Semantics of syntactically incorrect language

As anyone who has talked with a language-learner knows, syntactically incorrect sentences often succeed in expressing a proposition. This is true even in the case of formal languages.

Formal semantics, say of the Tarski sort, has difficulties with syntactically incorrect sentences. One approach to saving the formal semantics is as follows: Given a syntactically incorrect sentence, we find a contextually appropriate syntactically correct sentence in the vicinity (and what counts as vicinity depends on the pattern of errors made by the language user), and apply the formal semantics to that. For instance, if someone says “The sky are blue”, we replace it with “The sky is blue” in typical contexts and “The skies are blue” in some atypical contexts (e.g., discussion of multiple planets), and then apply formal semantics to that.

Sometimes this is what we actually do when communicating with someone who makes grammatical errors. But typically we don’t bother to translate to a correct sentence: we can just tell what is meant. In fact, in some cases, we might not even ourselves know how to translate to a correct sentence, because the proposition being expressed is such that it is very difficult even for a native speaker to get the grammar right.

There can even be cases where there is no grammatically correct sentence that expresses the exact idea. For instance, English has a simple present and a present continuous, while many other languages have just one present tense. In those languages, we sometimes cannot produce an exact grammatically correct translation of an English sentence. One can use some explicit markers to compensate for the lack of, say, a present continuous, but the semantic value of a sentence using these markers is unlikely to correspond exactly to the meaning of the present continuous (the markers may have a more determinate semantics than the present continuous). But we can imagine a speaker of such a language who imitates the English present continuous by a literal word-by-word translation of “I am” followed by the other language’s closest equivalent to a gerund, even when such translation is grammatically incorrect. In such a case, assuming the listener knows English, the meaning may be grasped, but nobody is capable of expressing the exact meaning in a syntactically correct way. (One might object that one can just express the meaning in English. But that need not be true. The verb in question may be one that does not have a precise equivalent in English.)

Thus we cannot account for the semantics of syntactically incorrect sentences by applying semantics to a syntactically corrected version. We need a semantics that works directly for syntactically incorrect sentences. This suggests that formal semantics are necessarily mere approximate models.

Similar issues, of course, arise with poetry.

What numbers could be

Benacerraf famously argued that no set theoretic reduction can capture the natural numbers. While one might conclude from this that the natural numbers are some kind of sui generis entities, Benacerraf instead opts for a structuralist view on which different things can play the role of different numbers.

The argument that no set theoretic reduction captures the negative numbers is based on thinking about two common reductions. On both, 0 is the empty set ⌀. But then the two accounts differ in how the successor sn of a number n is formed:

1. sn = n ∪ {n}

2. sn = {n}.

On the first account, the number 5 is equal to the set {0, 1, 2, 3, 4}. On the second account, the number 5 is equal to the singleton {{{{{⌀}}}}}. Benacerraf thinks that we couldn’t imagine a good argument for preferring one account over another, and hence (I don’t know how this is supposed to follow) there can’t be a fact of the matter about why one account—or any other set-theoretic reductive account—is correct.

But I think there is a way to adjudicate different set-theoretic reductions of numbers. Plausibly, there is reference magnetism to simpler referrents of our terminology. Consider an as consisting of a set of natural numbers, a relation <, and two operations + and ⋅, satisfying some axioms. We might then say that our ordinary language arithmetic is attracted to the abstract entities that are most simply defined in terms of the fundamental relations. If the only relevant fundamental relation is set membership ∈, then we can ask which of the two accounts (a) and (b) more simply defines <, + and ⋅.

If simplicity is brevity of expression in first order logic, then this can be made a well-defined mathematical question. For instance, on (a), we can define a < b as a ∈ b. One provably cannot get briefer than that. (Any definition of a < b will need to contain a, b and ∈.) On the other hand, on (b), there is no way to define a < b as simply. Now it could turn out that + or ⋅ can be defined more simply on (b), in a way that offsets (a)’s victory with <, but it seems unlikely to me. So I conjecture that on the above account, (a) beats (b), and so there is a way to decide between the two reductions of numbers—(b) is the wrong one, while (a) at least has a chance of being right, unless there is a third that gives a simpler reduction.

In any case, on this picture, there is a way forward in the debate, which undercuts Benacerraf’s claim that there is no way forward.

I am not endorsing this. I worry about the multiplicity of first-order languages (e.g., infix-notation FOL vs. Polish-notation FOL).

Semantic determinacy and indeterminacy

There are arguments that our language is paradoxically indeterminate. For instance, Wittgenstein-Kripke arguments for underdetermination of rules by cases, Quine’s indeterminacy of translation arguments, or Putnam’s model-theoretic arguments.

There are also arguments that our language is paradoxically determinate. First order logic shows that there is a smallest number of grains of sand that’s still a heap.

In other words, there are cases where we want determinacy, and we find indeterminacy threatening, and cases where we want indeterminacy, and we find determinacy puzzling. I wonder if there is any relevant difference between these cases other than the fact that we have different intuitions about them.

If we are to go with our intuitions, we need to bite the bullet on, or refute, both sets of arguments, in their respective cases. But if we embrace determinacy everywhere or embrace indeterminacy everywhere, then it’s neater: we only need to bite the bullet on, or refute, one family of arguments.

I find embracing determinacy everywhere rather attractive.

Sex as an iconic partially self-representing gesture

“Iconic representational gestures” are like a gestural onomatopoeia: their physical reality resembles in some way what they signify. For instance, blowing a kiss signifies a kiss, running a finger across a throat signifies a killing, and a baptism signifies cleansing from sin.

An interesting special case of iconic representational gestures is one where the physical reality of the gesture itself itself accomplishes a part of what it represents. A slap in the face is an iconic gesture that represents the punishment that the other party deserves for bad behavior and is itself physically a part of the punishment. Intercourse is an iconic gesture that signifies a union of persons and its physical reality constitutes the physical part of that union. And, on views on which Christ’s body is present in the Eucharist, the reception of the Eucharist is also such an iconic gesture representing union with Christ and physically effecting an aspect of that union. We can call such gestures partially self-representing.

Now, normally meaning gets attached to symbolic acts like words and gestures through other symbolic acts (you point to a “zebra” and say “Let’s call that ‘zebra’”). This threatens to lead to a regress of symbolic acts. The regress can only be arrested by symbolic acts that have an innate meaning. Now, while there is often an element of conventionality even in iconic representational gestures, just as there is in onomatopoeia, nonetheless I think our best candidate for symbolic acts that have an innate meaning is iconic representational gestures. Moreover, if the gesture has an innate meaning, it is plausible that it was used at least as long as humankind has been around.

If we think about the best candidates for such gestures, we can speculate that perhaps pointing or punching has been around as long as humans have been around. But that’s speculation. But it’s not speculation that sex has been around as long as humans have been around. Thus, sex is an excellent candidate for a gesture that has the following features:

• iconic representational

• partially self-representing

• innate meaning.

Moreover, given that the physical aspect of sex is a thorough biological union, it is very reasonable to think that this innate meaning is a thorough personal union. But, as Vincent Punzo has noted in his work on sex, a thorough personal union needs to include a normative commitment for life. And that is marriage. Thus, sex signifies marriage.

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?

Syntax and semantics

One of the things that I've been puzzled by for a long time is the distinction between syntax and semantics. Start with this syntactically flawed bit of English:

1. Obama equals.

It is syntactically flawed, because "to equal" is a transitive verb, and a sentence that applies an intransitive verb to a single argument is ungrammatical, just as an atomic sentence in First Order Logic that applies a binary predicate to one argument is ungrammatical. (I leave open the further question whether "Obama equals Obama" is grammatically correct; maybe the arguments of the English "equals" have to be quantities.) This is a matter of syntax. But now consider this more complicated bit of language:

2. Let xyzzing be sitting if the temperature is more than 34 degrees and let it be equalling otherwise. Obama xyzzes.

My second sentence makes perfect sense when the temperature is 40 degrees, but is ungrammatical in exactly the same way that (1) is when the temperature is 30 degrees. Its grammaticality is, thus, semantically dependent.

One might object that the second sentence of (2) is syntactically correct even when the temperature is 30 degrees. It's just that it then has a semantic value of undefined. This move is similar to how we might analyze this bit of Python code:

def a(f): print(f(1))
def g(x,y): x+y
def h(x): 2*x
a(h if temperatureSensor()>34 else g)

This code will crash with

TypeError: <lambda>() takes exactly 2 arguments (1 given)

when the temperature sensor value is, say, 30. But the behavior of a program, including crashing, is a matter of semantics. The Python standard (I assume) specifies that the program is going to crash in this way. I could catch the TypeError if I liked with try/except, and make the program politely print "Sorry!" when that happens instead of crashing. There is no syntactic problem: print(f(1)) is always a perfectly syntactically correct bit of code, even though it throws a TypeError whenever it's called with f having only one argument.

I think the move to say that it is the semantic value of the second sentence of (2) that depends on temperature, not its grammaticality, is plausible. But this move allows for a different way of attacking the distinction between syntax and semantics. Once we've admitted that the second sentence of (2) is always grammatical but sometimes has the undefined value, we can say that (1) is grammatically correct, but always has the semantic value of undefined, and the same is true for anything else that we didn't want to consider grammatically correct.

One might then try to recapture something like the syntax/semantics distinction by saying things like this: an item is syntactically incorrect in a given context provided that it's a priori that its semantic value in that context is undefined. This would mean that (2) is syntactically correct, but the following is not:

3. Let xyzzing be sitting if Fermat's Last Theorem is false and let it be equalling otherwise. Obama xyzzes.

For it's a priori that Fermat's Last Theorem is true. I think, though, that a syntax/semantics distinction that distinguishes (2) from (3) is too far from the classical distinction to count as an account of it.

It may, however, be the case that even if there is no general distinction between syntax and semantics, in the case of particular languages or families of languages one can draw a line in the sand for convenience of linguistic analysis. But as a rule of thumb, nothing philosophically or semantically deep should rely on that line.

Now it's time to be a bit of a hypocrite and prepare my intermediate logic lecture, where instilling the classical distinction between syntax and semantics is one of my course objectives. But FOL is a special case where the distinction makes good sense.

System-relativity of proofs

There is a generally familiar way in which the question whether a mathematical statement has a proof is relative to a deductive system: for a proof is a proof in some system L, i.e., the proof starts with the axioms of L and proceeds by the rules of L. Something can be provable in one system—say, Euclidean geometry—but not provable in another—say, Riemannian geometry.

But there is a less familiar way in which the provability of a statement is relative. The question whether a sentence p is provable in a system L is itself a mathematical question. Proofs are themselves mathematical objects—they are directly the objects in a mathematical theory of strings of symbols and indirectly they are the objects of arithmetic when we encode them using something like Goedel numbering. The question whether there exists a proof of p in L is itself a mathematical question, and thus it makes sense to ask this question in different mathematical systems, including L itself.

If we want to make explicit both sorts of relativity, we can say things like:

1. p has (does not have) a proof in a system L according to M.

Here, M might itself be a deductive system, in which case the claim is that the sentence "p has (does not have) a proof in L" can itself be proved in M (or else we can talk of the Goedel number translation of this), or M might be a model in which case the claim is that "p has a proof in L" is true in that model.

This is not just pedantry. Assume Peano Arithmetic (PA) is consistent. Goedel's second incompleteness theorem then tells us that the consistency of PA cannot be proved in PA. Skipping over the distinction between a sentence and its Goedel number, let "Con(PA)" say that PA is consistent. Then what we learn from the second incompleteness theorem is that:

2. Con(PA) has no proof in PA.

Now, statement (2), while true, is itself not provable in PA.[note 1] Hence there are non-standard models of PA according to which (2) is false. But there are also models of PA according to which (2) is true, since (2) is in fact true. Thus, there are models of PA according to which Con(PA) has no proof and there are models of PA according to which Con(PA) has a proof.

This has an important consequence for philosophy of mathematics. Suppose we want to de-metaphysicalize mathematics, move us away from questions about which axioms are and are not actually true. Then we are apt to say something like this: mathematics is not about discovering which mathematical claims are true, but about discovering which mathematical claims can be proved in which systems. However, what we learn from the second incompleteness theorem is that the notion of provability carries the same kind of exposure to mathematical metaphysics, to questions about the truth of axioms, as naively looking for mathematical truths did.

And if one tries to de-metaphysicalize provability by saying that what we are after in the end is not the question whether p is provable in L, but whether p is provable in L according to M, then that simply leads to a regress. For the question whether p is provable in L according to M is in turn a mathematical question, and then it makes sense to ask according which system we are asking it. The only way to arrest the regress seems to be to suppose that at some level that we simply are talking of how things really are, rather than how they are in or according to a system.

Maybe, though, one could say the following to limit one's metaphysical exposure: Mathematics is about discovering proofs rather than about discovering what has a proof. However, this is a false dichotomy, since by discovering a proof of p, one discovers that p has a proof.

Reference magnetism and anti-reductionism

According to reference magnetism, the meanings of our terms are constituted by requiring the optimization of desiderata that include the naturalness of referents (or, more generally, by making the joints in language correspond to joints in the world, as much as possible) and something like charity (making as many real-world uses as possible be correct).

Suppose we measure naturalness by the complexity of expression in fundamental terms—terms that correspond to perfectly natural things. (In particular, we can't talk of what cannot be expressed in fundamental terms, since reference magnetism would presumably not permit reference to what is infinitely unnatural.) Consider the reductionist thesis that the vocabulary of microphysics is the only fundamental vocabulary about the natural world. If this thesis is true, then our ordinary terms like "conscious" or "intention" or "wrong" are going to be cashed out in terms of extremely complex sentences, often of a functional sort. But I suspect that once these expressions are sufficiently complex, then there will be many non-equivalent variants of them that will fit our actual uses about as well and are about as complex. Consequently, we should expect that the meaning of terms terms like "conscious", "intention" and "wrong" to be highly underdetermined.

If we have reason to resist this underdetermination, we need to embrace an anti-reductionism on which the terms of microphysics are not the only fundamental ones, or else have another measure of naturalness.

Two kinds of pantheism

There are two kinds of pantheism. One might call them: reductive pantheism and world-enhancing pantheism.

Reductive pantheism says that the world is pretty much like it seems to us scientifically (though it might opt for a particular scientific theory, such as a multiverse one), and that God is nothing but this world. In so doing, one will be trying to find a place for the applicability of divine attributes for the world.

World-enhancing pantheism, however, says that there is more to the world than meets the eye. There is something numinous pervading us, our ecosystem, our solar system, our galaxy, our universe and all reality, with this mysterious world being a living organism that is God. World-enhancing pantheism paints a picture of a divinized world.

World-enhancing pantheism is a genuine religious view, one that leads to distinctive (and idolatrous!) practices of worshipful reverence for the world around us. Reductive pantheism, on the other hand, is a philosophers' abstraction.

It is an interesting question which version of pantheism is Spinoza's. His influence on the romantics is surely due to their taking him to be a world-enhancing pantheist, and he certainly sometimes sounds like one. But it is not clear to me that he is one. Though it may be that Spinoza has managed to do both: we might say that under the attribute of extension, we have a reductive pantheism, but the availability of the attribute of thought allows for a world-enhancing pantheism.

World-enhancing pantheism is idolatrous, while reductive pantheism is just a standard atheistic metaphysics with an alternate semantics for the word "God".

Norms, assertion and marriage

Cappelen has recently defended (see a chapter here) a "no-assertion" theory on which there is no special speech act of "assertion' associated with a special norm or set of norms. Rather, there is a multitude of contextual norms that saying can fall under.

Cappelen's most powerful argument applies to any theory on which being governed by a norm like the ones discussed under the head of "the norms of assertion"—standard proposals include: don't say the false, don't say what you don't believe, don't say what you don't know, etc.—is necessary for a speech act to be an assertion. The argument starts by noting that we can imagine people who engage in a practice that is governed by a different one of the proposed norms of assertion. For instance, instead of having a large number of token speech acts governed by Williamson's prefered "don't say what you don't know" norm, they may have a practice of speech acts governed by, e.g., "don't say what you don't believe." And such a group of people is surely no less asserting than we are, assuming of course there is such a thing as assertion. Moreover, there may be such variation within our very own community, as the apparent counterexamples to particular norms of assertion show.

I want to consider several answers. I think each of them is capable of defense, and then draw some interesting connections with debates on marriage.

1. Family resemblance. On this proposal, while there is no such thing as the norm of assertion, there is a family resemblance between all the proposed norms of assertion, and any practice governed by a norm that sufficiently resembles a member of the family counts as a practice of assertion. This allows that there is such a thing as assertion, but its boundaries are vague. This is a way of partially letting Cappelen have what he wants, but without giving up on assertion altogether. I think solutions like this should be a last resort.

2. The truth norm is triumphant. If we adopt the truth norm as constitutive of assertion, we can give a coherent story about what happens in communities where they have practices governed by other norms. Namely, in those communities, they are making assertions, but the content of their assertion is not the proposition most obviously indicated by their words. For instance, take the community K where there is a common practice participants in which are allowed to say "s" only when they know that s. The truth-normer can say that when a member of K is participating in that practice and says "s", she is in fact asserting the proposition that she knows that s. The move is to reinterpret the content of the speech act, and still make it an assertion, but of a slightly different content. There will of course have to be adjustments made elsewhere, too. Thus when A says "s" and B says "That's false", B is not referring to the content of A's assertion (which on this view is the proposition that A knows that s), but B is referring to the content of what A asserted knowledge of, namely that s. Thus, B is asserting that she knows that what A asserted to have knowledge of is false. Such adjustments are awkward but can be done, and the truth-normer, and only the truth-normer, can reinterpret speech under the other norms as assertion of a more complex content. This makes it easier for her to bite the bullet on Cappelen's objection, because she does not have to say that the members of these other communities are bereft of assertion. She might even say that there can be vagueness as to the proposition that someone is expressing, and that this vagueness may correspond to a vagueness as to what norm the saying is falling under.

It seems that this move privileges the truth norm—only the truth-normer can reinterpret speech acts governed by the other norms as assertions of a different content.

3. Focal meaning. A distinctively Aristotelian view is to accept a version of the family resemblance view, but insist that one norm is focal, and speech acts governed by the other norms are assertions in a derivative sense, in virtue of the resemblance between them and the focal norm. I think working out the details of this view will require something like the "norm magnet" view from below.

4. Natural norm bundles. Consider this hypothesis. Some types of activites are strongly natural to humans, in the sense that engaging in these activities is a constitutive part of natural human flourishing. For instance, while it is possible to lead an on-balance flourishing human life without ever eating (one might always be fed intravenously), such a life would lack an aspect of natural human flourishing (admittedly, not one of the most important ones). Among the strongly natural activities, one may hypothesize, there is the engagement in certain social practices. Perhaps requesting is a strongly natural social practice. A human who never requests anything of another is lacking a constitutive part of natural human flourishing—she is failing to flourish as the kind of dependent rational being she is. Likewise, it is pretty plausible that asserting is a strongly natural social practice. Baseball and asserting-in-English are not—one can live a fully flourishing life without participating in baseball or asserting in English.

How does this help? Well, a lot of people think that to make sense of semantic phenomena, we need the notion of "reference magnets", which are "natural" entities. Rabbits are reference magnets, while undetached rabbit parts aren't, and so saying "Gavagai!" around rabbits, barring some defeating data, refers to rabbits, not undetached rabbit parts. It could be that just as there are reference magnets, there are practice magnets. Reference magnets can be pretty strong. Someone who thinks that our language of mid-sized objects refers to spacetime regions may nonetheless be referring to a rabbit with the word "rabbit", even if she thinks she's referring to a spacetime region. Likewise, a community engaging in a practice that "looks" sufficiently similar to a practice magnet is actually engaging in the practice magnet.

Suppose that the real norm of assertion is the knowledge norm. We might have a community who think that they are engaging in a practice governed by, say, the belief norm, and their censure and praise is in line with the belief norm, but in fact their practice is similar enough to assertion that it gets pulled by the practice magnetism of assertion. Thus, the members of this community are in fact governed by the knowledge norm (assuming that's the real norm of assertion), even though their actions, especially their censure and praise, is a better fit for a different norm.

The notion of a practice magnet by itself is enough to show a weakness in Cappelen's argument. But practice magnetism is still a pretty murky idea. The notion of strongly natural practices may, however, help here. It could be that it is strongly natural practices that function as practice magnets. Any sufficiently similar behavior gets pulled in under the governance of the norms associated with the strongly natural practice.

This magnetism seems spooky (though maybe no more so than reference magnetism). Maybe, though, we can somewhat de-spookify it as follows, though to non-Aristotelians what I will say will be just as bad. When we assert, we are engaging in a strongly natural norm-governed social practice. But individually or as a community we need not have much grasp of what in fact are the norms of assertion, and we might in fact have quite mistaken ideas as to what the norms are. What makes it be that the practice has the norms of the practice of assertion is that it is a practice that arises in the right way out of our tendency to fulfill our nature in such-and-such respect (here, ostend to a particular tendency built into our nature). Notice that this suggestion no longer makes the norms be what ultimately constitutes the practice.

---

Now, here is something that I find quite interesting about all this. Analogous questions come up for marriage. While Cappelen's "no-assertion" theory is controversial and I suspect very much a minority position among philosophers, analogous views of marriage are, I think, going to be quite common among philosophers and other thinkers. These views will hold that there is no such thing as "the norms of marriage", and that there are simply multiple contextually defined practices. To those in the grip of such a "no-marriage" view, the idea that there is a fact of the matter as whether "the" nature of marriage allows this or that is unintelligible. At most there is a family of social practices, much as in suggestion 1.

The idea that there is such a thing as "the norms of marriage", to be investigated by moral and social philosophers, is intrinsically no more problematic than the idea that there is such a thing as "the norm (or norms) of assertion", to be investigated by philosophers of language and epistemologists. Each of the solutions on the side of assertion that I considered as options 2-4 has an analogue in the case of marriage.

I find particularly plausible the idea that marriage is a strongly natural practice with a strong practice magnetism. This allows someone who thinks that the norm of marriage requires monogamy to say that a man in a polygynous culture erroneously thinks that it is possible and permissible to marry more than one person simultaneously. The potential polygamist really is referring to marriage, even if he gets wrong what the norms are.

It may of course be that if you get the norms sufficiently wrong, the practice magnatism no longer pulls you in. Thus, someone who thought that "marriage" is what happens between people whenever they co-write a physics paper, and that the norms of marriage are the norms of co-authorship, would probably be using the word "marriage" in a different sense from us. And her drive for what she calls "marriage" would not be a species of the drive for marital flourishing.

(I do want to say, however, that in fact one of the norms of marriage is that it needs a commitment that is based upon a sufficient understanding of what norms one will have to live by. What kind of understanding is sufficient is a hard question. So we could have a case of someone whose understanding of the norms is sufficiently defective that unbeknownst to him, he is unable to enter into a valid marriage, as he lacks a sufficient understanding of the norms that is needed for the right kind of commitment to them, while at the same time he really wants to get married. That is a tragic situation.)

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.

Non-semantic definitions of truth

Here is a good reason to think that Tarski-style attempts at a definition of truth that do not make use of semantic concepts are going to fail. Such attempts are likely to make use of concepts like predicate and name. But these concepts are semantic concepts. A predicate is something can be applied to a name, and a name is something to which a predicate can be applied, and application is a semantic concept. Moreover, the definition of truth is going to have to presuppose an identification of the application function for the language (which takes a predicate and one or more names or free variables, and generates well formed formula, say by taking the predicate, appending a parenthesis, then appending a comma-delimited list of the names/variables, and then a parenthesis). But there is a multitude of functions from linguistic entities to linguistic entities, and to say which of them is application will be to make a semantic statement about the language.

Minimalism about truth

Consider the claim:

1. "Snow is white" is true because snow is white.

Say that a minimalist about truth is someone who thinks that statements like (1) fully explain all that calls out for explanation in the concept of truth.

Such a minimalist is wrong. It is clear that there is something fishy about (1) as a full explanation because the explanandum is about an object—the sentence "Snow is white"—which the explanans does not mention. In this regard, (1) is like the puzzling:

2. Fred smoked a cigarette because Maxine called up Patrick.

The explanandum is about Fred but the explanans has nothing about Fred. Claim (2) might be true—but if so, it is incomplete. It might be partially completed by adding that Patrick is a notorious gossip and Fred and Maxine had a deal that Fred would quit smoking while Maxine would quit gossiping.

Sometimes we do not notice that an explanation is incomplete because the additional facts are obvious: "Fred was jealous because Maxine kissed Patrick" needs nothing added if we know that Fred and Maxine are married and we know some facts of human psychology. But even so, the explanation is incomplete, enthymematic. And a sure sign of an anthymematic explanation is that the explanans does not mention the subject of the explanandum.

How to complete (1)? Maybe:

3. "Snow is white" is true because "Snow is white" says that snow is white, and snow is white.

Of course, normally we all know that "Snow is white" says that snow is white and so the first conjunct of the explanans is left off. Bu tit is needed, as is evident in cases where we do not understand the quoted phrase right away:

4. "Snieg jest bialy" is true because snow is white.

And once we relize that (1) is enthymematic for something like (3), we can see why (1) doesn't solve all the puzzles in the vicinity of "truth". For the obvious question after seeing (3) is: "Why does 'Snow is white' say that snow is white?" And here a correspondence theory may reappear as a side-effect of solving this problem of meaning (this observation is not original—I recall it in, I think, Ayer and in Davidson).

Semantics

I am reading Tarski's "The Semantic Conception of Truth" and came across this paragraph which I just had to blog:

It is perhaps worth while saying that semantics as it is conceived in this paper (and in former papers of the author) is a sober and modest discipline which has no pretensions of being a universal patent-medicine for all the ills and diseases of mankind, whether imaginary or real. You will not find in semantics any remedy for decayed teeth or illusions of grandeur or class conflicts. Nor is semantics a device for establishing that everyone except the speaker and his friends is speaking nonsense.

(Sorry if there are typos—I am writing this with vim over ssh from my Treo.)

A dilemma for naturalists: Second horn

For the definitions, see yesterday's post.

Horn 2: Truth is nomically coextensive with a Natural property.

We will now generate a problem for the naturalist from the following very plausible claims:

1. If naturalism holds, any syntactic property of utterances is a Natural property.
2. If naturalism holds, then Utterance is, of nomic necessity, a Natural kind.
3. Nomically necessarily, if naturalism holds, and s is any sentence in an interpreted typed first-order language L such that (a) the types coincide with Natural kinds, (b) quantification is restricted to within a type and (c) all predication is of Natural properties, then s expresses a proposition which is either true or false.

Now, let L be a rich enough subset of technical English that satisfies the condition in (3). Let T be the Natural property that is of nomic necessity coextensive with Truth. Let s be any sentence-type of the form:

For all utterances u, if P(u), then u does not have T,

where P is an explicit statement of a finitely-expressible Natural property of an utterance sufficient to nomically entail that all and only utterances of type s satisfy P. (I'll construct P in a moment. Technically, P(u) in the above should be in right-angle brackets.)

Now, if naturalism holds, then for any finite sentence type in L, there is a nomically possible world in which naturalism also holds and where that sentence type is uttered exactly once. Let w be a world where s is uttered exactly once. If u is the utterance of s in w, then at w, u satisfies P and only utterances of s satisfy P. But then u is true if and only if u is false, since T is coextensive with truth at w. And this is absurd.

To construct P, we presumably can use Goedel numbers as in the proof of the diagonal lemma in Goedel's theorem. Or perhaps more simply, we can use what I call "modified Goedel numbers". A "numeric expression" is a literal number in a sentence, e.g., "44.58" or "-1909". The modified Goedel number of a sentence with no numeric expressions is -1. If a sentence s constains a numeric expression, we let s* be the sentence with its first numeric expression replaced by 0, and let n(s) be the numeric value of that first numeric expression of s. If the Goedel number of s* is equal to n(s), then the modified Goedel number of s is n(s). Otherwise, the modified Geodel number of s is -1. Then, it's really easy to construct P. We let N be a numeric expression of the Goedel number of "For all utterances u, if 0 equals the modified Goedel number of u, then u does not have T", and then let P(u) be "N equals the modified Goedel number of u" (where here N is expanded out—it should be in right-angle brackets, I guess). Since P(u) expresses a syntactic property of u, it follows that it expresses a Natural property of u.

The argument can be modified by replacing (1) with the weaker claim that enough basic syntactic properties for computing the modified Goedel number of a sentence are of nomic necessity coextensive with Natural properties.

Hence, absurdity also follows from the second horn of the dilemma.

Therefore, naturalism is false.

A dilemma for naturalists: First horn

This argument is a dilemma, of which I will only give the first horn today. First we need two definitions. Say that a property is "Natural" provided that it either occurs in a correct scientific account of the world or else can be constructed, in a finite number of steps, from ingredients that occur in correct scientific accounts of the world. Say that two properties A and B are "nomically coextensive" provided that in any world that has the same laws as our world, the two properties have the same extension (i.e., are had by the same particulars). Finally, "Truth" shall be the property that an utterance (a token of a sentence) has in virtue of being true.

Suppose naturalism is true. The dilemma is this: Either Truth is nomically coextensive with a Natural property or it is not.

Horn 1: Truth is not nomically coextensive with any Natural property.

There are now two kinds of problems for the naturalist. The first is the question of how we got to have the concept of Truth. A naive account of concept formation is that we observe particulars, and then from these particulars we abstract the simplest, i.e., most natural (in Lewis's sense), properties that these particulars have in common. Thus, we observe a bunch of massive objects, and the simplest property they all have in common is Mass, and so we form the concept of mass. This naive account may need to be modified in various ways. For instance, perhaps we do not need to make all the observations ourselves—our cultural or even genetic forebears might have made some of them. We might not always opt for the very simplest property that the particulars have in common, as there may be explanatory constraints on the properties beside simplicity. Thus, we might from a bunch of particulars extract not just the simplest property that they have in common, but the simplest property that they have in common which explains some common phenomenon (such as the phenomenon of being observed by us in a particular way). Furthermore, given a bunch of concepts abstracted from particulars, we can form new concepts through various combinations. Further modifications are needed to take care of determinables (like mass of x grams).

But however we modify it, I think the following will hold: If naturalism holds, we will only ever arrive at concepts of Natural properties. It is only Natural properties that enter into genuine explanations or that are causally efficacious if naturalism holds, after all. And the simplest property that a bunch of instances has in common will never be a non-Natural property, since, given naturalism, a non-Natural property is either not at all a natural property (in Lewis's sense) or is formed as an infinitely describable combination of natural properties. But if we can only ever arrive at concepts of Natural properties, and Truth is not one of those (since it's not nomically coextensive to one of those) then we do not have the concept of Truth.

The second problem is that if Truth is not nomically coextensive with a Natural property, none of our cognitive faculties could have been evolutionarily selected for truth. (Here I am sliding a bit between Truth as a property of utterances and truth as a property of belief-tokens. But just about all I say about utterances holds for belief-tokens, too.) For only Natural properties stand in causal relations if naturalism holds, and only properties that stand in causal relations occur in evolutionary explanations. But it seems to be at least partly definitory of our doxastic faculties that they are for the sake of truth. A part of what distinguishes a belief from a desire, for instance, is that a belief is something that ought only occur when it is true. If Truth is not Natural, and if naturalism holds, then our doxastic faculties do not have this normativity, and we do not have beliefs at all (including the belief in naturalism).

Moreover, if Truth is not nomically coextensive with a Natural property, not only will it fail to be the case that our doxastic faculties are selected for generating correct beliefs, but neither will there be another evolutionary explanation, say using exaptation, of any truth-tendency in our beliefs. For all scientific explanations are of the possession of Natural properties or at best (and I am sceptical of this) of the possession of properties nomically coextensive with Natural ones. But then scepticism threatens, as in Plantinga's argument.

One might hold, however, that while Truth simpliciter is not nomically coextensive with a Natural property, Truth in some limited realm, such as the truth of simple claims about mastodons and potatoes is nomically coextensive with a Natural property (the argument in the second horn of the dilemma will not contradict this). Fine. But the belief in naturalism will not fall within this limited realm. Hence the belief in naturalism still undercuts itself.

Tomorrow, we'll look at the second horn.

The impossibility of a naturalistic semantics

The stuff below may be old hat. But it's fun.

A naturalistic semantics would give an account of the truth of a naturalistically acceptable sentence (i.e., sequence of symbols) in scientific terms (I am not asking for the naturalistic semantics to give an account of the truth of non-naturalistic sentences, though that might be letting the naturalist off too easy). It would, thus, give a naturalistically acceptable predicate (perhaps a very logically complex one) T such that a naturalistically acceptable sentence s satisfies T if and only if s expresses a truth. Thus, for instance, T will be such that "A dog is running" satisfies T if and only if a dog is running.

A complete naturalistic semantics is impossible, and its impossibility can be shown in a way parallel to the proof of Goedel's first incompleteness theorem. (I am now thinking of ways of generalizing the incompleteness theorem to something very, very general. This is just one application.) Any syntactically permissible combination of naturalistically acceptable terms, logical constants, and quantification over naturalistically acceptable entities (and that should include sentences, since we can model these mathematically as sequences of symbols) should be a naturalistically acceptable sentence. Let P be any naturalistically acceptable predicate such that the sentence

1. Every sentence s satisfying P fails to satisfy T

is in fact the one and only sentence that satisfies P. (For instance, P might simply specify the time and place at which (1) is written.) Then (1) is a naturalistically acceptable sentence, and so (1) is true iff (1) satisfies T. It follows from the fact that (1) is the one and only sentence that satisfies P that (1) is true iff (1) is false, which is absurd.

I think a case can be made from this that there is no naturalistically acceptable property equivalent to truth. This is a good argument against naturalism.

(A challenge is to show that this does not lead to a paradox for the non-naturalist. I think there is a principled way in which one can count as nonsense sentences that directly or indirectly talk of their own truth. But (1) doesn't do that—it talks of the sentence s's satisfying T, where T is some natural predicate, not truth.)