the existence, activities, properties and/or relations of concreta.

Finally, add this:

3. Every being is either concrete or abstract.
4. There exists a necessary being.
5. There is a world where no one of the contingent concrete beings of our world exists.

One might try to give the number three as an example of a necessary being to support (4).

Now, let N be the necessary being of (4). If N is essentially concrete, we get to conclude that there is a concrete necessary being. If N is essentially abstract, then N is grounded in the existence, activities, properties and/or relations of concreta. If some concreta are necessary, we conclude that there is a concrete necessary being. So suppose all concreta are contingent. Then the beings that N is grounded in don't exist at the world mentioned in (5), which violates the conjunction of (1), (2) and the necessity and abstractness of N. So, no matter what, it follows from (1)-(5) that:

6. There is a necessary concrete being.

Grounding graphs, new take

In a previous post, I looked at the idea of grounding graphs as global entities. But I think there is a more natural way of looking at them. There are two main views about grounding. On the truthmaker view, true propositions are grounded in entities that make them true. On the propositional view, true propositions are grounded in other true propositions. But I think a more natural approach is to say that propositions are grounded in graphs.

A candidate grounding graph for a proposition p is a directed graph G satisfying the following properties:

1. all the vertices of G are true propositions
2. p is a vertex of G
3. all the vertices of G other than p are ancestors of p
4. p is not the only vertex of G.

The grounding relation is then a relation between a proposition p and a candidate grounding graph for p. For instance, the proposition <The sky is blue or (roses are red and violets are blue)> is grounded in a graph with two vertices, one of which is <The sky is blue> and the other being the target proposition, with one arrow from the former to the latter. But it is also grounded in a more complex graph with four vertices: <Roses are red>, <Violets are blue>, <Roses are red and violets are blue>, and the target propositions, with arrows from the first two propositions to the third, and an arrow from the third to the target.

Define a proposition as fundamental provided that it is true but has no grounding graph. Say that a grounding graph for p is a candidate grounding graph for p that in fact grounds p. A vertex is initial provided that it has no ancestors and is final provided it has no children. A candidate grounding graph has exactly one final vertex. Say that G* extends G provided that (a) G* has the same final vertex as G and (b) every vertex of G that has a parent in G has exactly the same parents in G* as in G. The following are important properties of grounding graphs: Say that a graph where every initial vertex is fundamental is a fundamental graph.

• Acyclicity: Every grounding graph is acyclic.
• Extensibility: If G is a grounding graph for p, then there is a fundamental extension of G that is also a grounding graph for p.
• Adjoining: If G₁ is a grounding graph for p, and G₂ is a grounding graph for some initial vertex q of G₁ such that G₂ has a fundamental extension whose only vertex in common with G₁ is q, then the graph whose vertex collection is the union of the vertex collections of G₁ and G₂ and whose arrow collection is the union of the arrow collections of G₁ and G₂ is also a grounding graph for p.
• Truncation: If G is a grounding graph for p, then any subgraph of G that is a candidate grounding graph and that has the property that if it contains any one of G's arrows to q then it contains all of G's arrows to q is a grounding graph.

The following is very controversial but very helpful:

• Well-foundedness: No grounding graph contains an infinite chain of arrows.

This is compatible with some grounding graphs being infinite. For instance, we could have a fundamental grounding graph for an infinite conjunction. There, the infinite conjunction will have infinitely many parents. Moreover, there may be arbitrarily long chains in the graph—the first parent might be fundamental, the second might have a chain of length two to a fundamental ancestor, and so on.

I think that if we reject well-foundedness, we should reject acyclicity. For the most plausible putative counterexamples will be infinitely nested propositions like p₁&(p₂&(p₃&...)). But if we accept such propositions, we will also accept p&(p&(p&...)), and these will be cyclically grounded if the former will be non-well-foundedly grounded. But we shouldn't reject acyclicity, so we should accept well-foundedness, and if there are such infinitely nested propositions, we should ground them all at once in the symmetric conjunction p₁&p₂&... which then is grounded in each of its conjuncts.

Finally, we want to say something about how this interacts with logic. Say that p is free of q provided that is a fundamental grounding graph for p that does not contain q.

• Disjunction introduction: If p is free of (p or q), then the following is a grounding graph: p→(p or q).