What Grammatical Structures Say and the Linguistic Theory of Logical Truth

In the last chapter of Philosophy of Logic, Quine discusses the “linguistic theory of logical truth.” This is the theory that “[a] sentence is logically true if true by virtue purely of its grammatical structure. […] It is language that makes logical truths true – purely language, and nothing to do with the nature of the world.” (2nd ed., 95) Quine offers a few different reasons not to buy into this doctrine, most of them familiar from “Two Dogmas” and “Truth by Convention”. The freshest argument, I think, is the one he offers in the paragraph immediately following the previous quote. Here it is, in full:

Granted, grammatical structure is linguistic; but so is lexicon. The lexicon is used in talking about the world; but so is grammatical structure. A logical truth, staying true as it does under all lexical substitutions, admittedly depends upon none of those features of the world that are reflected in lexical distinctions; but may it not depend on other features of the world, features that our language reflects in grammatical constructions rather than its lexicon? It would be pointless to protest that grammar varies from language to language, for so does lexicon. Perhaps the logical truths owe their truth to certain traits of reality which are reflected in one way by the grammar of our language, in another way by the grammar of another language, and in a third way by the combined grammar and lexicon of a third language. (ibid., my italics)

Logical truths are about the world, or are true because they “reflect” features of the world, because their grammatical structures are about the world or reflect features of the world. In what sense could a grammatical structure possibly be about the world, say anything about the world, or “reflect” features of the world? First, we should note that grammatical structures are not about the world in the same way that sentences, names, or predicates are. Grammatical structures as such aren’t true or false like (truth-apt) sentences. By all appearances, grammatical structures don’t refer to anything in the world; Tarski’s definition of truth gets along just fine without assigning semantic values to grammatical structures. Nor do they have (Fregean) senses on any theory that I know of. Nor, intuitively, are they meaningful. If a person were to speak or write down a grammatical structure – say, by speaking or writing a sequence of particles and schematic variables for grammatical categories – I can’t see why anyone would want to say that she, or her utterance or inscription, meant anything.

We might want to say that grammatical structures say something about the world in a different sense – viz., in the sense that sentences with the same non-logical constants but different grammatical structures have different truth-conditions. “(Ax)(Cat(x))” says something different from “~(Ax)(Cat(x))” because of the difference in grammatical structure between the two. We might say that a negation symbol says that the negated sentence is false, a universal quantifier over a variable says that the sentence in the scope of the quantifier is true for all values of the bound variable, and so on.* In this way, by specifying what all of the particles or logical constants say, we can state more or less precisely what an entire grammatical structure, paired with a particular sentence instantiating it, says about the world. But two observations are in order. First, it is not clear how we should construe what the grammatical structures of atomic sentences say.** Second, and more importantly, the worldliness of a grammatical structure, in this sense, is dependent on the worldliness of the non-logical constants in the sentence instantiating it. For instance, the grammatical structure of “~(Cat(Dora))” says something about the world because “(Cat(Dora))” says something about the world – it is either true or false depending on the actual features of the thing called “Dora”. The grammatical structures of uninterpreted schemata say nothing about the world, because the talk of truth, falsity, and values of variables in our sketchy specification of what the particles say presupposes an interpretation of the lexical items in the sentence. When we admit, with Quine, that a logical truth “admittedly depends upon none of those features of the world that are reflected in lexical distinctions”, then the grammatical structure of a logical truth cannot derive its worldliness from the worldliness of the “lexical distinctions” marked by the non-logical constants in a sentence instantiating the structure. Briefly, since the worldliness of grammatical structures derives from the worldliness of the terms in the sentences instantiating them, and since the putative worldliness of logical truths does not depend on the worldliness of these terms, the grammatical structures of logical truths have nothing from which to derive their putative worldliness.

There might be some way of being about the world or reflecting features of the world that I haven’t grasped yet. Perhaps we should understand Quine as saying that we might as well postulate a sui generis mode of being about the world specific to grammatical structures. I can only say in response to this that we might as well not, both for the sake of not multiplying features of the world beyond necessity and for the sake of keeping “about the world” intelligible. Lastly, someone might say that the grammatical structure of logical truths such as “it is raining or it isn’t raining” is about the world because it reflects the fact about the world that things, in general, are or aren’t the case. But this begs the question against the proponent of the linguistic theory of logical truth. What is at issue is whether this fact is about the world.

* - It is much easier to fill in the “and so on” for a formalized language than for a natural language. What does a particle like “if” say? It seems we need a worked-out semantics for conditionals to fill out a proposal like this. If you aren’t satisfied by my hand-waving here, then that probably goes to show that it is even more difficult to make Quine’s argument come out sound.
** - This probably isn’t such a big deal, since the only atomic sentence that is a logical truth is “x = x”, and Quine seems to reckon “=” a particle.

The latest Philosophers' Carnival...

... is here.

My post, On What There Is and What We Can Perceive, is included under epistemology.

And now...

Immanuel Kant






















and my girlfriend's old roommate's dog, Sally Monster.

Quantifiers and the Grammatical Definition of Logical Truth

The grammatical definition of logical truth, discussed here, is probably inadequate for all sorts of interesting languages, including English. The definition, from Quine's Philosophy of Logic, is this:

"a logical truth is a truth that cannot be turned false by substituting for lexicon. When for its lexical elements we substitute any other strings belonging to the same grammatical categories, the sentence is true." (2nd ed., 58)

For Quine - and I think he's right on this count - we treat a class of words as a grammatical category, as opposed to a class of particles yielding new grammatical constructions, just in case the category is big enough. For instance, in a language with lots of intransitive verbs, we treat those as comprising a grammatical category. If an L-structure has an infinite stock of variables, we treat variables (or argument-terms, more generally) as a grammatical category; if it has three variables, we might do well to treat each as a particle.

I take it that English has an infinite - or at least a very large - stock of quantifiers. This is because I think that, in English, the quantifiers translated by "(E_)" and "(A_)" in first-order logic belong to the same grammatical category as expressions such as "There are many", "There are ten", "There are one million", and "There are innumerable". If the literature on quantifiers in natural language says otherwise, please correct me. Also, there are usually an infinite number of quantifier-expressions in languages that support generalized quantification, right? Anyway, if quantifiers all belong to the same grammatical category, and we assume the grammatical definition of logical truth, then I can't think of a single logical truth containing a quantifier. For instance, "If Steve and Janice are cats, then there are some cats" would fail to be a logical truth, since "If Steve and Janice are cats, then there are innumerable cats" - gotten by "substituting for lexicon" - is false.

I imagine the Quinean response to all of this would be to say that, given a prior commitment to standard FOL, we should translate "There are n Fs" as "The class of all Fs has cardinality n." But it seems obvious to me that the average English speaker does not, as a matter of linguistic anthropology, commit herself to the existence of the class of all Fs in uttering "There are n Fs." The nominalist cannot properly respond, "No, there is no such thing as the set of all Fs." And besides, what if we substitute "self-member" for "F"?

Quine on Grammatical Structure and Logical Truth

In Philosophy of Logic, Quine offers the following definition of "logical truth": "a logical truth is a truth that cannot be turned false by substituting for lexicon. When for its lexical elements we substitute any other strings belonging to the same grammatical categories, the sentence is true." (2nd ed., 58)

Later on, considering whether to strengthen FOL to allow for adverbial modification of predicates, Quine claims that, on the definition of "logical truth" lately quoted:

the sentence

(5) ~(Ex)(x walks rapidly . ~(x walks)),

or 'Whatever walks rapidly walks', would qualify as logically true. (76)

This is interesting. The grammatical definition of logical truth is both epistemologically interesting and clears up a lot of confusions I have about the relationship between formal logic and natural language. I still don't know whether it adequately captures all of the intuitive cases of logical truth.

Really, my only observation here is that it seems that the grammatical definition does not make (5) a logical truth. This is because adverbs can sometimes alienate the predicates they modify. An adverb A alienates a predicate F in a sentence token S iff removal of A from S would change the truth-value of the clause of which F is a part. Briefly, A alienates F (in a certain context) if something can be F A'ly without being F simpliciter. Consider the following cases of adverbial alienation:

(1) Tim indirectly told John about Sally.
(2) Paul is coming home shortly.
(3) Sue allegedly stole the watch.
(4) Esther nearly won the tennis match.

We can imagine cases in which (1), (2), (3), and (4) are true, but their non-adverbialized counterparts aren't. There doesn't seem to be anything syntactically unusual about these adverbs. By all appearances, (1), (2), (3), and (4) have the same grammatical structure, respectively, as the following:

(1`) Tim excitedly told John about Sally.
(2`) Paul is coming home currently.
(3`) Sue actually stole the watch.
(4`) Esther barely won the tennis match.

But if (1), (2), (3), and (4) can be turned from truth to falsehood by transformation into (1`), (2`), (3`), and (4`) then, by the grammatical definition, none of these are logically true.

Poem

“For example, imagine the plight of the autophenomenologist who sets out to study the intentional objects that accompanied his engagement in wildly abandoned sex; he would end up studying the intentional objects of someone engaged in sex while simultaneously performing epoche – hardly the same experience at all.” Daniel Dennett, “Two Approaches to Mental Images.” Brainstorms, 185.

Preamble
Speedy Onan
(Blessed in his age)
Spills his delicate seed
Beside his wife’s behind,
But out of sight.

Ha! El-Shaddai takes note;
In time, Gehinnom’s guestbook’s listless pages, catalyzed,
flip and crack with age.
Which is funny, since
the whole thing was not to give birth.

Father Onan, give us tension,
Give us strength.

Let us create.

I.
Let me see past the walls,
Through the houses, possibilities
Rows on rows apart,
Through your blouse, skirt, pants,
And past the place where I am not,

Or back past the evening,
Through the steps I took,
Past where we were alone, a bit,
Through the outrageous veil of tame actuality,

Or through all time and laws of space –
In short, past all the physical supports –
Locks on our bodies, nothing more –
Past anything other than suggestions of motion,

Also, past your faults of height,
Your too-much hair,
Past here and there a bit more or less of this or that,
Past where you were wholly there –
Accidents all, accidents all –:

Past the fact that once nothing’s in the way of you and me,
then that’s in the way,
you alone
my thoughts’ mundane, gnomish fixture.

Then… then, perhaps what I really like
Is the way that
What I have stripped the scene to
Rests on my clear, glowing, pulsing eyelids;
Through the wall-less, naked, possible, timeless, spaceless, barely suggestive mental act,
I can still see those in back.
Still I can see my eyelids.

II.
My better half sends half-flaccid, unfathomable tendrils,
Dredged up from old-time psychology,
Jerking – explode, collapse, explode, collapse –
Through tensed muscles,
Out into the mental-lexical silk
of semantic space:

humey, human, predicament, boobs, david,
david, uriah, you, rapist,
ending,
reference, inscrutable reference.


Oh, uhu, awa, t-t, my God.
What was that twitch in my leg?
Huh – what did that amount to?
{Khaw-khem.} Yet is there a place on earth,
For all the things to which our dreams give birth?

On What There Is and What We Can Perceive

Our knowledge of the physical world depends on our perceptual faculties being just about as sensitive as they are. Consider an ideally rational animal, or ideally rational community of inquirers, with only, say, our auditory and proprioceptive faculties. Call it A. It is hard to imagine how A could develop anything we should consider a complete physics. Perhaps I’m wrong – I’m only going on conceivability here – but it sure seems as if the ontology of the theory A would possess at the end of inquiry just could not be the same as any ontology human beings will reach at the end of inquiry.*

A’s theory and ontology would be, in some sense, inadequate. This is because we have all of the evidence A has and more, and A’s theory is inadequate to account for all of our evidence. My intuition is that this inadequacy has some sort of normative force; that is, in some sense, A should adopt a theory that is able to account for our evidence as well as our own completed physics.

Here’s my question: What reason do we have, if any, not to believe that, for all we know, we are in A’s position? More precisely, what reason do we have, if any, not to believe that, for all we know, there is some (perhaps uninstantiated) perceptual faculty, such that our theories should account for the evidence this faculty would furnish for us, were we to have it, in the same sense that A’s theory should account for the evidence furnished by our visual faculties? To be sure, we have pretty good evidence to suspect that none of the animals on Earth have such a perceptual faculty. The question is whether, for all we know, something could have such a faculty, with the world and its contents remaining otherwise entirely as they are.

The only reason I can think of – if it is true – would be furnished by a certain sort of “instrumentalist” meta-ontology. I take instrumentalism to be the view that ontology O is to be preferred over ontology P just in case O is better suited to our purposes than P. Once we have all the evidence in hand, we answer these sorts of “big picture” ontological questions (Carnap might have called them external questions) by recourse to our purposes in theorizing. Suppose our purposes are just to develop a theory with all of the virtues of descriptive adequacy, explanatory power, and simplicity that accounts as well as possible for all of the way things appear to us.** In that case, it is clear that no such troublesome perceptual faculty could exist, because no perceptual faculties other than our own can have any normative bearing on our choice of ontology. Note that this sort of meta-ontology makes it difficult to hold on to the initial intuition I had about A’s epistemic situation – namely, that A should adopt a theory that is able to account for evidence outside of her own perceptual purview. If we think that ontologies are beholden, at last, only to the way things appear to the community of inquiry, then the percepts of outsiders, real or imagined, can have no rational bearing on what exists, in our language.

It’s not clear which I like more – this version of instrumentalism or the offending intuition. Is there some adjudicating evidence that I’ve left out of court? Can anyone point me in the direction of some relevant literature?
____________________
* - I’m assuming here that neither we nor A ever experience any sort of divine revelation or anything like that.
** - This description of “our” purposes is really unsatisfactory to me, but I can’t do any better yet. For what it’s worth, it probably beats Carnap in ESO, where he suggests that the purpose for which he uses the language of serious theory is “the purpose of communicating factual knowledge.”

"Single-horned 'Unicorn' deer is found in Italy"

So much for unicorns being essentially fictional.

Poetic Acts

Poetic acts are acts with potentially limitless reference. Interpretations of poetic acts are legitimated by the recognition of some thoroughgoing mapping between the act and its scene, on the one hand, and some other domain. Poetic acts refer to whatever it is that they map onto in these other domains, or whatever those things refer to. The legitimation of an interpretation is both epistemic and ontic. That is, we learn of these interpretations just as they come into existence – by grasping them mentally. An interpretation can become legitimate, even though the domain might not have had a configuration appropriate for the interpretation at the time of the act; as the world changes, as our concepts and knowledge change, so do the possible interpretations of poetic acts. The reference of a poetic act is potentially limitless because there is no (obvious) limit to the mappings that might be grasped by human minds, or the domains into which these mappings might be projected.

I think that the aesthetic evaluation of poetic acts is, in large part, a matter of aesthetically evaluating their referents, with extra weight given to more detailed mappings and mappings with greater salience. If we take a sports game, religious ritual, or dance performance to be a poetic act, then, whatever the other aesthetic characteristics of the game or performance, it seems that it grows more beautiful the more beautiful its referents, or more profound the more profound its referents, or more tasteless the more tasteless its referents. One explanation of what is sometimes unsettling (to me) about competitive sports when I have war or human conflict on my mind is that there is such a clear analogy between the sport and these domains.

There is something more to say about the attractiveness of poetic acts and the attractiveness of the religious life. Stay tuned.