Drum: "Change, Meaning, and Information" (1957)

This 1957 article is one of those countless papers by people in linguistics trying helplessly to say something non-trivial about the relationship between information theory and linguistics. The central claim of the paper is that there are several kinds of predictability or redundancy at play in language, and that "meaning" should be seen as one of them.

More precisely, certain sentences are predictable on account of their content, so that

the "meaning"—the "what we know about it"–in a message has some direct effect upon the amount of information transmitted. (p. 166)

However, this notion is dangerously confused and conflated with the idea that more predictability equals more meaning at several points in the paper.

At any rate, Dale places semantics among several "levels" of redundancy, including a historically interesting division between "formal" and "contextual" syntax:

In the sentence, "The chair is __________," many words could be correctly used in terms of grammar; but in terms of the context of the sentence, only a very few apply. "The chair is clock" is correct grammatically, though it is nonsense in even the broadest contexts. (p. 167)

Further,

A person knowing English grammar but not word meanings, might very well write "The chair is clock," but would never do so if he knew the meanings involved. (The relationship of syntactic to contextual redundancy is much that of validity in logic to truth, incidentally). (p. 168)

A strange counterfactual, but quite revealing.

Literature on the meaning of "only if"

I've been looking for some empirical studies of A only if B constructions. In the theoretical literature on natural language semantics, there is a number of models, but I want to know more about how they are actually understood. Fortunately, there seem to be some facts about that out there, too.

What Does It Mean, Allegedly?

The problematic issue with the only if construction is that it is supposed to be logically equivalent to a number of related constructions, even though non-logicians sometimes disagree with this. According to the classical convention, the following sentences thus all mean the same:

• It only thunders if it rains.
• If it thunders, it rains.
• If it doesn't rain, it doesn't thunder.

On the other hand, if we reverse the implication, we change the truth conditions:

• It only rains if it thunders.
• If it rains, it thunders.
• It it doesn't thunder, it doesn't rain.

If this was just a mere convention about logical language, all would be fine. The problem is, however, that these sentence forms are not used in the same situations, and they do not integrate equally well into all reasoning patterns in spite of their (alleged) equivalence.

The Performance Problem

One difference between the If A, then B and A only if B forms is that if form is generally more difficult to use in a modus tollens inference than the only if. At least, this is what Carlos Santamaría and Orlando Espino say (Santamaría and Espino 2002, p. 42). They're referring to three studies, including one by Jonathan Evans and M. A. Beck (Evans and Beck 1981).

The problematic case is thus the following inference:

If it thunders, it rains.
It doesn't rain.
–––––––––––––––––
It doesn't thunder.

This (clasically valid) inference should be performed more readily when served in this alternative, and supposedly equivalent formulation:

It only thunders if it rains.

I doesn't rain.

–––––––––––––––––––––

It doesn't thunder.

Cognitively, or perhaps in terms of actual natrual language semantics, this seems to indicate that A only if B works more like the contrapositive If not B, then not A than like its positive translation, If A, then B. Or at least, it seems to issue a conversational warrant closer to it.

It would be interesting to know if this alternative formulation comes with a corresponding decrease—are we trading of willingness to perform the straightforward modus ponens inference for higher rates of modus tollens? This would imply that the following inference generally is less accepted:

It only thunders if it rains.
It thunders.
–––––––––––––––––––––
It rains.

If the only if formulation really does works like a contrapositive, then this inference should appear to us like a modus tollens inference in terms of plausibility and difficulty. I do not know right now whether such an effect can actually be measured or not.

The Issue of Time

Another interesting proposal that Santamaría and Espino cite, also coming from Evans and Beck, is that there is a systematic interaction between our conception of temporal order and the choice of form.

Thus, even though If A, then B, and A only if B are supposedly logically equivalent, we get different patterns of acceptability or naturalness depending on whether A or B happened first. For A preceding B, we then (perhaps) have:

• If you bought on Tuesday, you're paying on Wednesday.
• (?) You bought on Tuesday only if you paying on Wednesday.

And for B preceding A:

• (?) If you're paying on Wednesday, you bought on Tuesday.
• You're only paying on Wednesday if you bought on Tuesday.

Of course, much clearer intuitions can be produced if we ruffle up the tenses a bit. But this, I think, relatively fair example to start the discussion from.

So, Causality?

Note that the issue of before/after interfaces with the concept of causality, which is notoriously bound up with implication, even if logicians and statisticians hate to admit this fact.

Possibly, the the only way we can really justify an inference from a later effect to a prior cause in the form of If EFFECT, then CAUSE is to objectify the cause and the effect by thinking about the observation of the effect and the deduction of the cause. In this way, we would straighten out the temporal sequence so that EFFECT could in fact precede CAUSE.

If this is true, it has a quite important consequence for the psychology of reasoning: We would then only be able to understand abductive inference by effectivly embedding a cause/effect relationship in a different and larger cause/effect relationship—namely the only in which a real or imagined person reasons from fire (the logical "cause") to smoke (the logical "effect").

Hintikka: "Quantifiers in Natural Languages" (1977)

In this paper, Jaakko Hintikka yet again presents his "game-theoretical semantics" and its extension to games with imperfect information. However, he also proposes his "any-thesis" — a conjecture stating that any is grammatical in exactly the contexts in which it means something different than every.

The paper was originally published in the second-ever issue of Linguistics and Philosophy (1977), but was reprinted in the anthology Game-Theoretical Semantics (1979).

Ordering Principles

The most notable new idea in the paper is the set of ordering principles that Hintikka introduces in section 10. These are principles that govern whether, say, the modality player or the quantification player should move first when we encounter sentences like some people might not be nice. These principles can essentially be translated into rules about the relative scope of various operations.

The reason he introduces these principles is that he wants to account for the fact that any sometimes behaves like an existential quantifier rather than a universal:

• I can do anything. (universal)
• I can't do anything. (existential)

He explains this by introducing an ordering principle that requires any to "scope out" over a negation whenever there is one. Using this principle, "not any" will then be equivalent with "every not," which then again is equivalent to "not some" which we were looking for. The change of quantifier type is, in other words, achieved by swapping negation and quantification.

One possible problem with this approach is double negation. Hintikka doesn't explicitly discuss this, but somewhere in his system, he needs to bar the quantifier from scoping out over both negations in sentences like

• It's not true that we haven't done anything about the crisis.

Otherwise this sentence would come out as equivalent to We have done everything about the crisis instead of We have done something about the crisis.

The Any-Thesis

So, Hintikka's self-titled any-thesis can be put as follows: the quantifier any is grammatical if and only if it appears under a negation, or under some other operation that it can scope out of with a resulting change of meaning. Beyond that, it is synonymous with every.

This explains distributions like the following:

• *I know anything.
• I don't know anything.

If we take If A, then B to be equivalent with Not A, or B, then it further explains the following pattern:

• If I have any medical issues, the test comes out positive.
• *If the test comes out positive, I have any medical issues.

In both cases, any can scope out over the negation and thus effectively change its meaning from every to some.

The Status of the Context

But then things get a little hairy. We can note that some tenses apparently interact with the universal quantifier in a meaning-changing way, while others don't:

• *I have been anywhere.
• ?*I am going anywhere.
• ?*I went anywhere.
• *I am anywhere.
• ?I will go anywhere.
• I would go anywhere.

The asterisks here are based on my estimates. I'm quite unsure what native speakers would think about them.

Hintikka has a conjecture about this (sec. 18). He thinks that when we're thinking about the future or about counterfactual scenarios, we're dealing with a special kind of modal logic in which the domain of quantification changes from world to world. We consequently get a logical difference between sentence pairs like these:

• I every scenario, everybody wins.
• Everybody wins in every scenario.

To see this, consider for instance a model in which some possible worlds contains one more loser than the actual world. In that case, the first sentence might be false, while the second true. (I am here assuming that entities have all properties in scenarios where they don't exist.)

So, since such pairs are no longer equivalent in such a grow/shrink logic, it makes a difference whether a universal quantification comes before or after a necessity modal. This thus introduces a difference in meaning and accounts for the (possible) grammaticality of I will steal anything.

An Alternative Approach

However, I would explain these facts in terms of how odd it is to use a free choice operation in a context that essentially excludes any real choice. This would also explain the context-sensitivity that there seems to be about acceptability of any.

As far as I can tell, at least, we are more inclined to accept present tense uses of any when the sentence can be interpreted as offering a real choice. Thus, according to my personal intuitions (and some googling), we have:

• I buy anything if the price is right.
• *I have any problems if the test is positive.
• I switch off any device that consumes electricity.
• *I experienced any emotion that is humanly possible.

This seems to support a free-choice reading of any over the scoping-out story.

Methodology: Some Quotes

In this paper as elsewhere, Hintikka is pretty dismissive of asking other people for their opinions about sample sentences. He thus brushes off "different speakers' more or less confused uneducated intuitions" (p. 91) as misrepresenting "what a truly competent speaker would do" (p. 90).

In footnote 13, he writes:

I have been amazed time and again by linguists who claim that they are dealing with competence and not performance and then go on to base their theories on people's uneducated and unanalysed reactions to complicated sentences. (p. 115)

It's difficult to see what the object of semantics is, then, if only the intuitions of trained logicians really count as data. With the right training, people can come to see whatever sentence meaning we want them to see.

Methodology: Examples

Just to illustrate the problem, let me briefly cite a couple of the sentences that Hintikka takes to be good, grammatical English sentences with definite meaning:

• Every townsman admires a friend and every villager envies a cousin who have met each other. (p. 89)
• Every actor of each theatre envies a film star, every review of each critic mentions a novelist, and every book by each chess writer describes a grand master, of whom the star admires the grand master and hates the novelist while the novelist looks down on the grand master. (p. 97)
• Some product of some subdivision of every company of every conglomerate is advertised in some page of some number of every magazine of every newspaper chain. (p. 97)
• Every girl has not been dated by John. (p. 101)
• If Jane wins, anybody who has bet on her is happy. (p. 113)

How thin is the line between "complicated sentences" and word salad? Well, compare these "grammatical" sentences to the ones that Hintikka stars as ungrammatical:

• If Jane has won any match, she has won any match. (p. 100)
• John must pick any apple. (p. 101)
• If everyone loses, anyone loses. (p. 110)
• Mary hopes that Jane will win any match. (p. 112)
• Mary believes that Jane will win any match. (p. 112)

According to Hintikka's methodology, if we ask an average English speaker about these sentences, we would get nothing but "confused uneducated intuitions." Instead, we should look for a "consistent, general set of rules of semantic interpretation" (p. 91) that can accommodate the entailments that we, we educated logicians, think the sentences ought to have.

That sounds like a recipe for injecting the theory into the data, even quite openly and deliberately. I find it difficult to see how any rational discussion could follow if we value our own speculative and introspective intuitions about foreign languages over the judgments of other people.