Lambda the Ultimate

If you notice, the author offers several other formulations in perfectly good English that make a clear distinction between the two formulae.

Aren't they less natural (i.e., less probability to use them)?

The author is simply cautioning against the sloppy use of one particular homophonous reading of the two formulae that seems to equate them, though they are logically distinct.

Do not different languages encourage sloppiness in different classes of situations? For that we just need a lemma that word count (or other complexity metric) ratio for accurate/sloppy formulations in a specific situation in different languages differs dramatically.

Aren't they less natural

Nope. "It is necessary that A implies B" is perfectly natural, and not really much longer.

In fact, upon further reflection, the author's statement is not even entirely true.

For example, I can say "Necessarily, if A then B" versus "if A then necessarily B". If I'm consistent with these readings, there is no ambiguity.

I think the big problem is assuming that modal logic makes any sense in ordinary language. ;-)

Do not different languages encourage sloppiness in different classes of situations?

No. Usage habits in the history of modal logic or trying to shoehorn ordinary language into a precise formal one create the sloppiness here.

"Mistah Whorf, he dead."

Do not different languages encourage sloppiness in different classes of situations?

No. Usage habits in the history of modal logic or trying to shoehorn ordinary language into a precise formal one create the sloppiness here.

I'm sure that's part of it, but I also think it's likely that Andris is right in general. This is exactly the sort of thing which Sussmann & Wisdom are talking about in the preface of SICM, which I posted about yesterday. They mention "uncovering puns and flaws in reasoning" by expressing physical formulae in computational form. They're not only critiquing the flaws in expressing such things in natural language, but also in traditional mathematical notation.

Footnote 2 in that preface gives an example of a confusing pun in classical notation. At the very least, things like this cause confusion amongst students. However, it goes beyond students — Sussman and Wisdom wrote "We quickly learned that many things we thought we understood we did not in fact understand". I don't see how it can be denied that languages, both formal and informal, play a role both as a barrier to understanding in some cases, and in the improved understanding that comes from using more precise language. Where that fits in the Sapir-Whorf spectrum between weak and strong linguistic determinism is a separate question. However, in general, this demonstrates a fairly strong effect of the languages we use on our thought processes.

They mention "uncovering puns and flaws in reasoning" by expressing physical formulae in computational form.

Do you see how easily we have slipped from claims about human language to a completely different notion of language, computation?

S-W is normally a claim about human language; I think it is fair to say that a specialized notation, such as mathematical notation, and computational languages such as PLs are quite different animals, since the semantic range of both the latter is a priori much more limited and specific.

So there is a logical error drawing conclusions about HL based on other completely different entities that just happen to share the ordinary name of "language".

Is this a property of English per se? No, because even though we use the term "notation" for some of these entities, we still insist upon equating them with language, by analogy.

So this is the result of sloppy thinking on our part, and on a tradition of usage in the specific fields we are discussing, rather than an inherent problem with our language.

Which is exactly my point to Andris. ;-)

So there is a logical error drawing conclusions about HL based on other completely different entities that just happen to share the ordinary name of "language".

Wikipedia's article on Oligosynthetic language (reached from article about Benjamin Whorf in one step) is quite interesting in this light:

The fact that no existing language, living or dead, has been demonstrably shown to exhibit oligosynthetic properties has led some linguists to regard true oligosynthesis as impossible (or at any rate, wildly impractical) for productive use by human beings.

So either computer scientists and logicians are not human beings, or they use their languages impractically, or their languages are misnomer.

So either computer scientists and logicians are not human beings, or they use their languages impractically, or their languages are misnomer.

Two things:

1) When did Wikipedia move from being informative to being authoritative?

2) Once again you have taken a statement intended to describe HLs, and used it to make a logical inference about a particular usage of language, i.e. logical or computational notation.

No cookie for you! ;-)

Do you see how easily we have slipped from claims about human language to a completely different notion of language, computation?

No, computation is not the point — it's merely being held up as a means of testing for errors which can arise from ambiguites in formal languages, or natural languages, or even non-linguistic thought processes.

Despite the differences between natural languages and formal languages, in the area we're discussing, the same kinds of issue arise with both: when the language itself has ambiguities and can hide subtle errors, we're more likely to make those errors and not notice them.

So there is a logical error drawing conclusions about HL based on other completely different entities that just happen to share the ordinary name of "language".

There is a logical error in assuming that the only similarity is in the sharing of the name. There are certainly other similarities, and they may well be significant enough for some of the same principles to apply to both. For example, both rely heavily on common features such as abstraction via names. As a result, both can suffer from problems with ambiguity in either or both the sense and referent of names, for example. Such ambiguities may mislead us whether we're using natural or formal languages.

Sussmann and Wisdom pointed out that they found the classical formal language of mechanics wanting, and that their understanding was incomplete partly as a result of that. I think it goes without saying that if that's the case, that natural language wasn't helping them overcome these difficulties either. (I'm also assuming that both of them are reasonably good examples of human intellectual capabilities.)

You seem to be saying that with sufficiently rigorous and logical thought on our part, we can avoid such errors. However, the same is true of something like the Wason test, which is claimed to demonstrate that our innate logical abilities are biased in certain ways. The fact that we can overcome such things by being careful and rigorous doesn't change the fact that we have a handicap.

when the language itself has ambiguities and can hide subtle errors, we're more likely to make those errors and not notice them

There is a second conflation of ideas here that I'm trying to separate. That is the distinction between usage of language and the language itself as a system.

As in the modal logic case, there were certain usages of sentences as readings of formulae that resulted in ambiguity. As I showed in that case, this was not a property of the language (English) itself, but a particular usage conditioned by traditions in the particular field and by sloppy formulation of precise ideas.

In general, this is the case; particular formulations within a language can lead to ambiguities in the expression of a particular idea, but this does NOT mean that the language itself prevents, discourages, distorts or obscures more precise and unambiguous formulations.

To the extent that language is inherently ambiguous, this is a property of ALL language rather than of a particular instance.

The fact that we can overcome such things by being careful and rigorous doesn't change the fact that we have a handicap.

We may have a handicap, but it is not a problem with our specific language, but rather with our general reasoning abilities or perhaps language systems in general.

Again, I think we are being too fluid calling particular frames of reasoning different "languages" and then using that to prove S-W.