Tuesday, September 27, 2011

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:
  1. 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 ∃nnx? 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:

  1. 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".


Heath White said...

Kripke's response to Tarski-style level theories was to point out that claims like "What Alex believes is true" have no definite level; it depends on what Alex believes. The meaning of the sentence seems pretty clear, however. I'm not sure how this view handles that kind of case?

Alexander R Pruss said...

The move I suggest in the post is that what proposition "What Alex believes is true" involves quantifier-indexicality. (I assume that "What Alex believes is true" involves an implicit quantification over propositions.) The level of the quantification depends on the context, and what Alex in fact believes is a part of the context. Thus, if Alex in fact believes a second order truth, this is a third order truth.

Serious problem that just occurred to me: This indexical move doesn't seem to work well in modal contexts. "Possibly, Alex believes a truth." Which level is that? (This is particularly hard if the statement is false.)

Perhaps better: bite the bullet. Admit that "believes" is genuinely analogical between the different levels. Take really seriously the needed revisions to modal intuitions about our beliefs, such as that you don't get to say "Possibly, Alex believes a truth" without specifying a level.

On Wittgensteinian grounds you might think that what it is to believe a tautology is a different sort of thing from what it is to believe an empirical claim. Well, is it that implausible that what it is to believe a first-order claim is different from what it is to believe a second-order claim, and so there is no single "believes" predicate that covers all levels--there is only a family of analogical predicates.

This is revisionary, but doesn't require abandoning plausible rules of logic, just plausible claims about beliefs and the like.