This is one of a series of posts setting out my work on the Nature of Representation. You can view the whole series by following this link.
I turn now to another logical concept: generality. Specifically, I am interested in unrestricted generality.
For starters, we do seem able to express thoughts that range over everything without restriction. Our ability to do so is, furthermore, significant to us. The philosophical thesis of physicalism is that absolutely everything is physical, not that everything-around-here is physical. Without the ability to talk about absolutely everything, such theses would be ineffable. The moral rule is that all people should be treated fairly, not that all people-who-meet-some-further-condition should be so treated. The “all” still needs to be absolutely unrestricted in force in order to capture the intended thought, and this illustrates that even explicitly restricted quantification (“all people”) presupposes that we are not missing out on absolutely anything in the underlying domain from which the restricted class is selected.
But against this, there are formal results that show there are deviant, restricted interpretations of the generality of “everything” and “all” on which our interpretee Sally doesn’t quantify over absolutely everything. It turns out that we can construct such restricted interpretations in ways that pass most tests we can muster for what makes an interpretation correct. For example, we can find a deviant interpretation that agrees with the correct, unrestricted, interpretation of Sally on the truth value of every thought she can think. And we can choose the interpretation so it agrees with the distribution of truth values over thoughts not just in the actual world, but at every possible situation. Further, such deviant interpretations diverge from the correct interpretation only on the domain of the quantifiers. So they assign the same denotation to singular concepts, predictive concepts and the propositional connectives.
Because of the matching truth values/truth conditions, you won’t be able to rule out the deviant interpretation of Sally on the grounds that it makes her thoughts any less reliable than the original interpretation. Because of way the interpretations on everything-except-quantifiers, side-constraints on the interpretation of singular terms and the like won’t help at all. The challenge for us is to explain how, despite the existence of such “skolemite” interpretations, Sally manages to generalize unrestrictedly, in the way that philosophy and morality presupposes she can.
Aside. I’m here skating over a number of technical issues in constructing the construction of these deviant “skolemite” interpretations. So let me just flag, in lieu of getting into details, that it may be that the challenge is restricted to those whose conceptual repertoire is expressively limited in some ways. While the technicalities (involving higher order quantifiers and modal resources) are interesting in their own right, and I have things to say on the topic, I don’t find it at all credible that our ability to quantify restrictedly depends on our ability to think thoughts whose logical form goes beyond that of the first-order extensional calculus. So I’m happy setting these aside.
While I’m flagging side-notes, let me also highlight the point just made that skolemite interpretations aren’t ruled out by imposing side-constraints on the denotation of singular or predicative concepts or propositional connectives. While we’re talking about radical interpretation specifically here, notice that this feature means that the skolemite challenge is relevant to theorists who start from a quite different place, since there’s a laser-like focus on the denotation of quantifiers here—the problems won’t come out in the wash just because you have some interesting things to say about the denotation of concepts in other categories. End Aside.
Following the pattern of the last post, I will make some architectural assumptions about the way that quantificational thoughts figure in our psychology, and explore what radical interpretation will predict under those assumptions. The first three architectural assumptions I will make are just as before—that belief states are structured, that facts about syntax are grounded prior to questions of content arising, and that we can pick out inferential dispositions interrelating belief-types, again prior to questions of content being determined. I will continue to use the Peacockian idea that certain core inferences are treated as primitively compelling.
The fourth architectural assumption is about the character of the inferential role associated with the quantifier concept everything which I write as q. I’ll concentrate on just one of these rules:
- From qx:Fx derive Fa
Our interpretee, Sally, endorses an instance of this for every individual concept a that she possesses. We’ll come back in a moment to the question of whether this is all the relevant facts relating Sally to tokens of this inference-type.
If the story rolled on from this point as it did for conjunction, we might expect that we’d find radical interpretation predicting something along the same lines as the Peacockian “determination theory” for conjunction, that is, the semantic value for q will be the quantifier, whatever it is, that make all instances of the above instances of a valid type. Just as before, Radical interpretation will approximate an interpretative constraint of this kind. Sally’s rationality, which includes maximizing justified beliefs, which ceteris paribus includes interpreting q so as to make the inferences justification-preserving. Making them valid looks just the ticket.
If this is all we can extract from the story, we’d be in trouble. There are every-so-many ways of choosing restricted quantifiers as an interpretation of q to make all instances of the above truth-preserving. The ones that are constructed by the skolem procedure are among them, since the skolemite interpretation’s restricted domain includes every object for which the subject has an individual concept a.
In some special restricted cases, we have devices that allow us to construct concepts for every member of a restricted domain (as in our procedure for constructing numerals for natural numbers). But that’s not so in the general case. It’s not even so in very large restricted domains, for example for quantifiers over the real numbers, over space-time points, or sets.
Aside. One reaction in the literature has been to double down on the idea of making instances of the above scheme valid, and argue that more “instances” are relevant than one might at first think. Thus, one might argue that Sally is disposed to find compelling not just instances of the above scheme for singular concepts she currently has available, but also for potential singular terms not currently within her ken. I think, though, that this is ultimately not a productive approach. Although the general idea that Sally’s endorsement of the inference pattern above is “open-ended” when she’s using an absolutely unrestricted quantifier is a good one, I do not think that this is best factored into the theory of how denotation is fixed by an insistence that those extra token inferences be interpreted so as to be instances of a valid type. (I consider the approach at length elsewhere, but briefly, the problem is that consistently with this constraint, one can interpret the agent as deploying a contingent and contextually flexible restricted quantifier whose domain “expands and contracts” in sync with the agent’s available singular conceptual resources. In a slogan: using counterfactual pegs to fix quantifier domains only end up constraining the counterfactual domains. End Aside.
Radical Interpretation can explain how finding tokens of the elimination rule above primitively compelling can fix a truly unrestricted interpretation of our quantificational concepts. It can do so by appeal to the epistemological character of even a single token instance of the inference-type.
Take an interpretation where Sally uses a quantifier tacitly restricted to some skolemite domain S, and see what we think of the epistemological status:
From Everything (psst… in S) is beautiful. Derive Toby is beautiful.
Now, Toby is among the items in the skolemite domain S, we may assume. So the inference preserves truth, on the suggested interpretation. But what’s striking is that Sally is not interpreted as having or utilizing any information either way about this fact. To state the obvious: that’s not the usual way we would think of restricted quantification as going. For example, in order to justify my belief that “Toby should be treated fairly” by inference from my (justified) belief that “All people should be treated fairly”, I surely also need to have the justified belief that Toby is a person. After all, if I wasn’t justified in thinking that Toby was a person—if my evidence was that he’s my neighbour’s cat—then the justification for the derived claim would be undercut. The same goes for ordinary, tacitly restricted quantification. From “everyone has some marking to do” (contextually restricted to faculty members) I can’t be justified in believing “Toby has some marking to do” unless I have some justification for believing that Toby is a member of faculty. What’s striking and central about Sally’s deployment of an unrestricted quantifier is that her inference is not enthymetic in this way. She doesn’t pause to check whether or not Toby has this or that feature before inferring that he’s beautiful.
In sum: Sally’s justification for the belief that everything is beautiful transfers to Toby is beautiful without mediation. The lack of mediations explains why her acceptance of the inference rule is “open ended”, as theorists like McGee and Lavine have emphasized. But what matters for grounding facts about quantifier-meaning is not the way this open-endedness manifests in the piling up of accepted instances of the inference across counterfactual scenarios, but the lack of mediation in the epistemological structure of the inference, a feature that is already present the actual cases.
I propose the following piece of epistemology. Consider an elimination rule for a restricted quantifier—whether restricted explicitly (all people) or restrictedly tacitly (either by contextual mechanisms or in the way proposed by the skolemite construction. If deployments of that rule are to transfer justification, then that rule will have to include a side-premise, to the effect that the object in question has the feature that defines the restriction. This is not the case for an unrestricted quantifier.
This piece of epistemology then tells us why we wouldn’t be interpreting Sally as substantively rational if we interpreted her as using a skolemite quantifier—we’d be representing her as constantly engaging in inferences that involve enthymetic premises for which she has no justification.
This story gives us a satisfying resolution of long-standing skolemite puzzles about what grounds our ability to quantify unrestrictedly. Methodologically, it illustrates the virtue of thinking through what radical interpretation require in detail—in the case of conjunction, we only needed to make the primitively compelling inferences valid in order to pin down the denotation. That is insufficient here, since making valid the elimination rule (and indeed, the analogous introduction rule) wouldn’t eliminate the deviant interpretations.
I finish by running through the derivation of Sally’s q denoting the unrestrictedly general universal quantifier. First, we have the a posteriori assumption that c plays a distinctive cognitive role in Sally’s cognitive architecture, captured by the unmediated elimination rule. Second, we have substantive radical interpretation which tells us that the correct interpretation of q is one that maximizes (substantive) rationality of the agent. We add the “localizing” assumption, inferential role determinism for q, which says that the interpretation on which Sally is most rational overall is one on which the particular inferential dispositions captured by the rules just given for q are rational. Putting these three together we have the following: the correct interpretation of Sally is one that makes the inferential role associated with q most rational. And now we add the conclusion of the discussion we’ve just been having: that the way to make the inferential role that consists in the elimination rule without side-premises most rational (especially to make it most justification-preserving) is to make it denote the unrestricted quantifier.
Edited 12/9/17
J Robert G Williams 