All-false open futurism

On All-False Open Futurism (AFOF), any future tensed statement about a future contingent must be false. It is false that there will be a sea battle tomorrow, for instance.

Suppose now I realize that due to a bug, tomorrow I will be able to transfer ten million dollars from a client’s account to mine, and then retire to a country that won’t extradite me. A little angel says to me:

1. Your freely taking your client’s money without permission tomorrow entails your being a thief tomorrow.

I don’t want to be a thief, tomorrow or ever, so I am about to decide not to do it. But now a little devil convinces me of AFOF and says that while (1) is true, so is:

2. Your freely taking your client’s money without permission tomorrow entails your being a saint tomorrow.

Perhaps I am not very good at modal logic and the devil needs to explain. Given AFOF, it is necessarily false that I will freely take my client’s money without permission tomorrow, and a necessary falsehood entails everything. So, the devil adds, I might as well buy my plane tickets now.

The angel, however, grants AFOF for the sake of argument, but says that notwithstanding (2), the following holds:

3. Tomorrow it will be the case your taking your client’s money without permission entails your being a thief.

For the entailment holds always.

At this point, we have an interesting question. Given AFOF, should I guide my actions by the entailment between future-tensed claims in (2) or by the future-tensed entailment claim in (3)? The angel urges that the devil’s reasoning undercuts all rationality, while the angel’s reasoning does not, and hence is superior.

But the devil has one more trick up his sleeve. He notes that it is a contingent question whether there will be a tomorrow at all. For God might freely decide to end time before tomorrow. Thus, that there will be a tomorrow is false on AFOF. But (3) implies that there will be a tomorrow, and so (3) is false as well. I try to argue on the basis of Scripture that God has made promises that entail a future eternity, but the devil is a lot better at citing the Bible than I, and convinces me that God might transfer us to a timeless state or maybe eternal life is a supertask lasting from 8 to 9 pm tonight. And in any case, surely it should not depend on revelation whether the angel has a good argument not to take the client’s money. This is a problem for AFOF.

Maybe this is the way out. The angel could say this:

4. Necessarily, if there will be a tomorrow, then it will be true tomorrow that taking your client’s money without permission entails your being a thief.

But while this conditional is true on AFOF, if the devil has made his case that God hasn’t promised there will be a tomorrow, he can respond with:

5. Necessarily, if God hasn’t promised there will be a tomorrow and there will be a tomorrow, then it will be true tomorrow that taking your client’s money without permission entails your being a saint.

For the antecedent of the conditional here is necessarily false on AFOF, it being contingent that there will be a tomorrow absent a divine promise. And it seems that (5) is even more relevant to guiding action than (4), then.

Maybe the defender of AFOF can insist that the future must be infinite. But this does not seem plausible.

Entailment and Open Future views

This is probably an old thing that has been discussed to death, but I only now noticed it. Suppose an open future view on which future contingents cannot have truth value. What happens to entailments? We want to say:

1. That Jones will freely mow the lawn tomorrow entails that he will mow the lawn tomorrow

and to deny:

2. That Jones will freely mow the lawn tomorrow entails that he will not mow the lawn tomorrow.

Now, a plausible view of entailment is that:

3. p entails q if and only if it is impossible for p to be true while q is false.

But if future contingents cannot have truth value, then that Jones will freely mow the lawn tomorrow cannot be true, and hence by (3) it entails everything. In particular, both (1) and (2) will be true.

Presumably, the open futurist who believes future contingents cannot have truth value will give a different account of entailment, such as:

4. p entails q if and only if there is no history in which p is true and q is false.

But what is a history? Here is a possible story. For a time t, let a t-possibility be a maximal set of propositions that could all be true together at t. Given the open future view we are exploring, a t-possibility will not include any propositions reporting contingent events after t. If t₁ < t₂, and A₁ is a t₁-possibility while A₂ is a t₂-possibility, we can say that A₁ is included in A₂ provided that for any proposition p in A₁, the proposition that p was true at t₁ is a member of A₂. We can then say that a history h is a function that assigns a t-possibility h(t) to every time t such that h(t₁) is included in h(t₂) whenever t₁ < t₂.

(Technical note: Open theism implies a theory of tensed propositions, I assume. Thus if A is a t₁-possibility, then it is not a t₂-possibility if t₂ ≠ t₁, since any t-possibility will include the proposition that t is present.)

But what does it mean to say that a proposition p is true in a history h. Here is a plausible approach. Suppose t₀ is the present time. Given a proposition p that says that s, let p_t0 be the backdated proposition that at t₀ it was such that s (with whatever shifts of tense are needed in s to make this grammatical). Then p is true in h provided that there is a time t₁ > t₀ such that p_t0 is a member of h(t₁). In other words, a proposition p is true in h provided that eventually h settles its truth value.

This works nicely for letting us affirm (1) and deny (2). In every history in which it becomes true that Jones will freely mow the lawn it becomes true that Jones will mow the lawn, while this is not so if we replace the consequent with “Jones will not mow the lawn.” But what about statements that quantify over times? Consider:

5. Jones will mow the lawn, and for every time t at which Jones will mow the lawn, there will be a time t′ that is more than a year after t such that Jones will freely mow the lawn at t′.

This entails:

6. Jones will mow the lawn, and for every time t at which Jones will mow the lawn, there will be a time t′ that is more than a year after t such that Jones will mow the lawn at t′.

but does not entail:

7. Jones will not mow the lawn.

But there is no history h at which (5) is true by the above account of truth-at-a-history given our open future view. For let t₀ be the present and let p be the proposition expressed by (5). Then at any future time t and any history h, the proposition p_t0 is not a member of h(t). For if it were a member of h(t), it would be affirming the existence of an infinite number of future free mowings, and such a proposition cannot be true on our open future view. Since there is no history h at which (5) is true, by (4) we have it that (5) entails both (6) and (7), which is the wrong result.

What if instead of saying that future contingents lack truth value, we say that they are all false? This requires a slight modification to the account of p being true at a history. Instead of saying that p is true at h provided that there is some future time t such that p_t0 is in h(t), we need to say that there is some future time t such that p_t0 is in h(t′) for all t′ ≥ t. This gives the right truth values for (1) and (2), but it also makes (7) true.

I think the above open futurist accounts of entailment work nicely for statements with a single unbounded quantifier over times, but once we get alternating quantifiers like in (5), where the second conjunct is of the form ∀t∃t′ϕ, things break down.

Perhaps the open futurist just needs to be willing to bite the bullet and say that (5) entails (7)?

Intention and entailment

Suppose Alice intends to hit Bob with a stick. There are two ways that the stick could be involved in Alice’s intentions. First, Alice might not care that it is a stick she hits Bob with, but a stick happens to be ready to hand. In that case, her hitting Bob with a stick is a means to her hitting Bob.

Second, Alice might care about hitting Bob with a stick—perhaps she is punishing him for hitting a defenseless person with a stick and wants the punishment to match the crime. In that case, hitting Bob with a stick is not a means to her hitting Bob, as her hitting Bob does not figure in her intentions apart from the stick. But even in that case it seems right to say that Alice intends to hit Bob. For while it is false to say in general that

1. if p entails q and Alice intends p then Alice intends q

(even if one adds that Alice knows about the entailment, or makes the entailment relevant in the sense of relevance logic), it seems that the following special case is true:

2. if q is a specification of p and Alice intends q then Alice intends p.

Alice’s hitting Bob with a stick is a specification of Alice’s hitting Bob.

A similar point applies to conjunctions. If Alice intends to hit Bob with a stick and to insult him, she intends to hit Bob with a stick and she intends to insult him. But sometimes at least, hitting Bob with a stick and insulting him do not figure as independent intentions. Yet they are intended nonetheless. So we have another special case of (1):

3. if p is a conjunct of q and Alice intends q then Alice intends p.

It is an unhappy situation that some special cases of (1) are true, but (1) is not true in general, and I do not know how to specify which special cases are true.

Socratic perfection is impossible

Socrates thought it was important that if you didn't know something, you knew you didn't know it. And he thought that it was important to know what followed from what. Say that an agent is Socratically perfect provided that (a) for every proposition p that she doesn't know, she knows that she doesn't know p, and (b) her knowledge is closed under entailment.

Suppose Sally is Socratically perfect and consider:

1. Sally doesn’t know the proposition expressed by (1).

If Sally knows the proposition expressed by (1), then (1) is true, and so Sally doesn’t know the proposition expressed by (1). Contradiction!

If Sally doesn’t know the proposition expressed by (1), then she knows that she doesn’t know it. But that she doesn’t know the proposition expressed by (1) just is the proposition expressed by (1). So Sally doesn’t know the proposition expressed by (1). So Sally knows the proposition expressed by (1). Contradiction!

So it seems it is impossible to have a Socratically perfect agent.

(Technical note: A careful reader will notice that I never used closure of Sally’s knowledge. That’s because (1) involves dubious self-reference, and to handle that rigorously, one needs to use Goedel’s diagonal lemma, and once one does that, the modified argument will use closure.)

But what about God? After all, God is Socratically perfect, since he knows all truths. Well, in the case of God, knowledge is equivalent to truth, so (1)-type sentences just are liar sentences, and so the problem above just is the liar paradox. Alternately, maybe the above argument works for discursive knowledge, while God’s knowledge is non-discursive.

Causation and entailment

From time to time, one reads the sentiment that if E causes F, then that E occurs does not entail that F occurs. Here is a counterexample. God in a reverberating voice announces that you're going to be terrified. This causes you to be terrified. But that God announces something entails that the announced event will happen.

Here's another example, perhaps less compelling.  I just figured out, by induction, that I'm not going to be silent all day today.  But I am also the sort of the person who can't keep his mouth shut about what he knows.  So I tell you: "Today I am not going to be silent all day."  Then, my knowing that today I wasn't going to be silent all day caused me not to be silent all day.  But it also entailed it.  (Objection: Belief, not knowledge, enters into causal explanations.  Response: We certainly use knowledge talk in causal explanations.  In any case, this is why this example is less compelling than the first.)

Property entailment as the ground of modality

Consider a view on which all modality is grounded in property entailment: the relation between properties F and G expressed by "having F entails having G". Jubien has defended a view that might sound like this (though see comments at the end). One way to make this precise is to say that the theory of necessity is generated by the axiom schema

1. (x₁)...(x_n)(A(x₁,...,x_n)→B(x₁,...,x_n)),

where A expresses a relation that entails the relation expressed by B (I take relations and properties to be the same thing here, just that we use different words depending on the adicity), and where → is material implication, together with the rule of inference that

2. from a subproof of p that reiterates no assumptions other than instances of (1), we can derive Necessarily(p).

Here is one quick problem. This fits best with a Platonic metaphysics of properties (Jubien certainly does that). On a Platonic metaphysics of properties:

3. Necessarily(a is a circle → circularity exists).

Actually, standard Platonists will say that the consequent holds necessarily, independently of the antecedent, but we don't need this for the argument. But (3) cannot be proved via (1) and (2), unless the system is inconsistent. Why? Here is a simple way to see this. The axioms make no reference to circularity. There are instances of (1) that use the predicate "is a circle", but that's not good enough.[note 1] But then take any proof of (3) and replace "circularity" with a non-referring singular term. We will then get a proof that the referrent of the non-referring singular term exists, and given that the axioms are all true (this is uncontroversial), the only way we can get a proof of a falsehood is if the system is incoherent.

One might think that something could be done if existence is a property and there are entailment relations like that being a property entails existence. But that won't help unless we add to the axioms that circularity is a property. But the axioms are automatically necessary by (2), so now we are no longer just giving a property entailment account. We are adding axioms that directly force cerain essentialist claims, like that any property is essentially a property.

Alright, maybe that's unfair. Maybe any Platonist who has a property entailment view of modality will also have among the axioms the schema:

4. exists(P)

for every property P. But what about other very plausible necessities, like:

5. Necessarily(circle(a) → circularity is a property).

We had better not make existence entail propertyhood, as then Obama becomes a property.

Of course, we can add things like (5) to our axiom scheme. But the theory is now really swelling, and it is no longer true that it grounds modality in property entailment. It grounds modality in provability from a whole bunch of axiom schemata, one of which is the property entailment one.

My fairly quick glance at Jubien doesn't show him discussing this. But it does show him discussing a related issue, Kripkean arguments that a certain particular table must be made of wood. Jubien says that the table has a "table-essence" being this table and being this table entails being made of wood. So he could handle (5) by saying that circularity has an "object-essence" (I think it had better be an obejct essence, in his terminology), being circularity, and that being circularity entails being a property. Fine so far, but what about the following necessities:

6. Necessarily(exists(circularity) → circularity has being circularity).

So what this shows is that on the proposed account we need additional necessity-generating axioms governing object essences. When Jubien introduces object essences, he explicitly says that they have modal properties, such as that an object necessarily has its object essence. So perhaps Jubien is not someone we should describe as giving an entailment view. For maybe he has an axiom schema that generates (6), as well as the Platonist schema that generates (5). Now as long as the theory of necessity was generated by (1) and (2), it was a pretty cool partial reduction. But once we had to add the schemata (4) and something generating (6), we start to wonder: what guarantees that no further schemata need to be added? All the axioms are automatically necessary, after all, and once we have a plurality of schemata, we have failed to explain what they all have in common—what makes them all be necessary. The account becomes in effect disjunctive.

Here is a different kind of problem. Consider this claim:

7. Necessarily(wrongs(x,y) → agent(x)).

Only agents can wrong. But (7) isn't an instance of (1). Now, maybe, "wrongs(x,y)" is an abbreviation for something one of whose conjuncts ascribes to x a monadic property that entails being an agent. But what if it's not like that? And is it really plausible that all relations that entail a monadic property in one of the relata are abbreviations for stuff that includes a monadic property attribution? (Here is an example every Platonist should accept: Instantiates(a,circularity) entails circle(a).)

So, it seems, the system needs to be extended to include entailments between relations of different adicities. Moreover, it needs to be extended to include entailments between relations of the same adicity but with the relata reorganized:

8. Necessarily(wrongs(x,y) → iswrongedby(y,x)).

We no longer have just entailment between relations, then, at the base of the system. We have what one might call "twisted entailment". Specifically, if f is a function from {1,...,m} to {1,...,n}, we can say that the n-adic property P f-twistedly-entails the m-adic property Q provided that:

9. Necessarily(A(x₁,...,x_n) → B(x_f(1),...,x_f(m))),

where A and B express P and Q respectively.

The view is still non-trivial. But it is messy, and it is difficult to see what motive one has for believing it rather than just giving up on the grounding project altogether—or going for my view. :-) After all, f-twisted-entailment is not such a natural property as entailment.

[Fixed definition of twisted entailment.]

Ockham on the paradoxes of entailment

While looking through an edition of Ockham, I came across this interesting text:

Other rules are given:
(10) From an impossibility anything follows.
(11) What is necessary follows from everything.
Therefore this follows: 'You are a donkey, therefore you are God'. This also follows: 'You are white, therefore God is triune'. But these consequences are not formal ones and they should be used much, nor, indeed are they used much. (Summa totius logicae III, III, C. XXXVI)

What is interesting to me is (a) that the consequences are not formal ones (don't follow in relevance logic?) and especially (b) that these rules shouldn't be used much. Why not?

Presumably, the only time you establish an impossibility in a sound argument is as part of a reductio, and if you do that, you don't use (10) next—you close the subproof and use reductio ad absurdum.

And using (11) leads to arguments that are not as perspicuous as they could be. For if you've established necessarily(p), it is more perspicuous to conclude p from necessarily(p) by axiom M of modal logic than to do something like:

12. necessarily(p).
13. 2+2=4.
14. p. (By (12), (13) and rule (11))

However, while (10) and (11) are useless considered as rules of inference, as true propositions they can be quite useful, and do in fact occur in philosophical discussion, for instance in providing counterexamples (suppose you say that x depends on y if and only if exists(x) entails exists(y); then if numbers are necessary beings, everything depends on the number 49, which may seem to be absurd). So we need to distinguish between (10) and (11) as truths and (10) and (11) as rules of inference.

This distinction is needed anyway for modus ponens; for if modus ponens is simply the universally quantified truth:

15. For all p and q, if (p is true and it is true that if p, then q), then q is true,

then to apply modus ponens given the truth of P and of if P then Q, you will need to do universal instantiation on (15) to get:

16. If (P is true and it is true that if P, then Q), then Q is true.

And then using the antecedent of (16) as a premise to get to the conclusion Q, one will have to use modus ponens, which lands one in a vicious regress. (This is an argument of Sextus Empiricus against the very idea of rules of logic. But Sextus confuses truths qua truths and rules. Not that I know exactly how to draw the distinction either.)

Some liar paradoxes without truth

Let "@" be the name of the actual world.

1. The proposition expressed by (1) in English is not entailed by the proposition that @ is actual.
2. The proposition expressed by (2) in English is not compossible with the proposition that @ is actual.
3. The proposition expressed by (3) in English is not necessary.
4. The proposition expressed by (4) in English is not known by anybody.
5. The proposition expressed by (5) in English cannot be known by anybody.

That (1) and (2) are paradoxical is obvious. That (3) is paradoxical is easy to see. For if (3) is false, then (3) is necessarily true. If (3) is true, then then it is only contingently true. But the argument that if (3) is false, then (3) is necessarily true works in all worlds. So in no world is (3) false. So (3) cannot be contingently true.

The paradoxicality of (4) is a bit more fun, though I am less sure of it. If (4) is false, then (4) is known by somebody and hence true. So, (4) cannot be false. But now that we have a logically sound argument for (4), we know (4)—or at least we could, and then we can consider the argument in the possible world where we do know it. But if we know (4), then (4) is false.

What about (5)? Well, if (5) can be known by anybody, it can be true and known. But it cannot be both true and known. So, (5) cannot be known by anybody. But this is a good argument for the truth of (5), so even if we don't know (5), somebody can know it on the basis of this argument. But then (5) is false.