Inferentialism and the fictitious isolated hydrogen atom

This is another attempt at an argument against inferentialism about logical constants.

Given a world w, let w^* be a world just like w except that it has added to it an extra spatiotemporally disconnected island universe containing exactly one hydrogen atom with a precisely specified wavefunction ψ₀. Suppose that in in the actual world there is no such isolated hydrogen atom. Now, given a nice first-order language L describing our world, let L^* be a language whose constants are the same as the constants of L with an asterisk added to every logical constant, name and predicate. Given a sentence ϕ of L, let ϕ^* be the corresponding sentence of L^*—i.e., the sentence with all of L’s logical constants asterisked.

Let the rules of inference of L^* be the same as those of L with asterisks added as needed.

Let the semantics of L^* be as follows:

• Every predicate P^* in L^* means the same thing as P in L.

• Every name a^* in L^* means the same thing as a in L.

• Any sentence ϕ^* in L^* without quantifiers means the same thing as ϕ in L.

• But if ϕ^* has a quantifier, then ϕ^* means that ϕ would be true if there were an extra spatiotemporally disconnected island universe containing exactly one hydrogen atom with wavefunction ψ₀.

Thus, ϕ^* is true in world w if and only if ϕ is true in w^*.

Observe that because L contains only names for things that exist in the actual world, and hence not for the extra hydrogen atom or its components, an atomic sentence P(a₁,...,a_n) in L is true if and only if the corresponding sentence P^*(a₁^*,...,a_n^*) is true in L^*.

Logical inferentialism tells us that the logical constants of L^* mean the same thing as those of L, modulo asterisks. After all, modulo asterisks, we have the same inferences, the same meanings of names, and the same meanings of predicates. But this is false: for if ∃^* in L^* were an existential quantifier, then it would be true that there exists an isolated hydrogen atom with wavefunction ψ₀. But there is none such.

Identity and quantification

One way of posing the question of diachronic identity is to ask for an explanation of facts like

1. The xs compose the very same object at t₁ as the ys compose at t₂

where we do not use sameness or identity or similar concepts in the explanation.

This task turns out to be quite easy. The following is logically equivalent to (1) and does not use sameness or identity:

2. There is an object z such that the xs compose z at t₁ and the ys compose z at t₂.

This is a variant of the point made here.

Ways of being and quantifying

Pluralists about ways of being say that there are multiple ways to be (e.g., substance and accident, divine being and finite being, the ten categories, or maybe even some indefinitely extendible list) and there is no such thing as being apart from being according to one of the ways of being. Each way of being comes with its own quantifiers, and there is no overarching quantifier.

A part of the theory is that everything that exists exists in a way of being. But it seems we cannot state this in the theory, because the "everything" seems to be a quantifier transcending the quantifiers over the particular ways of being. (Merricks, for instance, makes this criticism.)

I think there is a simple solution. The pluralist can concede that there are overarching unrestricted quantifiers ∀ and ∃, but they are not fundamental. They are, instead, defined in terms of more fundamental way-of-being-restricted quantifiers in the system:

1. ∀xF(x) if and only if ∀_BWoBb∀_bxF(x)

2. ∃xF(x) if and only if ∃_BWoBb∃_bxF(x).

The idea here is that for each way of being b, there are ∀_b and ∃_b quantifiers. But, the pluralist can say, one of the ways of being is being a way of being (BWoB). So, to use Merricks’ example, to say that there are no unicorns at all, one can just say that no way of being b is such that a unicorn b-exists.

Note that being a way of a being is itself a way of being, and hence BWoB itself BWoB-exists.

The claim that everything that exists exists in a way of being can now be put as follows:

3. ∀x(x = x → ∃_BWoBb∃_by(x = y)).

Of course, (3) will be a theorem of the appropriate ways-of-being logic if we expand out "∀x" in accordance with (1). So (3) may seem trivial. But the objection of triviality seems exactly parallel to worrying that it is trivial on the JTB+ account of knowledge that if you know something, you believe it. Whether we have triviality depends on whether the account of generic existence or knowledge, respectively, is stipulative or meant to be a genuine account of a pre-theoretic notion. And nothing constrains the pluralist to making (1) and (2) be merely stipulative.

Suppose, however, your motivations for pluralism are theological: you don’t want to say that God and humans exist in the same way. You might then have the following further theological thought: Let G be a fundamental way of being that God is in. Then by transcendence, G has to be a category that is special to God, having only God in it. Moreover, by simplicity, G has to be God. Thus, the only way of being that God can be in is God. But this means there cannot be a fundamental category of ways of being that includes divine and non-divine ways of being.

However, note that even apart from theological considerations, the BWoB-quantifiers need not be fundamental. For instance, perhaps, among the ways of being there might be being an abstract object, and one could hold that ways of being are abstract objects. If so, then ∀_BWoBbG(b) could be defined as ∀_BAb(WoB(b)→G(b)), where BA is being abstract and WoB(x) says that x is a way of being.

Coming back to the theological considerations, one could suppose there is a fundamental category of being a finite way of being (BFWoB) and a fundamental category of being a divine way of being (BDWoB). By simplicity, BDWoB=God. And then we could define:

4. ∀_BWoBbF(b) if and only if ∀_BDWoBbF(b) and ∀_BFWoBbF(b).

5. ∃_BWoBbF(b) if and only if ∃_BDWoBbF(b) or ∃_BFWoBbF(b).

Note that we can rewrite ∀_BDWoBbF(b) and ∃_BDWoBbF(b) as just F(God).

World shuffling and quantifiers

Let ψ be a non-trivial one-to-one map from all worlds to all worlds. (By non-trivial, I mean that there is a w such that ψ(w)≠w.) We now have an alternate interpretation of all sentences. Namely, if I is our “standard” method of interpreting sentences of our favorite language, we have a reinterpretation I_ψ where a sentence s reinterpreted under I_ψ is true at a world w if and only if s interpreted under I is true at ψ(w). Basically, under I_ψ, s says that s correctly describes ψ(actual world).

Under the reinterpretation I_ψ all logical relations between sentences are preserved. So, we have here a familiar Putnam-style argument that the logical relations between sentences do not determine the meanings of the sentences. And if we suppose that ψ leaves fixed the actual world, as we surely can, the argument also shows that truth plus the logical relations between sentences do not determine meanings. Moreover, can suppose that ψ is a probability preserving map. If so, then all probabilistic relations between sentences will be preserved, and hence the meanings of sentences are not determined by truth and the probabilistic and logical relations between sentences. This is all familiar ground.

But here is the application that I want. Apply the above to English with its intended interpretation. This results in a language English^* that is syntactically and logically just like English but where the intended interpretation is weird. The homophones of the English existential and universal quantifiers in English^* behave logically in the same way, but they are not in fact the familiar quantifiers. Hence quantifiers are not defined by their logical relations. I’ve been looking for a simple argument to show this, and this is about as simple as can be.

Existential quantifiers aren't defined by their logic

Start with a first order language L describing the concrete objects of our world and expand L to a language L^* by adding a new name “obump”. Given an interpretation I of L in a model M, create a model M^* such that:

• The domain of M^* is the domain of M with one more object, o, in its domain.

• The non-unary relations of M^* are the same as those of M, except that I(=) is replaced by a new relation J = I(=)∪{(o, o)}.

• The unary relations of M^* are all and only the relations R^* for R a unary relation of M, where R^* = R ∪ {o} if R is equal to I(F) either for a physical predicate F of L such that I(trump)∈I(F) or for a mental predicate F of L such that I(obama)∈I(F) and R^* = R otherwise.

Define the interpretation I^* of L^* in M^* as follows: I^*(a)=I(a) for any name other than obump, I^*(obump)=o, I^*(F)=(I(F))^* for a unary F, and I^*(F)=I(F) for any non-unary F.

We can now give a semantics for L^*: If I_w is the intended interpretation of L in a world w, then the intended interpretation of L^* in w is given by I_w^*. We can define validity for L in an analogous way.

The symbols ∃ and ∀ of L^* have the same logic as the same symbols of L. But the ∃ of L^* is not really an existential quantifier. For if it were, then it would be true that there exists an entity that has all the mental properties of Obama and all the physical properties of Trump, which is false. Thus, logic is not sufficient to make a symbol be an existential quantifier.

Presentism and theoretical simplicity

It's oft stated that Ockham's razor favors the B-theory over the A-theory, other things being equal. But the theoretical gain here is small: the A-theorist need only add one more thing to her ideology over what the A-theorist has, namely an absolute "now", and it wouldn't be hard to offset this loss of parsimony by explanatory gains. But I want to argue that the gain in theoretical simplicity by adopting B-theoretic eternalism over presentism is much, much larger than that. In fact, it could be one of the larger gains in theoretical simplicity in human history.

Why? Well, when we consider the simplicity of a proposed law of nature, we need to look at the law as formulated in joint-carving terms. Any law can be formulated very simply if we allow gerrymandered predicates. (Think of "grue" and "bleen".) Now, if presentism is true, then a transtemporally universally quantified statement like:

1. All electrons (ever) are negatively charged

should be seen as a conjunction of three statements:

2. All electrons have always been negatively charged, all electrons are negatively charged and all electrons will always be positively charged.

But every fundamental law of nature is transtemporally universally quantified, and even many non-fundamental laws, like the laws of chemistry and astronomy, are transtemporally universally quantified. The fundamental laws of nature, and many of the non-fundamental ones as well, look much simpler on B-theoretic eternalism. This escapes us, because we have compact formulations like (1). But if presentism is true, such compact formulations are mere shorthand for the complex formulations, and having convenient shorthand does not escape a charge of theoretical complexity.

In fact, the above story seems to give us an account of how it is that we have scientifically discovered that eternalist B-theory is true. It's not relativity theory, as some think. Rather it is that we have discovered that there are transtemporally quantified fundamental laws of nature, which are insensitive to the distinction between past, present and future and hence capable of a great theoretical simplification on the hypothesis that eternalist B-theory is true. It is the opposite of what happened with jade, where we discovered that in fact we achieve simplification by splitting jade into two natural kinds, jadeite and nephrite.

Technical notes: My paraphrase (2) fits best with something like Prior's temporal logic. A competitor to this are ersatz times, as in Crisp's theory. Ersatz time theories allow a paraphrase of (1) that seems very eternalist:

3. For all times t, at t every electron is negatively charged.

However, first, the machinery of ersatz times is complex and so while (3) looks relatively simple (it just has one extra quantifier beyond (1)), if we expand out what "times" means for the ersatzist, it becomes very complex. Moreover on standard ersatzist views, the laws of nature become disjunctive in form, and that is quite objectionable. For a standard approach is to take abstract times to be maximal consistent tensed propositions, and then to distinguish actual times as times that were, are or will be true.

Deep Thoughts XXXVII

Everybody is somebody.

Particularizers instead of haecceities

A haecceity of x is a property that, necessarily, x and only x has. For instance, it might be the property of being identical with x. If a particularly strong converse to the essentiality of origins holds, a good choice for a haecceity would be a complete history of the coming-into-existence of x. Haecceities are a useful tool. For instance, they let one replace de re modality with de dicto. For another, they help explain what God deliberates about when he deliberates which individuals to create.

There is a different tool that can do some of the same work: a particularizer. We can think of an x-particularizer as equivalent to the second order property of being instantiated by x. Thus, if A is an x-particularizer, then necessarily a property Q has A if and only if x has Q. I will occasionally read "Q has A" as "A particularizes Q". The main trick to using particularizers is to note that, necessarily, x exists if and only if x instantiates some property. Thus, if A is an x-particularizer, then, necessarily, x exists if and only if some property has A.

Suppose that any two distinct things differ in some property and that particularizers exist necessarily.

Then we can use particularizers for de re modals. Suppose A is an x-particularizer. Then, Q is an essential property of x if and only if necessarily: if any property has A, then Q has A. If we have an abundant account of properties, we can then account for more complex modals. And we can likewise account for God's creative deliberation about individuals: God deliberates about which particularizers should be instantiated.

A particularly neat thing about particularizers is that with some generalization they allow us to reduce quantification over particulars to quantification over properties. We need the primitive predicate P where P(A) if and only if A is a particularizer. If A is a property, I will use A(y) to abbreviate: y has A. Use E(A) to abbreviate ∃B(A(B)). If A is a particularizer of x, then E(A) holds if and only if x exists. Use A~B to abbreviate ∀C(A(C) iff B(C)). If A and B are particularizers, then A~B means that they are co-particularizers—i.e., there is an x such that they are both x-particularizers. Suppose now we want to say that there are exactly two dogs. Let D be the property of being a dog. We say:

• ∃A∃B(P(A)&P(B)&A(D)&B(D)&~(A~B)&∀C((P(C)&C(D))→(A~C or B~C)).

I.e., there are particularizers that (a) particularize doghood, (b) are not co-particularizers, and (c) any particularizer that particularizes doghood is a co-particularizer of one of them.

If we want to deal with relations, and not just unary properties, then we need to generalize the notion of particularizers. One way to do this would to be suppose a primitive "multiplication" operation that forms an n-ary particularizer A₁A₂...A_n out of a sequence A₁,A₂,...,A_n, where an n-ary relation B has A₁A₂...A_n if and only if x₁,x₂,...,x_n stand in B, where A_i is an x_i-particularizer.

Instead of names of particulars, we will then work with names of their particularizers. Note that if in a Fregean way we think of quantifiers as corresponding to second-order properties, then particularizers will correspond to quantifiers (and remember the Montague way of thinking of names as quantifiers—this all fits neatly together).

Abundant Platonists who think that for every predicate there is a corresponding property should not balk at the existence of particularizers. We can define a particularizer either in terms of an entity x, as the property of being instantiated by x, or in terms of a haecceity H, as the property of having an instantiator in common with H. Likewise, we can define a haecceity in terms of a particularizer. If A is a particularizer, then the property of having all the properties that are particularized by A will make a fine haecceity. Or we can take particularizers to be primitive, whether we have abundant or sparse Platonism.

The above shows that we could do without first-order quantification and without talking of particulars. Now I think that nobody should simplify their ontology by getting rid of objects. Yet the above shows that we can do so. How to resist this simplifying reduction? I think the best way is to say that it does not sit well with the fundamentality of claims such as "I exist" and "I am conscious." For on the above reduction, these claims end up being reducible to E(A) and A(consciousness), where A is a me-particularizer. But only someone with an ontology on which "I exist" or "I am conscious" can resist the reduction in this way.

Fun with substitutional quantification

Stipulate that "x strongly believes p" iff x believes p and it is not the case that x believes not-p. Consider the argument:

1. For anything that Freddie believes, there is a possible world where Sally strongly believes it.
2. Freddie believes the negation of Sally's deepest held belief.
3. Therefore, there is a possible world where Sally strongly believes the negation of Sally's deepest held belief.

Isn't it fun to derive an impossibility from two propositions whose conjunction is possible?

We learn from this that if we are to read (1) substitutionally, we need a substitutional quantification in which we are only allowed to substitute names. In that case, (3) does not follow from (1) and (2), because if "Xyzzy" is the name of the negation of Sally's deepest held belief, then instead of (3) all we get to conclude is:

4. There is a possible world where Sally strongly believes Xyzzy.

But there is no contradiction here, because in the relevant possible world, Xyzzy isn't the negation of Sally's deepest held belief. But still, wasn't (1)-(3) fun?

Is quantification substitutional?

Consider:

1. Possibly, something exists which could not be referred to with a linguistic expression.

If quantification is at base substitutional, then (1) is false. But the negation of (1) is:

2. Necessarily, everything can be referred to with a linguistic expression.

Call this "referential universalism". Now, there presumably are worlds where there is no language. Could there be entities that could exist only in such worlds? If so, then, most likely (2) would be false (some such individuals could be referred to in a cross-worldly way by appropriate definite descriptions, but there is little reason to think they all could be). So, referential universalism is not particularly plausible.

The substitutionist could affirm that referential universalism is a trivial truth. In English, some names, like "Alex", ambiguously refer to multiple entities, and are disambiguated contextually. Presumably, there is an extension English* of English which has the name "Ting" that is much more ambiguous—it can refer to anything at all. Thus, "George loves Sally" is appropriately translated by "Ting loves Sally" as well as by "George loves Ting", in different contexts. But then (2) is a trivial truth—"Ting" can refer to anything at all.

It is a fine question how to allow for ambiguous reference and remain a substitutionist. One way is not open: take substituents to be pairs consisting of an ambiguous referring term and a referent. For if one did that, one is doing objectual quantification over referents. So, probably, what one needs to do is to substitutionally quantify over pairs <e,c> where e is an ambiguously referring expressing and c is a description of a context (it can't just be a context as then we'd be objectually quantifying over contexts). But then our substitutionist becomes committed to the highly non-trivial truth:

3. Necessarily, everything is such that in some context there is a linguistic expression that unambiguously refers to it.

Now, maybe it will be said that I haven't offered an argument against (2) or (3). True. But I now make this move. Look: (2) and (3) are trivially true when read substitutionally. Our understanding of (2) and (3) as non-trivial truths shows that we do not, in fact, read their quantifiers substitutionally, and hence substitutionism is false.

None of this affects the claim that there is a perfectly good substitutional quantifier--only the claim that all quantification is to be understood in terms of it.

Liar paradox with only quantification

The following remark is inspired by Williamson's "Everything" piece. Here is a liar paradox that uses no direct reference (as in "This sentence is false"), and indeed where the only funny business going on in it is a quantification over all sentences:

No actually tokened written sentence is true if it both ends with a decimal number which is the MD5 checksum of all of that sentence minus its last sequence of non-space symbols and if the MD5 checksum of all of that sentence minus its last sequence of non-space symbols is 187835884982830523138282294681725949791.

The paradox relies on the extremely likely claim (probability about 1−2⁻¹²⁸, I suppose) that nobody ever tokens a different written sentence satisfying the condition after the "if". Take my word for it that the sentence above does satisfy the condition.

Note that "that sentence" is not directly referential—it is, rather, a bound variable, bound by the quantification over sentences.

What should we do? Well, I think we can should either reject quantification over sentences, or reject something like compositionality. Neither is an appealing prospect, though I've got other reasons to be suspicious of compositionality and its relatives.

If one says that one should reject quantification over sentences, but allow quantification over sentence tokens, then I'll offer the following variant:

No actually written sentence token is true if it both ends with a decimal number which is the MD5 checksum of all of that sentence token minus its last sequence of non-space symbols and if the MD5 checksum of all of that sentence token minus its last sequence of non-space symbols is 127533944667835603647534200477710876898.

This yields interesting arguments. If one allows compositionality, then one should reject quantification over all sentences or all sentence tokens. I think this forces one to be an irrealist about sentences and sentence tokens. Or one can just disallow compositionality, and thus deny that the items in block quotes are bona fide sentences, expressive of propositions.