Do stipulations change the language?

Technical and legal writing often contains stipulations. The stipulations change the meanings of words already in the language and sometimes introduce neologisms. It seems, however, that technical and legal writing in English is still writing in English. After all, the stipulations are given in English, and stipulation is a mechanism of the English language, akin to macros in some computer programming languages. But we can now suppose that there is a pair of genuinely distinct natural languages, A and B, such that the grammatical structure of A is a subset of the grammatical structure of B, so that if we take any sentence of A, we can translate it word-by-word or word-by-phrase to a sentence of B. Now we can imagine Jan is a speaker of B and as a preamble she goes through all the vocabulary of A and stipulates its meaning in B. She then speaks just like a speaker of A.

When Jan utters something that sounds just like a sentence of A, and means the same thing as the sentence of A, is she speaking B or A? It seems she is speaking B. Stipulation is a mechanism of B, after all, and she is simply heavily relying on this mechanism.

Of course, there probably is no such pair of natural languages. But there will be partial cases of this, particularly if A is restricted to, say, a technical subset, and if we have a high tolerance for artificial-sounding sentences of B. And we can imagine that eventually a human language will develop (whether "naturally" or by explicit construction) that not only allows the stipulation of terms, but has highly flexible syntax, like some programming languages. At this point, they will be able to speak their extensible language, but with one preamble sound just like speakers of French and with another just like speakers of Mandarin. But the language itself wouldn't be a superset of French or Mandarin. And eventually the preamble could be skipped. The language could have a convention where by adopting a particular accent and intonation, one is implicitly speaking within the scope of a preamble made by another speaker, a preamble that stipulated which accent and intonation counted as a switch to the scope of that preamble. Then all we would need to do is to have a speaker (or a family of speakers) give a French-preamble and another speaker give a Mandarin-preamble. As soon as any speaker of our flexible language starts accenting and intoning as in French or Mandarin, their language falls under the scope of the preamble. (The switch of accent and tone will be akin to importing a module into a computer program.) But it's important to note that the production of a preambles should not be thought of as a change in the language any more than saying "Let's call the culprit x" changes English. It's just another device within the old language.

What's the philosophical upshot of these thought experiments? Maybe not that much. But I think they confirm some thoughts about language that people have had already. First, the question of when a language is being changed and when one is simply making use of the flexible facilities of the original question is probably not well-defined. Second, given linguistic flexibility, the idea of context-free sentences and of lexical meaning independent of context is deeply problematic. Stipulative preambles are a kind of context, and any sentence can have its meaning affected by them. There might be default meanings in the absence of some marker, but the absence of a marker is itself a marker. Third, we get further confirmation of the point here that syntax is in general semantically fraught, since it is possible to make the choice of preamble be conditional on how the world is. Fourth, this line of thought makes more plausible the idea that in some important sense we are all speaking subsets of the same language (cf. universal grammar).

This post is based on a line of inquiry I'm pursuing: What can we learn about language from computer languages?

Top-down and bottom-up syntax

There are two fundamentally different approaches to syntax. One way starts at the bottom, with fundamental building blocks like names, variables and predicates, and thinks of a sentence as built up out of these by applying various operators. Thus, we get "The cat is on the mat and the dog is beside the mat" from elements like "the cat", "is on", "the mat", "the dog" and "is beside", by using operators like conjunction and binary-predication:

1. "The cat is on the mat and the dog is beside the mat" = conjunction(binary-predication("is on", "the cat", "the mat"), binary-predication("is beside", "the cat", "the mat")).

We can then parse the sentence back down into the elements it came from by inverting the operators (and if the operators are many-to-one there will be parsing ambiguity).

The other approach starts at the top with a sentence (or, more generally, well-formed formula) and then parses it by using parsing relations like conjoins (e.g., "p and q" conjoins "p" and "q") or binarily-applies (e.g., "the cat is on the mat" binarily-applies "is on" to "the cat" and "the mat").

There are four reasons I know of for preferring the top-down approach.

A. The possibility of multiple ways of expressing the same structure. For instance, "p and q" conjoins "p" and "q", but it's not the only way of conjoining these: "p but q" also conjoins "p" and "q". The bottom-up approach can handle this by having multiple conjunction operators like conjoin-with-and, conjoin-with-but and conjoin-with-and-also, but then we need to introduce a higher order property of these operators that says that they are conjunctions. Moreover, we should not suppose separate operators in cases where the meaning is the same, and sometimes the meaning will be exactly the same.

B. Partial sense. There is no way of forming the sentence

2. 2+2=5 and the borogove is mimsy

in the bottom-up approach, because "borogove" is not a noun of English and "is mimsy" is not a predicate of English, so there is nothing to plug into a unary-predication operator to form the second conjunct. But on the top-down approach, we can do a first step of parsing the sentence: (2) conjoins "2+2=5" and "the borogove is mimsy". And we know that one conjunct is false, so we conclude that (2) isn't true before we even start asking whether the second conjunct makes sense.

C. Ungrammatical sentences. The bottom-up approach has no way of making sense of ungrammatical sentences like a non-native speaker's

3. Jane love Bob.

For there is no predicate F such that the sentence is equal to binary-predicate(F, "Jane", "Bob"), so there is no way of parsing. But the top-down approach is not committed to all sentences coming from application of specified predicates. But the top-down approach can say that (3) binarily-applies "loves" to "Jane" and "Bob", school-marmish opinions to the contrary notwithstanding. The bottom-up approach can handle ungrammatical sentences in two different ways. One way is to suppose that any particular ungrammatical sentence is in fact a mistaken version of a grammatically correct sentence. Maybe that's true for (3), but I doubt that this is tenable for the full range of understandable but grammatically incorrect sentences. The second is to include a range of ungrammatical operators, such as binary-predicate-dropping-suffix-s. This is not satisfactory—there are too many such.

D. Extensibility. It's an oversimplification to think that a sentence that applies a predicate is formed simply out of the predicate and its arguments by means of a predication operator. There are other elements that need to be packaged up into the sentence, such as emphasis, degree of confidence, connotation, etc. These may be conveyed by tone of voice, context or choice of "synonym". One could handle this in two ways on the bottom-up view. One way is to add additional argument slots to the predication operators, slots for emphasis type, confidence, connotation, etc. This is messy, because as we discover new features of our language, we will have to keep on revising the arity of these operators. The second approach is to suppose that a sentence is formed by applying additional operators, such an emphasis operator or a confidence operator, after applying, say, the last predication operator. Thus, a particular instance of "Socrates is wise" might be the result of:

4. confidence(emphasis(predication("Socrates, "is wise"), 3.4), .98).

But now we can't take the resulting sentence and directly parse it into subject and predicate by simply inverting the predication operator. We first have to invert the confidence operator, and then we have to invert the emphasis operator. In other words, parsing requires a large number of other operators to invert. But on the top-down approach, this is easy. For if S is our confidenced and emphasized token of "Socrates is wise", then applies(S, "is wise", "Socrates"). No need to invert several additional operators to say that. If we are interested in the other features of S, however, then we can see what other parsing predicates, such as has-confidence, can be applied to S. But that's optional. Because we are not parsing in principle by inverting compositional operators, we don't need to worry about the other operators when we don't care about that aspect of the communicative content.

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There is also a down-side to the top-down approach. Because of point C, we have no way of codifying its parsing predicates like binarily-applies for natural languages. That, I think, is exactly how it should be.

Names, quantifiers, Aristotelian logic and one-sided relations

This is going to be a pretty technically involved post and it will be written very badly, as it's really just notes for self. Start with this objection to Aristotelian logic. A good logical system reveals the deep logical structure of sentences. But Aristotelian logic takes as fundamental sentences like:

1. Everyone is mortal.
2. Socrates is mortal.

In so doing, Aristotelian logic creates the impression that (1) and (2) have similar logical form, and it is normally taken to be that modern quantified logic has shown that (1) and (2) have different logical forms, namely:

3. ∀x(Mortal(x))
4. Mortal(Socrates).

I shall show, however, that there is a way of thinking about (1) and (2), as well as about (3) and (4), that makes them have the same deep logical form, as Aristotelian logician makes it seem. (This is a very surprising result for me. Until I discovered these ideas this year, I had a strong antipathy to Aristotelian logic.) Moreover, this will give us some hope of understanding the medieval idea of one-sided relations. The medievals thought, very mysteriously, that creation is a one-sided relation: we are related to God by the created by relation, but God is not related to us by the creates relation.

Now to the technical stuff. Recall Tarski's definition of truth in terms of satisfaction. I think the best way to formulate the definition is by means of a substitution sequence. A substitution sequence s is a finite sequence of variable-object pairs, which I will write using a slash. E.g., "x₁"/Socrates,"x₂"/Francis,"x₃"/Bucephalus is a substitution sequence. The first pair in my example consists of the variable letter "x₁", a linguistic entity (actually in the best logic we might have slot identifiers instead of variable letters) and Socrates—not the name "Socrates" (which is why the quotation marks are as they are). We then inductively define the notion of a substitution sequence satisfying a well-formed formula (wff) under an interpretation I. An interpretation I is a function from names and predicates to objects and properties respectively. And then we have satisfaction simpliciter which is satisfaction under the intended interpretation, and that's what will interest me. So henceforth, I will be the intended interpretation. (I've left out models, because I am interested in truth simpliciter.) We proceed inductively. Thus, s satisfies a disjunction of wffs if and only if it satisfies at least one of the wffs, and so on, the negation of a wff if and only if it does not satisfy the wff, and so on.

Quantifiers are a little more tricky. The sequence s satisfies the wff ∀xF iff for every object u, the sequence "x"/u,s (i.e., the sequence obtained by prepending the pair "x"/u" at its head) satisfies F. The sequence s satisfies ∃xF iff for some object u, the sequence "x"/u,s satisfies F.

What remains is to define s's satisfaction of an atomic wff, i.e., one of the form P(a₁,...,a_n) where a₁,...,a_n are a sequence of names or variables. The standard way of doing this is as follows. Let u₁,...,u_n be a sequence of objects defined as follows. If a_i is a variable "x", then we let u_i be the first object u occuring in s paired with the variable "x". If for some i there is none such pair in s, then we say s doesn't satisfies the formula. If a_i is a name "n", then we let u_i=I("n"). We then say that s satisfies P(a₁,...,a_n) if and only if u₁,...,u_n stand in I(P).

Now notice that while the definition of satisfaction for quantified sentences is pretty neat, the definition of satisfaction for atomics is really messy, because it needs to take into account the question of which slot of the predicate has a variable in it and which one has a name.

There is a different way of doing this. This starts with the Montague grammar way of thinking about things, on which words are taken to be functors from linguistic entities to linguistic entities. Let us ask, then, what kind of functors are represented by names. Here is the answer that I think is appealing. A name, say "Socrates", is a functor from wffs with an indicated blank to wffs. In English, the name takes a wff like "____ likes virtue" and returns the wff (in this case sentence) "Socrates likes virtue". (The competing way of thinking of names is as zero-ary functors. But if one does it this way, one also needs variables as another kind of zero-ary functor, which I think is unappealing since variables are really just a kind of slot, or else one has a mess in treating atomics differently depending on which slots are filled with names and which with variables.) We can re-formulate First Order Logic so that a name like "Socrates" is (or at least corresponds to) a functor from wff-variable pairs to new wffs. Thus, when we apply the functor "Socrates" to the wff "Mortal(x)" and the variable "x", we get the wff (sentence, actually) "Mortal(Socrates)". And the resulting wff no longer has the variable "x" freely occurring in it. But this is exactly what quantifiers do. For instance, the universal quantifier is a functor that takes a wff and a variable, and returns a new wff in which the variable does not freely occur.

If we wanted the grammar to indicate this with particular clarity, instead of writing "Rides(Alexander, Bucephalus)", we would write: "Alexanderx Bucephalusy Rides(x,y)". And this is syntactically very much like "∀x∀y Rides(x,y)".

And if we adopted this notation, the Tarski definition of satisfaction would change. We would add a new clause for the satisfaction of a name-quantified formula: s satisfies nxF, where "n" is a name, if and only if "x"/I("n"),s satisfies F. Now once we got to the satisfaction of an atomic, the predicate would only be applied to variables, never to names. And so we could more neatly say that s satisfies P(x₁,...,x_n) if and only if every variable occurs in the substitution sequence and u₁,...,u_n stand in I(P) where u_i is the first entity u occurring in s in a pair of the form "x_i"/u.  Neater and simpler, I think.

Names, thus, can be seen as quantifiers. It might be thought that there is a crucial disanalogy between names and the universal/existential quantifiers, in that there are many names, and only one universal and only one existential quantifier. But the latter point is not clear. In a typed logic, there may be as many universal quantifiers as types, and as many existential ones as types, once again. And the number of types may be world-dependent, just as the number of objects.

If I am right, then if we wanted to display the logical structure of (1) and (2), or of (3) and (4) for that matter, we would respectively say:

5. ∀x Mortal(x)
6. Socratesx Mortal(x).

And there is a deep similarity of logical structure—we simply have different quantifiers. And so the Aristotelian was right to see these two as similar.

Now, the final little bit of stuff. Obviously, if "m" and "n" are two names, then:

7. "mx ny F(x,y)" is true iff "ny mx F(x,y)" is true,

just as:

8. "∀x∀yF(x,y)" is true iff "∀y∀xF(x,y)" is true.

But the two sentences in (8), although they are logically equivalent, arguably express different propositions. And I submit that so do the two sentence in (7). And we even have a way of marking the difference in English, I think. Ordinarily, what the left hand side in (7) says is that u has the property P_xnyF(x,y) while the right hand side in (8) says that v has the property P_ymxF(x,y), where u and v are what "m" and "n" respectively denote, and P_xH(x) is the (abundant) property corresponding to the predicate H (the P-thingy is like the lambda functor, except it returns a property, not a predicate). These are distinct claims.

The medievals then claim that in the case of God we have this. They say that "Godx ny F(x,y)" is true in virtue of "ny Godx F(x,y)" being true. It is to the referent of "n" that the property P_y Godx F(x,y) is attributed, and the sentence that seems to attribute a property to God is to be analyzed in terms of the one that attributes a property to the referent of "n".

Grammar, types and tokens

Question: Does grammar primarily govern the relations between word-types or those between word-tokens?

Answer: Grammar does not primarily govern the relations between word-tokens. In human spoken and written languages, there are are tokens corresponding to parts of sentence types like the subject or the predicate. Thus, a token of the sentence "Paris is full of snow" contains a token of "Paris" and a token of "is full of snow". But this is a mere accident of our languages. We can easily imagine languages where the grammatical parts of a sentence type do not correspond to physical parts of a sentence token. For instance, we could imagine that a language can only be spoken via Goedel numbering. In such a language, we can still have a complex grammar, and there will be tokens of sentences—e.g., Arabic numeral expressions of Goedel numbers—but there need be no tokens of individual words. We could, I suppose, stipulate that a word is tokened when a sentence containing it is tokened, but that only gives us acts of tokening and not tokens. And, ontologically, it is not clear that there would be a separate act of tokening for each of the parts—maybe one could just say the sentence as a whole, without thinking about the parts. (I am a coarse-grained action theorist.)

One can imagine languages where the only tokens are tokens of sentences, but where there is still a Montague grammar. But in such a language, the arguments of the functors do not correspond to tokens.

Since we want the phenomenon of grammar to be as uniform as we can make it across imaginable languages, we should not take grammar to govern the relations between tokens, or even potential tokens, because some languages just don't have enough of these.

But strictly speaking we should not take grammar to govern the relations between types. For in a language where there are no tokens corresponding to, say, individual nouns, but only sentence tokens, the grammar may still take account of nouns. But these nouns won't be types, because a type is the sort of thing that is supposed to have a token. Rather, in such a language we would introduce abstract entities to play the role of types, but these abstract entities wouldn't actually be types, since there would be no type-token relation defined for them. We could call these entities "linguistic items". The grammar of the language would then specify how linguistic items can combine into other linguistic items, in the standard Montague grammar way. And then some special, distinguished linguistic items—for instance, sentences—would have the additional property of being expressible by a token. And these linguistic items would also be types.

So in fact the answer to the question is "neither". Grammar governs the relations between linguistic items. But these items need be neither types nor tokens. (This undercuts Goodman-Quine attempts to do grammar at the token level.) Then there is something other than grammar, which we might call coding, which governs the relation between the special linguistic items that are expressible by a token and their tokens. And of course there will be semantics/pragmatics (I do not distinguish these, though of course most do).

So what are these linguistic items? One option, inspired by Rob Koons: Carefully delineated social practices. I am not sure this will work, but it might. Second option: Don't worry! Just do your grammar, coding and semantics/pragmatics in terms of linguistic items, and Ramseyfy. I also don't know if this will work, but it might.

Some naive thoughts on syntax

I am neither a linguist nor a philosopher of language, so what I will say is naive and may be completely silly.

It seems to be common to divide up the task of analyzing language between syntax and semantics. Syntax determines how to classify linguistic strings into categories such as "sentence", "well-formed formula", "predicate", "name", etc. If the division is merely pragmatic, that's fine. But if something philosophical is supposed to ride on the division, we should be cautious. Concepts like "sentence" and "predicate" are ones that we need semantic vocabulary to explain—a sentence is the sort of thing that could be true or false, or maybe the sort of thing that is supposed to express a proposition. A predicate is the sort of thing that can be applied to one or more referring expressions.

If one wants syntax to be purely formal, we should see it as classifying permissible utterances into a bunch of formal categories. As pure syntactitians, we should not presuppose any particular set of categories into which the strings are to be classified. If we are not to suppose any specific semantic concepts, the basic category should be, I think, that of a "permissible conversation" (it may well be that the concept of a "conversation" is itself semantic—but it will be the most general semantic concept). Then, as pure syntactitians, we study permissible conversations, trying to classify their components. We can model a permissible conversation as a string of characters tagged by speaker (we could model the tagging as colors—we put what is spoken by different people in different colors). Then as pure syntactitians, we study the natural rules for generating permissible conversations.

It may well be that in the case of a human language, the natural generating rules for speakers will make use of concepts such as "sentence" and "well-formed formula", but this should not be presupposed at the outset.

Here is an interesting question: Do we have good reason to suppose that if we restricted syntax to something to be discovered by this methodology, the categories we would come up with would be at all the familiar linguistic categories? I think we are not in a position to know the answer to this. The categories that we in fact have were not discovered by this methodology. They were discovered by a mix of this methodology and semantic considerations. And that seems the better way to go to generate relevant syntactic categories than the road of pure syntax. But the road that we in fact took does not allow for a neat division of labor between syntax and semantics, since many of our syntactic categories are also natural semantic ones, and their semantic naturalness that goes into making them linguistic relevant.

Ungrammatical sentences

I think there is something important to be learned in the philosophy of language from the fact that grammatically wrong sentences often succeed in clearly expressing propositions. (Maybe something along the lines of the claim that speaker meaning is the only meaning there is.)