Substitutions and Models, Part 1: Bolzano, Quine, Tarski, and Boolos

Posted on January 10, 2013 by | 12 Comments

Let ‘\Rightarrow‘ stand for a (not-yet analyzed) relation of logical consequence. According to the substitutional analysis of quantification, \Delta \Rightarrow P iff every substitution scheme \Delta is true under is also one P is true under. According to the model-theoretic analysis, \Delta \Rightarrow P iff every model on which \Delta is true is also a model on which P is true. The substitutional account is generally associated with Bolzano, and the model-theoretic with Tarski. Tarski’s objections to Bolzano’s account are generally thought to tell against the substitutional account and in favor of the model-theoretic. And what goes for logical consequence goes for logical truth — especially as P‘s being a logical truth is generally thought equivalent to it’s being a consequence of the empty set.

Enough of background. Now for the plot: Quine, in Philosophy of Logic, has an argument that — provided the background language is rich enough for elementary arithmetic (say, Peano Arithmetic) — the two accounts of logical truth are equivalent. This is prima facie surprising, and for two reasons: First, Tarski’s objection against Bolzano’s account shows that the two aren’t equivalent in general, and so it’s surprising that they would collapse for languages of this expressive power. Second, certain objections raised by e.g. John Etchemendy (The Concept of Logical Consequence) against Tarski’s official account (or, at least, Etchemendy’s reading of it) seem easily parlayed into an argument that the two are not equivalent for any first-order language.

To thicken the plot, in “On Second-Order Logic,” George Boolos argues that, although Quine is right about the two accounts of logical truth, the result does not extend to logical consequence, even in first-order languages. This is surprising, too: given compactness results, we ought to be able to move from logical truth to logical consequence.

I want to get to the bottom of these issues in a series of four posts. Here’s my plan of attack. In this post, I’ll sketch the two theories and lay out these three prima facie puzzles in more detail. The next post focuses on Quine’s side of things, first solving some of the puzzles and then examining Quine’s argument — and the scope of the premises it requires — in more detail. The third post will be dedicated to understanding Boolos’s argument and seeing how it interacts with the central puzzle to be outlined here. A final post will look at philosophical issues kicked up in the first three.

Here’s this post’s plan of attack. In section one, I’ll lay out the substitutional account in detail and give Tarski’s argument against it. Section two explains the model-theoretic account. Section three gives an argument (derived from Etchemendy’s) that the substitutional account of logical truth predictably delivers different results for first-order languages than the model-theoretic. And in section four, I’ll outline the argument that, if the accounts of logical truth are equivalent, then so are the accounts of logical consequence.

Section 1: The Substitutional Account

Suppose L is an interpreted language. Let a substitution scheme be a function from syntactically simple terms of L to any expressions of L at all that could be substituted without turning sentences into non-sentences. (If there are bound variables, we’ll also need some restrictions to keep them from interfering with each other, but we can ignore that here.) So, for instance, a syntactically simple predicate like `is tired’ (I’m pretending copula are syntactically embedded — just work with me here) could be taken to another simple one like `is hungry’, or to a complex one like `is about to eat every Taco in the tri-state area’. And a connective such as `and’ could be taken to another simple connective, such as `or’, or to a complex connective, such as ‘unless I hear by tomorrow that _______, John is going to think that _________’.

Some substitution schemes will take some simple expressions to themselves. In this case, say that it preserves those expressions.

One final bit of terminology. If P is a sentence and S a substitution scheme, let S(P) be the result of replacing all of P’s simple expressions with their values given by S. And if \Delta is a set of sentences, let S(\Delta) be the set of S(P) for P \in \Delta.

OK, now we’re ready for our definitions of logical consequence (and, by extension, logical truth). If E is a set of expressions, then say that P is an E substitutional consequence of \Delta, written \Delta |\!\!\models_{E} P, iff for every E-preserving substitution scheme S, if all of S(\Delta) are true, then S(P) is true, too.

Now, this doesn’t quite give us a definition of logical consequence; it gives us instead a definition of something like `logical consequence relative to a set of preserved expressions’. The idea, though, is that if we can pick the right expressions, we’ll have our definition of logical consequence: logical consequence is substitutional consequence under substitutions that preserve those. Those are the expressions we call the `logical constants’. Of course, for an arbitrary language, it’s no easy going deciding which expressions are the logical constants. But for, say, first-order languages, the idea is that the logical constants are those that are treated as logically special in our textbooks and so on (you know, quantifiers, truth-functional connectives, etc.)

That, anyway, is the idea. But in “On the Concept of Logical Consequence,” Tarski argued against the view more-or-less as follows. Suppose that L is really impoverished. Suppose, for instance, that along with logical vocabulary, it only has one predicates and two names (each of which means what they do in English):

Names Predicates
Washington was president.
Lincoln

Now consider the argument:

  1. Lincoln was president.
  2. Therefore, Washington was president.

Intuitively, this shouldn’t count as a valid argument. But — given the extreme impoverishment of the language — it does. Any substitution scheme that preserves the logical terms won’t have any way of substituting in something for (1) that makes it true while substituting in something for (2) that makes it false. The problem, essentially, is that the only predicate the language has doesn’t divide between Lincoln and Washington, and the language doesn’t let us talk about anything else.

Consider, for instance, the scheme that sends `Lincoln’ to `Washington’, `Washington’ to `Lincoln’, and `was president’ to `was not president’. This makes the conclusion false — but also makes the premise false! And no other substitution scheme does the job either.

The problem is apparent: it’s simply the impoverishment of the language that’s to blame. If we had the predicate `had a beard’ in the language, the argument would count as invalid: the scheme that sends `was president’ to `had a beard’ (and kept the names the same) would make the premise true and the conclusion false. But since our language doesn’t have such a predicate, the problem remain.

Notice the nature of the problem. The complaint isn’t that the substitutional account delivers the wrong verdict about any arguments we ever actually encounter. We speak a very rich language (English, at least for readers of this blog, and perhaps others as well). The problem is that the account delivers the wrong verdict about arguments in some possible language. So if we’re going to let this worry move us to a different account of logical consequence, notice that we must be implicitly endorsing a principle like,

UNIVERSAL: If an account of logical consequence is correct, it must deliver the right results for any possible language.

For those who accept UNIVERSAL, the natural “fix” behind the basic Bolzanian idea is to move to the model-theoretic account of logical consequence.

Section 2: The Model-Theoretic Account

The basic idea behind the model-theoretic account is that there are a range of things, models, with a certain structure, and a relation of `truth in’ defined between these models and sentences of L. If the language is first-order, we can be a bit more precise: a model is an ordered pair \langle D, I\rangle of a domain D and an interpretation function I. D is just any old set of things. I is a function from names of L to objects in D and predicates of L to subsets of D (or subsets of D^{n}, for n-placed predicates) that provides the `extensions’ of those predicates on the model — provides the things in the domain that, according to the model, are the way the predicate says they are. We also need a definition for `truth in’ — thanks to the necessity of dealing with quantifiers and variables, this is a fairly complex recursive one in terms of satisfaction, but I’m not going to make hay about the details here, since it’s pretty easy to see which sentences should be true on a model given that we know what terms’ denotations and predicates’ extensions are.

For instance, a model for our simple language from above would give us a set of things D, and then tell us which of those things got assigned `Lincoln’, which got assigned `Washington’, and which were in the extension of `was president’. Then a sentence like, say, `Lincoln was president’ would be true iff the thing assigned to `Lincoln’ on the model was in the extension assigned to `was president’ on that model, and a sentence like `Something was not president’ would be true iff there were things in the domain that weren’t in the extension of `was president’.

A model is called a model of a sentence iff the sentence is true on it, and a model of a set of sentences iff all the sentences in the set are true on it. If every model of \Delta is also a model of P, we write \Delta \models P, and say that P is a model-theoretic consequence of \Delta.

The model-theoretic account of consequence insists that logical consequence is model-theoretic consequence. It’s fairly easy to see that the model-theoretic account won’t suffer from the problem that plagued the substitutional. Consider again the argument:

  1. Lincoln was president.
  2. Therefore, Washington was president.

For it to be valid, every model of 1 must be a model of 2. But there are models of 1 that aren’t models of 2. Just take a model with a domain of two things, say a and b which assigns `Lincoln’ to the first, `Washington’ to the second, and assigns the set \{a\} to the predicate `was president’.

Section 3: Inequivalence

The foregoing is enough to show that the two accounts are inequivalent; that is, it’s enough to show that the following does not hold for arbitrary languages L:

(EQ) \Delta |\!\!\models P \textrm{ iff } \Delta \models P

We might think that’s not a huge deal: perhaps for suitably expressive languages EQ does hold. (This, in effect, is what Quine will argue.)

But here’s a simple argument, inspired by one John Etchemendy gives against (his reading of) Tarski in The Concept of Logical Consequence. The (adapted) argument shows that for any first-order language, EQ fails.* It goes a little something like this.

Let L be a first-order language. Presumably, its quantifier ranges over more than one object. Now consider the `counting sentence’

(E2) \exists x\exists y(x \ne y)

This sentence is true. Is it a logical truth? Or — more to the point — do we have |\!\!\models (E2)? Do we have \models (E2)?

Well, note first off that \not\models (E2). There are always models with just one thing in the domain; on the model-theoretic conception, then, this does not count as a logical truth.

But the substitutional account disagrees. All of the terms in (E2) are logical terms, so any of the substitutions we care about for evaluating consequence will be ones that keep (E2) the same. (That is: they’re substitutions S where S(E2) = (E2)!) Since (E2) is in fact true, and the substitutions don’t turn it into any different sentence, it remains true under all substitutions — and so |\!\!\models (E2).

Maybe we can fiddle with this. Here’s one way: allow that `preservation’ of quantifiers can include trading in simple quantifiers for complex `restricted’ ones — allow \exists x, to be taken, say, to \exists x(Fx \wedge \underline{\ \ \ \ \ })‘. (If we want to preserve duality — that is, if we want \exists x P to be equivalent to \neg\forall x \neg P — we’ll need to insist that substitution schemes have to `mesh’ in a certain way with how they treat their existential and universal quantifiers.) Once we revise our account, we no longer have |\!\!\models (E2). If `F‘ is a predicate that in fact applies to only one thing, the following counts as a false substitution instance:

(E2*) \exists x(Fx \wedge (\exists y(Fy \wedge (x \ne y)))

But this modification brings model-theoretic and substitutional consequence into agreement on (E2) only by making them disagree about the argument form:

  1. Fa
  2. Therefore, \exists x Fx

This is straightforwardly valid on the model-theoretic account. But on the (new, modified) substitutional account, the following is a fair substitution instance:

  1. Fa
  2. Therefore, \exists x(\neg Fx \wedge Fx)

(We get it by the substitution scheme that takes \exists x to \exists x(\neg Fx \wedge \underline{\ \ \ \ \ }).) And this disagreement about valid arguments will turn into a disagreement about logical truth, since the model-theoretic account will, and the substitutional account will not, reckon the conditional with 1 as antecedent and 2 as consequent a logical truth.

It is possible to further modify the substitutional account to make it march in step with the model-theoretic one, at least given that L’s quantifiers in fact range over infinitely many objects. The modifications basically impose large scale constraints on the substitution instances — constraints that, for instance, if you substitute \exists x(Fx \wedge \underline{\ \ \ \ \ }) for \exists x, you have to also substitute names of things that satisfy F for names, and predicate expressions that are satisfied only by F-things for predicates. However motivated these restrictions might be (and they don’t seem very), they don’t help the prima facie puzzle. For the prima facie puzzle is that the two accounts disagree, but Quine is going to argue that they do agree, contra what’s already happened. And Quine’s argument isn’t predicated on any fancy-schmancy jazzing up of the old substitutional account; it’s predicated on the boring substitutional account of the sort described back in Section 1. So what gives?

Section 4: Consequence and Truth

The puzzle in Section 3 admits a simple solution, but I’ll wait until the next post to give it. Here, I want to present a deeper puzzle. At the end of the next post, we ought (if Quine does his work right) to be convinced that, for expressive enough languages, the following restricted version of EQ holds:

(EQ-R) |\!\!\models P \textrm{ iff } \models P

In the final post, Boolos is going to give us a general argument that, even for these more expressive languages (especially for them, in fact), EQ fails. All the languages in question are first-order. And there is a fairly straightforward argument that EQ holds for a given first-order language iff EQ-R holds for that language.

The argument stems from three properties of first-order logic. The first is compactness. I’ll give a generic version of it here:

(COMPACTNESS) \Delta \Rightarrow P iff there is a finite subset \Gamma \subseteq \Delta where \Gamma \Rightarrow P.

(In other words, no valid first-order arguments require infinitely many premises in order to be valid.) The second theorem is a kind of generalized deduction theorem, which I’ll write as follows:

(DEDUCTION) Q_{1}, Q_{2}, \ldots, Q_{n} \Rightarrow P iff \Rightarrow (Q_{1} \wedge Q_{2} \wedge \ldots \wedge Q_{n}) \supset P

Finally, we have

(MONOTONICITY) If \Delta \Rightarrow P and \Delta \subseteq \Gamma, then \Gamma \Rightarrow P.

That is: adding premises to an argument can’t take validity away. For all of these results, we can consider model-theoretic and substitutional versions by replacing ‘\Rightarrow‘ with ‘|\!\!\models‘ or ‘\models‘, respectively.

DEDUCTION and MONOTONICITY are relatively trivial, and can be seen to hold by inspecting the definitions of each kind of consequence, plus (for DEDUCTION) the truth-table for the material conditional, \supset. COMPACTNESS is not at all trivial, although in the model-theoretic setting it’s very well established. The substitutional variant is probably going to show itself as the cause of all the bother when all is said and done, but let’s not get ahead of ourselves. Let’s just go through the argument.

I’m actually going to just go through one half of the argument: I’ll argue that, if \Delta |\!\!\models P, then \Delta \models P. But it is trivial to reverse the argument, because all of the premises (other than EQ-R, which is a biconditional) apply equally well to both kinds of consequence. One bit of notation will help: if \Gamma is any finite set, then I take the privilege of rewriting it as \{Q_{1}, Q_{2}, \ldots, Q_{n}\}. OK, here goes.

  1. Suppose \Delta |\!\!\models P
  2. By COMPACTNESS, for some finite \Gamma \subseteq \Delta, \Gamma |\!\!\models P
  3. Rewriting, \{Q_{1}, Q_{2}, \ldots, Q_{n}\} |\!\!\models P
  4. By DEDUCTION, |\!\!\models (Q_{1} \wedge Q_{2} \wedge \ldots \wedge Q_{n}) \supset P
  5. By EQ-R, \models (Q_{1} \wedge Q_{2} \wedge \ldots \wedge Q_{n}) \supset P
  6. By DEDUCTION, \{(Q_{1} \wedge Q_{2} \wedge \ldots \wedge Q_{n}\} \models P
  7. Since \{Q_{1}, \ldots, Q_{n}\} \subseteq \Delta, by MONOTONICITY, \Delta \models P

The argument works just as well in the other direction, and EQ follows. But if both Quine and Boolos are right, EQ doesn’t follow. Something’s gone wrong. COMPACTNESS for substitutional consequence looks like the best bet. But how? What sentences follow substitutionally from an infinite set of sentences but no finite subset?

This is the central puzzle. We’ll have to wait a few more posts before hitting the bottom of this mess.

* Well, maybe just almost any — there might be some wiggle room if some first-order languages have quantifiers semantically restricted to range over just one object.

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12 responses to “Substitutions and Models, Part 1: Bolzano, Quine, Tarski, and Boolos”

  1. Günther | June 30, 2013 at 2:41 am | Reply

    dear jason,

    first of all: i found your post really interesting (unfortunately i haven’t been able to find the other three announced posts. maybe you have written a paper in the meantime?!) i am not entirely sure, however, if the counterexample you provide against the “new, modified” account of logical truth (which allows for something like “domain-variation”) is really a counterexample, if evaluated on `quinian grounds’.

    first: according to quine, in order to check if some sentence $\phi$ is a logical consequence of some finite set of sentences $\phi_1, … \phi_n$, we have to check whether the conditional $\phi_1 \wedge … \wedge \phi_n \rightarrow \phi$ is a logical truth

    so in your example, we have to check whether

    (1) $Fa \rightarrow \exists x Fx$

    is a logical truth. Now if $a$ here is supposed to indicate some $name$, then we no longer stand on `quinian grounds’, for in a quinian `standard language’, names are eliminated in favour of predicates being true of just one object. so if $a$ is supposed to stand for a name, the `real’ logical form of (1) would be something like this:

    (2) $\exists x \forall y((Ay \leftrightarrow x=y) \wedge Fx) \rightarrow \exists x Fx$

    which is indeed a logical truth according to the `new, modified’ account of logical truth. (your `counter-model’ would lead to

    (3) $\exists x (\neg Fx \wedge \forall y(\neg Fy \rightarrow (Ay \leftrightarrow x=y) \wedge Fx) \rightarrow \exists x (\neg Fx \wedge Fx)$

    which will be true (in any interpreted language).

    If, on the other hand, we look at

    (4) $Fy \rightarrow \exists x Fx$

    we have no $sentence$ to begin with (let alone a logically true sentence), for (4) now contains the free variable $y$. this formula, however, is $satisfied$ by all objects as values for $y$, so we should expect that

    (5) $\forall y (Fy \rightarrow \exists x Fx)$

    is a logical truth – and it $is$, even according to the substitutional account. taking as the “domain-formula” (to which quantifiers are to be restricted) again the formula “$\neg Fx$ (as you suggest) leads to

    (6) $\forall y (\neg Fy \rightarrow (Fy \rightarrow \exists x(\neg Fx \wedge Fx)))$

    which reduces to

    (7) $\forall y (\neg Fy \wedge Fy \rightarrow \exists x(\neg Fx \wedge Fx))$

    and which will in fact be true (again, in any interpreted language).

    so, unless i am completely wrong (which might well be the case), it seems to me that, given the quinian presuppositions, your counterexample won’t hurt someone who insists on a quinian “standard langauge”.

    cheers,

    günther

    • jason | July 1, 2013 at 6:31 am | Reply

      Hi Günther — I’m afraid I had lost track of this project. I’ll get the next post up later today.

      As for my comment — you’re right, of course, that Quine has a rejoinder to my rejoinder, in that he won’t recognize the name-using existential instantiation as an inference in any language he’s giving a logic for. That by itself doesn’t quite fix the problem, because so far as we’ve said, the model theory is first-order. So we’ll have \vDash \exists x(x=x), but on the proposed substitution scheme, we do not have \vert\vDash \exists x(x=x), because we can substitute in a quantifier with an unsatisfied restrictor. As we’ll see in the next post, though, Quine has an even more direct response to the worry that doesn’t involve tinkering with the substitution scheme.

      Really, in this post I wanted to just point out a prima facie puzzle with Quine’s claim, so we can see why it might be surprising and might need to be thought through. He’s clever enough that no simple counterexample is likely to stand up to scrutiny in the long run!

      • Günther | July 5, 2013 at 5:28 pm |

        hi jason!

        thanks for your reply and the new post! i’m looking forward to read it! (i’m sorry for not writing back earlier – i have been out of town for a couple of days and haven’t been able to.)

        as to your “counter-counter-example”: i am still not totally convinced. 🙂 if i understand you correctly, your point is that

        (1) \exists x (x = x)

        is not a logical truth according to the (new, modified) quinian definition, because we can use a restricting “domain-predicate” \phi(x) which is not satisfied by any object. So

        (2) \exists x (\phi(x) \wedge x = x)

        would be an admissible substitution instance for (1), which is false, because by assumption \neg \exists x \phi(x). hence, (1) can not be a logical truth according to the substitutional account.

        if that’s the argument, however, i am not sure if i would accept it. i would simply require such a domain predicate to be non-empty! this may sound ad hoc. but i think its not more ad hoc than the requirement, on the standard account, that the domain on which a sentence is to be evaluated has to be non-empty. for \exists x (x=x) will not be a logical truth on the standard account either if we admit of empty domains.

        to be “fair”, it seems to me that we should apply the same standards in both cases. so if domains are not allowed to be empty (as is usually the case on the standard model-theoretic account), we should not be allowed to substitute non-empty restricting formulas as well.

        best, günther

      • jason | July 8, 2013 at 8:26 am |

        Insisting that the domain-predicate be non-empty will do the trick, but it would be anathema to Quine’s substitutional approach. I’ve not been making much of this, but it’s very important to Quine (though I don’t know why) that the substitutional account is grammatical: he takes logical constants to form their own grammatical category, and then treats logical truths as truths that remain truths on substitutions that preserve that category.

        Even adding a domain-restricting predicate would break that picture, but the account would remain a largely syntactic account: the only place semantics would enter is in the truth (or falsity) of substitution instances. But adding a constraint that the restricting predicate be non-empty is dragging in a whole new (subsentential) semantic component to the account that I doubt Quine would be happy with.

  2. Pingback: Substitutions and Models, Part 2: Quine | Centre for Metaphysics and Mind

  3. Günther | July 12, 2013 at 2:32 pm | Reply

    hi jason,

    i think you’re right with your point concerning grammaticality: if “S is a logical truth” means something like “S is true just in virtue of its logico-grammatical structure”, then it will not quite help to consider a class of related sentences which arise from S by substitutions PLUS restricting quantifiers to a domain-predicate. I am just curious to see in which directions quine’s account might be modified whilst still preserving his “anti-model-theoretic” spirit. (where model theory is conceived of as something like a branch of set theory). besides, a syntactical account like this one is pretty close to what proof-theoristst do with their “syntactic interpretations”. here, as in quine, the idea is to “simulate” model-theory by using syntactic translations instead of models.

    as to your second and third post: i’m not quite sure if i understood everything correctly, i have to read them again, but maybe you can help me with two (maybe related) points:

    the first thing is this: we know (due to (weak) completeness and the hilbert-bernays-strenghtening of the löwenheim-theorem) that the usual modeltheoretic and quine’s substitutional account agree with respect to single sentences (or, of course, finte sets of them). what the boolos-argument then shows is that there is a certain set B (your Tr(LB)) (the set of truths of his second language M – your LB) which is satisfiable in the usual, but not in the quinian, substitutional sense (in the sufficiently rich language L – your LA). the boolos set B (or the set of gödel-numbers of the sentences in B) is of course highly non-recursive (it isn’t even definable-in-L(or -M)), so what this shows is that the hilbert-bernays-löwenheim-theorem does not further generalize to include arbitrary, infinite sets of sentences. (although it can be further strenghtened to include infinite, yet recursive sets.) so we have two strengthenings of the löwenheim-theorem: one with respect to arbitrary, infinite sets of sentences but without “constructibility” (downward löwenheim-skolem); the other one with respect to “constructibility”, but which does not apply to arbitrary infinite sets of sentences (hilbert-bernays-löwenheim). again, this last point is exactly what i take boolos to have established. (from this it follows, of course, that the general consequence relation (applying to arbitrary, infinite sets of sentences) induced by quines “substitutional models” is not equivalent to the usual one as well).
    but at the end of your first post you claim that the problem is compactness. the problem i have is not that i think that compactness is not a problem (it surely must be, for otherwise your argument at the end of the first post would be correct!) the problem is that i do not quite see how the boolos-argument (or your reconstruction of it) shows that compactness fails. what it shows is that a certain satisfiable set B is not quine-satisfiable (which, again, is enough to see that it is inadequate). but in order to see that his substitutional notion of satisfiability (or, what is the same, his consequence-relation) is not compact, we would have to show that some finite subset of B (in fact, some single sentence) is not quine-satisfiable as well. so i would be curious to see how you might argue for this.

    the second question is concerned with your overall proof strategy: why do you invoke peano arithmetic (or anything related to derivability, for that matter)? it seems to me that all we need is a semantic version of tarski’s theorem, namely: the set of (codes of) arithmetical truths is not arithmetically definable (plus some basic recursion theory). this is a matter only of the expressive strength of the languages in question – not of anything related to provability-in-PA (or provability-in-Q, or whatever). so why bother about questions concerning representability-in-PA of certain predicates, relations, and so on?

    cheers, günther

  4. Günther | July 12, 2013 at 2:42 pm | Reply

    by the way: i am currently trying to write a paper on the metamathematics of quine’s concepts of logical truth and consequence, in particular boolos’ argument and how a quinian might respond. so i would be happy if you had some pointers where the issue is discussed. (most papers i read on the subject only mention the boolos-argument in a footnote or so.)

  5. jason | July 12, 2013 at 10:44 pm | Reply

    Well, this discussion should probably be over at the third post, not this one. But since we’re here already I’ll see what I can say.

    I was planning on addressing your first and third question explicitly in a fourth post, but that hasn’t been coming together really well, so instead of waiting for some future date I’ll mention a few things about them now.

    On the first question, you seem to have already spotted this, but I’ll go through it anyway in case any lurkers are hanging around and hoping for more clarification. Compactness says that if every finite subset of S is satisfiable, then so is S. So I need to show that for some S, every finite subset of S is substitutionally satisfiable even though S is not. Here, Tr(LB) just is the set. (My reconstruction of) Boolos’s argument shows that Tr(LB) isn’t substitutionally satisfiable, even though it IS (model-theoretically) satisfiable. But since it’s model-theoretically satisfiable, any finite subset T of Tr(LB) is also model-theoretically satisfiable, in which case the conjunction C of that set is M-T satisfiable. Then we run through Quine’s argument to conclude that C is substitutionally satisfiable, and therefore that T is as well.

    On the third question: I honestly don’t know anywhere post-Boolos’s-paper that this is talked about. (You might have a better picture of the literature than I do!) These posts came into being in part because I was trying to figure out what was going on in the two arguments — especially as Boolos’s shows up in a highly compressed appendix, and is done in such a way to be pretty opaque to those not already very comfortable with that kind of technicality.

    What Quine seems to want to say in Philosophy of Logic is (roughly) that we should just define \Delta |\vDash \phi as saying ‘there is a finite subset \Gamma \subseteq \Delta such that |\vDash \bigwedge\Gamma \rightarrow \phi. Boolos complains that this seems ad hoc, and in a way that undercuts Quine’s theoretical elegance argument for logic’s being first-order. (That argument uses of the coincidence of substitutional and model-theoretic conceptions in the first-order case as a reason to think “Hooray first-orderness!”, but Boolos’s point is (roughly) that this is a false elegance if one of the consequence relations is cooked up precisely to generate the coincidence.) Also, at least as I’ve done it, the definition relies on sets of sentences in a way that undercuts Quine’s “hooray for doing without set theory” mentality — although perhaps this can be finessed. As far as I know, though, that’s where the back-and-forth burns out.

  6. jason | July 12, 2013 at 10:57 pm | Reply

    As for your second question: A few parts to the answer. The first is just historical: When I was first thinking through this I was wondering what Quine meant by “strong enough for number theory,” and so had syntactic stuff on the brain.

    The second (and perhaps more interesting) answer is that it has to do with Quine’s reliance on the canonical number-predicates. I had to show that if there was a substitution instance that made all of Tr(LB) true, then Tr(LA) had its own truth-predicate. But in the first instance what we get is not that the substitution of “G” that would make Tr(LB) true would apply to all and only the Godel numbers of Tr(LA) sentences, but that they would apply to certain images of those numbers. To complete the reductio, I had to show that the `imaging’ itself was expressible in the language and could be `undone,’ so that afterwords we got a genuine truth predicate for LA. My way of doing it was to note that it was clearly computable, rely on the Church-Turing thesis to conclude it was recursive, and then rely on the capturability of recursive functions via Godel coding. I don’t say this is the only way; maybe we could have done without it.

    But there’s also a pedagogical point in all this. I could have just said “it’s well known that / Godel showed that / etc. that recursive functions can be expressed arithmetically, and so…” Then I could have done without all the material about provability. But this is a bit misleading, since the way Godel showed it was via provability in a system like Q or PA. Then, since (or so long as) Q or PA is true, we get the semantic results about expressibility coming along for the ride. Since part of the motivation for these posts was to make some tough material at least a little bit more accessible for those not already in the know — which I may well have failed in! — I wanted to at least summarize the critical foundational principles that Boolos’s argument is ultimately resting on.

  7. Günther | July 17, 2013 at 10:51 pm | Reply

    hello jason!

    thanks for your reply! i am still not sure though if i got the problem right. you seem to believe that quine wants to DEFINE logical consequence (applying to arbitrary premiss-sets) as meaning “there is some finite subset…” (so he can bypass the compactness problem). however, i couldn’t find anything in quines philosophy of logic (second edition), where he actually defines logical consequence in this way (or even considers it informally). i couldn’t find any passage in his methods of logic either where he takes into account arbirary sets of sentences in connection with the problem of defining logical consequence and/or satisfiability. which is quite weird, given his first-order-obsession and the fact that there are no interesting finitely axiomatized theories. (well, recursively axiomatized theories can be dealt with by quine’s definitions i think; for it seems that we can extend the hilbert-bernays-theorem to recursive infinite sets; but the general case remains.) so it seems to me that boolos is just contemplating on a “possible way out” – not on a definition quine actually gave. (it would be interesting for me to have a passage in quine’s published writings, where he explicitly deals with infinite sets of sentences – so if you know of any: please let me know!)

    as to your more “syntactically-minded” proof, i was just wondering whether there was some other point to that besides the pedagogical one. but apparently, that’s not the case. 🙂

    and by the way: you clearly did not fail! i got some inspiration for my own to-be-written paper on the issue. (i find the surrogate-numeral-notation for quinean languages particularly neat!) the motivation for writing such a paper was essentially the same as yours: to make the boolos-argument and its implications more accessible and maybe discuss possible responses by a quinian. (as i’ve said: i’ve read some papers on quine’s defintions, but they all seem to avoid talking about the boolos argument.)
    so it would be nice if you could send me your full name, so i can refer to you in the acknowledgements (in case the paper ever gets finished). if you don’t wanna have your full name visible here – just send me an email (guenther.eder@univie.ac.at)

    cheers, günther

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