Epistemic possibility and the Liar

Here’s a fun Liar paradox involving epistemic possibility. Say that a proposition p is epistemically possible if it is consistent with all you know.

Construct a sentence G such that:

0. G is true if and only if G is not epistemically possible.

E.g., “The proposition expressed by the first sentence in this post found in quotation marks is not epistemically possible.”

Now, you only know truths, and truth is consistent with truth. Thus:

1. If G is true, then it is consistent with everything you know.

But G is true if and only if it is not epistemically possible. So:

2. If G is true, then it is not consistent with everything you know.

Hence:

3. G is not true.

But now that you’ve seen this argument, you surely are in a position to know G not to be true. Suppose you exploit this and indeed come to know G not to be true. But then we have a contradiction. For if you know G not to be true, then G is not epistemically possible, and hence by (0), it must be that G is true.

The last thought

Entertain this thought:

1. There is a unique last thought that anyone ever thinks and it’s not true.

Or, more briefly:

2. The last thought is not true.

There are, of course, possible worlds where there is no last thought, because (temporal) thoughts go on forever, and worlds where there is a tie for last thought, and worlds where the last thought is “I screwed up” and is true. But, plausibly, there are also worlds where there is a unique last thought and it’s not true—say, a world where the last thought is “I see how to defuse the bomb now.”

In other words, (1) seems to be a perfectly fine, albeit depressing, contingent thought.

Is there a world where (1) is the last thought? You might think so. After all, it surely could be the case that someone entertains (1) and then a bomb goes off and annihilates everyone. But supposing that (1) is the last thought in w, then (1) either is or is not true in w. If it is true in w, then it is not, and if it is not true, then it is true. Now that’s a paradoxical last thought!

Over the last week, I’ve been thinking of a paradox about thoughts and worlds, inspired by an argument of Rasmussen and Bailey. I eventually came to realize that the paradox (apparently unlike their argument) seems to be just a version of the Liar Paradox, essentially the one that I gave above.

But we shouldn’t stop thinking just because we have hit upon a Liar. (You don’t want your last thought to be that you hit upon a Liar!) Let’s see what more we can say. First, the version of the Liar in (1) is the Contingent Liar: we only get paradoxicality in worlds where the last thought is (1) or something logically equivalent to (1).

Now, consider that (1) has unproblematic truth value in our world. For in our world, there is no last thought, given eternal life. And even if there were no eternal life, and there was a last thought, likely it would be something that is straightforwardly true or false, without any paradox. Now an unproblematic thought that has truth value has a proposition as a content. Let that proposition be p. Then we can see that neither p nor anything logically equivalent to it can be the content of the last thought in any world.

This is very strange. If you followed my directions, as you read this blog post, you began your reading by entertaining a thought with content p. It surely could have happened that at that exact time, t, no one else thought anything else. But since a thought with content p cannot be the last thought, it seems that some mysterious force would be compelling people to think something after t. Granted, Judaism, Christianity and Islam, there is such a mysterious force, namely God: God has promised eternal life to human beings, and this eternal life is a life that includes thinking. But we could imagine someone thinking a thought with content p at a time when no one else is thinking in a world where God has made no such promises.

So what explains the constraint that neither p nor anything logically equivalent to it can be the content of the last thought in a possible world? After all, we want to maintain some kind of a reasonable rearrangement or mosaic principle and it’s hard to think of one that would let one require that a world where a thought with content p happens at a time t when no one else is thinking, then a thought must occur later. Yet classical logic requires us to say this.

I think what we have to say is this. Take a world w₁ without any relevant divine promises or the like, where after a number of other thoughts, Alice finally thinks a thought with content p at a time when no one else is engaging in any mental activity, and then she permanently dies at t before anyone else can get to thinking anything else. Then at w₁ there will be other thoughts after Alice’s death. Now take a world w₂ that is intrinsically just like w₁ up to and including t, and then there is no thought. I think it’s hard to avoid saying that worlds like w₁ and w₂ are possible. This requires us to say that at w₂, Alice does not think a thought with content p before death, even though w₂ is intrinsically just like w₁ up to and including the time of her death.

What follows is that whether the content of Alice’s thought is p depends on what happens after her (permanent) death. In other words, we have a particularly controversial version of semantic externalism on which facts about the content of mental activity depend on the future, even in cases like p where the proposition does not depend on the identities of any objects or natural kinds other than perhaps ones (is thought a natural kind?) that have already been instantiated. Semantic externalism extends far!

The lastness in (1) and (2) functions to pick out a unique thought in some worlds without regard for its content. There are other ways of doing so:

• the most commonly thought thought

• the least favorite thought of anybody

• the one and only thought that someone accepts with credence π/4.

Each of these leads to a similar argument for a very far-reaching semantic externalism.

An improved paradox about thoughts and worlds

Yesterday, I offered a paradox about possible thoughts and pluralities of worlds. The paradox depends on a kind of recombination principle (premise (2) in the post) about the existence of thoughts, and I realized that the formulation in that post could be objected to if one has a certain combination of views including essentiality of origins and the impossibility of thinking a proposition that involves non-qualitative features (say, names or natural kinds) in a world where these features do not obtain.

So I want to try again, and use two tricks to avoid the above problem. Furthermore, after writing up an initial draft (now deleted), I realized I don’t need pluralities at all, so it’s just a paradox about thoughts and worlds.

The first trick is to restrict ourselves to (purely) qualitative thoughts. Technically, I will do this by supposing a relation Q such that:

1. The relation Q is an equivalence (i.e., reflexive, symmetric, and transitive) on worlds.

We can take this equivalence relation to be qualitative sameness or, if we don’t want to make the qualitative thought move after all, we can take Q to be identity. I don’t know if there are other useful choices.

We then say that a Q-thought is a (possible) thought θ such that for any world there aren’t two worlds w and w′ with Q(w,w′) such that θ is true at one but not the other. If Q is qualitative sameness, then this captures (up to intensional considerations) that θ is qualitative. Furthermore, we say that a Q-plurality is a plurality of worlds ww such that there aren’t two Q-equivalent worlds one of which is in ww and the other isn’t.

The second trick is a way of distinguishing a “special” thought—up to logical equivalence—relative to a world. This is a relation S(w,θ) satisfying these assumptions:

2. If S(w,θ) and S(w,θ′) for Q-thoughts θ and θ′, then the Q-thoughts are logically equivalent.

3. For any Q-thought θ and world w, there is a thought θ′ logically equivalent to θ and a world w such that S(w,θ′).

4. For any Q-thought θ and any Q-related worlds w and w′, if S(w,θ), there is a thought θ′ logically equivalent to θ such that S(w′,θ′).

Assumption (2) says that when a special thought exists at a world, it’s unique up to logical equivalence. Assumption (3) says that every thought is special at some world, up to logical equivalence. In the case where Q is identity, assumption (4) is trivial. In the case where Q is qualitative sameness, assumption (4) says that a thought’s being special is basically (i.e., up to logical equivalence) a qualitative feature.

We get different arguments depending on what specialness is. A candidate for a specialness relation needs to be qualitative. The simplest candidate would be that S(w,θ) iff at w the one and only thought that occurs is θ. But this would be problematic with respect (3), because one might worry that many thoughts are such that they can only occur in worlds where some other thoughts occur.

Here are three better candidates, the first of which I used in my previous post, with the thinkers in all of them implicitly restricted to non-divine thinkers:

1. S(w,θ) iff at w there is a time t at which θ occurs, and no thoughts occur later than t, and any other thought that occurs at t is entailed by θ

2. S(w,θ) iff at w the thought θ is the favorite thought of the greatest number of thinkers up to logical equivalence (i.e., there is a cardinality κ such that for each of κ thinkers θ is the favorite thought up to logical equivalence, and there is no other thought like that)

3. S(w,θ) iff at w the thought θ is the one and only thought that anyone thinks with credence exactly π/4.

On each of these three candidates for the specialness relation S, premises (2)–(4) are quite plausible. And it is likely that if some problem for (2)–(4) is found with a candidate specialness relation, the relation can be tweaked to avoid the relation.

Let L be a first-order language with quantifiers over worlds (Latin letters) and thoughts (Greek letters), and the above predicates Q and S, as well as a T(θ,w) predicate that says that the thought θ is true at w. We now add the following schematic assumption for any formula ϕ = ϕ(w) of L with at most the one free variable w, where we write ϕ(w′) for the formula obtained by replacing free occurrences of w in ϕ with w′:

6. Q-Thought Existence: If ∀w∀w′[Q(w,w′)→(ϕ(w)↔︎ϕ(w′))], there is a thought θ such that ∀w(T(θ,w)↔︎ϕ(w)).

Our argument will only need this for one particular ϕ (dependent on the choice of Q and S), and as a result there is a very simple way to argue for it: just think the thought that a world w such that ϕ(w) is actual. Then the thought will be actual and hence possible. (Entertaining a thought seems to be a way of thinking a thought, no?)

Fact: Premises (1)–(6) are contradictory.

Eeek!!

I am not sure what to deny. I suppose the best candidates for denial are (3) and (6), but both seem pretty plausible for at least some of the above choices of S. Or, maybe, we just need to deny the whole framework of thoughts as entities to be quantified over. Or, maybe, this is just a version of the Liar?

Proof of Fact

Let ϕ(w) say that there is a Q-thought θ such that S(w,θ) and but θ is not true at w.

Note that if this is so, and Q(w,w′), then S(w′,θ′) for some θ′ equivalent to θ by (4). Since θ is a Q-thought it is also not true at w′, and hence θ′ is not true at w′, so we have ϕ(w′).

By Q-Thought Existence (6), there is a Q-thought that is true at all and only the worlds w such that ϕ(w) and by (3) there is a Q-thought ρ logically equivalent to it and a world c such that S(c,ρ). Then ρ is also true at all and only the worlds w such that ϕ(w).

Is ρ true at c?

If yes, then ϕ(c). Hence there is a Q-thought θ such that S(c,θ) but θ is not true at w. Since S(c,ρ), we must have θ and ρ equivalent by (2), so ρ is is not true at c, a contradiction.

If not, then we do not have ϕ(c). Since we have S(c,ρ), in order for ϕ(c) to fail we must have ρ true at c, a contradiction.

Non-formal provability

A simplified version of Goedel’s first incompleteness theorem (it’s really just a special case of Tarski’s indefinability of truth) goes like this:

• Given a sound semidecidable system of proof that is sufficiently rich for arithmetic, there is a true sentence g that is not provable.

Here:

• sound: if s is provable, s is true

• semidecidable: there is an algorithm that given any provable sentence verifies in a finite number of steps that it is provable.

The idea is that we start with a precisely defined ‘formal’ notion of proof that yields semidecidability of provably, and show that this concept of proof is incomplete—there are truths that can’t be proved.

But I am thinking there is another way of thinking about this stuff. Suppose that instead of working with a precisely defined concept of proof, we have something more like a non-formal or intuitive notion of proof, which itself is governed by some plausible axioms—if you can prove this, you can prove that, etc. That’s kind of how intuitionists think, but we don’t need to be intuitionists to find this approach attractive.

Note that I am not explicitly distinguishing axioms.

The idea is going to be this. The predicate P is not formally defined, but it still satisfies some formal constraints or axioms. These can be formulated in a formal language (Brouwer wouldn’t like this) that has a way of talking about strings of symbols and their concatenation and allows one to define a quotation function that given a string of symbols returns a string of symbols that refers to the first string.

One way to do this is to have a symbol ′α′ for any symbol α in the original language which refers to α, and a concatenation operator +, so one can then quote αβγ as ′α′ + ′β′ + ′γ′. I assume the language is rich enough to define a quotation function Q such that Q(x) is the quotation of a string x.

To formulate my axioms, I will employ some sloppy quotation mark shorthand, partly to compensate for the difficulty of dealing with corner quotes on the web. Thus, ′αβγ′ is shorthand for ′α′ + ′β′ + ′γ′, and as needed I will allow substitution inside the quotation marks. If there are nested quotation marks, the inner substitutions are resolved first.

1. For all sentences ϕ and ψ, if P(′ϕ↔︎∼ψ′) and P(′ϕ′), then P(′∼ϕ′).

2. For all sentences ϕ and ψ, if P(′ϕ↔︎∼ψ′) and P(′ψ→ϕ′), then P(′ϕ′).

3. For all sentences ϕ, we have P(′P(′ϕ′)→ϕ′).

4. If ϕ has a formal intuitionistic proof from sufficiently rich axioms of concatenation theory, then P(′ϕ′).

Here, (1) and (2) embody a little bit of facts about proof, both of which facts are intuitionistically and classically acceptable. Assumption (3) is the philosophically heaviest one, but it follows from its being axiom that if ϕ is provable, then ϕ, together with the fact that all axioms count as provable. That a formal intuitionistic proof is sufficient for provability is uncontroversial.

Using similar methods to those used to prove Goedel’s first incompleteness theorem, I think we should now be able to construct a sentence g and the prove, in a formal intuitionistic proof in a sufficiently rich concatenation theory, that:

5. g ↔︎  ∼ P(′g′).

But these facts imply a contradiction. Since 5 can be proved in our formal way, we have:

6. P(′g↔︎∼P(′g′)′). By 4.

7. P(′P(′g′)→g′). By 3.

8. P(′g′). By 6, 7 and 2.

9. P(′∼g′). By 6, 8 and 1.

Hence the system P is inconsistent in the sense that it makes both g and  ∼ g are provable.

This seems to me to be quite a paradox. I gave four very plausible assumptions about a provability property, and got the unacceptable conclusion that the provability property allows contradictions to be proved.

I expect the problem lies with 3: it lets one ‘cross levels’.

The lesson, I think, is that just as truth is itself something where we have to be very careful with the meta- and object-language distinction, the same is true of proof if we have something other than a formal notion.

[Thesis:] April Fool's Philosophy Post Generator

[This post works better if you have Javascript enabled.]

The thesis that [thesis] has not received much of a defense[literature type]. But here is an argument for it:

1. This argument is valid.

2. Therefore, [thesis].

Let's see why this argument is not only valid but sound.

First, let’s see that it’s valid. Suppose for a reductio that it is invalid. But whether an argument is valid or not cannot be a contingent matter. Thus if, the argument is invalid, it is necessarily invalid. But if it is necessarily invalid, then necessarily its first premise is false (since the premise says that the argument is valid). But any argument which has a necessarily false premise is automatically valid. (An argument is valid if and only if it is impossible for the premises to be true and the conclusion false. This is trivially satisfied if it is impossible for a premise to be true.) But that would contradict the assumption that it’s invalid. So, the argument must be valid.

But if the argument (1)–(2) is valid, it’s also automatically sound. For a valid argument is sound provided its premises are true. But the only premise of the argument is (1), the statement that the argument is valid. If the argument is valid, then that premise is true, and so the argument is sound.

But the conclusion of a sound argument is true. Therefore, [thesis].

Being mistaken about what you believe

Consider:

1. I don’t believe (1).

Add that I am opinionated on what I believe:

2. For each proposition p, I either believe that I believe p or believe that I do not believe p.

Finally, add:

3. My beliefs are closed under entailment.

Now I either believe (1) or not. If I do not believe (1), then I don’t believe that I don’t believe (1), by closure. But thus, by (2), I do believe that I do believe (1). Hence in this case:

4. I am mistaken about what I do or do not believe.

Now suppose I do believe (1). Then I believe that I don’t believe (1), by closure and by what (1) says. So, (4) is still true.

Thus, we have an argument that if I am opinionated on what I believe and my beliefs are closed under entailment, then I am mistaken as to what I believe.

(Again, we need some way of getting God out of this paradox. Maybe the fact that God’s knowledge is non-discursive helps.)

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.

Logical closure accounts of necessity

A family of views of necessity (e.g., Peacocke, Sider, Swinburne, and maybe Chalmers) identifies a family F of special true statements that get counted as necessary—say, statements giving the facts about the constitution of natural kinds, the axioms of mathematics, etc.—and then says that a statement is necessary if and only if it can be proved from F. Call these “logical closure accounts of necessity”. There are two importantly different variants: on one “F” is a definite description of the family and on the other “F” is a name for the family.

Here is a problem. Consider:

1. Statement (1) cannot be proved from F.

If you are worried about the explicit self-reference in (1), I should be able to get rid of it by a technique similar to the diagonal lemma in Goedel’s incompleteness theorem. Now, either (1) is true or it’s false. If it’s false, then it can be proved from F. Since F is a family of truths, it follows that a falsehood can be proved from truths, and that would be the end of the world. So it’s true. Thus it cannot be proved from F. But if it cannot be proved from F, then it is contingently true.

Thus (1) is true but there is a possible world w where (1) is false. In that world, (1) can be proved from F, and hence in that world (1) is necessary. Hence, (1) is false but possibly necessary, in violation of the Brouwer Axiom of modal logic (and hence of S5). Thus:

2. Logical closure accounts of necessity require the denial of the Brouwer Axiom and S5.

But things get even worse for logical closure accounts. For an account of necessity had better itself not be a contingent truth. Thus, a logical closure account of necessity if true in the actual world will also be true in w. Now in w run the earlier argument showing that (1) is true. Thus, (1) is true in w. But (1) was false in w. Contradiction! So:

3. Logical closure accounts of necessity can at best be contingently true.

Objection: This is basically the Liar Paradox.

Response: This is indeed my main worry about the argument. I am hoping, however, that it is more like Goedel’s Incompleteness Theorems than like the Liar Paradox.

Here's how I think the hope can be satisfied. The Liar Paradox and its relatives arise from unbounded application of semantic predicates like “is (not) true”. By “unbounded”, I mean that one is free to apply the semantic predicates to any sentence one wishes. Now, if F is a name for a family of statements, then it seems that (1) (or its definite description variant akin to that produced by the diagonal lemma) has no semantic vocabulary in it at all. If F is a description of a family of statements, there might be some semantic predicates there. For instance, it could be that F is explicitly said to include “all true mathematical claims” (Chalmers will do that). But then it seems that the semantic predicates are bounded—they need only be applied in the special kinds of cases that come up within F. It is a central feature of logical closure accounts of necessity that the statements in F be a limited class of statements.

Well, not quite. There is still a possible hitch. It may be that there is semantic vocabulary built into “proved”. Perhaps there are rules of proof that involve semantic vocabulary, such as Tarski’s T-schema, and perhaps these rules involve unbounded application of a semantic predicate. But if so, then the notion of “proof” involved in the account is a pretty problematic one and liable to license Liar Paradoxes.

One might also worry that my argument that (1) is true explicitly used semantic vocabulary. Yes: but that argument is in the metalanguage.

First-orderism

It’s notoriously hard to characterize the physical precisely enough to attack or defend physicalism. To attack physicalism, however, it is enough to attack a characterization broader than physicalism, and to defend physicalism, a narrower characterization will do.

Here’s a suggestion along the “broader” lines. We can characterize reductive physicalism over-broadly as:

• Reductive first-orderism: All facts about the concrete (variants: contingent, spatiotemporal) features of the world reduce to first-order facts.

This may take in some theories other than what one intuitively counts as reductive physicalism, but if the object is ciriticism, that’s all we need.

Note how this characterization nicely shows how paradigm examples of magic would violate reductive physicalism: for paradigm examples of magic involve causation irreducibly by virtue of the meaning of a spell, gesture, etc., and meaning is a higher-order property. It also shows why irreducible Aristotelian teleology has no place in a reductive physicalist story: for teleological properties are second-order (I think).

Moreover, if we see reductive physicalism in the above way, it’s also easy to see that it’s false, by an argument of Leon Porter. For any first-order fact can be expressed in a first-order language. But, famously, the property of truth cannot be reduced to properties expressible in a first-order language (Tarski’s indefinability of truth; or, more simply, note that if you could express something equivalent to the property of truth in a first-order language, you could express the Liar Sentence in a first-order language). And some concrete objects, namely inscriptions, have this property of truth. Hence, some concrete objects have a property that cannot be expressed in first-order terms, contrary to reductive first-orderism.

Some paradoxes of reference

Liar-like:

• one plus the biggest integer that can be expressed in English in fewer than fifty words
• one; two; three; one plus the biggest integer mentioned in this list
• one; two; three; one plus the last integer mentioned in this list
• one plus the last integer mentioned in this list; two; three; one plus the first integer mentioned in this list
• one plus this integer

Truthteller-like:

• one; two; three; the biggest integer mentioned in this list
• one; two; three; the last integer mentioned in this list
• the last integer mentioned in this list; two; three; the first integer mentioned in this list
• this integer
• the square of this integer

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?

Quick puzzle

You will be paid oodles of money if and only if your next decision is irrational. What should you do?

(There are standard cases in the literature where you lose a lot unless you become irrational, say by taking an irrationality potion. The case at hand differs significantly from those, because it concerns the next decision. You can't just rationally decide to take an irrationality potion, because doing so would constitute your next decision and that would be rational.)

More against neo-conventionalism about necessity

Assume the background here. So, there is a privileged set N of true sentences from some language L, and N includes, among other things, all mathematical truths. There is also a provability-closure operator C on sets of L-sentences. And, according to our neo-conventionalist, a sentence p of L is necessarily true just in case p∈C(N).

Moreover, this is supposed to be an account of necessity. Thus, N cannot contain sentences with necessity operators and C must have the property that applying C to a set of sentences without necessity operators does not yield any sentence of the form Lp, where L is the necessity operator (It may be OK to yield tautologies like "Lp or ~Lp" or conjunctions of tautologies like that with sentences in the input set, etc.) If these conditions are not met, then we have an account of necessity that presupposes a prior understanding of necessity.

Now consider an objection. Then not only is L(1=1) true, but it is necessarily true. But now we have a problem. For C(N) by the conditions in the previous paragraph contains no Lp sentences. Hence it doesn't contain the sentence "L(1=1)".

But this was far too quick. For the neo-conventionalist can say that "L(1=1)" is short for something like "'1=1'∈C(N)". And the constraints on absence of necessity operators is compatible with the sentence "'1=1'∈C(N)" itself being a member of C(N).

This means that the language L must contain a name for N, say "N", or some more complex rigidly designating term for it (say a term expressing the union of some sets). Let's suppose that "N" is in L, then. Now, sentences are mathematical objects—finite sequences of symbols in some alphabet. (Or at least that seems the best way to model them for formal purposes.) We can then show (cf. this) that there is a mathematically definable predicate D such that D(y) holds if and only if y is the following sentence:

• "For all x, if D(x), then ~(x∈N)."

But if y is this sentence, then y is a mathematical claim. If this mathematical claim isn't true, then y is a member of N. But then y is true. On the other hand, if y is true, then being a mathematical claim it is a member of N, and hence y is false. (This is, of course, structurally like the Liar. But it is legitimate to deploy a version of the Liar against a formal theory whose assumptions enable that deployment. That's what Goedel's incompleteness theorems do.)

To recap. We have an initial difficulty with neo-conventionalism in that no sentences with a necessity operator ends up necessary. That difficulty can be overcome by replacing sentences with a necessity operator with their neo-conventionalist analyses. But doing that gets us into contradiction.

(It's perhaps formally a bit nicer to formulate the above in terms of Goedel numbers. Then we replace Lp with n∈C*(N*) where n is the Goedel number of p, and C* and N* are the Goedel-number analogues of C and N. Diagonalization then yields a contradiction.)

One place where I imagine pushback is my assumption that C doesn't generate Lp sentences. One might think that C embodies the rule of necessitation, and hence in particular it yields Lp for any theorem p. But I think necessitation presupposes necessity, and so it is illegitimate to use rules that include necessitation to definite necessity. However, this is a part of the argument that I am not deeply confident of.

A practical liar paradox in two words

I saw a woman with a tattoo that said only: CAVEAT LECTOR.

A surprise exam

Intermediate Logic

Instructor: Alexander R. Pruss

Date: An unexpected day between February 17 and February 21, 2014, inclusive.

Name: _________________________

Closed book.  Answer all questions.  Be careful not to follow any of these directions.

Section A. Multiple choice. Circle exactly one option for each of these questions.

1. This sentence is not true.

(a) false

(b) true

(c) neither true nor false

2. Which of the following is not the answer you are circling?

(a) Elephant

(b) Frog

3. The sentence displayed at Question 4 is true.

(a) false

(b) true

4. The sentence displayed at Question 3 is not true.

(a) false

(b) true

5. Which of the following is the capital of China?

Section B. Short essay. Write a one page essay.

6. Give a valid deductive argument that dialetheism is incorrect without using premises or rules of inference that can together yield the law of noncontradiction.

Section C. Reflection. No writing necessary.

7. Reflect on the implications of this sentence for your grade: "If this sentence is true, you failed."

Liar and truthteller questions

Here are some fun questions:

1. Is the answer to this question negative?
2. Is the answer to this question positive?
3. What is the answer to this question?

The last one is due to my six-year-old.

Hypochondria

Let's say that when you feel that you have a disease but you don't have it, you have the disease of hypochondria. But what if you feel that you have hypochondria and you don't feel that you have any other disease? :-)

Propositions and the liar

This is an attempt to provide a metaphysical backing to hierarchical theories of truth like those of Tarski, thereby skirting the liar paradox. The details of the construction of propositions can be done in more than one way—the following is more an example of the structure of a theory than a theory.

Step 1: Assume an abundant theory of Armstrong-like first-order states of affairs, namely states of affairs that exist if and only if they obtain. (Intuitively, we will need enough states of affairs so that each first-order true proposition represents a state of affairs.) Now, say that the first-order propositions are ordered pairs (s,v) where s is a first-order state of affairs and v is 0 or 1. We can then define truth for first-order propositions very simply. For any first-order proposition (s,v), the proposition is true if and only if v=1. We can also define truth at a world w as follows: (s,v) is true at w if and only if either s obtains at w and v=1 or s does not obtain at w and v=0.

On the view I am defending, ordered pairs are mere abstractions—they are not first-class members of our ontology. That is a nominalist or perhaps Aristotelian moment in the story. Ordered pairs are not in the domain of ordinary objectual quantification. We need another kind of quantifier, ∃₁x, to handle quantification over first-order propositions. This is like the quantifiers in this post. (Actually, I think states of affairs aren't first-class members of our ontology either. They, too, will be some form of logical construction. This is part of why I am not happy with the details.)

Now, there is one technical issue here. Since what states of affairs exist differs between worlds, likewise what propositions there are differs between worlds. The solution here is to adopt a counterpart theory for first-order propositions. Given worlds w₁ and w₂, the proposition (s,v) has a counterpart in w₂. If s obtains at w₂, then the counterpart is just (s,v). If s does not obtain at w₂, then the counterpart of (s,v) is (~s,1−v), where ~s is the negation of s—a state of affairs that obtains if and only if s does not.

Step 2: We've defined a quantifier ∃₁x over first-order propositions and a truth predicate for them. We do not say that first-order propositions exist simpliciter. Facts about first-order propositions and their truth supervene on first-order facts about the world and are grounded in them. Now, we repeat the process. Define, abstractly, the notion of second-order states of affairs—these are states of affairs that are partly about first-order propositions. In so doing, we're introducing a new quantifier over the second-order states of affairs. And then define a second-order proposition as, again, a part (s,v) where s is a second-order state of affairs and v is 0 or 1. Do everything else as before, defining a quantifier ∃₂x over second-order propositions.

Steps 3 and so on: Repeat.

Imagine the process completed. What have we done? It is tempting to say:

1. We have defined nth order propositions for all n, and a truth predicate for each level.

And then it is tempting to ask:

2. But what, then, about truths about propositions of any order whatsoever, and a truth predicate for them?

But the temptation should be resisted. We should not say (1), since (1) quantifies in a way not allowed by the theory. We have the quantifiers ∃₁x, ∃₂x, .... We do not have a quantifier over these quantifiers. (Can't we do ∃n∃_nx? No: the subscript in "∃₁x" is just a marker that cannot be replaced by a variable.)

Likewise, there isn't a single truth predicate over all the levels. Rather, what we defined are analogical truth predicates. That's another Aristotelian moment in the theory.

For convenience, we can define cumulative quantifiers and cumulative truth predicates. Thus, we can define ∃₃*xFx as the disjunction: ∃₁xFx or ∃₂xFx or ∃₃xFx.

The liar disappears. Never in the process do we get to define a paradoxical sentence. The sentence of the form "The proposition expressed by this sentence is not true" needs a level of quantification. But when we fix the level of quantification, we get something like:

3. ∀_n*x (if this sentence expresses x, then x is not true)

But there is no nth or lower order proposition expressed by (3), and so (3) is trivially true.

We then need a thesis about the level of quantifiers in ordinary language. Perhaps the thesis is that we charitably raise the level of quantifiers in ordinary language sentences until we get something meaningful and relevant. Or perhaps we have quantifier-indexicality: which quantifier level is used in a sentence depends on the length of the chain of truth-nesting that goes on in the grounding of this sentence.

But what about the usual sorts of tricks for breaking out hierarchies, like taking an infinite set of sentences with different levels of quantifiers and then asserting that all the sentences in the set are true? We don't get to do that. For corresponding to the levels of quantifiers and propositions, there are levels of expression relations between sentences a propositions. When we try to say that all the sentences in a set are true, we mean something like: "the proposition expressed by each sentence in the set is true." But there is no univocal sense of "the proposition expressed" that holds for all the sentences.

But what if a clever person then goes on and defines infinite order propositions, with a new quantifier over them, just as I did it for all the finite orders. That game can, indeed, go on. But it does not generate a paradox. The paradox always appears to loom one step away, but when we get there, it's further off--it's like "tomorrow is always a day away".

Reflections on Horwich's minimalism

Horwich's minimalism is a theory of truth generated by the axiom:

1. If p is true, then p is a proposition

and the axiom schema obtained by taking all sentences s of extensions of English and substituting them into:

2. <s> is true if and only if s

(with this understood in the same extension of English as s was), with the exception of those sentences that lead to paradox (e.g., "This sentence is false"). Here, <s> denotes the proposition that s.

Here are some issues. None of them are fatal. But they all mean that minimalism isn't quite as simple as it initially seems.

Issue 1: What is an extension of English? We need to include sentences of extensions of English because no doubt there are propositions that no sentence of English can express. Now, an extension of English had better not change the meanings of "is true" or "if and only if"—for if that is allowed to change, then some instances of (2) will become false. Presumably, then, what makes L an extension of English is that for any linguistic element e of English, e is also a linguistic element of L, and it has the same meaning (semantic value, etc.) in L as it does in English. Thus, Horwich's minimalism in its description of the axiom schema presupposes the concept of meaning (semantic value, etc.). To avoid circularity, the concept of meaning had better not depend on that of truth.

Issue 2: Nitpicky stuff. Strictly speaking, (2) generates bad orthography. Suppose s is "Snow is white." Then we are told that <Snow is white.> is true if and only if Snow is white. But the last "Snow" should not be capitalized. This can be easily handled—we specify that when substituting s in, we adjust the first letter's case as needed. There is also that odd looking period after the first "white"; again, we can specify that it is to be omitted. A slightly less easy case is where s is "This sentence is short." Suppose s is true. But now consider:

3. <This sentence is short> is true if and only if this sentence is short.

But the second occurrence of "this sentence" refers not to s but to (3), and that sentence is not short, so (3) is false. Presumably, we handle this by not allowing sentences with indexicals or demonstratives. This requires the substantive assumption that any proposition that can be expressed with indexicals/demonstratives can be expressed without them. Next, let s be "u if and only if v" (for some u and v). Then (2) yields:

4. <u if and only if v> if and only if u if and only if v.

But this is wrong or badly ambiguous. Maybe then we're supposed to use an extension of English that has grouping parentheses, and then replace (2) with:

5. <s> is true if and only if (s).

Issue 3: Contingent liar. Axioms normally are supposed to not vary between worlds. But there are contingent liar sentences, like "The sentence on Alex's board is false", which is paradoxical when it is the unique sentence on Alex's board but need not be paradoxical when written on Jon's board (unless we have something like "The sentence on Jon's board is false" as the only sentence on Alex's board). This means that dropping those instances of (2) (or of (5)) that generates the paradox requires dropping different instances in different worlds, thereby making the axioms of truth differ from world to world.

There are two ways for axioms to differ between worlds. In the weak sense, whether p is an axiom varies between worlds, but p is true at all worlds. In the strong sense, the truth value varies between worlds, too. This is the kind of variation that we'd need to get out of the contingent liar. And this just doesn't fit with what we understand by "axiom", I think.

Syntactic self-reference without diagonal lemma or Gödel numbers

For the proof of Goedel's incompleteness theorem and in work on the Liar Paradox it is usual to use the Diagonal Lemma to secure self-reference. The challenge of self-reference is this. Given a predicate Q, find a syntactically definable predicate P such that

1. (s)(P(s) → R(s))

is provably the one and only sequence of symbols satisfying P. Then (1) says that Q holds of itself. (To get the (Strengthened) Liar Paradox, just make R(s) say that s is not true.) But the proof of the diagonal lemma is hard to understand.

I find the following way of securing self-reference easier to understand. Start with a language that has nestable quotation marks, which I'll represent with ‘...’, and some string manipulation tools. I'll use straight double quotation marks for meta-language quotation. Add to the language a new symbol "@" which is ungrammatical (i.e., no well-formed formula may contain it). For any sequence of symbols s, we define two new sequences of symbols N(s) and Q(s) by the following rules. If s contains no quoted expressions or contains imbalanced opening and closing quotation marks, N(s) and Q(s) are just "@". If s contains a quoted expression, Q(s) is the first quoted expression, without its outermost quotation marks (but with any nested quotations being included), and N(s) is the result of taking s and replacing that first quoted occurrence of Q(s), as well as its surrounding single quotation marks, with "@". Thus:

2. Q("abc‘def‘ghi’’+jkl")="def‘ghi’"
3. N("abc‘def‘ghi’’+jkl")="abc@+jkl".

It is easy to see that Q and N are syntactically defined. Now, let M(s) be equal to N(s) if N(s)=Q(s) and let M(s) be an empty sequence "" otherwise. Again, M(s) is syntactic. Now consider this sentence:

4. (s)(‘(s)(@=M(s) → R(s))’=M(s) → R(s)).

It is easy to prove (given a bit of string manipulation resources) that the only sequence s that satisfies the antecedent of the conditional is (4) itself. So we have constructed the syntactic predicate P(s). It is: ‘(s)(@=M(s) → R(s))’=M(s).

One can also adapt this to work with Goedel numbers and hence presumably for use in proving incompleteness.

[Removed a nasty typo.]